Calabi-Yau Manifolds, Hermitian Yang-Mills Instantons, and...

Research ArticleCalabi-Yau Manifolds Hermitian Yang-Mills Instantons andMirror Symmetry

Hyun Seok Yang1 and Sangheon Yun2

1Center for Quantum Spacetime Sogang University Seoul 121-741 Republic of Korea2Institute for the Early Universe Ewha Womans University Seoul 120-750 Republic of Korea

Correspondence should be addressed to Hyun Seok Yang ya19683gmailcom

Received 24 October 2016 Revised 9 April 2017 Accepted 30 July 2017 Published 9 November 2017

Academic Editor Anna Cimmino

Copyright copy 2017 Hyun Seok Yang and Sangheon Yun This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited The publication of this article was funded by SCOAP3

We address the issue of why Calabi-Yau manifolds exist with a mirror pair We observe that the irreducible spinor representation ofthe Lorentz group Spin(6) requires us to consider the vector spaces of two forms and four forms on an equal footing The doublingof the two-form vector space due to the Hodge duality doubles the variety of six-dimensional spin manifolds We explore howthe doubling is related to the mirror symmetry of Calabi-Yau manifolds Via the gauge theory formulation of six-dimensionalRiemannian manifolds we show that the curvature tensor of a Calabi-Yau manifold satisfies the Hermitian Yang-Mills equationson the Calabi-YaumanifoldTherefore the mirror symmetry of Calabi-Yaumanifolds can be recast as the mirror pair of HermitianYang-Mills instantons We discuss the mirror symmetry from the gauge theory perspective

1 Introduction

String theory predicts [1] that six-dimensional Riemannianmanifolds have to play an important role in explaining ourfour-dimensional world They serve as an internal geom-etry of string theory with six extra dimensions and theirshapes and topology determine a detailed structure of themultiplets for elementary particles and gauge fields throughthe Kaluza-Klein compactification This program initiatedby a classic paper [2] tries to make contact with a low-energy phenomenology in our four-dimensional world Inparticular a Calabi-Yau (CY) manifold is a (compact) Kahlermanifold with vanishing Ricci curvature and so a vacuumsolution of the Einstein equationsTheyhave a prominent rolein superstring theory and have been a central focus in bothcontemporarymathematics andmathematical physics As theholonomy group of CYmanifolds is 119878119880(3) the compactifica-tion onto a CY manifold in heterotic superstring theory pre-servesN = 1 supersymmetry in four dimensions One of themost interesting features in the CY compactification is thattype II superstring theories compactified on two distinct CYmanifolds lead to an identical effective field theory in four

dimensions [1 3] This suggests that CY manifolds exist witha mirror pair (119872 ) where the number of vector multipletsℎ11(119872) on 119872 is the same as the number of hypermultipletsℎ21() on and vice versa Here ℎ푝푞(119872) = dim119867푝푞(119872) isthe Hodge number of a CYmanifold119872This duality betweentwo CY manifolds is known as the mirror symmetry [3]While many beautiful properties of the mirror symmetryhave been discovered and it has been even proven for somecases it is fair to say that we are still far away from a deepunderstanding for the origin of mirror symmetry

Mirror symmetry is a correspondence between two topo-logically distinct CY manifolds that give rise to exactly thesame physical theory To recapitulate the mirror symmetrylet 119872 be a compact CY manifold The only nontrivialcohomology of the CY manifold is contained in 11986711(119872)and 11986721(119872) besides the one-dimensional cohomologiesℎ00(119872) = ℎ33(119872) = ℎ30(119872) = ℎ03(119872) = 1 These coho-mology classes parameterize CY moduli It is known [3]that every 11986711(119872) on one hand is represented by a realclosed (1 1)-form which forms a Kahler class represented bythe Kahler form of a CY manifold 119872 The elements in

HindawiAdvances in High Energy PhysicsVolume 2017 Article ID 7962426 27 pageshttpsdoiorg10115520177962426

2 Advances in High Energy Physics11986711(119872) infinitesimally change the Kahler structure of theCY manifold and are therefore called Kahler moduli (Instring theory these moduli are usually complexified byincluding119861-field)On the other hand11986721(119872)parameterizesthe complex structure moduli of a CY manifold 119872 It isthanks to the fact that the cohomology class of (2 1)-forms isisomorphic to the cohomology class 1198671

휕(119879119872) the first Dol-

beault cohomology group of119872with values in a holomorphictangent bundle 119879119872 that characterizes infinitesimal complexstructure deformations Hence the mirror symmetry of CYmanifolds is the duality between two different CY 3-fold 119872and such that the Hodge numbers of119872 and satisfy therelations [3] ℎ11 (119872) = ℎ21 () ℎ21 (119872) = ℎ11 () (1)

or in a more general formℎ푝푞 (119872) = ℎ3minus푝푞 () (2)

where the Hodge number ℎ푝푞 of a CY manifold satisfiesthe relations ℎ푝푞 = ℎ푞푝 and ℎ푝푞 = ℎ3minus푝3minus푞 As we havementioned above the only nontrivial deformations of a CYmanifold are generated by the cohomology classes in11986711(119872)and 11986721(119872) where ℎ11(119872) is the number of possible (ingeneral complexified) Kahler forms and ℎ21(119872) is thedimension of the complex structure moduli space of 119872Mirror symmetry suggests that for each CY 3-fold 119872 thereexists another CY 3-fold whose Hodge numbers obey therelation (1)

From a physical point of view two CY manifolds arerelated by mirror symmetry if the corresponding N = 2superconformal field theories are mirror [4] Two N = 2superconformal field theories are said to be mirror if they areequivalent as quantum field theories The mirror symmetrywas interpreted mathematically by Kontsevich in his 1994ICM talk as an equivalence of derived categories dubbed thehomological mirror symmetry [5] The homological mirrorsymmetry states that the derived category of coherent sheaveson a Kahler manifold should be isomorphic to the Fukayacategory of a mirror symplectic manifold [6] The Fukayacategory is described by the Lagrangian submanifold of agiven symplectic manifold as its objects and the Floer homol-ogy groups as their morphisms Hence the homologicalmirror symmetry formulates the mirror symmetry as anequivalence between certain aspects of complex geometry ofa CY manifold and certain aspects of symplectic geometryof a mirror CY manifold in all dimensions The geometricapproach to mirror symmetry was also unveiled in [7] thatmirror symmetry is a geometric version of the Fourier-Mukaitransformation along a dual special Lagrangian tori fibrationon a mirror CY manifold which interchanges the symplecticgeometry and the complex geometry of a mirror pair

In this paperwewill explore the gauge theory formulationof six-dimensional Riemannian manifolds to address theissue why CY manifolds exist with a mirror pair In order

to simplify an underlying argumentation we will focus onorientable six-dimensional manifolds with spin structure Ingeneral relativity the Lorentz group appears as the structuregroup acting on orthonormal frames of the tangent bundle ofa Riemannian manifold [8] On the frame bundle a Rieman-nian metric on spacetime manifold 119872 is replaced by a localorthonormal basis 119864퐴 (119860 = 1 119889) of the tangent bundle119879119872 Then Einstein gravity can be formulated as a gaugetheory of Lorentz group where spin connections play a roleof gauge fields and Riemann curvature tensors correspondto their field strengths On a six-dimensional Riemannianmanifold 119872 for example local Lorentz transformations areorthogonal rotations in Spin(6) and spin connections 120596퐴퐵 =120596푀퐴퐵119889119909푀 are the spin(6)-valued gauge fields from the gaugetheory point of view (we will use large letters to indicate aLie group 119866 and small letters for its Lie algebra g) Thenthe Riemann curvature tensor 119877퐴퐵 = 119889120596퐴퐵 + 120596퐴퐶 and 120596퐶퐵

precisely corresponds to the field strength of gauge fields120596퐴퐵 in Spin(6) gauge theory Since the Lie group Spin(6) isisomorphic to 119878119880(4) the six-dimensional Euclidean grav-ity can be formulated as an 119878119880(4) Yang-Mills gauge the-ory Via the gauge theory formulation of six-dimensionalRiemannian manifolds we want to identify gauge theoryobjects corresponding to CY manifolds and address theirmirror symmetry from the perspective of Yang-Mills gaugetheory To understand why there exists a mirror pair ofCY manifolds in particular we will employ the followingwell-known propositions for a 119889-dimensional Riemannianmanifold119872

(A) The Riemann curvature tensors 119877퐴퐵 are spin(119889)-valued two forms inΩ2(119872) = Λ2119879lowast119872

(B) There exists a global isomorphism between 119889-dimensional Lorentz groups and classical Lie groups

Spin (3) cong 119878119880 (2) Spin (4) cong 119878119880 (2)퐿 times 119878119880 (2)푅 Spin (5) cong 119878119901 (4) Spin (6) cong 119878119880 (4)

(3)

(C) There is an isomorphism between the Clifford algebraC119897(119889) in 119889-dimensions and the exterior algebra Λlowast119872of cotangent bundle 119879lowast119872 over 119872 [9 10] (the spaceof the Clifford algebra C119897(119889) is isomorphic as avector space to the vector space of the exterioralgebra Λlowast119872 This is not however an isomorphismof associative algebras because the product inΛlowast119872 isanticommutative while that in C119897(119889) is not due to thecentral term in the Dirac algebra (5))

C119897 (119889) cong Λlowast119872 = 푑⨁푘=0

Ω푘 (119872) (4)

whereΩ푘(119872) = Λ푘119879lowast119872For the isomorphism (C) between the vector spaces the

ldquovolume operatorrdquo Γ푑+1 equiv plusmn119894푑(푑minus1)2Γ1 sdot sdot sdot Γ푑 in the Clifford

Advances in High Energy Physics 3

M Spin(6) SU(4)

CY3 HYM

CY3 HYM

Spin(6) SU(4)

lArrrArr

lArrrArr

M

９3

９3A rarr

B rarr A rarr

B rarr （９－

（９－

Figure 1 Mirror symmetry

algebra C119897(119889) corresponds to the Hodge-dual operator lowast Ω푘(119872) rarr Ω푑minus푘(119872) in the exterior algebra Λlowast119872 whereΓ퐴 (119860 = 1 119889) are 119889-dimensional Dirac matrices obeyingthe Dirac algebra Γ퐴 Γ퐵 = 2120575퐴퐵I2[1198892] (5)

It is amusing to note that the Clifford algebra from a modernviewpoint can be thought of as a quantization of the exterioralgebra [10] in the same way that the Weyl algebra is aquantization of the symmetric algebra In particular theClifford map (4) implies that the Lorentz generators 119869퐴퐵 equiv(14)[Γ퐴 Γ퐵] inC119897(119889) are in one-to-one correspondence withtwo forms in the spaceΩ2(119872) And the representation spaceof the Clifford algebra is a spinor vector spacewhose elementsare called fermions and essential ingredients in StandardModel It may also be worthwhile to remark that any physicalforce is represented by two forms in the exterior algebrataking values in a classical Lie algebra In addition recall thatthe representation of Clifford algebra in even dimensionsis reducible and its irreducible representations are givenby chiral fermions Then the isomorphism (C) implies thatthere must be a corresponding irreducible decomposition oftwo forms in Λlowast119872 This fact in our case has a nontrivialconsequence for the Riemann curvature tensors 119877퐴퐵 =(12)119877퐴퐵퐶퐷119890퐶and119890퐷 since the spin(119889) Lie algebra indices (119860 119861)and the form indices (119862119863) must have an identical structurein a representation space of the Lorentz symmetry accordingto the isomorphism (C) Our principal concern is then topin down a geometrical consequence of the rudimentary fact(A) after implementing the isomorphisms (B) and (C) to six-dimensional CY manifolds

Let us briefly state the result summarized in the Figure 1in advance Compared to the four-dimensional case [11ndash14]some acute changes arise First of all there are two sources oftwo forms on an orientable six-dimensionalmanifold119872 Oneis of course usual two forms in Ω2(119872) and the other is theHodge duality of four forms in Ω4(119872) Therefore the vectorspace of two forms is doubled in six dimensionsΛ2 (119872) equiv Ω2 (119872) oplus lowastΩ4 (119872) (6)

The doubling of two forms is resonant with the fact thatthe irreducible representation of Lorentz symmetry is givenby the chiral Lorentz generators 119869퐴퐵plusmn equiv (12)(I8 plusmn Γ7)119869퐴퐵Definitely it corresponds to themixture of two forms and fourforms in Λ2(119872) according to the correspondence (Γ7 harr lowast)Since we need to take an irreducible representation of Lorentzsymmetry this demands us to think of the irreducible com-ponents of Riemann curvature tensors as a sum of the usualcurvature tensors 119877퐴퐵 and dual curvature tensors defined by

퐴퐵 equiv (lowast119866)퐴퐵 = 119889퐴퐵 + 퐴퐶 and 퐶퐵 where 119866퐴퐵 is a 4-formtensor taking values in spin(6) cong 119904119906(4) Lie algebra [14]Moreover it is necessary to impose the torsion-free conditionfor both spin connections 120596퐴퐵 and 퐴퐵 which leads to thesymmetry property of the curvature tensors 119877퐶퐷퐴퐵 = 119877퐴퐵퐶퐷

and 퐶퐷퐴퐵 = 퐴퐵퐶퐷 This is another reason why two kinds ofindices ([119860119861] [119862119863])must be treated symmetrically althoughthey belong to different vector spaces To summarize theHodge duality admits two independent types of curvaturetensors (119877퐴퐵퐶퐷 oplus 퐴퐵퐶퐷) and they have to be decomposedaccording to the irreducible representation of spin(6) cong 119904119906(4)Lie algebra In the end the duplication of curvature tensorsleads to the doubling for the variety of six-dimensional spinmanifolds

It might be stressed that the doubling of six-dimensionalspinmanifolds is an inevitable consequence of the elementaryfacts (A B C) It should be instructive to apply the foregoingpropositions (A B C) to four manifolds to grasp theirsignificance [12ndash14] although the four-dimensional situationis in stark contrast to the six-dimensional case In four dimen-sions the Lorentz group Spin(4) is isomorphic to 119878119880(2)퐿 times119878119880(2)푅 whose Lie algebras 119904119906(2)퐿푅 consist of chiral Lorentzgenerators 119869퐴퐵plusmn equiv Γplusmn119869퐴퐵 with Γplusmn = (12)(I4 plusmn Γ5) for chiraland antichiral representationsThe splitting of the Lie algebraspin(4) cong 119904119906(2)퐿 oplus 119904119906(2)푅 is precisely isomorphic to thecanonical decomposition of the vector space Ω2(119872) of twoforms Ω2 (119872) = Ω2

+ (119872) oplus Ω2minus (119872) (7)

whereΩ2plusmn(119872) equiv 119875plusmnΩ2(119872) and 119875plusmn = (12)(1 plusmn lowast) That is the

six-dimensional vector space Ω2(119872) of two forms splitscanonically into the sum of three-dimensional vector spacesof self-dual and anti-self-dual two forms One can apply thecanonical splitting of the two vector spaces to Riemann cur-vature tensors simultaneously according to (4) It results inthe well-known decomposition of the curvature tensor119877 intoirreducible components [15 16] schematically given by

119877 = (119882+ + 112119904 119861119861푇 119882minus + 112119904) (8)

where 119904 is the scalar curvature 119861 is the traceless Ricci tensorand 119882plusmn are the (anti-)self-dual Weyl tensors An impor-tant lesson from the four-dimensional example is that theirreducible (chiral) representation of Lorentz symmetry cor-responds to the canonical split (7) of two forms with theprojection operator 119875plusmn = (12)(1 plusmn lowast) We observe that thesame analysis in six dimensions brings about amore dramatic

4 Advances in High Energy Physics

result due to the fact that 6 = 2 + 4 The doubling of six-dimensional spin manifolds will be important to understandwhy CY manifolds arise with a mirror pair

The gauge theory formulation of six-dimensional spinmanifolds also leads to a valuable perspective for the dou-bling The first useful access is to identify a gauge theoryobject corresponding to a CY 3-fold in the same sense thata gravitational instanton (or a hyper-Kahler manifold) canbe identified with an 119878119880(2) Yang-Mills instanton in fourdimensions [11ndash13] An obvious guess goes toward a six-dimensional generalization of the four-dimensional Yang-Mills instantons known as Hermitian Yang-Mills (HYM)instantons Indeed this relationship has been well-knownto string theorists and mathematicians under the name ofthe Donaldson-Uhlenbeck-Yau (DUY) theorem [17 18] Wequote a paragraph in [19] (Page 221) to clearly summarize thispicture

The point of intersection between the Calabiconjecture and the DUY theorem is the tangentbundle And herersquos why once you have provedthe existence of CYmanifolds you have not onlythose manifolds but also their tangent bundlesbecause every manifold has one Since the tan-gent bundle is defined by the CY manifold itinherits its metric from the parent manifold (inthis case the CY) The metric for the tangentbundle in other words must satisfy the CYequations It turns out however that for thetangent bundle the Hermitian Yang-Mills equa-tions are the same as the CY equations providedthe background metric you have selected is theCY Consequently the tangent bundle by virtueof satisfying the CY equations automaticallysatisfies the Hermitian Yang-Mills equationstoo

If a CY manifold119872 can be related to a HYM instanton anatural question immediately arises Since a CY manifold119872has a mirror manifold there will be a mirror CYmanifold obeying themirror relation (1)This in turn implies that theremust be amirror HYM instanton which can be derived fromthe mirror CY manifold Thus we want to understand therelation between the HYM instanton and its mirror instantonfrom the gauge theory perspective Since the Lorentz groupSpin(6) is isomorphic to 119878119880(4) the chiral and antichiralrepresentations 4 and 4耠 of Spin(6) are equivalent to thefundamental and antifundamental representations 4 and 4 of119878119880(4) Recall that the fundamental representation 4 of 119878119880(4)is a complex representation and so its complex conjugate 4is an inequivalent representation different from 4 Thereforegiven a CY manifold 119872 one can embed the HYM instantoninherited from119872 into two different representations But thissituation is equally true for the mirror CY manifold Thusthere is a similar doubling for the variety of HYM instantonsas occurred to CY manifolds as summarized in Figure 1

It may be interesting to compare this situation withthe four-dimensional case [13 14] In four dimensions thepositive and negative chirality spinors of Spin(4) are givenby 119878119880(2)퐿 and 119878119880(2)푅 spinors 2퐿 and 2푅 respectively In

this case it is necessary to have two independent 119878119880(2)factors to be compatible with the splitting (7) because theirreducible representation of 119878119880(2) is real It is interestingto see how (A B C) take part in the conspiracy First aCY 2-fold can be mapped to a self-dual or 119878119880(2)퐿 instantonwhich lives in the chiral representation 2퐿 while a mirrorCY 2-fold is isomorphically related to an anti-self-dual or119878119880(2)푅 instanton in the antichiral representation 2푅 Forthis correspondence the 119878119880(2) gauge group of Yang-Millsinstantons is identified with the holonomy group of CY 2-fold This picture is generalized to six dimensions in aninteresting way In six dimensions the canonical splitting (7)is applied to the enlarged vector space (6) asΛ2 (119872) = Ω2

+ (119872) oplus Ω2minus (119872) (9)

where the decomposition Ω2plusmn(119872) is dictated by the chiral

splitting 119869퐴퐵 = 119869퐴퐵+ oplus 119869퐴퐵minus according to the isomorphism(C) From the gauge theory perspective the splitting (9) isalso compatible with the fundamental and antifundamentalrepresentations of the gauge group 119878119880(4) cong Spin(6) becausethe chiral representation of Spin(6) is identified with thefundamental representation of 119878119880(4) After all we will getthe picture that the HYM instanton on 119879119872 embedded inthe fundamental representation 4 is mirror to the HYMinstanton on 119879 in the antifundamental representation 4This structure is summed up in Figure 1 where CY3 refersto a CY 3-fold 119872 and CY3 its mirror And HYM denotesa HYM instanton on119872 in the complex representation either3 or 3 of 119878119880(3) sub 119878119880(4) and HYM its mirror on in theopposite complex representation

The purpose of this paper is to understand the structurein Figure 1 To the best of our knowledge there is noconcrete work to address the mirror symmetry based on thepicture in Figure 1 although the mirror symmetry has beenextensively studied so far We will show that CY manifoldsandHYM instantons exist withmirror pairs as a consequenceof the doubling (6) of two forms in six dimensions Itis arguably a remarkable consequence of the mysteriousClifford isomorphism (C)

This paper is organized as follows In Section 2 we formu-late 119889-dimensional Euclidean gravity as a Spin(119889) Yang-Millsgauge theoryThe explicit relations between gravity and gaugetheory variables are established In particular we constructthe dual curvature tensors 퐴퐵 equiv (lowast119866)퐴퐵 = 119889퐴퐵+퐴퐶and퐶퐵

that are necessary for an irreducible representation of LorentzsymmetryWe observe that the geometric structure describedby dual spin connections 퐴퐵 and curvature tensors 퐴퐵

is exactly parallel to the usual one described by (120596퐴퐵 119877퐴퐵)and so clarify why the variety of orientable spin manifolds isdoubled

We apply in Section 3 the gauge theory formulation tosix-dimensional Riemannian manifolds For that purpose wedevise a six-dimensional version of the rsquot Hooft symbolswhich realizes the isomorphism between spin(6) Lorentzalgebra and 119904119906(4) Lie algebra As the spin(6) Lorentz alge-bra has two irreducible spinor representations there areaccordingly two kinds of the rsquot Hooft symbols depending


on the chirality of irreducible spin(6) representations Ourconstruction of six-dimensional rsquot Hooft symbols is newto the best of our knowledge Using this construction weimpose the Kahler condition on the rsquot Hooft symbols Thisis done by projecting the rsquot Hooft symbols to 119880(3)-valuedones and so results in the reduction of the gauge group from119878119880(4) to 119880(3) After imposing the Ricci-flat condition thegauge group in Yang-Mills gauge theory is further reduced to119878119880(3) This result is utilized to show that six-dimensional CYmanifolds can be recast as HYM instantons in 119878119880(3) Yang-Mills gauge theory We elucidate why the canonical splitting(9) of six-dimensional spin manifolds corresponds to thechiral representation of Spin(6) It turns out that this splittingis equally applied to CYmanifolds as well asHYM instantons

In Section 4 we apply the results in Section 3 to CYmanifolds to see how the mirror symmetry between themcan be explained by the doubling of six-dimensional spinmanifolds We observe that it is always possible to find apair of CY manifolds such that their Euler characteristics indifferent chiral representations obey the mirror relation (1)This implies that a pair of CYmanifolds in the opposite chiralrepresentation are mirror to each other as indicated by thearrow (hArr) in Figure 1

In Section 5 we revisit the relation betweenCYmanifoldsand HYM instantons to discuss the mirror symmetry froma completely gauge theory perspective We show that apair of HYM instantons embedded in different complexrepresentations 4 and 4 correspond to a mirror pair of CYmanifolds as summarized in Figure 1This result is consistentwith the mirror symmetry because the integral of the thirdChern class 1198883(119864) for a vector bundle 119864 is equal to the Eulercharacteristic of tangent bundle 119879119872 when 119864 = 119879119872 and thethird Chern class has a desired sign flip between a complexvector bundle 119864 in the fundamental representation and itsconjugate bundle 119864 in the antifundamental representationTherefore we confirm the picture in Figure 1 that the mirrorsymmetry between CY manifolds can be understood asa mirror pair of HYM instantons in holomorphic vectorbundles

Finally we recapitulate in Section 6 the results obtainedin this paper and conclude the paper with a few speculativeremarks

In Appendix A we fix the basis for the chiral repre-sentation of Spin(6) and the fundamental representation of119878119880(4) and list their structure constants In Appendix B wepresent an explicit construction of the six-dimensional rsquotHooft symbols and their algebraic properties in each chiralbasis

2 Gravity as a Gauge Theory

In this section we consider the gauge theory formulation ofRiemannian manifolds taking values in an irreducible spinorrepresentation of the Lorentz group [13 14] This section is toestablish the notation for the doubled variety of Riemannianmanifolds but more detailed exposition will be deferred tothe next section On a Riemannianmanifold119872 of dimension119889 the spin connection is a spin(119889)-valued one form and canbe identified in general with a Spin(119889) gauge field In order to

make an explicit identification between the spin connectionsand the corresponding gauge fields let us first consider the119889-dimensional Dirac algebra (5) where Γ퐴 (119860 = 1 119889) areDiracmatricesThen the spin(119889) Lorentz generators are givenby 119869퐴퐵 = 14 [Γ퐴 Γ퐵] (10)

which satisfy the following Lorentz algebra[119869퐴퐵 119869퐶퐷]= minus (120575퐴퐶119869퐵퐷 minus 120575퐴퐷119869퐵퐶 minus 120575퐵퐶119869퐴퐷 + 120575퐵퐷119869퐴퐶) (11)

The spin connection is defined by 120603 = (12)120603퐴퐵119869퐴퐵 whichtransforms in the standard way as a Spin(119889) gauge field underlocal Lorentz transformations120603 997888rarr 120603耠 = Λ120603Λminus1 + Λ119889Λminus1 (12)

where Λ = 119890(12)휆119860119861(푥)퐽119860119861 isin Spin(119889)In even dimensions the spinor representation is reducible

and its irreducible representations are given by positive andnegative chiral representations In next section we will pro-vide an explicit chiral representation for the six-dimensionalcase The Lorentz generators for the chiral representation aregiven by 119869퐴퐵 = (119869퐴퐵+ 00 119869퐴퐵minus

) (13)

where 119869퐴퐵plusmn = Γplusmn119869퐴퐵 and Γplusmn = (12)(I2[1198892] plusmn Γ푑+1) Thereforethe spin connection in the chiral representation takes theform 120603 = 12120603퐴퐵119869퐴퐵 = (120596(+) 00 120596(minus)

)= 12 (120596(+)

퐴퐵119869퐴퐵+ 00 120596(minus)퐴퐵119869퐴퐵minus

) (14)

Here we used a sloppy notation for 120603 which must beunderstood as 120603 = (12)120603AB119869AB where A = (119860 119860耠) B =(119861 119861耠) and 120603퐴퐵1015840 = 120603퐴1015840퐵 = 0 For a notational simplicity wewill use this notation since it will not introduce too muchconfusion Note that the spin connections 120596(+) and 120596(minus) areconsidered as independent since they resulted from the dou-bling of one form due to the Hodge duality as will be shownlater

Now we introduce a Spin(119889) gauge field defined by

A = 119860푎T푎 = (119860(+)푎119879푎 00 119860(minus)푎 (119879푎)lowast)

T푎 = (119879푎 00 (119879푎)lowast) isin spin (119889) (15)


where119860(plusmn)푎 = 119860(plusmn)푎푀 119889119909푀 (119886 = 1 119889(119889minus1)2) are one-form

connections on 119872 We will take definition (15) by adoptingthe group isomorphism (3) The Lie algebra generators arematrices obeying the commutation relation[T푎 T푏] = minus119891푎푏푐

T푐 (16)

where 119879푎 and (119879푎)lowast are generators in a representation 119877 andits conjugate representation119877 respectivelyThe identificationwe want to make is then given by120603 = 12120603퐴퐵119869퐴퐵 cong A = 119860푎

T푎 (17)

Then the Lorentz transformation (12) can be interpreted as ausual gauge transformation

A 997888rarr A耠 = ΛAΛminus1 + Λ119889Λminus1 (18)

where Λ = 119890휆119886(푥)T119886 isin Spin(119889) The Riemann curvature tensoris defined by [8]

R = 12R퐴퐵119869퐴퐵 = 119889120603 + 120603 and 120603= 12 (119889120603퐴퐵 + 120603퐴퐶 and 120603퐶퐵) 119869퐴퐵= 12 (119877(+)


) (19)

where 119877(plusmn)퐴퐵 = (12)(120597푀120596(plusmn)

푁퐴퐵 minus 120597푁120596(plusmn)푀퐴퐵 + 120596(plusmn)

푀퐴퐶120596(plusmn)푁퐶퐵 minus120596(plusmn)

푁퐴퐶120596(plusmn)푀퐶퐵)119889119909푀and119889119909푁 Or in terms of gauge theory variables

it is given by

F = 119865푎T푎 = 119889A +A andA= (119889119860푎 minus 12119891푏푐푎119860푏 and 119860푐) T푎

= 12 (119865(+)푎119879푎 00 119865(minus)푎 (119879푎)lowast) (20)

where119865(plusmn)푎 = (12)(120597푀119860(plusmn)푎푁 minus120597푁119860(plusmn)푎

푀 minus119891푎푏푐119860(plusmn)푏푀 119860(plusmn)푐

푁 )119889119909푀and119889119909푁As we outlined in Section 1 in addition to the usual

curvature tensor 119877퐴퐵 = 119889120596퐴퐵 + 120596퐴퐶 and 120596퐶퐵 we need tointroduce the dual curvature tensor defined by퐴퐵 equiv (lowast119866)퐴퐵 = 119889퐴퐵 + 퐴퐶 and 퐶퐵 (21)

where 119866퐴퐵 is a (119889 minus 2)-form tensor taking values in spin(119889)Lie algebra One may consider the dual spin connection퐴퐵 equiv (lowast120579)퐴퐵 as the Hodge dual of a (119889 minus 1)-form 120579퐴퐵in spin(119889) Lie algebra It is useful to introduce the adjointexterior differential operator 120575 Ω푘(119872) rarr Ω푘minus1(119872) definedby 120575 = (minus1)푑푘+푑+1 lowast 119889lowast (22)

where the Hodge-dual operator lowast Ω푘(119872) rarr Ω푑minus푘(119872)obeys the well-known relationlowast2120572 = (minus1)푘(푑minus푘) 120572 (23)

for 120572 isin Ω푘(119872) Using the adjoint differential operator 120575 thespin(119889)-valued (119889 minus 2)-form 119866퐴퐵 in (21) can be written as119866퐴퐵 = (minus)푑minus1 120575120579퐴퐵 + 120579퐴퐶 and 120579퐶퐵 (24)

where we devised a simplifying notation120572and120573 equiv lowast ((lowast120572) and (lowast120573)) isin Ω푝+푞minus푑 (119872) (25)

for 120572 isin Ω푝(119872) and 120573 isin Ω푞(119872) Using the nilpotency of theadjoint differential operator 120575 that is 1205752 = 0 one can derivethe (second) Bianchi identity120575119866퐴퐵 + (minus)푑minus1 (120579퐴퐶 and119866퐶퐵 minus 119866퐴퐶 and 120579퐶퐵) = 0 (26)

It may be compared with the ordinary Bianchi identity ingeneral relativity written as119889119877퐴퐵 + 120596퐴퐶 and 119877퐶퐵 minus 119877퐴퐶 and 120596퐶퐵 = 0 (27)

Let us also introduce dual vielbeins 119890퐴 equiv (lowastℎ)퐴 whereℎ퐴 isin Ω푑minus1(119872) in addition to the usual vielbeins 119890퐴 (119860 =1 119889) which independently form a local orthonormalcoframe at each spacetime point in 119872 We combine the dualone-form 119890퐴 with the usual coframe 119890퐴 to define a matrix ofvielbeins

E = E퐴Γ퐴 = ( 0 119890(+)퐴120574퐴119890(minus)퐴120574퐴 0 ) (28)

where 119890(plusmn)퐴 equiv 12 (119890퐴 plusmn 119890퐴) (29)

The coframe basis 119890(plusmn)퐴 isin Γ(119879lowast119872) defines dual vectors119864(plusmn)퐴 = 119864(plusmn)푀

퐴 120597푀 isin Γ(119879119872) by a natural pairing⟨119890(plusmn)퐴 119864(plusmn)퐵 ⟩ = 120575퐴퐵 (30)

The above pairing leads to the relation 119890(plusmn)퐴푀 119864(plusmn)푀퐵 = 120575퐴퐵 Since

we regard the spin connections 120596(+) and 120596(minus) as independentlet us consider two kinds of geometrical data on a spinmanifold119872 dubbed A and B classes

A (119890(+)퐴 120596(+)퐴퐵)

B (119890(minus)퐴 120596(minus)퐴퐵) (31)

We emphasize that the geometric structure of a 119889-dimensional spin manifold can be described by either typeAor type B but they should be regarded as independent eventopologically In other words we can separately consider aRiemannian metric for each class given by1198891199042plusmn = 120575퐴퐵119890(plusmn)퐴 otimes 119890(plusmn)퐵 = 120575퐴퐵119890(plusmn)퐴푀 119890(plusmn)퐵푁 119889119909푀 otimes 119889119909푁equiv 119892(plusmn)

푀푁 (119909) 119889119909푀 otimes 119889119909푁(32)


or ( 120597120597119904)2

plusmn

= 120575퐴퐵119864(plusmn)퐴 otimes 119864(plusmn)

퐵 = 120575퐴퐵119864(plusmn)푀퐴 119864(plusmn)푁

퐵 120597푀 otimes 120597푁equiv 119892푀푁(plusmn) (119909) 120597푀 otimes 120597푁 (33)

In order to recover general relativity from the gaugetheory formulation of gravity it is necessary to impose thetorsion-free condition that is119879(plusmn)퐴 = 119889119890(plusmn)퐴 + 120596(plusmn)퐴

퐵 and 119890(plusmn)퐵 = 0 (34)

As a result the spin connections are determined by vielbeinsthat is 120596(plusmn) = 120596(plusmn)(119890(plusmn)) from which one can deduce the firstBianchi identity 119877(plusmn)

퐴퐵 and 119890(plusmn)퐵 = 0 (35)

where the curvature tensors 119877(plusmn)퐴퐵 are defined by (19) It is not

difficult to see that (35) leads to the symmetry property for theRiemann curvature tensors 119877(plusmn)

퐴퐵 equiv (12)119877(plusmn)퐴퐵퐶퐷119890(plusmn)퐶 and 119890(plusmn)퐷119877(plusmn)

퐴퐵퐶퐷 = 119877(plusmn)퐶퐷퐴퐵 It may be convenient to introduce the

torsion matrix T defined by

T = 119889E + 120603 and E + E and 120603 = ( 0 119879(+)퐴120574퐴119879(minus)퐴120574퐴 0 ) (36)

where we have defined the inverted spin connection 120603 equiv(12)120603퐴퐵119869퐴퐵 = (12) ( 휔(minus)119860119861퐽119860119861minus 0

0 휔(+)119860119861퐽119860119861+

) It is straightforward toshow that119889T = 12 ( 0 119877(+)

퐴퐵 and 119890(+)퐵120574퐴119877(minus)퐴퐵 and 119890(minus)퐵120574퐴 0 ) (37)

and so the first Bianchi identity (35) is automatic because ofthe torsion-free condition T = 0 Similarly using definition(19) it is easy to derive the second Bianchi identity DR equiv119889R + 120603 andR minusR and 120603 = 0 whose matrix form reads as

DR = (119863(+)119877(+) 00 119863(minus)119877(minus)) = 0 (38)

where119863(plusmn)119877(plusmn) equiv 119889119877(plusmn) + 120596(plusmn) and 119877(plusmn) minus 119877(plusmn) and 120596(plusmn) (39)

In terms of gauge theory variables it can be stated as DF equiv119889F + A and F minus F and A = 0 or 119863(plusmn)119865(plusmn) equiv 119889119865(plusmn) + 119860(plusmn) and 119865(plusmn) minus119865(plusmn) and 119860(plusmn) = 0To sum up a 119889-dimensional orientable Riemannian

manifold admits a globally defined volume form whichleads to the isomorphism between Ω푘(119872) and Ω푑minus푘(119872) Inparticular it doubles the two-form vector space which leadsto the enlargement for the geometric structure of RiemannianmanifoldsThe Hodge duality lowast Ω푘(119872) rarr Ω푑minus푘(119872) is thusthe origin of the doubling for the variety of Riemannian

manifolds One is described by (119890퐴 120596퐴퐵 119877퐴퐵) and the otherindependent construction is given by (119890퐴 퐴퐵 퐴퐵) conglowast(ℎ퐴 120579퐴퐵 119866퐴퐵) According to the isomorphism (C) they aredecomposed into two irreducible representations of Lorentzsymmetry In the next section we will apply the irreducibledecomposition to six-dimensional spin manifolds to see whythe variety of Riemannian manifolds is doubled

3 Spinor Representation of Six-DimensionalRiemannian Manifolds

We will apply the gauge theory formulation in the previoussection to six-dimensional Riemannian manifolds For thispurpose the Spin(6) Lorentz group for Euclidean gravity willbe identified with the 119878119880(4) gauge group in Yang-Mills gaugetheory A motivation for the gauge theory formulation of six-dimensional Euclidean gravity is to identify a gauge theoryobject corresponding to a CY manifold and to understandthe mirror symmetry of CYmanifolds in terms of Yang-Millsgauge theory Because our gauge theory formulation is basedon identification (17) we will restrict ourselves to orientablesix-dimensional manifolds with spin structure and considera spinor representation of Spin(6) in order to scrutinize therelationship

Let us start with the Clifford algebraC119897(6) whose genera-tors are given by

C119897 (6) = I8 Γ퐴 Γ퐴퐵 Γ퐴퐵퐶 Γ7Γ퐴퐵 Γ7Γ퐴 Γ7 (40)

where Γ퐴 (119860 = 1 6) are six-dimensional Dirac matricessatisfying the algebra (5) and Γ퐴1퐴2sdotsdotsdot퐴119896 equiv (1119896)Γ[퐴1Γ퐴2 sdot sdot sdot Γ퐴119896]assumes the complete antisymmetrization of indices Γ7 equiv119894Γ1 sdot sdot sdot Γ6 is the chiral matrix given by (A6) Accordingto the isomorphism (4) the Clifford algebra (40) can beisomorphically mapped to the exterior algebra of a cotangentbundle 119879lowast119872

C119897 (6) cong Λlowast119872 = 6⨁푘=0

Ω푘 (119872) (41)

where the chirality operator Γ7 corresponds to the Hodge-dual operator lowast Ω푘(119872) rarr Ω6minus푘(119872)

The spinor representation of Spin(6) can be constructedby 3 fermion creation operators 119886lowast푖 (119894 = 1 2 3) and thecorresponding annihilation operators 119886푗 (119895 = 1 2 3) (seeAppendix 5A in [1]) This fermionic system can be repre-sented in a Hilbert space 119881 of dimension 8 with a Fock vac-uum |Ω⟩ annihilated by all the annihilation operators Thestates in 119881 are obtained by acting the product of 119896 creationoperators 119886lowast푖1 sdot sdot sdot 119886lowast푖119896 on the vacuum |Ω⟩ that is

119881 = 3⨁푘=0

10038161003816100381610038161003816Ω푖1 sdotsdotsdot푖119896⟩ = 3⨁

푘=0

119886lowast푖1 sdot sdot sdot 119886lowast푖119896 |Ω⟩ (42)

The spinor representation of the algebra (40) is reducible andhas two irreducible spinor representations Indeed theHilbertspace 119881 splits into the spinors 119878plusmn of positive and negativechirality that is 119881 = 119878+ oplus 119878minus each of dimension 4 If the


Fock vacuum |Ω⟩ has positive chirality the positive chiralityspinors of Spin(6) are states given by119878+ = ⨁

푘 even

10038161003816100381610038161003816Ω푖1 sdotsdotsdot푖119896⟩ = |Ω⟩ + 10038161003816100381610038161003816Ω푖푗⟩ equiv 4 (43)

while the negative chirality spinors are those obtained by119878minus = ⨁푘 odd

10038161003816100381610038161003816Ω푖1 sdotsdotsdot푖119896⟩ = 1003816100381610038161003816Ω푖⟩ + 10038161003816100381610038161003816Ω푖푗푘⟩ equiv 4 (44)

As spin(6) Lorentz algebra is isomorphic to 119904119906(4) Lie algebrathe positive and negative chirality spinors of spin(6) can beidentified with the fundamental representation 4 and theantifundamental representation 4of 119904119906(4) respectively [1] Asa result the chiral spinor representations 119878+ and 119878minus of Spin(6)are identifiedwith the fundamental representations 4 and 4 of119878119880(4)

One can form a direct product of the fundamentalrepresentations 4 and 4 in order to classify the Cliffordgenerators in (40)

4 otimes 4 = 1 oplus 15 = Γ+ Γ퐴퐵+ (45)

4 otimes 4 = 1 oplus 15 = Γminus Γ퐴퐵minus (46)

4 otimes 4 = 6 oplus 10 = Γ퐴+ Γ퐴퐵퐶+ (47)

4 otimes 4 = 6 oplus 10 = Γ퐴minus Γ퐴퐵퐶minus (48)

where Γplusmn equiv (12)(I8 plusmn Γ7) are the projection operators ontothe space of definite chirality and Γ퐴1퐴2sdotsdotsdot퐴119896plusmn equiv ΓplusmnΓ퐴1퐴2 sdotsdotsdot퐴119896 Note that 15 in (45) and (46) is the adjoint representation of119878119880(4) and 6 and 10 in (47) and (48) are the antisymmetricand symmetric representations of 119878119880(4) respectively SeeAppendix A for the Lie algebra generators in the chiralrepresentation of Spin(6) and the fundamental representationof 119878119880(4) It is important to notice that Γ퐴퐵+ isin 15 and Γ퐴퐵minus isin 15are independent of each other that is [Γ퐴퐵+ Γ퐶퐷minus ] = 0 andthis doubling of the Clifford basis is parallel to the doublingof two forms according to the Clifford isomorphism (41) aswill be clarified below

We want to find the irreducible decomposition of Rie-mann curvature tensors under the Lorentz symmetry as thesix-dimensional version of (8) As we noticed before thereare two kinds of Lorentz generators given by the irreduciblecomponents Γ퐴퐵plusmn = ΓplusmnΓ퐴퐵 which correspond to the chiral andantichiral representations of Lorentz algebra spin(6) cong 119904119906(4)Recall that (19) takes the following split of curvature tensorsR = (12)R퐴퐵119890퐴 and 119890퐵

R퐴퐵 = 119877(+)퐴퐵 oplus 119877(minus)

퐴퐵 = (119865(+)푎퐴퐵 119879푎

1 oplus 119865(minus)푎퐴퐵 119879푎

2 ) = F퐴퐵 (49)

where 119877(plusmn)퐴퐵 = ΓplusmnR퐴퐵 and both 119879푎

1 and 119879푎2 obey the119904119906(4) Lie algebra defined by (16) The doubling of 119904119906(4)

Lie algebra in four-dimensional representations 1198771 and 1198772

on the right-hand side was considered in parallel to thespinor representation on the left-hand side Since the Lorentzgenerators 119869퐴퐵plusmn = Γplusmn119869퐴퐵 are in one-to-one correspondence

with two forms in the vector space (9) we identify thefollowing map 119869퐴퐵+ larrrarr 119865(+)

퐴퐵 119869퐴퐵minus larrrarr 119865(minus)퐴퐵 (50)

Since the role of the chiral operator Γ7 is parallel with theHodge-dual operator lowast Ω푘(119872) rarr Ω6minus푘(119872) the chiralLorentz generators 119869퐴퐵plusmn = (14)(I8 plusmn Γ7)Γ퐴퐵 = (14)(Γ퐴퐵 ∓(1198944)120576퐴퐵퐶퐷퐸퐹Γ퐶퐷퐸퐹) correspond to the canonical split of theenlarged vector space (6) Therefore the two forms 119865(plusmn) =(12)119865(plusmn)

퐴퐵 119890(plusmn)퐴 and 119890(plusmn)퐵 in (49) must be understood as theelement of the irreducible vector space in (9) that is119865(plusmn) isin Ω2

plusmn (119872) (51)

As a result R퐴퐵 on the left-hand side of (49) has twice asmany components as the usual Riemann curvature tensor

Let us summarize the gauge theory formulation in Sec-tion 2 Suppose that 119869퐴퐵lowast and 119879푎

lowast are Lie algebra generatorsin an irreducible representation 119877lowast of Spin(6) and 119878119880(4)respectively First consider an 119878119880(4) gauge field 119861 = 119861푎119879푎

lowast =lowast119862 in the representation 119877lowast obtained by taking the Hodgeduality of a four-form 119862 = 119862푎119879푎

lowast and make the followingidentification = 12 퐴퐵119869퐴퐵lowast cong 119861 = 119861푎119879푎

lowast120579 = 12120579퐴퐵119869퐴퐵lowast cong 119862 = 119862푎119879푎lowast (52)

where = (12)퐴퐵119869퐴퐵lowast is the dual spin connection and = lowast120579 Then the dual curvature tensors (21) and (24) arerespectively written as119865 = 119889119861 + 119861 and 119861 = lowast119867119867 = (minus)푑minus1 120575119862 + 119862and119862 (53)

where119867 is a four-form field strength whose Hodge duality isthe field strength 119865 in 119878119880(4) gauge theory The nilpotency ofexterior differentials 1198892 = 1205752 = 0 immediately leads to theBianchi identity 119889119865 + 119861 and 119865 minus 119865 and 119861 = 0 lArrrArr120575119867 + (minus)푑minus1 (119862 and119867 minus 119867and119862) = 0 (54)

Hence the geometric structure described by the dual variables(퐴퐵 퐴퐵) will be exactly parallel to the usual one describedby (120596퐴퐵 119877퐴퐵)

Thus it is natural to put the two geometric structures onan equal footing Moreover the irreducible representation ofthe Clifford algebra C119897(6) suggests that the curvature tensorsin (51) are given by the combination119865(plusmn) = 12 (119865 plusmn 119865) = 12 (119865 plusmn lowast119867) (55)


One may note that on an orientable (spin) manifold theduplication of curvature tensors always happens by theHodge duality The combination (55) can be understoodas follows One may regard the Riemann tensor R퐴퐵 =(12)R퐴퐵퐶퐷119869퐶퐷 as a linear operator acting on the Hilbertspace 119881 in (42) As R퐴퐵 contains two gamma matrices itdoes not change the chirality of the vector space119881Thereforewe can represent it in a subspace of definite chirality as either119877(+)퐴퐵 119878+ rarr 119878+ or119877(minus)

퐴퐵 119878minus rarr 119878minusThe former case119877(+)퐴퐵 119878+ rarr119878+ takes values in 4 otimes 4 in (45) with a singlet being removed

while the latter case119877(minus)퐴퐵 119878minus rarr 119878minus takes values in 4otimes4 in (46)

with no singlet This implies two independent identificationsdefined by

A 12119877(+)퐴퐵퐶퐷119869퐶퐷+ equiv 119865(+)푎

퐴퐵 (119879푎 oplus 0) B 12119877(minus)

퐴B퐶퐷119869퐶퐷minus equiv 119865(minus)푎퐴퐵 (0 oplus (119879푎)lowast) (56)

where class A(B) acts on the subspace 119878+ (119878minus) of positive(negative) chirality See Appendix A for the irreduciblerepresentation of Spin(6) and 119878119880(4) Because classes A andB in (56) are now represented by 4times4matrices on both sideswe can take a trace operation for the matrices which leads tothe following relations

A 119877(+)퐴퐵퐶퐷 = minus119865(+)푎

퐴퐵 Tr (119879푎119869퐶퐷+ ) equiv 119865(+)푎퐴퐵 120578푎퐶퐷 (57)

B 119877(minus)퐴퐵퐶퐷 = minus119865(minus)푎

퐴퐵 Tr ((119879푎)lowast 119869퐶퐷minus ) equiv 119865(minus)푎퐴퐵 120578푎퐶퐷 (58)

Here we have introduced a six-dimensional analogue of the rsquotHooft symbols defined by120578(plusmn)푎퐴퐵 = minusTr (119879푎

plusmn119869퐴퐵plusmn ) (59)

where we used a bookkeeping notation 120578(+)푎퐴퐵 equiv 120578퐴퐵 120578(minus)푎퐴퐵 equiv120578퐴퐵 and 119879푎+ equiv 119879푎 119879푎

minus equiv (119879푎)lowast They serve as a complete basisof the vector space 15 in (45) and (46) An explicit expressionof the six-dimensional rsquot Hooft symbols and their algebra arepresented in Appendix B

Note that 119865(plusmn)푎 = (12)119865(plusmn)푎퐴퐵 119890(plusmn)퐴 and 119890(plusmn)퐵 in (56) are the

field strengths of 119878119880(4) gauge fieldsThus we introduce a pairof 119878119880(4) gauge fields (119860(+) 119860(minus)) whose field strengths aregiven by 119865(plusmn) = 119889119860(plusmn) + 119860(plusmn) and 119860(plusmn) (60)

The 119878119880(4) gauge field 119860(+) (119860(minus)) is nothing but the spinconnection resident in the vector space 119878+ (119878minus) of positive(negative) chirality that is120596(plusmn) = 12120596(plusmn)

퐴퐵119869퐴퐵plusmn cong 119860(plusmn) = 119860(plusmn)푎119879푎plusmn (61)

Using (B7) the field strengths can be written as 119865(plusmn)푎퐴퐵 =119877(plusmn)

퐴퐵퐶퐷120578(plusmn)푎퐶퐷 = 120578(plusmn)푎퐶퐷 119877(plusmn)퐶퐷퐴퐵 One can apply again the same

expansion to the index pair [119860119861] of the Riemann tensor

119877(plusmn)퐶퐷퐴퐵 That is one can expand the 119878119880(4) field strengths in

terms of the chiral bases in (59)

A 119865(+)푎퐴퐵 = 119891푎푏

(++)120578푏A퐵 (62)

B 119865(minus)푎퐴퐵 = 119891푎푏

(minusminus)120578푏퐴퐵 (63)

As was pointed out in (41) the Clifford algebra (40) isisomorphic to the exterior algebra Λlowast119872 as vector spaces sothe rsquot Hooft symbol in (59) has a one-to-one correspondencewith the basis of two forms in Ω2

plusmn(119872) = Ω2(119872) oplus lowastΩ4(119872)depending on the chirality for a given orientation Conse-quently the six-dimensional Riemann curvature tensors canbe expanded as follows

A 119877(+)퐴퐵퐶퐷 = 119891푎푏

(++)120578푎퐴퐵120578푏퐶퐷 (64)

B 119877(minus)퐴퐵퐶퐷 = 119891푎푏

(minusminus)120578푎퐴퐵120578푏퐶퐷 (65)

Note that the index pairs [119860119861] and [119862119863] in the curvaturetensor 119877(plusmn)

퐴퐵퐶퐷 have the same chirality structure because of thesymmetry property 119877(plusmn)

퐴퐵퐶퐷 = 119877(plusmn)퐶퐷퐴퐵

The Riemann curvature tensor in six dimensions has225 = 15times15 components in total which is the number of theexpansion coefficients119891푎푏

(plusmnplusmn) in each class Because the torsion-free condition has been assumed for the curvature tensorsthe first Bianchi identity 119877(plusmn)

퐴[퐵퐶퐷]= 0 should be imposed

which leads to 120 constraints for each class After all the cur-vature tensor has 105 = 225 minus 120 independent componentswhich must be equal to the number of remaining expansioncoefficients in class A or B after solving the 120 constraints120576퐴퐵퐶퐸퐹퐺119877(plusmn)

퐷퐸퐹퐺 = 0 (66)

It is worthwhile to notice that the curvature tensor automat-ically satisfies the symmetry property 119877(plusmn)

퐴퐵퐶퐷 = 119877(plusmn)퐶퐷퐴퐵 after

dictating the first Bianchi identity (66) Therefore one cansplit the 120 constraints in (66) into the 105 = (15 times 14)2conditions imposing the symmetry 119877(plusmn)

퐴퐵퐶퐷 = 119877(plusmn)퐶퐷퐴퐵 and the

extra 15 conditionsThese extra conditions can bemanifest byconsidering the tensor product of 119878119880(4) [20]

15 otimes 15 = (1 + 15 + 20 + 84)푆 oplus (15 + 45 + 45)퐴푆

(67)

where the first part with 120 components is symmetric andthe second part with 105 components is antisymmetric It isobvious from our construction that 119891푎푏

(plusmnplusmn) isin 15 otimes 15 The 84components in the symmetric part is the number ofWeyl ten-sors in six dimensions and the 21 = 20+1 components refer toRicci tensorsThe remaining 15 components in the symmetricpart are removed by the first Bianchi identity (66) afterexpelling the antisymmetric components in (67)

One can easily solve the symmetry property 119877(plusmn)퐴퐵퐶퐷 =119877(plusmn)

퐶퐷퐴퐵 with the coefficients satisfying119891푎푏(++) = 119891푏푎

(++)119891푎푏(minusminus) = 119891푏푎

(minusminus) (68)


which results in 120 components for each chirality belongingto the symmetric part in (67) Now the remaining 15 condi-tions can be reduced to the equations120576퐴퐵퐶퐷퐸퐹119877(plusmn)

퐶퐷퐸퐹 = 0 (69)

It is obvious that (69) gives rise to a nontrivial relation only forthe coefficients satisfying (68) Finally using (B9) and (B10)(69) can be reduced to the 15 constraints119889푎푏푐119891푏푐

(++) = 119889푎푏푐119891푏푐(minusminus) = 0 (70)

for each sector In the end 119891푎푏(plusmnplusmn) have 105 independent

components for each chirality which precisely match withthe independent components of Riemann curvature tensorsin class A or B (It may be worthwhile to recall the four-dimensional situation [12 13] In four dimensions the firstBianchi identity gives rise to 16 constraints Thus Riemanncurvature tensors have 20 = 36 minus 16 independent compo-nents And the 16 constraints split into 15 ones for 119877퐴퐵퐶퐷 =119877퐶퐷퐴퐵 and one more constraint which reads as 120575푎푏119891푎푏

(++) =120575 푎푏119891 푎푏(minusminus) The last constraint is responsible for the equality of

the Ricci scalar 119904 in the chiral and antichiral sectors in (8)The constraints in (70) correspond to the six-dimensionalanalogue of the last one)

Let us introduce the following (projection) operatoracting on 6 times 6 antisymmetric matrices defined by119875퐴퐵퐶퐷

plusmn equiv 14 (120575퐴퐶120575퐵퐷 minus 120575퐴퐷120575퐵퐶) plusmn 18120576퐴퐵퐶퐷퐸퐹119868퐸퐹= 119875퐶퐷퐴퐵plusmn (71)

where 119868 equiv I3 otimes 1198941205902 Because any 6 times 6 antisymmetric matrixof rank 4 spans a four-dimensional subspace R4 sub R6 theoperator (71) in this case can be written in the four-dimensional subspace as119875퐴퐵퐶퐷

plusmn equiv 14 (120575퐴퐶120575퐵퐷 minus 120575퐴퐷120575퐵퐶) plusmn 14120576퐴B퐶퐷(119860 119861 119862119863) isin R4 (72)

so it reduces to the projection operator for such rank 4matrices that is 119875퐴퐵퐸퐹

plusmn 119875퐸퐹퐶퐷plusmn = 119875퐴퐵퐶퐷

plusmn 119875퐴퐵퐸퐹plusmn 119875퐸퐹퐶퐷

∓ = 0 (73)

Note that 119868퐴퐵 is a 6times6 antisymmetric matrix of rank 6 In thiscase the operator (71) does not act as a projection operatorbut acts as 119875퐴퐵퐶퐷

plusmn 119868퐶퐷 = (12 plusmn 1) 119868퐴퐵 (74)

In general one can deduce by a straightforward calculationthe following properties119875퐴퐵퐸퐹


plusmn + 18119868퐴퐵119868퐶퐷119875퐴퐵퐸퐹plusmn 119875퐸퐹퐶퐷

∓ = minus18119868퐴퐵119868퐶퐷 (75)

After a little algebra one can classify the rsquot Hooft symbolsin (59) into the eigenspaces of the operator (71)119897(+)푎퐴퐵 equiv 12057813퐴퐵 = 11989421205821 otimes 1205902 12057814퐴퐵 = 11989421205822otimes I2 1radic3 (1205788퐴퐵 minus radic212057815퐴퐵) = minus 11989421205823 otimes 1205902 1205786퐴퐵= 11989421205824 otimes 1205902 1205787퐴퐵 = minus 11989421205825 otimes I2 12057811퐴퐵 = 11989421205826otimes 1205902 12057812퐴퐵 = minus 11989421205827otimes I2 2radic3 (minus121205783퐴퐵 + 1radic31205788퐴퐵 + 1radic612057815퐴퐵) = minus 11989421205828otimes 1205902

(76)

119898(+) 푎퐴퐵 equiv 1205781퐴퐵 = 11989421205822 otimes 1205901 1205782퐴퐵 = minus 11989421205822 otimes 1205903 1205789퐴퐵= minus 11989421205825 otimes 1205901 12057810퐴퐵 = 11989421205825 otimes 1205903 1205784퐴퐵 = 11989421205827otimes 1205901 1205785퐴퐵 = minus 11989421205827 otimes 1205903 (77)

119899(+)0퐴퐵 equiv 1205783퐴퐵 + 1radic31205788퐴퐵 + 1radic612057815퐴퐵 = 12119868퐴퐵 = 12 I3otimes 1198941205902 (78)

where 119886 = 1 8 and 119886 = 1 6 are 119904119906(4) indicesin the entries of 119897(+)푎퐴퐵 and 119898(+) 푎

퐴퐵 respectively They obey thefollowing relations 119875퐴퐵퐶퐷

minus 119897(+)푎퐶퐷 = 119897(+)푎퐴퐵 119875퐴퐵퐶퐷+ 119897(+)푎퐶퐷 = 0119875퐴퐵퐶퐷

minus 119898(+) 푎퐶퐷 = 0119875퐴퐵퐶퐷

+ 119898(+) 푎퐶퐷 = 119898(+) 푎

퐴퐵 119875퐴퐵퐶퐷minus 119899(+)0퐶퐷 = minus12119899(+)0퐴퐵 119875퐴퐵퐶퐷+ 119899(+)0퐶퐷 = 32119899(+)0퐴퐵

(79)

Thus the (projection) operators (71) decompose the vectorspace 15 into their eigenspaces as 15 = 8 oplus 6 oplus 1

Similarly one can also classify the rsquot Hooft symbols in(B2) into the eigenspaces of the operator (71)

l(minus)푎퐴퐵 equiv minus12057813퐴퐵 = 11989421205821 otimes 1205902 12057814퐴퐵 = 11989421205822otimes I2 1radic3 (minus1205788퐴퐵 + radic212057815퐴퐵) = minus 11989421205823 otimes 1205902 minus 1205789퐴퐵


= 11989421205824 otimes 1205902 minus 12057810퐴퐵 = minus 11989421205825 otimes I2 1205784퐴퐵 = 11989421205826otimes 1205902 1205785퐴퐵 = minus 11989421205827otimes I2 2radic3 (121205783퐴퐵 + 1radic31205788퐴퐵 + 1radic612057815퐴퐵) = minus 11989421205828otimes 1205902 119898(minus) 푎

퐴퐵 equiv minus1205781퐴퐵 = 11989421205822 otimes 1205901 1205782퐴퐵 = minus 11989421205822 otimes 1205903minus 1205786퐴퐵 = minus 11989421205825 otimes 1205901 minus 1205787퐴퐵 = 11989421205825 otimes 1205903 12057811퐴퐵= 11989421205827 otimes 1205901 12057812퐴퐵 = minus 11989421205827 otimes 1205903 119899(minus)0퐴퐵 equiv minus1205783퐴퐵 + 1radic31205788퐴퐵 + 1radic612057815퐴퐵 = 12119868퐴퐵 = 12 I3otimes 1198941205902 (80)

The same properties such as (79) also hold for the above rsquotHooft symbols

The geometrical meaning of the (projection) operatorsin (71) can be understood as follows Consider an arbitrarytwo-form vector space (we will indicate the superscript (+) or(minus) only when we refer to a quantity belonging to a definitechirality class we will often omit the superscript whenever itis not necessary to specify the chirality class)119865 = 12119865푀푁119889119909푀 and 119889119909푁 = 12119865퐴퐵119890퐴 and 119890퐵 isin Ω2 (119872) (81)

and introduce the 15-dimensional complete basis of twoforms inΩ2

plusmn(119872) for each chirality of spin(6) Lorentz algebra119869푎+ equiv 12120578푎퐴퐵119890(+)퐴 and 119890(+)퐵 isin Ω2+ (119872) 119869푎minus equiv 12120578푎퐴퐵119890(minus)퐴 and 119890(minus)퐴 isin Ω2minus (119872) (82)

It is easy to derive the following identity using (B9) and(B10) 119869푎plusmn and 119869푏plusmn and 119869푐plusmn = 12119889푎푏푐vol (119892(plusmn)) (83)

where vol(119892(plusmn)) = radic119892(plusmn)1198896119909 The Hodge-dual operator lowast Ω푘(119872) rarr Ω6minus푘(119872) is an isomorphism of vector spaceswhich depends upon a metric 119892(plusmn) and the orientation of 119872The nowhere vanishing volume form in (83) guarantees thatthere exists a set of nondegenerate 2-form vector spaces on119872Ωplusmn = 12119868퐴퐵119890(plusmn)퐴 and 119890(plusmn)퐵= 119890(plusmn)1 and 119890(plusmn)2 + 119890(plusmn)3 and 119890(plusmn)4 + 119890(plusmn)5 and 119890(plusmn)6 (84)

This two-form vector space can be wedged with the Hodgestar to construct a diagonalizable operator on Λ2(119872) =Ω2(119872) oplus lowastΩ4(119872) as followslowastΩplusmn

equiv lowast (∙ and Ωplusmn) Ω2 (119872) ∙andΩplusmn997888997888997888997888rarr Ω4 (119872) lowast997888rarr Ω2 (119872) (85)

by lowastΩplusmn(120572) = lowast(120572 and Ωplusmn) for 120572 isin Ω2(119872) After a little

inspection the 15 times 15 matrix representing lowastΩplusmnis found

to have the eigenvalues 2 1 and minus1 with the eigenspaces ofdimensions 1 6 and 8 respectively On any six-dimensionalorientable spin manifold 119872 the space of 2-form Ω2

+(119872) inthe positive chirality space can thus be decomposed into threesubspaces Ω2

+ (119872) = Λ21 oplus Λ2

6 oplus Λ28 (86)

which coincides with the decomposition in (79) The spacesΛ21 and Λ2

6 are locally spanned byΛ21 = Ω+Λ26 = 1198691+ 1198692+ 1198694+ 1198695+ 1198699+ 11986910+ (87)

and Λ28 by Λ2

8 = 1198696+ 1198697+ 11986911+ 11986912+ 11986913+ 11986914+ 119870+ 119871+ (88)

with 119870+ equiv (1radic3)(1198698+ minus radic211986915+ ) and 119871+ equiv (2radic3)(minus(12)1198693+ +(1radic3)1198698+ +(1radic6)11986915+ ) A similar decomposition can be donewith the negative chirality basis 119869푎minus

Note that the entries of Λ21 Λ2

6 and Λ28 coincide with

those of 119899(plusmn)0퐴퐵 119898(plusmn) 푎퐴퐵 and 119897(plusmn)푎퐴퐵 respectively One can quickly

see that this coincidence is not an accident Consider theaction of the projection operator (71) on the two-form (81)which is given by119875퐴퐵퐶퐷

plusmn 119865퐶퐷 = 12 (119865퐴퐵 plusmn 14120576퐴퐵퐶퐷퐸퐹119865퐶퐷119868퐸퐹) (89)

or in terms of form notation2119875plusmn119865 = 119865 plusmn lowast (119865 and Ωplusmn) = 119865 plusmn lowastΩplusmn119865 (90)

It is easy to see that 119865 isin Λ28 if 119875+119865 = 0 so it satisfies theΩ-anti-self-duality equationlowast (119865 and Ωplusmn) = minus119865 (91)

whereas119865 isin Λ26 satisfies theΩ-self-duality equation119875minus119865 = 0

that is lowast (119865 and Ωplusmn) = 119865 (92)

It is not difficult to show [1] that the set 119897(plusmn)푎퐴퐵 119899(plusmn)0퐴퐵 can beidentified with 119906(3) generators which are embedded in119904119900(6) cong 119904119906(4) In general an element of 119880(3) group can berepresented as 119880 = exp(119894 8sum

푎=0

120579푎120582푎) equiv 119890Θ (93)


where 1205820 = I3 is a 3 times 3 unit matrix 120582푎 (119886 = 1 8) arethe 119904119906(3) Gell-Mann matrices and 120579푎rsquos are real parametersfor 119880 to be unitary The 3 times 3 anti-Hermitian matrix Θconsists ofmatrix elementswhich are complex numbersΘ푖푗 =minus(Θ푗푖)lowast (119894 119895 = 1 2 3) and it can easily be embedded intoa 6 times 6 real matrix in 119904119900(6) Lie algebra by replacing Θ푖푗 =ReΘ푖푗+119894 ImΘ푖푗 by the 2times2 realmatrix Θ퐴퐵 = I2 sdotReΘ푖푗+1198941205902 sdotImΘ푖푗 A straightforward calculation (see (B17)) shows thatthe resulting 6 times 6 antisymmetric real matrix Θ퐴퐵 can bewritten asΘ퐴퐵 = 2 (1205790119899(plusmn)0퐴퐵 + 120579푎119897(plusmn)푎퐴퐵 ) = 119875퐴퐵퐶퐷

minus Θ퐶퐷 + 31205790119899(plusmn)0퐴퐵 (94)

Note that 119880(3) is the holonomy group of Kahler mani-folds That is the projection operators in (71) can serve toproject a Riemannian manifold whose holonomy group is119878119874(6) into a Kahler manifold with 119880(3) holonomy Let usshow that it is indeed the case Suppose that 119872 is a complexmanifold Let us introduce local complex coordinates 119911훼 =1199091+1198941199092 1199093+1198941199094 1199095+1198941199096 120572 = 1 2 3 and their complex con-jugates 119911훼 120572 = 1 2 3 in which a complex structure 119869 takesthe form 119869훼훽 = 119894120575훼훽 119869훼훽 = minus119894120575훼훽 [1] Note that relative tothe real basis 119909푀 119872 = 1 6 the complex structure isgiven by 119869 = 119868 = I3 otimes 1198941205902 which was already introducedin (71) We further impose the Hermitian condition on thecomplex manifold119872 defined by 119892(119883 119884) = 119892(119869119883 119869119884) for any119883119884 isin Γ(119879119872) This means that the Riemannian metric 119892 onthe complex manifold119872 is a Hermitian metric that is 119892훼훽 =119892훼훽 = 0 119892훼훽 = 119892훽훼 The Hermitian condition can be solvedby taking the vielbeins as119890푖훼 = 119890푖훼 = 0119864훼

푖 = 119864훼

푖= 0 (95)

where a tangent space index119860 = 1 6 has been split into aholomorphic index 119894 = 1 2 3 and an antiholomorphic index119894 = 1 2 3 This in turnmeans that 119869푖푗 = 119894120575푖푗 119869푖푗 = minus119894120575푖푗 Thenone can see that the two-formΩplusmn in (84) is aKahler form thatisΩplusmn(119883 119884) = 119892(plusmn)(119869119883 119884) and it is given byΩplusmn = 119894119890(plusmn)푖 and 119890(plusmn)푖 = 119894119890(plusmn)푖훼 119890(plusmn)푖

훽119889119911훼 and 119889119911훽= 119894119892(plusmn)

훼훽119889119911훼 and 119889119911훽 (96)

where 119890(plusmn)푖 = 119890(plusmn)푖훼 119889119911훼 is holomorphic one form and 119890(plusmn)푖 =119890(plusmn)푖훼

119889119911훼 is antiholomorphic one form It is also easy to seethat the condition for a Hermitian manifold (119872 119892(plusmn)) to beKahler that is 119889Ωplusmn = 0 is equivalent to the one where thespin connection 120596(plusmn)

퐴퐵 is 119880(3)-valued that is120596(plusmn)푖푗 = 120596(plusmn)

푖푗= 0 (97)

Therefore the spin connection after theKahler condition (97)can be written as the form (94)

All the above results can be clearly understood by theproperties of Spin(6) and 119878119880(4) groups Introducing complexcoordinates onR6 means that one has to consider the Lorentzsubgroup 119880(3) sub 119878119880(4) acting on C3 sub C4 and so onedecomposes the 4 and 4 of 119878119880(4) as 4 = 11 oplus 3minus13 and4 = 1minus1oplus313 under119880(3) = 119880(1)times119878119880(3)where the subscriptsdenote 119880(1) charges Using the branching rule of 119878119880(4) sup119880(1) times 119878119880(3) [20] one can get the following decompositionsafter removing 119878119880(4) singlets

4 otimes 4 minus 1 = (3 otimes 3)0oplus (3minus43 oplus 343)= (8 oplus 1)0 oplus (3minus43 oplus 343)


(98)

The spin connection 120596퐴퐵 isin 15 can be decomposed accordingto the above branching rule as120596푖푗 isin 3minus43120596푖푗 isin 343120596푖푗 minus 13120575푖j120596푘푘 isin 80120596푖푖 isin 10

(99)

Hence the Kahler condition (97)means that spin connectionsin 3minus43 and 343 decouple from the theory and only thecomponents in 80 and 10 survive It is now obvious why wecould have such decompositions in (76)ndash(80) in which 119897(plusmn)푎퐴퐵 isin80119898(plusmn) 푎

퐴퐵 isin (3minus43 oplus 343) and 119899(plusmn)0퐴퐵 isin 10One can rephrase the Kahler condition (97) using the

gauge theory formalism From the identification 120596(plusmn) equivΓplusmn120603 = 119860(plusmn)푎119879푎plusmn in (61) we get the relation120596(plusmn)

퐴퐵 = 119860(plusmn)푎120578(plusmn)푎퐴퐵 119860(plusmn)푎 = minus2Tr (120596(plusmn)119879푎plusmn) (100)

We will focus on type A case as the same analysis can beapplied to type B case If (119872 119892(+)) is a Kahler manifold (97)means that119860(+)1 = 119860(+)2 = 119860(+)4 = 119860(+)5 = 119860(+)9 = 119860(+)10 = 0 (101)

because 120578푎푖푗 = 0 only for 119886 = 1 2 4 5 9 10 otherwise 120578푎푖푗 = 0See (B17) This result is consistent with the branching rule(99) that is 119898(+) 푎

퐴퐵 isin (3minus43 oplus 343) In other words 119860(+) 푎 =0 and so the gauge fields take values in 119906(3) Lie algebraaccording to the result (94) Then the 119878119880(4) structure con-stants 119891푎푏푐 in Table 1 guarantee that the corresponding fieldstrengths also vanish that is119865(+)푎 = 12119891푎푏

(++)120578푏퐴퐵119890(+)퐴 and 119890(+)퐵 = 0 (102)


Table 1 The nonvanishing structure constants 119891푎푏푐119886 119887 119888 119891푎푏푐

1 2 3 11 4 7 121 5 6 minus121 9 12 121 10 11 minus122 4 6 122 5 7 122 9 11 122 10 12 123 4 5 123 6 7 minus123 9 10 123 11 12 minus124 5 8 radic324 9 14 124 10 13 minus125 9 13 125 10 14 126 7 8 radic326 11 14 126 12 13 minus127 11 13 127 12 14 128 9 10 12radic38 11 12 12radic38 13 14 minus1radic39 10 15 radic2311 12 15 radic2313 14 15 radic23for 119886 = 1 2 4 5 9 10 Thus we get 119891 푎푏

(++) = 0 for forall119887 =1 15 This immediately leads to the conclusion that119865(+)푎 = 119889119860(+)푎 minus 12119891푎푏푐119860(+)푏 and 119860(+)푐 = 119891푎푏(++)119869푏+isin Λ2

8 oplus Λ21 (103)

where 119865(+)푎 119886 = 0 1 8 are the field strengths of 119880(3)gauge fields As will be shown below 119865(+)0 isin Λ2

1 is the fieldstrength of the119880(1) part of119880(3) spin connections and119865(+)푎 isinΛ2

8 119886 = 1 8 belong to the 119878119880(3) part In particular as119865(+)푎 isin Λ28 they satisfy the Ω-anti-self-duality equation (91)

known as the HYM equation [17 18]119865(+)푎 = minus lowast (119865(+)푎 and Ω+) 119886 = 1 8 (104)

It is well-known [1] that the Ricci-tensor of a Kahlermanifold is the field strength of the 119880(1) part of the 119880(3)spin connection Therefore the Ricci-flat condition can bestated as 119865(+)0 = 0 One can explicitly check it as followsRecall that 119865(+)푎

퐴퐵 = 119891푎푏(++)120578푏퐴퐵 Using the result (102) one

can see that the nonzero components of 119891푎푏(++) run only over

(119886 119887) isin 3 6 7 8 11 12 13 14 15 Thereby the constraint(70) becomes nontrivial only for those values As a resultthe number of independent components of 119891푎푏

(++) is given by(9 times 10)2 minus 9 = 36 The Ricci-flat condition 119877(+)퐴퐵 equiv 119877(+)

퐴퐶퐵퐶 =119891푎푏(++)120578푎퐴퐶120578푏퐵퐶 = 0 further constrains the coefficients A close

inspection shows that out of 21 equations 119877(+)퐴퐵 = 0 only 9

equations are independent and after utilizing the constraint(70) the equations for the Ricci-flatness can be succinctlyarranged as 1198913푎

(++) + 1radic31198918푎(++) + 1radic611989115푎

(++) = 0 (105)

The above condition means that119865(+)0퐴퐵 = (1198913푎

(++) + 1radic31198918푎(++) + 1radic611989115푎

(++)) 120578푎퐴퐵= 119865(+)3퐴퐵 + 1radic3119865(+)8

퐴퐵 + 1radic6119865(+)15퐴퐵 = 0 (106)

If one introduces a gauge field defined by119860(+)0 equiv 120596(+)퐴퐵119899(+)0퐴퐵 = 119860(+)3 + 1radic3119860(+)8 + 1radic6119860(+)15 (107)

one can show that the field strength in (106) is given by119865(+)0 = 119889119860(+)0 (108)

after using the fact that the 119880(3) structure constants 119891푎푏푐

satisfy the following relation1198913푎푏 + 1radic31198918푎푏 + 1radic611989115푎푏 = 0 (109)

The relation (109) is easy to understand because the 119880(1)part among the 119880(3) structure constants has to vanish Thisestablishes the result that the Ricci-flatness is equal to thevanishing of the 119880(1) field strength That is 119865(+)0 = 119889119860(+)0 isinΛ2

1 has a trivial first Chern classThe same result can be obtained for type B case The

Kahler condition (97) can similarly be solved by119860(minus)1 = 119860(minus)2 = 119860(minus)6 = 119860(minus)7 = 119860(minus)11 = 119860(minus)12 = 0 (110)

Note that the entries of 119880(3) generators for type B case aredifferent from those for type A case The Ricci-flat condition119877(minus)퐴퐵 equiv 119877(minus)

퐴퐶퐵퐶 = 119891푎푏(minusminus)120578푎퐴퐶120578푏퐵퐶 = 0 leads to the equationminus1198913푎(minusminus) + 1radic31198918푎

(minusminus) + 1radic611989115푎(minusminus) = 0 (111)

It is equivalent to the vanishing of119880(1) field strength that is119865(minus)0 = 119889119860(minus)0 = 0 where the 119880(1) gauge field is defined by119860(minus)0 equiv 120596(minus)퐴퐵119899(minus)0퐴퐵 = minus119860(minus)3 + 1radic3119860(minus)8 + 1radic6119860(minus)15 (112)

This fact can be derived by using the fact that the 119880(3)structure constants 119891푎푏푐 for type B case satisfy the followingrelation minus1198913푎푏 + 1radic31198918푎푏 + 1radic611989115푎푏 = 0 (113)


where 119886 119887 now run over 3 4 5 8 9 10 13 14 15 Hence onecan see that CYmanifolds for typeB case also obey the HYMequations 119865(minus)푎 = minus lowast (119865(minus)푎 and Ωminus) 119886 = 1 8 (114)

Note that the HYM equations for a vector bundle 119864over a CY manifold 119872 define a set of differential equationssatisfied by the gauge fields of the vector bundle 119864 rarr 119872In general the connection of a vector bundle 119864 rarr 119872 overa CY manifold 119872 is not related to the spin connection ofthe CY manifold 119872 unless 119864 = 119879119872 For example the 119866-bundle over a CY manifold 119872 is such a case However inour case we have adopted the so-called standard embedding119864 = 119879119872 where 119879119872 is the tangent bundle of a CY manifold119872 Then the gauge fields of 119864 = 119879119872 are the spin connectionson the tangent bundle 119879119872 of the CY manifold 119872 Sincethe tangent bundle 119879119872 is defined by the CY manifold 119872itself the HYM instanton in this case is inherited from theCY manifold Therefore the HYM instanton for the tangentbundle 119879119872 cannot be identified with an ordinary Yang-Millsinstanton on a fixed background manifold since the Yang-Mills connection of 119879119872 is directly determined by the CYmanifold119872

In summary the Kahler condition (97) projects the rsquotHooft symbols to119880(3)-valued ones in 10oplus80 and results in thereduction of the gauge group from 119878119880(4) to 119880(3) The Ricci-flatness is equivalent to the condition119865(plusmn)0 = 119889119860(plusmn)0 = 0 isin 10so the gauge group is further reduced to 119878119880(3) Remainingspin connections are 119878119880(3) gauge fields that belong to 80 andsatisfy the HYM (104) or (114) As a Kahler manifold with thetrivial first Chern class is a CY manifold we see that the CYcondition is equivalent to theHYM equations whose solutionis known as HYM instantons [1] Consequently we find thata six-dimensional CY manifold automatically satisfies theHYM equations in 119878119880(3) Yang-Mills gauge theory but theconverse is not generally true

4 Mirror Symmetry of Calabi-Yau Manifolds

In this section we want to explore the geometrical propertiesof six-dimensional Riemannian manifolds in the irreduciblerepresentations A and B In Section 2 we have introduceddual vielbeins 119890퐴 = (lowastℎ)퐴 and dual spin connections퐴퐵 = (lowast120579)퐴퐵 in addition to usual ones (119890퐴 120596퐴퐵) The dualgeometric structure described by (119890퐴 퐴퐵) cong (ℎ퐴 120579퐴퐵) isbasically originated from the Hodge duality of the exterior

algebra Λlowast119872 on an orientable manifold119872 According to thechiral structure of irreducible representations in (45)ndash(48)we have associated two geometric structures (119890(+)퐴 120596(+)

퐴퐵) and(119890(minus)퐴 120596(minus)퐴퐵) on a spin manifold119872 where119890(plusmn)퐴 = 12 (119890 plusmn 119890)퐴 = 12 (119890 plusmn lowastℎ)퐴 120596(plusmn)

퐴퐵 = 12 (120596 plusmn )퐴퐵 = 12 (120596 plusmn lowast120579)퐴퐵 (115)

This means that there are two independent ways to charac-terize a six-dimensional spin manifold Accordingly we canconsider two kinds of Riemannian manifolds depicted by themetrics 1198891199042A = 119890(+)퐴 otimes 119890(+)퐴1198891199042B = 119890(minus)퐴otimes(minus)퐴 (116)

where A and B refer to their chirality class Each of themetrics defines their own spin connections 120596(plusmn)

퐴퐵 = 120596(plusmn)퐴퐵(119890(plusmn))

through the torsion-free condition (34) Generally speakingthe six-dimensional spin manifolds described by A and B

metrics (116) are independent of each other so the variety issimply doubled due to the Hodge duality on Λlowast119872

The spin connections can take arbitrary values as far asthey satisfy the integrability condition (35) Their symmetryproperties can be characterized by decomposing them intothe irreducible subspaces under 119878119874(6) group

=otimes oplusABC isin otimes = = oplus (117)

where = 20 is a completely antisymmetric part of spinconnections defined by 120596[퐴퐵퐶] = (13)(120596퐴퐵퐶 +120596퐵퐶퐴 +120596퐶퐴퐵)In six dimensions the spin connections120596[퐴퐵퐶]maybe furtherdecomposed into (imaginary) self-dual (sd) and anti-self-dual (asd) parts that is120596[퐴퐵퐶] = (120596푠푑

[퐴퐵퐶] isin 10) oplus (120596푎푠푑[퐴퐵퐶] isin 10) (118)

The above decomposition may be shaky because = 20 isalready an irreducible representation of 119878119874(6) It is just fora heuristic comparison with the irreducible 119878119880(4) represen-tation Note that 6 is coming from the antisymmetric tensorin 4 times 4 in (47) or 4 times 4 in (48) Thus under 119878119880(4) groupone can instead get the following decomposition of the spinconnections [20]

= = oplus = oplus oplus = otimesABC isin otimes = = (119)

Hence notice that the irreducible representation of 119878119880(4) forspin connections is more refined than the irreducible spinorrepresentation of 119878119874(6) Given a metric 1198891199042 = 119890퐴 otimes 119890퐴 one can determine the

spin connection 120596퐴퐵 using the torsion free condition 119879퐴 =119889119890퐴 + 120596퐴퐵 and 119890퐵 = 0 Because we are dictating an irreducible


spinor representation of local Lorentz symmetry for the iden-tification (17) it is necessary to specify which representationis chosen to embed the spin connection120596퐴퐵 One can equallychoose either the positive or negative chiral representationThis situation may be familiar with a supersymmetric solu-tion in supergravity To be specific consider the supersymme-try transformation of six-dimensional gravitino Ψ푀 given by120575Ψ푀 = 119863푀120578 where aDirac operator119863푀 = 120597푀+120596푀 acts on achiral spinor 120578Then a background geometry obeying 120575Ψ푀 =119863푀120578 = 0 must satisfy a well-known condition [119863푀 119863푁]120578 =(12)119877푀푁푃푄119869푃푄120578 = 0 In this case the representation isdetermined by an unbroken supersymmetry generated bythe chiral spinor 120578 Hence the corresponding 119878119880(4) gaugefields are also identified according to themap (61) dependingon the chiral representation chosen by the supersymmetricbackground geometry Whenever a metric is known in aspecific chiral representation one can determine the coeffi-cients 119891푎푏

(++) in (64) or 119891푎푏(minusminus) in (65) through the explicit cal-

culation of Riemann curvature tensors Since the geometricstructures described by the data (119890(+)퐴 120596(+)

퐴퐵) and (119890(minus)퐴 120596(minus)퐴퐵)

are completely independent of each other one can attributethem to two different Riemannian manifolds

So let us denote the geometric structures (119890(+)퐴 120596(+)퐴퐵)

and (119890(minus)퐴 120596(minus)퐴퐵) by A and B respectively according to

(31) Suppose that the geometric structures (AB) describea pair of six-dimensional spin manifolds (119872+119872minus) Sinceeach manifold can be described by either A-type or B-type the geometric data for the pair are given by either(119872+(A)119872minus(B)) or (119872+(B)119872minus(A))This pairing has shownup in Figure 1 In general the manifolds (119872+119872minus) in thepair are assumed to be different even topologically Giventheir metrics for the pair one can determine the coefficients(119891푎푏

(++) 119891푎푏(minusminus)) in (64) and (65) Since the pair consist of

independent manifolds it is possible to arrange the pairsuch that the coefficients (119891푎푏

(++) 119891푎푏(minusminus)) obey some relation

for example (1) To be specific let us choose the embedding(119872+(A)119872minus(B)) Thus one CY manifold 119872+ = 119872 isdescribed by themetric 1198891199042A = 119890(+)퐴otimes119890(+)퐴 of typeAwhile theother CYmanifold119872minus = is described by the metric 1198891199042B =119890(minus)퐴 otimes 119890(minus)퐴 of type B The Euler characteristic 120594(119872) of asix-dimensional Riemannian manifold119872 is defined by120594 (119872) = 6sum

푟=0

(minus)푟 119887푟 (120)

where 119887푟 = sum푝+푞=푟 ℎ푝푞(119872) is the 119903th Betti number A mirrorpair of CY manifolds (119872 ) obeys the property ℎ푝푞(119872) =ℎ3minus푝푞() This property leads to an important result that theEuler characteristic of themirrormanifold has an oppositesign that is 120594() = minus120594(119872) Thus the mirror symmetryimplies that everyCYmanifold has a partnerwith an oppositeEuler characteristic We will use this fact to identify a mirrorCY manifold

Recall that the CY manifold 119872 is of type A while theother CY manifold is of type B Thus the spin connectionof 119872 () acts on the spinor vector space 119878+ = 4 (119878minus =4) of positive (negative) chirality Since classes A and B are

completely independent and defined in the different vectorspaces one can choose the pair (119872 ) such that their Eulercharacteristics satisfy the relation 120594(119872) = minus120594() and so themirror relation (1) Let us explain why this is possible

Every complex vector bundle 119864 of rank 119899 has an underly-ing real vector bundle 119864R of rank 2119899 obtained by discardingthe complex structure on each fiber Then the top Chernclass of a complex vector bundle 119864 is the Euler class of itsrealization [21] 119888푛 (119864) = 119890 (119864R) (121)

where 119899 = rank119864Therefore the Euler characteristic 120594(119872) of119872 for a tangent bundle 119864R = 119879119872 is given by the integral ofthe top Chern class 120594 (119872) = int

푀119888푛 (119864) (122)

Recall that if 119864 is a complex vector bundle then there existsa dual or conjugate bundle 119864 with an opposite complexstructure whose 119895th Chern class is given by [9 21]119888푗 (119864) = (minus1)푗 119888푗 (119864) (123)

The Euler characteristic 120594(119872) for a six-dimensionalRiemannian manifold119872 is given by120594 (119872) equiv minus 127 sdot 31205873

int푀

120576퐴1퐴2 sdotsdotsdot퐴6119877퐴1퐴2and 119877퐴3퐴4

and 119877퐴5퐴6= minus 1210 sdot 31205873sdot int푀

1198896119909120576푀1푀2 sdotsdotsdot푀6120576퐴1퐴2sdotsdotsdot퐴6119877푀1푀2퐴1퐴2119877푀3푀4퐴3퐴4

119877푀5푀6퐴5퐴6 (124)

On one hand for type A in (57) where 119877(+)퐴퐵 = 119865(+)푎120578푎퐴퐵 it is

given by120594+ (119872)= minus 127 sdot 31205873int푀

120576퐴1퐴2sdotsdotsdot퐴6119877(+)퐴1퐴2

and 119877(+)퐴3퐴4

and 119877(+)퐴5퐴6= minus 1210 sdot 31205873

int푀

(120576퐴1퐴2sdotsdotsdot퐴6120578푎퐴1퐴2120578푏퐴3퐴4120578푐퐴5퐴6) 119865(+)푎

and 119865(+)푏 and 119865(+)푐= minus 1961205873int푀

119889푎푏푐119865(+)푎 and 119865(+)푏 and 119865(+)푐(125)

where (B9) was used On the other hand for type B in (58)where 119877(minus)

퐴퐵 = 119865(minus)푎120578푎퐴퐵 the Euler characteristic (124) can bewritten as120594minus ()= 127 sdot 31205873

int푀

120576퐴1퐴2 sdotsdotsdot퐴6119877(minus)퐴1퐴2

and 119877(minus)퐴3퐴4

and 119877(minus)퐴5퐴6= 1961205873

int푀

119889푎푏푐119865(minus)푎 and 119865(minus)푏 and 119865(minus)푐 (126)


where (B10) was used (in order to define the Euler charac-teristic for type B it is considered that the flip of chiralitycorresponds to the parity transformation (see Appendix A)and so the orientation reversal That is the reason for the signflip of 120594minus() But there is some ambiguity for the choice ofsign because the six-dimensional Euler characteristic needsnot be positive unlike the four-dimensional case This signambiguity is insignificant since it can be compensated withthe redefinition 119877(minus)

퐴퐵 rarr minus119877(minus)퐴퐵 Hence one may keep the

same sign convention for 120594+(119872) and 120594minus() In any case themirror pair (119872 )will be defined by the condition 120594+(119872) =minus120594minus() The doubling of geometric variety guarantees thefreedom to arrange a mirror pair (119872 ) to satisfy the rela-tion 120594+(119872) = minus120594minus()) It is straightforward to represent theabove Euler characteristics in terms of the chiral bases (62)and (63) For type A where 119865(+)푎 = 119891푎푏

(++)119869푏+ 120594+(119872) using theidentity (83) reads as120594+ (119872)= minus 11921205873

int푀

1198896119909radic119892(+)119889푎푏푐119889푑푒푓119891푎푑(++)119891푏푒

(++)119891푐푓

(++) (127)

Similarly 120594minus() for type B where 119865(minus)푎 = 119891푎푏(minusminus)119869푏minus can be

written as120594minus ()= 11921205873int푀

1198896119909radic119892(minus)119889푎푏푐119889푑푒푓119891푎푑(minusminus)119891푏푒

(minusminus)119891푐푓

(minusminus) (128)

Recall that two irreducible spinor representations ofSpin(6) can be identified with the fundamental and antifun-damental representations of 119878119880(4) By choosing a complexstructure the Spin(6) tangent bundle 119879119872 reduces to a 119880(3)vector bundle 119864 In order to utilize the relation (122) let usconsider the 119880(3) sub 119878119880(4) subbundle 119864 such that 119879119872 otimes C =119864 oplus 119864 Note that 119880(3) does not mix an underlying complexstructureThus we embed classA into the119880(3) vector bundle119864 over 119872 Similarly by considering the complexification119879 otimes C = 119865 oplus 119865 class B is embedded into the 119880(3) vectorbundle 119865 over It is important to recall that the curvaturecoefficients 119891푎푏

(++) and 119891푎푏(minusminus) are determined by completely

independent metrics 119892(+) on 119872 and 119892(minus) on respectivelyTherefore it is always possible to find a pair (119872 ) such thatthe Euler characteristics (127) for typeA and (128) for type Bhave a precisely opposite sign that is120594+(119872) = minus120594minus() For aCYmanifold119872whose holonomy is 119878119880(3) the structure con-stants 119889푎푏푐 take values only in the 119904119906(3) sub 119906(3) Lie algebra Inthis case the Euler characteristic 120594(119872) is given by [1]120594 (119872) = 2 (ℎ11 (119872) minus ℎ21 (119872)) (129)

Considering the definition of the Hodge number ℎ푝푞(119872) =dim119867푝푞(119872) ge 0 the sign flip of the Euler characteristics120594+(119872) = minus120594minus() can be explained if the pair (119872 ) satisfythe mirror relation ℎ11 (A) = ℎ21 (B) ℎ11 (B) = ℎ21 (A) (130)

Indeed themirror relation (130) is the only way to explain thesign flip of the Euler characteristic

The mirror symmetry (130) can be further clarified byusing the fact that the Euler characteristic 120594(119872) of a spinmanifold 119872 is related to the index of the Dirac operator on119872 [2] Denote the Dirac index for fermion fields in a rep-resentation 119877 by index(119877) The Euler characteristic 120594(119872) isthen given by 120594 (119872) = index (119877) minus index (119877) (131)

where 119877 is the complex conjugate representation of 119877 Let 4be the fundamental representation of 119878119880(4) and 4 its complexconjugate that is the antifundamental representation Thenindex(4) = minusindex(4) since in six dimensions the complexconjugation exchanges positive and negative chirality zeromodes while also exchanging 4 and 4 Under the 119878119880(3)representation 4 = 1 oplus 3 and 4 = 1 oplus 3 where index(1) =index(1) = 0 so the Euler characteristic (131) is given by [2]120594 (119872) = index (4) minus index (4) = 2 index (4)= index (3) minus index (3) = 2 index (3) (132)

Then the identity (132) immediately implies the relation120594+(119872) = minus120594minus() for a pair of spin manifolds embedded inthe opposite chirality representations 4 and 4 (or 3 and 3 forCY manifolds) This result is consistent with the mirrorsymmetry (130) since A cong 3 and B cong 3

5 Mirror Symmetry from Gauge Theory

In Section 3 the six-dimensional Euclidean gravity has beenformulated as 119878119880(4) cong Spin(6) Yang-Mills gauge theory Itwas shown that a Kahlermanifold is described by the reduced119880(3) sub 119878119880(4) gauge symmetry After imposing the Ricci-flat condition on the Kahler manifold the gauge group inthe Yang-Mills theory is further reduced to 119878119880(3) After all aCY manifold 119872 from the gauge theory point of view can bedescribed by 119878119880(3) connections supported on 119872 satisfyingthe HYM equations And the mirror symmetry says that aCY manifold has a mirror pair satisfying the relation (1)Therefore there must be a corresponding HYM instantonwhich can be derived from amirror CYmanifold obeying themirror relation (130) In this section we will identify the mir-ror HYM instanton from the gauge theory approach and thenclarify the mirror symmetry between CY manifolds fromthe gauge theory formulation

Suppose that themetric of a six-dimensional Riemannianmanifold119872 is given by1198891199042 = 119892푀푁 (119909) 119889119909푀119889119909푁 (133)

Let 120587 119864 rarr 119872 be an 119878119880(4) bundle over119872 whose curvatureis defined by119865 = 119889119860 + 119860 and 119860= 12 (120597푀119860푁 minus 120597푁119860푀 + [119860푀 119860푁]) 119889119909푀 and 119889119909푁 (134)


where 119860 = 119860푎푀(119909)119879푎119889119909푀 is a connection one-form vector

space on the vector bundle 119864 The generators 119879푎 of 119904119906(4) Liealgebra satisfy the commutation relation (16) with normal-ization Tr119879푎119879푏 = minus(12)120575푎푏 Consider the 119878119880(4) Yang-Millsgauge theory defined on the Riemannian manifold 119872 withthe metric (133) whose action is given by119878푌푀 = minus 121198922

푌푀

int푀

1198896119909radic119892119892푀푃119892푁푄Tr119865푀푁119865푃푄 (135)

Using the projection operator (71) and the identity (75) it iseasy to derive the following formula(119875plusmn119865)2 = (119875퐴1퐵1퐴2퐵2

plusmn 119865퐴2퐵2) (119875퐴1퐵1퐴3퐵3plusmn 119865퐴3퐵3)= 14 (119865퐴1퐵1 plusmn 14120576퐴1퐵1퐴2퐵2퐴3퐵3119865퐴2퐵2119868퐴3퐵3)2

= 12119865퐴퐵119865퐴퐵 plusmn 18120576퐴퐵퐶퐷퐸퐹119865퐴퐵119865퐶퐷119868퐸퐹+ 18 (119868퐴퐵119865퐴퐵)2= 119875퐴퐵퐶퐷plusmn 119865퐴퐵119865퐶퐷 + 18 (119868퐴퐵119865퐴퐵)2

(136)

One can rewrite the action (135) using the above identity as119878푌푀 = minus 141198922푌푀sdot int

푀1198896119909radic119892Tr [(119865퐴퐵 plusmn 14120576퐴퐵퐶퐷퐸퐹119865퐶퐷119868퐸퐹)2

minus 12 (119868퐴퐵119865퐴퐵)2 ∓ 12120576퐴퐵퐶퐷E퐹119865퐴퐵119865퐶퐷119868퐸퐹] (137)

The above action can be written in a more compact form as119878푌푀 = minus 141198922푌푀

int푀

1198896119909radic119892sdot Tr [(119865퐴퐵 plusmn lowast (119865 and Ω)퐴퐵)2 minus 12 (119868퐴퐵119865퐴퐵)2]plusmn 11198922

푌푀

int푀

1198896119909Tr119865 and 119865 and Ω (138)

where Ω is the two-form vector space of rank 6 defined by(84)

Using the fact

Tr119865 and 119865 = 119889Tr(119860 and 119865 minus 13119860 and 119860 and 119860) equiv 119889119870 (139)

one can see that the last term in (138) is a topological termthat is

Tr119865 and 119865 and Ω = 119889 (119870 and Ω) (140)

if and only if the two-form Ω is closed that is 119889Ω =0 In other words when 119872 is a Kahler manifold the

last term in (138) depends only on the topological classof the Kahler-form Ω and the vector bundle 119864 over 119872Note that (138) except the second term is very similar tothe Bogomolrsquonyi equation for Yang-Mills instantons whoseaction is bounded by a topological term Indeed we can applythe Bogomolrsquonyi argument to (138) thanks to the identity(18)120576퐴퐵퐶퐷퐸퐹119868퐶퐷119868퐸퐹 = 119868퐴퐵 More precisely it is easy to seethat a solution obeying the Ω-self-duality equations119865퐴퐵 plusmn lowast (119865 and Ω)퐴퐵 = 0 (141)

automatically satisfies the condition119868퐴퐵119865퐴퐵 = 0 (142)

Therefore the minimum action can be achieved by theconfiguration satisfying (141) and is given by the lasttermmdashthe topological termmdashin (138) Note that we havealready encountered the above self-duality equations in (91)and (92) They can be summarized as the so-called DUYequations [17 18] 119865(20) = 119865(02) = 0 (143)119865 and Ω2 = 0 (144)

The first equation states that the 119878119880(4) gauge field is aconnection on a holomorphic vector bundle and the lastcondition corresponds to the stability of the holomorphicvector bundle in algebraic geometry It is straightforward toshow [1 22] that a solution of the self-duality equations (141)automatically satisfies the Yang-Mills equations of motion119892푀푁119863푀119865푁푃 = 0 (145)

on a Kahler manifoldLet us analyze the HYM equations (141) We observed

in Section 3 that the rsquot Hooft symbols in (59) realizes theisomorphism between irreducible spin(6) Lorentz algebraand 119904119906(4) Lie algebra and provides a complete basis of twoforms in Ω2

plusmn(119872) For instance one may expand the 119878119880(4)field strengths 119865푎

퐴퐵 (119886 = 1 15) using the basis (59) likeeither (62) or (63) A question is how to realize the doublingof CY manifolds from the 119878119880(4) gauge theory approachThe crux for this question is that the 119873-dimensional fun-damental representation of 119878119880(119873) for 119873 greater than two isa complex representation whose complex conjugate is oftencalled the antifundamental representation And the complexconjugate representation N is an inequivalent representationdifferent from the original one N In particular the positiveand negative chirality representations of Spin(6) cong 119878119880(4)coincide with the fundamental (4) and the antifundamental(4) representations of 119878119880(4) Therefore we have a freedom toembed the solutions of Yang-Mills gauge theory in a specificrepresentation This freedom is basically related to the exis-tence of two independent bases of two forms 120578푎퐴퐵 and 120578푎퐴퐵according to the isomorphism (B) and (C)

Thereby we will identify the 119878119880(4) field strength 119865푎퐴퐵 in

the fundamental representation 4 with type A in (62) andin the antifundamental representation 4 with type B in (63)


For the antifundamental representation 4 the Lie algebragenerators are given by (119879푎)lowast = minus(1198942)120582lowast

푎 and they obey thesame Lie algebra as 119879푎[(119879푎)lowast (119879푏)lowast] = minus119891푎푏푐 (119879푐)lowast (146)

But one can see from (A9) that the symmetric structureconstants have an opposite sign that is

Tr 119879푎 119879푏 119879푐 = minus 1198942119889푎푏푐Tr (119879푎)lowast (119879푏)lowast (119879푐)lowast = 1198942119889푎푏푐 (147)

It turns out that this sign flip is correlated with the oppositesign in (A5) According to the tensor product (67) one candecompose the coefficients 119891푎푏

(plusmnplusmn) into a symmetric part andan antisymmetric part119891푎푏

(plusmnplusmn) = 119891(푎푏)(plusmnplusmn) + 119891[푎푏]

(plusmnplusmn) (148)

Although it is not necessary to impose the symmetry property(68) for a general vector bundle 120587 119864 rarr 119872 we will imposethe symmetric prescription that is 119891[푎푏]

(plusmnplusmn)= 0 because we are

interested in the gauge theory formulation of six-dimensionalRiemannian manifolds where 119864 = 119879119872 and the bundleconnections are identified with spin connections (of coursethe symmetric condition (149) greatly reduces the numberof field strengths (225 rarr 120) According to the relation(B5) for the tangent bundle 119864 = 119879119872 it is easy to derivethe identities [119865(plusmn)

퐴퐵 119869퐴퐵plusmn ] = 0 and (14)119865(plusmn)퐴퐵 119869퐴퐵plusmn =minus(18)119891푎푏

(plusmnplusmn)120575푎푏 + (1198942)119889푎푏푐119891푎푏(plusmnplusmn)119879푐

plusmn) Then119865(plusmn)푎퐴퐵 = 119891푎푏

(plusmnplusmn)120578(plusmn)푏퐴퐵 (149)

where we have omitted the symmetrization symbol withrespect to 119886 harr 119887 for brevity

Now let us consider the HYM equations on a Kahlermanifold 119872 Recall that the HYM equation (141) can beresolved by decomposing the Yang-Mills field strengths (149)into the eigenspaces of the Hodge operator (85) And weshowed that the decomposition (86) is equivalent to thebranching (98) of 119878119880(4) under the 119880(3) subgroup since the119878119880(4) gauge group is reduced to 119880(3) by the backgroundKahler class Ω (It might be obvious from the expansion(149) which intertwines the 119878119880(4) index 119886 and 119878119874(6) indices119860 119861 Note that the different choice of background Kahlerclasses can be parameterized by the homogeneous space119878119880(4)119880(3) = C1198753 Also (B17) implies that the spaceC1198753 = 119878119880(4)119880(3) can be identified with the space ofcomplex structure deformations [9] This coincidence mightpresage themirror symmetry)Therefore the Yang-Mills fieldstrengths obeying (141) take values in 119906(3) Lie algebra thatis 119886 119887 run over 3 6 7 8 11 12 13 14 15 for the fundamentalrepresentation 4 and 3 4 5 8 9 10 13 14 15 for the antifun-damental representation 4 To be specific 120578푎퐴퐵 isin 119897(+)푎퐴퐵 119899(+)0퐴퐵 for 4 and 120578푎퐴퐵 isin 119897(minus)푎퐴퐵 119899(minus)0퐴퐵 for 4 with the rsquot Hooft symbols119897(plusmn)푎퐴퐵 and 119899(plusmn)0퐴퐵 defined in Section 3 As the background Kahler

class Ω determines a particular 119880(3) sub 119878119880(4) subgroup and4 and 4 belong to two different representations the Kahlerclasses in the representations 4 and 4 should be attributed todifferent Kahler manifolds

Hence let us consider the 119878119880(4) gauge theory defined ontwo different Kahler manifolds119872 and whose backgroundKahler classes are respectively given byΩ+ = 119899(+)0퐴퐵 119890(+)퐴 and 119890(+)퐵Ωminus = 119899(minus)0퐴퐵 119890(minus)퐴 and 119890(minus)퐵 (150)

Given a fixed Kahler class the HYM equations will be solvedby119880(3) connectionsThe stability equation (142) for each caseis then reduced to the following equations119868퐴퐵119865(+)푎

퐴퐵 = 119891푎푏(++)120578푏퐴퐵119868퐴퐵 = 0 lArrrArr1198913푎

(++) + 1radic31198918푎(++) + 1radic611989115푎

(++) = 0119868퐴퐵119865(minus)푎퐴퐵 = 119891푎푏

(minusminus)120578푏퐴퐵119868퐴퐵 = 0 lArrrArr1198913푎(minusminus) minus 1radic31198918푎

(minusminus) minus 1radic611989115푎(minusminus) = 0

(151)

By applying the exactly same argument as Section 3 onecan conclude that the above equations are equivalent to thevanishing of the first Chern-class that is119865(+)0 = 119889119860(+)0 = 0119865(minus)0 = 119889119860(minus)0 = 0 (152)

where the 119880(1) gauge fields 119860(plusmn)0 are defined by (107) and(112) One can also see from (91) that the 119878119880(3) basis 119897(plusmn)푎퐴퐵 definitely picks up the +-sign in (141) and its solution is givenby 119865(plusmn)푎

퐴퐵 = 119891푎푏(plusmnplusmn)119897(plusmn)푏퐴퐵 119886 = 1 8 (153)

Consequently we found that the HYM instanton inher-ited from a CY manifold is described by the 119878119880(3) con-nections with the trivial first Chern class This means thatthe tangent bundle 119879119872 of a CY manifold 119872 gives rise to119878119880(3) connections in a stable holomorphic vector bundle [1]This is exactly the statement of the DUY theorem [17 18]for a particular case of the vector bundle 119864 = 119879119872 over aCY manifold 119872 Now it becomes clear what is the mirrorrelation for theHYM instantonsThemirror symmetry of CYmanifolds can be understood as the relationship between twokinds ofHYM instantons in 119878119880(4) gauge theory embedded inthe fundamental representation 4 and the antifundamentalrepresentation 4 Each representation has its own cohomol-ogy classes taking values in the holomorphic vector bundles119864 over119872 and119865 over It should be remarked that we intendto construct the complex vector bundles 119864 and 119865 via thetangent bundles 119879119872otimesC = 119864oplus119864 and 119879otimesC = 119865oplus119865 so thevector bundles 119864 and 119865 are independent of each other Since


the Kahler class Ωplusmn reduces the gauge group to 119880(3) theunderlying complex structures are not mixed under gaugetransformations

Since we want to understand the mirror symmetrybetween CYmanifolds in terms of 119878119880(4) gauge theory it willbe useful to calculate the Chern classes of the vector bundleto elucidate the mirror symmetry between HYM instantonsIt was already shown that the first Chern class 1198881(119864) of theholomorphic vector bundle satisfying (141) is trivial that is1198881(119864) = 0 Also we have shown that the last term in (138) is atopological invariant which contains the second Chern class1198882(119864) After using (141) one can derive the inequality181205872

int푀plusmn

Tr119865(plusmn) and 119865(plusmn) and Ωplusmn ge 0 (154)

where 1198882 (119881plusmn) = 181205872Tr119865(plusmn) and 119865(plusmn) (155)

is the second Chern class of a complex vector bundle 119881+ = 119864over 119872+ = 119872 or 119881minus = 119865 over 119872minus = This is known asthe Bogomolov inequality [22 23] which is true for all stablebundles with 1198881(119881plusmn) = 0 Using the identification in (57) and(58) one may translate the above inequality into the one ingravity theoryminus 1161205872

int푀plusmn

119877(plusmn)퐴퐵 and 119877(plusmn)

퐴퐵 and Ωplusmn ge 0 (156)

Finally according to formula (122) we calculate theintegral of the third Chern class 1198883(119881plusmn) given by120594+ (119864) = minus 119894241205873

int푀Tr119865(+) and 119865(+) and 119865(+)

= minus 1961205873int푀

119889푎푏푐119865(+)푎 and 119865(+)푏 and 119865(+)푐

= minus 11921205873int푀


(++)119891푐푓

(++)120594minus (119865) = minus 119894241205873

int푀Tr119865(minus) and 119865(minus) and 119865(minus)

= 1961205873int푀

119889푎푏푐119865(minus)푎 and 119865(minus)푏 and 119865(minus)푐

= 11921205873int푀



(minusminus)

(157)

It might be remarked that the relative sign 1198883(119865) = minus1198883(119865) forthe third Chern classes of a complex vector bundle 119865 and itsconjugate bundle 119865 arises from the property (147) Since thecomplex vector bundles 119864 and 119865 are independently definedover two different Kahler manifolds 119872 and respectivelythe expansion coefficients119891푎푏

(plusmnplusmn) in (149) will also be separatelydetermined by them Hence it should be possible to constructa pair of complex vector bundles (119864 rarr 119872119865 rarr ) suchthat 120594+(119864) = minus120594minus(119865) One may notice that the sign flip in theEuler characteristic is also consistent with the general result

(122) In consequence the above Euler characteristics cor-rectly reproduce (127) and (128) for the CY manifold 119872and its mirror manifold This constitutes a gauge theoryformulation of mirror symmetry

In conclusion we have confirmed the picture depicted inFigure 1 that the mirror symmetry between CY manifoldscan be understood as the mirror pair of HYM instantonsin the fundamental representations 3 and 3 of 119878119880(3) gaugeconnections Since the existence of two different fundamentalrepresentations of 119878119880(4) cong Spin(6) is related to the doublingof the vector space in (6) according to the isomorphism (4)we see that the mirror symmetry of CY manifolds and HYMinstantons originates from the Hodge duality in the vectorspace Λlowast119872

6 Discussion

Thephysics on a curved spacetime becomesmore transparentwhen expressed in a locally inertial frame and it is even indis-pensable when one wants to couple spinors to gravity sincespinors in 119889-dimensions form a representation of Spin(119889)Lorentz group rather than119866119871(119889R) It is also required to takean irreducible spinor representation of the Lorentz symmetryThen one can apply the elementary propositions (A B C)in Section 1 to 119889-dimensional Riemannian manifolds to seetheir consequences It is especially interesting to apply themto six-dimensional CY manifolds Proposition (A) first saysthat Riemann curvature tensors 119877퐴퐵퐶퐷 carry two kinds ofindices the first group say [119860119861] belongs to Spin(6) indicesand the second group [119862119863] belongs to form indices inΩ2(119872)But proposition (C) requires that two groups must have anisomorphic structure as vector spaces After imposing thetorsion free condition that leads to the symmetry property119877퐴퐵퐶퐷 = 119877퐶퐷퐴퐵 the vector space structure for the two groupsshould be even identified For example the irreducible spinorrepresentation of the Lorentz group Spin(6) requires us toconsider the vector spaces Ω2(119872) and Ω4(119872) on equal foot-ing The doubling of the vector space (6) is realized as eithertwo independent chiral representations of the Lorentz groupSpin(6) or two independent complex representations 4 and4 of the gauge group 119878119880(4) We observed that the doublingof the vector spaces essentially brings about the doublingfor the variety of six-dimensional spin manifolds which isresponsible for the existence of themirror symmetry betweenCY manifolds

It may be worthwhile to compare the four and sixdimensions in perspective On a four-dimensional orientablemanifold the vector space of two-formΩ2(119872) is not doubledbecause theHodge duality of a two-form vector space is againa two-form vector space Instead the vector space Ω2(119872)splits canonically into two vector spaces as (7) This split isresonant with the self-duality of chiral Lorentz generators 119869퐴퐵plusmn

because they obey the relation119869퐴퐵plusmn = plusmn12120576퐴퐵퐶퐷119869퐶퐷plusmn (158)

Therefore the chiral Lorentz generators 119869A퐵plusmn have three inde-pendent components only Combining them together they


consist of six generators which match with the dimensionof Ω2(119872) Applying this fact to (19) one can see that 119877(plusmn)

퐴퐵contains 18 = 6 times 3 components and so 36 = 18 + 18components in total which is the number of componentsof Riemann curvature tensors 119877퐴퐵퐶퐷 before imposing thefirst Bianchi identity This situation is different from the six-dimensional case as is evident from the comparison of (6)and (7) This difference is originated from the fact that in sixdimensions there is another source of two-form vector spacecoming from the Hodge duality of four-form vector space Asa consequence 119877(plusmn)

퐴퐵 in (79) has 225 = 15 times 15 components insix dimensions and so 450 = 225 + 225 components in totalbefore imposing the first Bianchi identity After imposingthe first Bianchi identity in each class that totally comprises240 = 120 + 120 constraints the physical curvature tensors(119877(+)

퐴퐵 oplus 119877(minus)퐴퐵) have 210 = 105 + 105 components in total This

doubling for the variety of CYmanifolds is a core origin of themirror symmetry between CY manifolds

Via the gauge theory formulation of six-dimensionalEuclidean gravity we showed that HYM instantons canbe constructed in two different ways by embedding theminto the fundamental or antifundamental representation of119878119880(4) cong Spin(6) gauge group Since a CY manifold can berecast as a HYM instanton from the gauge theory point ofview (see the quotation in Section 1) and the chiral repre-sentation of Spin(6) corresponds to the fundamental repre-sentation of 119878119880(4) the structure in Figure 1 has been nicelyverified After all the mirror symmetry of CY manifolds canbe understood as the existence of the mirror pair of HYMinstantons by doubling the variety of six-dimensional spinmanifolds according to the Hodge duality (6)

Strominger et al recently proposed [7] that the mirrorsymmetry is a T-duality transformation along a dual specialLagrangian tori fibration on a mirror CY manifold The T-duality transformation along the dual three-tori introducesa sign flip in the Euler characteristic as even and oddforms exchange their role Note that the odd number ofT-duality operations transforms type IIB string theory totype IIA string theory and vice versa Hence types IIA andIIB CY manifolds will be mirror to each other because thesix-dimensional chirality will be flipped after the T-dualityand the ten-dimensional chirality is correlated with the six-dimensional one This result implies that the mirror symme-try in string theory originates from the two different chiralrepresentations of CY manifolds which is consistent withour picture

Our gauge theory formulation may be generalized toa general six-dimensional Riemannian manifold like thefour-dimensional case [12 13] because it is simply basedon the general propositions (A B C) in Section 1 Onemay consider for example the Strominger system [24 25]for non-Kahler complex manifolds The Strominger systemadmits a conformally balanced Hermitian form on a three-dimensional compact complex manifold 119872 a nowhere van-ishing holomorphic (3 0)-form and a HYM connection ona vector bundle 119864 over this manifold The consistency ofthe underlying physical theory imposes a constraint that thecurvature forms have to satisfy the anomaly equation As far

as the non-Kahler CY manifold admits a spin structure thegauge theory formulation for the Strominger system may bestraightforward as much as we have done in this paper Thusit may be interesting to formulate the mirror symmetry fornon-Kahlermanifolds from the gauge theory perspective andto generalize it to the casewithout spin structure for instancea manifold with SpinC structure only If there is a substantialprogress along this line it will be reported elsewhere

If we consider a CYmanifold119872 to be the HYM instantonof the tangent bundle 119879119872 this instanton will have their ownmoduli space given by their zero modes with the nonzeromodes providing various ldquoupliftingrdquoThen the following ques-tions naturally arise How is themoduli space ofHYM instan-tons related to theCYmoduli spaceAlso howdo the nonzeromodes of the instanton solution correspond to the CYdeformations

To discuss this issue let us consider an infinitesimaldeformation of the gauge field119860휇 + 120575119860휇 (159)

If we demand (143) for both the original gauge field andthe deformation then the deformation must satisfy 120597120575119860 =0 This means that 120575119860 isin 1198671(End119864) On a manifold of119878119880(3) holonomy and for the case 119864 = 119879119872 1198671(End119864)coincides with 11986721(119872) Therefore the bundle moduli (159)for the condition (143) of the holomorphic tangent bundle119879119872 correspond to the deformations of the complex structurecounted by 1198671(119872 119879119872) cong 11986721(119872) [2 3] However it isknown [26] that the HYM equations (143) and (144) do notfix any of the Kahler moduli Indeed the DUY theorem states[17 18] that for a fixed choice of Kahler moduli there existsa solution of the HYM equations if the holomorphic vectorbundle is slope-stable which is the case for the tangent bun-dle Therefore it is not possible to extract the Kahler modulifrom the bundle moduli (159) This implies that the modulispace of a CY manifold is not fully captured by the instantonmoduli space even for the tangent bundle since we fix theKahler moduli of a background CY manifold to define theHYM equations

It is well-known that the instanton moduli space hassingularities the so-called ldquosmall instantonrdquo singularitiesAlso the CY moduli space has singularities the conifoldpoints [3] Thus it will be interesting to understand howthese two kinds of singularities are related to each otherOur result implies that the singularities in the instantonmoduli space are related to the singularities (the conifoldpoints) of the CY moduli space because the tangent bundle119879119872 is defined by the CY manifold 119872 But the blown-upconifold singularities may arise in different ways from theinstanton picture of the CY manifold since it is known [27]that there is a natural complex structure on the resolutionbut not a natural Kahler structure while the deformation issymplectic in a natural way but not naturally complex Sinceonly the complex structure deformations of the CY manifoldare encoded in the bundle moduli (159) we speculate thatthe resolved conifold is realized from the instanton sidewhile the deformed conifold appears in the background CYmanifold Note that the deformed conifold is mirror to the


resolved conifold which is related to the conifold transitionsThus the instanton picture of CY manifolds implies that theHYM instanton for the tangent bundle 119879119872 over a deformedconifold 119872 is mirror to the HYM instanton for the tangentbundle119879over a resolved conifold We leave this problemfor the future work

Mirror symmetry provides an isomorphism betweencomplex geometry and symplectic geometrywhich relates thedeformation of complex structure on the complex geometryside to the counting of pseudo-holomorphic spheres on thesymplectic geometry side A very similar picture arises inemergent gravity that isomorphically relates the deformationof symplectic structure described by a NC119880(1) gauge theoryto the deformation of complex structure in Einstein gravityThe deformation of symplectic structure is represented byF = 119861 + 119865 where 119861 is an underlying symplectic structureon 119872 and 119865 = 119889119860 is identified with the curvature of linebundle 119871 rarr 119872 In order to allow singular 119880(1) gauge fieldssuch as 119880(1) instantons it is necessary to generalize the linebundle to a torsion free sheaf or an ideal sheaf NC119880(1) gaugefields are introduced via a local coordinate transformation120601 isin Diff(119872) eliminating 119880(1) gauge fields that is 120601lowast(F) =119861 known as the Darboux theorem in symplectic geometry Itwas claimed in [28 29] and recently shown in [30 31] that six-dimensional CYmanifolds are emergent fromNCHermitian119880(1) instantons Note that the NCHermitian119880(1) instantonscorrespond to 119880(1) connections in a stable holomorphic linebundle 119871 rarr 119872 or more generally a torsion free sheaf (anideal sheaf) from the commutative description [32] Whenwe conceive the emergent CY manifolds from the mirrorsymmetry perspective an interesting question is how to real-ize the mirror symmetry from the emergent gravity pictureIt turns out [33] that the emergent gravity picture providesa very nice result for the mirror symmetry

Appendix

A Spin(6) and 119878119880(4)We consider the six-dimensional Clifford algebra with theDirac matrices given by

Γ퐴 = ( 0 120574퐴120574퐴 0 ) 119860 = 1 6 (A1)

where 120574퐴 = (120574퐴)dagger Thus the Dirac matrices we have takenare Hermitian that is (Γ퐴)dagger = Γ퐴 We choose (120574푖)dagger =120574푖 (119894 = 1 5) and (1205746)dagger = minus1205746 We will use the followingrepresentation of Dirac matrices [34]

1205741 = ( 0 0 0 minus10 0 1 00 1 0 0minus1 0 0 0 )

1205742 = (0 0 0 1198940 0 119894 00 minus119894 0 0minus119894 0 0 0)1205743 = (0 0 1 00 0 0 11 0 0 00 1 0 0)1205744 = (0 0 minus119894 00 0 0 119894119894 0 0 00 minus119894 0 0) 1205745 = (1 0 0 00 1 0 00 0 minus1 00 0 0 minus1) = minus1205741120574212057431205744

(A2)

satisfying the Spin(5) Clifford algebra relation120574푖120574푗 + 120574푗120574푖 = 2120575푖푗 119894 119895 = 1 5 (A3)

The Lorentz generators for the irreducible (chiral) spinorrepresentation of Spin(6) are defined by119869퐴퐵plusmn equiv 12 (I8 plusmn Γ7) 119869퐴퐵 (A4)

where Γ7 = 119894Γ1 sdot sdot sdot Γ6 Note that 119869퐴퐵+ and 119869퐴퐵minus independentlysatisfy the Lorentz algebra (11) and commute each other thatis [119869퐴퐵+ 119869퐶퐷minus ] = 0 They also satisfy the anticommutationrelation119869퐴퐵plusmn 119869퐶퐷plusmn = minus12 (120575퐴퐶120575퐵퐷 minus 120575퐴퐷120575퐵퐶) Γplusmnplusmn 1198942120576퐴퐵퐶퐷퐸퐹119869퐸퐹plusmn (A5)

Because the chiral matrix Γ7 is given byΓ7 = (I4 00 minusI4) (A6)

where I4 is the 4times4 identitymatrix the generators of the chiralspinor representation in (A4) are given by 4 times 4 matricesThen the two independent chiral spinor representations ofSpin(6) are given by119869퐴퐵+ = 119869푖푗+ = 14 [120574푖 120574푗] 119869푖6+ = 1198942120574푖 119869퐴퐵minus = 119869푖푗minus = 14 [120574푖 120574j] 119869푖6minus = minus 1198942120574푖 (A7)


One can check that the generators 119869퐴퐵+ and 119869퐴퐵minus separatelyobey the Lorentz algebra (11)

One can exchange the positive chiral representation andthe negative chiral representation by a parity transformationwhich is a reflection 119909푀 rarr minus119909푀 of any one element of thefundamental six-dimensional representation of Spin(6) [1] inour case 1199096 rarr minus1199096 But they cannot be connected by any119878119874(6) rotations

The anti-Hermitian 4 times 4 matrices 119879푎 = (1198942)120582푎 119886 =1 15 with vanishing traces constitute the basis of 119878119880(4)Lie algebra The Hermitian 4 times 4 matrices 120582푎 are givenby

1205821 = (0 1 0 01 0 0 00 0 0 00 0 0 0)

1205822 = (((

0 minus119894 0 0119894 0 0 00 0 0 00 0 0 0)))

1205823 = (1 0 0 00 minus1 0 00 0 0 00 0 0 0)1205824 = (0 0 1 00 0 0 01 0 0 00 0 0 0)1205825 = (0 0 minus119894 00 0 0 0119894 0 0 00 0 0 0)1205826 = (0 0 0 00 0 1 00 1 0 00 0 0 0)1205827 = (0 0 0 00 0 minus119894 00 119894 0 00 0 0 0)

1205828 = 1radic3 (1 0 0 00 1 0 00 0 minus2 00 0 0 0)1205829 = (0 0 0 10 0 0 00 0 0 01 0 0 0)12058210 = (0 0 0 minus1198940 0 0 00 0 0 0119894 0 0 0) 12058211 = (0 0 0 00 0 0 10 0 0 00 1 0 0)12058212 = (0 0 0 00 0 0 minus1198940 0 0 00 119894 0 0) 12058213 = (0 0 0 00 0 0 00 0 0 10 0 1 0)12058214 = (0 0 0 00 0 0 00 0 0 minus1198940 0 119894 0) 12058215 = 1radic6 (1 0 0 00 1 0 00 0 1 00 0 0 minus3)

(A8)

The generators satisfy the following relation119879푎119879푏 = minus18120575푎푏I4 minus 12119891푎푏푐119879푐 + 1198942119889푎푏푐119879푐 (A9)

where the structure constants 119891푎푏푐 are completely antisym-metric while 119889푎푏푐 are symmetric with respect to all of theirindices Their values are shown up in the Tables 1 and 2 Wehave got these tables from [35]



1 1 8 1radic31 1 15 1radic61 4 6 121 5 7 121 9 11 121 10 12 122 2 8 1radic32 2 15 1radic62 4 7 minus122 5 6 122 9 12 minus122 10 11 123 3 8 1radic33 3 15 1radic63 4 4 123 5 5 123 6 6 minus123 7 7 minus123 9 9 123 10 10 123 11 11 minus123 12 12 minus124 4 8 minus12radic34 4 15 1radic64 9 13 124 10 14 125 5 8 minus12radic35 5 15 1radic65 9 14 minus125 10 13 126 6 8 minus12radic36 6 15 1radic66 11 13 126 12 14 127 7 8 minus12radic37 7 15 1radic67 11 14 minus127 12 13 128 8 8 minus1radic38 8 15 1radic68 9 9 12radic38 10 10 12radic38 11 11 12radic38 12 12 12radic38 13 13 minus1radic38 14 14 minus1radic39 9 15 minus1radic610 10 15 minus1radic611 11 15 minus1radic612 12 15 minus1radic613 13 15 minus1radic614 14 15 minus1radic615 15 15 minusradic23

B Six-Dimensional rsquot Hooft Symbols

The explicit representation of the six-dimensional rsquot Hooftsymbol 120578푎퐴퐵 = minusTr(119879푎119869퐴퐵+ ) is given by

1205781퐴퐵 = 12 (((((

0 0 0 1 0 00 0 1 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= 11989421205822 otimes 1205901

1205782퐴퐵 = minus12 (((((

0 0 1 0 0 00 0 0 minus1 0 0minus1 0 0 0 0 00 1 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= minus 11989421205822 otimes 1205903

1205783퐴퐵 = 12 (((((

0 1 0 0 0 0minus1 0 0 0 0 00 0 0 1 0 00 0 minus1 0 0 00 0 0 0 0 00 0 0 0 0 0)))))= 1198942 (23 I3 + 1radic31205828) otimes 1205902

1205784퐴퐵 = 12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 1 00 0 0 minus1 0 00 0 minus1 0 0 0)))))

= 11989421205827 otimes 1205901

1205785퐴퐵 = minus12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 minus10 0 minus1 0 0 00 0 0 1 0 0)))))

= minus 11989421205827 otimes 1205903

1205786퐴퐵 = 12 (((((

0 0 0 0 0 10 0 0 0 minus1 00 0 0 0 0 00 0 0 0 0 00 1 0 0 0 0minus1 0 0 0 0 0)))))

= 11989421205824 otimes 1205902


1205787퐴퐵 = minus12 ((((((

0 0 0 0 1 00 0 0 0 0 10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 minus1 0 0 0 0))))))

= minus 11989421205825 otimes I2

1205788퐴퐵 = minus 12radic3 ((((((

0 1 0 0 0 0minus1 0 0 0 0 00 0 0 minus1 0 00 0 1 0 0 00 0 0 0 0 minus20 0 0 0 2 0))))))= minus 1198942radic3 (minus23 I3 + 1205823 + 2radic31205828) otimes 1205902

1205789퐴퐵 = minus12 ((((((

0 0 0 0 0 10 0 0 0 1 00 0 0 0 0 00 0 0 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 0))))))

= minus 11989421205825 otimes 1205901

12057810퐴퐵 = 12 ((((((

0 0 0 0 1 00 0 0 0 0 minus10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 1 0 0 0 0))))))

= 11989421205825 otimes 1205903

12057811퐴퐵 = 12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 minus1 00 0 0 1 0 00 0 minus1 0 0 0))))))

= 11989421205826 otimes 1205902

12057812퐴퐵 = minus12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 10 0 minus1 0 0 00 0 0 minus1 0 0))))))

= minus 11989421205827 otimes I2

12057813퐴퐵 = 12 ((((((

0 0 0 1 0 00 0 minus1 0 0 00 1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205821 otimes 1205902

12057814퐴퐵 = 12 ((((((

0 0 1 0 0 00 0 0 1 0 0minus1 0 0 0 0 00 minus1 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205822 otimes I2

12057815퐴퐵 = 1radic6 ((((((

0 1 0 0 0 0minus1 0 0 0 0 00 0 0 minus1 0 00 0 1 0 0 00 0 0 0 0 10 0 0 0 minus1 0))))))= 119894radic6 (13 I3 + 1205823 minus 1radic31205828) otimes 1205902

(B1)

where I푛 is the 119899-dimensional identity matrix and (1205901 1205902 1205903)are the Pauli matrices and 120582푎 (119886 = 1 8) are the 119878119880(3)Gell-Mann matrices

Another six-dimensional rsquot Hooft symbol 120578푎퐴퐵 =minusTr((119879푎)lowast119869퐴퐵minus ) can be obtained similarly1205781퐴퐵 = minus 11989421205822 otimes 12059011205782퐴퐵 = minus 11989421205822 otimes 12059031205783퐴퐵 = minus 1198942 (23 I3 + 1radic31205828) otimes 12059021205784퐴퐵 = 11989421205826 otimes 12059021205785퐴퐵 = minus 11989421205827 otimes I21205786퐴퐵 = 11989421205825 otimes 12059011205787퐴퐵 = minus 11989421205825 otimes 12059031205788퐴퐵 = 1198942radic3 (23 I3 + 1205823 minus 2radic31205828) otimes 12059021205789퐴퐵 = minus 11989421205824 otimes 1205902


12057810퐴퐵 = 11989421205825 otimes I212057811퐴퐵 = 11989421205827 otimes 120590112057812퐴퐵 = minus 11989421205827 otimes 120590312057813퐴퐵 = minus 11989421205821 otimes 120590212057814퐴퐵 = 11989421205822 otimes I212057815퐴퐵 = 119894radic6 (13 I3 minus 1205823 minus 1radic31205828) otimes 1205902(B2)

In order to derive the algebras obeyed by the rsquot Hooftsymbols first note that either Spin(6) generators 119869퐴퐵plusmn or 119878119880(4)generators 119879푎

+ equiv 119879푎 and 119879푎minus equiv (T푎)lowast can serve as a complete

basis of any traceless Hermitian 4 times 4matrix119870 that is119870 = 15sum푎=1

119896plusmn푎119879푎plusmn = 12 6sum

퐴퐵=1

119870plusmn퐴퐵119869퐴퐵plusmn (B3)

Using definition (59) one can easily deduce that119879푎plusmn = 12120578(plusmn)푎퐴퐵 119869퐴퐵plusmn (B4)119869퐴퐵plusmn = 2120578(plusmn)푎퐴퐵 119879푎

plusmn (B5)

where 120578(+)푎퐴퐵 equiv 120578푎퐴퐵 and 120578(minus)푎퐴퐵 equiv 120578푎퐴퐵Then one can consider thefollowing matrix products(I) 119879푎

plusmn119879푏plusmn = 14120578(plusmn)푎퐴퐵 120578(plusmn)푏퐶퐷 119869퐴퐵plusmn 119869퐶퐷plusmn (II) 119869퐴퐵plusmn 119869퐶퐷plusmn = 4120578(plusmn)푎퐴퐵 120578(plusmn)푏퐶퐷 119879푎

plusmn119879푏plusmn (B6)

By applying (11) (A5) and (A9) to the above matrixproducts one can easily get the algebras obeyed by the six-dimensional rsquot Hooft symbols120578푎퐴퐵120578푏퐴퐵 = 120575푎푏 = 120578푎퐴퐵120578푏퐴퐵 (B7)120578푎퐴퐵120578푎퐶퐷 = 12 (120575퐴퐶120575퐵퐷 minus 120575퐴퐷120575퐵퐶) = 120578푎퐴퐵120578푎퐶퐷 (B8)14120576퐴퐵퐶퐷퐸퐹120578푎퐶퐷120578푏퐸퐹 = 119889푎푏푐120578푐퐴퐵 (B9)14120576퐴퐵퐶퐷퐸퐹120578푎퐶퐷120578푏퐸퐹 = 119889푎푏푐120578푐퐴퐵 (B10)120578푎퐴퐶120578푏퐵퐶 minus 120578푎퐵퐶120578푏퐴퐶 = 119891푎푏푐120578푐퐴퐵 (B11)120578푎퐴퐶120578푏퐵퐶 minus 120578푎퐵퐶120578푏퐴퐶 = 119891푎푏푐120578푐퐴퐵 (B12)119891푎푏푐120578푎퐴퐵120578푏퐶퐷= 12 (120575퐴퐶120578푐퐵퐷 minus 120575퐴퐷120578푐퐵퐶 minus 120575퐵퐶120578푐퐴퐷 + 120575퐵퐷120578푐퐴퐶) (B13)

119891푎푏푐120578푎퐴퐵120578푏퐶퐷= 12 (120575퐴퐶120578푐퐵퐷 minus 120575퐴퐷120578푐퐵퐶 minus 120575퐵퐶120578푐퐴퐷 + 120575퐵퐷120578푐퐴퐶) (B14)

119889푎푏푐120578푎퐴퐵120578푏퐶퐷 = 14120576퐴퐵퐶퐷퐸퐹120578푐퐸퐹 (B15)119889푎푏푐120578푎퐴퐵120578푏퐶퐷 = 14120576퐴퐵퐶퐷퐸퐹120578푐퐸퐹 (B16)

Finally we list the nonzero components of the rsquot Hooftsymbols in the basis of complex coordinates 119911훼 = 1199111 =1199091 + 1198941199092 1199112 = 1199093 + 1198941199094 1199113 = 1199095 + 1198941199096 and their complexconjugates 119911훼 where 120572 120572 = 1 2 3 We will denote 120578푎훼훽 = 120578푎

푧120572푧120573120578푎

훼훽= 120578푎

푧120572푧120573 and so forth in the hope of no confusion with the

real basis 120578112 = minus 1198944 120578212 = minus14 120578423 = minus 1198944 120578523 = minus14 120578913 = 1198944 1205781013 = 14 120578311

= 1198944 120578322

= 1198944 120578613

= 1198944 120578713

= minus14 120578811

= minus 1198944radic3 120578822

= 1198944radic3 120578833

= 1198942radic3 1205781123

= 1198944 1205781223

= minus14 1205781312

= 1198944 1205781412

= 14


1205781511

= 1198942radic6 1205781522

= minus 1198942radic6 1205781533

= 1198942radic6 (B17)

Here the complex conjugates are not shown up since they caneasily be implemented The corresponding values of 120578푎퐴퐵 canbe obtained from those of 120578푎퐴퐵 by flipping the sign for theentries 119886 = 1 3 4 6 8 9 11 13 15 as well as interchanging1199113 harr 1199113 for all entries Note that the first line in (B17) belongsto 119898(+) 푎

퐴퐵 in (77) with purely holomorphic or antiholomor-phic indices This result implies that the space of complexstructure deformations can be identified with the cosetspace 119878119880(4)119880(3) = C1198753 [9]

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors are grateful to Chanju Kim for collaborationat an early stage and very helpful discussions Hyun SeokYang thanks Richard Szabo for his warm hospitality andvaluable discussions during his visit to Heriot-Watt Uni-versity Edinburgh He also thanks Jungjai Lee and John JOh for constructive discussions and collaborations with arelated subject The work of H S Yang was supportedby the National Research Foundation of Korea (NRF)grant funded by the Korean government (MOE) (no NRF-2015R1D1A1A01059710)Thework of S Yunwas supported bythe World Class University Grant no R32-10130

References

[1] M B Green J H Schwarz and E Witten Superstring Theoryvol 1 Cambridge University Press Cambridge Uk 1987

[2] P Candelas G T Horowitz A Strominger and E WittenldquoVacuum configurations for superstringsrdquo Nuclear Physics Bvol 258 no 1 pp 46ndash74 1985

[3] K Hori S Katz A Klemm et al Mirror Symmetry AmericanMathematical Society and the Clay Mathematics InstituteCambridge Mass USA 2003

[4] E Witten ldquoMirror manifolds and topological field theoryrdquo inEssays on Mirror Manifolds S-T Yau Ed pp 120ndash158 Interna-tional Press Somerville Mass USA 1992

[5] M Kontsevich Homological algebra of mirror symmetry [ alg-geom9411018]

[6] ldquoSymplectic geometry and mirror symmetryrdquo in Proceedings ofthe 4th KIAS Annual International Conference K Fukaya Y-GOh K Ono and G Tian Eds World Scientific Seoul SouthKorea August 2000

[7] A Strominger S-T Yau and E Zaslow ldquoMirror symmetry is119879-dualityrdquo Nuclear Physics B vol 479 no 1-2 pp 243ndash259 1996

[8] W K Misner S Thorne and J A Wheeler Gravitation W HFreeman and Company New York NY USA 1973

[9] H B Lawson and M-L Michelsohn Spin Geometry PrincetonUniversity Press Princeton NJ USA 1989

[10] E Meinrenken Clifford Algebras and Lie Theory vol 58Springer Berlin Germany 2013

[11] J J Oh C Park and H S Yang ldquoYang-Mills instantons fromgravitational instantonsrdquo Journal of High Energy Physics vol 4article 87 2011

[12] J J Oh and H S Yang ldquoEinstein manifolds as Yang-MillsinstantonsrdquoModern Physics Letters A vol 28 no 21 2013

[13] J Lee J J Oh and H S Yang ldquoAn efficient representation ofEuclidean gravity Irdquo Journal of High Energy Physics vol 12article 25 2011

[14] H S Yang ldquoRiemannian manifolds and gauge theoryrdquo inProceedings of the Corfu Summer Institute 2011 PoS (CORFU rsquo11)Corfu Greece September 2011

[15] M F AtiyahN JHitchin and IM Singer ldquoSelf-duality in four-dimensional Riemannian geometryrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol362 no 1711 pp 425ndash461 1978

[16] A L Besse Einstein Manifolds Springer Berlin Germany 1987[17] S K Donaldson ldquoAnti self-dual Yang-Mills connections over

complex algebraic surfaces and stable vector bundlesrdquo Proceed-ings of the London Mathematical Society vol 50 no 1 pp 1ndash261985

[18] K Uhlenbeck and S-T Yau ldquoOn the existence of Hermitian-Yang-Mills connections in stable vector bundlesrdquo Communica-tions on Pure and Applied Mathematics vol 39 pp S257ndashS2931986

[19] S-T Yau and S Nadis The Shape of Inner Space Basic BooksNew York NY USA 2010

[20] R Slansky ldquoGroup theory for unified model buildingrdquo PhysicsReports vol 79 no 1 pp 1ndash128 1981

[21] R Bott and L W Tu Differential Forms in Algebraic Topologyvol 82 of Graduate Texts in Mathematics Springer New YorkNY USA 1982

[22] G Tian ldquoGauge theory and calibrated geometry Irdquo Annals ofMathematics vol 151 no 1 pp 193ndash268 2000

[23] M R Douglas R Reinbacher and S-T Yau ldquoBranes bundlesand attractors bogomolov and beyondrdquo Algebraic Geometryvol 4 2006

[24] A Strominger ldquoSuperstrings with torsionrdquo Nuclear Physics Bvol 274 no 2 pp 253ndash284 1986

[25] J Li and S-T Yau ldquoThe existence of supersymmetric stringtheory with torsionrdquo Journal of Differential Geometry vol 70no 1 pp 143ndash181 2005

[26] L B Anderson J Gray A Lukas and B Ovrut ldquoThe edgeof supersymmetry stability walls in heterotic theoryrdquo PhysicsLetters B vol 677 no 3-4 pp 190ndash194 2009

[27] I Smith R P Thomas and S-T Yau ldquoSymplectic conifoldtransitionsrdquo Journal of Differential Geometry vol 62 no 2 pp209ndash242 2002

[28] H S Yang ldquoTowards a background independent quantumgrav-ityrdquo Journal of Physics Conference Series vol 343 no 1 2012

[29] H S Yang ldquoQuantization of emergent gravityrdquo InternationalJournal of Modern Physics A vol 30 no 4-5 115 pages 2015

[30] H S Yang ldquoHighly effective action from large 119873 gauge fieldsrdquoPhysical Review D vol 90 Article ID 086006 2014


[31] H S Yang ldquoCalabi-Yau manifolds from noncommutative Her-mitian 119880(1) instantonsrdquo Physical Review D vol 91 no 10Article ID 104002 104002 pages 2015

[32] M Marino R Minasian G Moore and A Strominger ldquoNon-linear instantons from supersymmetric 119901-branesrdquo Journal ofHigh Energy Physics vol 1 article 5 2000

[33] H S Yang ldquoMirror symmetry in emergent gravityrdquo NuclearPhysics B vol 922 pp 264ndash279 2017

[34] G Arutyunov and S Frolov ldquoFoundations of the AdS5 times 1198785superstring Irdquo Journal of Physics A Mathematical and Generalvol 42 no 25 2009

[35] W Greiner and B Muller Quantum Mechanics SymmetriesSpringer-Verlag Berlin Germany 1993

Submit your manuscripts athttpswwwhindawicom

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Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


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Journal of



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ThermodynamicsJournal of

2 Advances in High Energy Physics11986711(119872) infinitesimally change the Kahler structure of theCY manifold and are therefore called Kahler moduli (Instring theory these moduli are usually complexified byincluding119861-field)On the other hand11986721(119872)parameterizesthe complex structure moduli of a CY manifold 119872 It isthanks to the fact that the cohomology class of (2 1)-forms isisomorphic to the cohomology class 1198671

휕(119879119872) the first Dol-

beault cohomology group of119872with values in a holomorphictangent bundle 119879119872 that characterizes infinitesimal complexstructure deformations Hence the mirror symmetry of CYmanifolds is the duality between two different CY 3-fold 119872and such that the Hodge numbers of119872 and satisfy therelations [3] ℎ11 (119872) = ℎ21 () ℎ21 (119872) = ℎ11 () (1)

or in a more general formℎ푝푞 (119872) = ℎ3minus푝푞 () (2)

where the Hodge number ℎ푝푞 of a CY manifold satisfiesthe relations ℎ푝푞 = ℎ푞푝 and ℎ푝푞 = ℎ3minus푝3minus푞 As we havementioned above the only nontrivial deformations of a CYmanifold are generated by the cohomology classes in11986711(119872)and 11986721(119872) where ℎ11(119872) is the number of possible (ingeneral complexified) Kahler forms and ℎ21(119872) is thedimension of the complex structure moduli space of 119872Mirror symmetry suggests that for each CY 3-fold 119872 thereexists another CY 3-fold whose Hodge numbers obey therelation (1)

From a physical point of view two CY manifolds arerelated by mirror symmetry if the corresponding N = 2superconformal field theories are mirror [4] Two N = 2superconformal field theories are said to be mirror if they areequivalent as quantum field theories The mirror symmetrywas interpreted mathematically by Kontsevich in his 1994ICM talk as an equivalence of derived categories dubbed thehomological mirror symmetry [5] The homological mirrorsymmetry states that the derived category of coherent sheaveson a Kahler manifold should be isomorphic to the Fukayacategory of a mirror symplectic manifold [6] The Fukayacategory is described by the Lagrangian submanifold of agiven symplectic manifold as its objects and the Floer homol-ogy groups as their morphisms Hence the homologicalmirror symmetry formulates the mirror symmetry as anequivalence between certain aspects of complex geometry ofa CY manifold and certain aspects of symplectic geometryof a mirror CY manifold in all dimensions The geometricapproach to mirror symmetry was also unveiled in [7] thatmirror symmetry is a geometric version of the Fourier-Mukaitransformation along a dual special Lagrangian tori fibrationon a mirror CY manifold which interchanges the symplecticgeometry and the complex geometry of a mirror pair

In this paperwewill explore the gauge theory formulationof six-dimensional Riemannian manifolds to address theissue why CY manifolds exist with a mirror pair In order

to simplify an underlying argumentation we will focus onorientable six-dimensional manifolds with spin structure Ingeneral relativity the Lorentz group appears as the structuregroup acting on orthonormal frames of the tangent bundle ofa Riemannian manifold [8] On the frame bundle a Rieman-nian metric on spacetime manifold 119872 is replaced by a localorthonormal basis 119864퐴 (119860 = 1 119889) of the tangent bundle119879119872 Then Einstein gravity can be formulated as a gaugetheory of Lorentz group where spin connections play a roleof gauge fields and Riemann curvature tensors correspondto their field strengths On a six-dimensional Riemannianmanifold 119872 for example local Lorentz transformations areorthogonal rotations in Spin(6) and spin connections 120596퐴퐵 =120596푀퐴퐵119889119909푀 are the spin(6)-valued gauge fields from the gaugetheory point of view (we will use large letters to indicate aLie group 119866 and small letters for its Lie algebra g) Thenthe Riemann curvature tensor 119877퐴퐵 = 119889120596퐴퐵 + 120596퐴퐶 and 120596퐶퐵

precisely corresponds to the field strength of gauge fields120596퐴퐵 in Spin(6) gauge theory Since the Lie group Spin(6) isisomorphic to 119878119880(4) the six-dimensional Euclidean grav-ity can be formulated as an 119878119880(4) Yang-Mills gauge the-ory Via the gauge theory formulation of six-dimensionalRiemannian manifolds we want to identify gauge theoryobjects corresponding to CY manifolds and address theirmirror symmetry from the perspective of Yang-Mills gaugetheory To understand why there exists a mirror pair ofCY manifolds in particular we will employ the followingwell-known propositions for a 119889-dimensional Riemannianmanifold119872

(A) The Riemann curvature tensors 119877퐴퐵 are spin(119889)-valued two forms inΩ2(119872) = Λ2119879lowast119872

(B) There exists a global isomorphism between 119889-dimensional Lorentz groups and classical Lie groups

Spin (3) cong 119878119880 (2) Spin (4) cong 119878119880 (2)퐿 times 119878119880 (2)푅 Spin (5) cong 119878119901 (4) Spin (6) cong 119878119880 (4)

(3)

(C) There is an isomorphism between the Clifford algebraC119897(119889) in 119889-dimensions and the exterior algebra Λlowast119872of cotangent bundle 119879lowast119872 over 119872 [9 10] (the spaceof the Clifford algebra C119897(119889) is isomorphic as avector space to the vector space of the exterioralgebra Λlowast119872 This is not however an isomorphismof associative algebras because the product inΛlowast119872 isanticommutative while that in C119897(119889) is not due to thecentral term in the Dirac algebra (5))

C119897 (119889) cong Λlowast119872 = 푑⨁푘=0

Ω푘 (119872) (4)

whereΩ푘(119872) = Λ푘119879lowast119872For the isomorphism (C) between the vector spaces the

ldquovolume operatorrdquo Γ푑+1 equiv plusmn119894푑(푑minus1)2Γ1 sdot sdot sdot Γ푑 in the Clifford


M Spin(6) SU(4)

CY3 HYM

CY3 HYM

Spin(6) SU(4)

lArrrArr

lArrrArr

M

９3

９3A rarr

B rarr A rarr

B rarr （９－

（９－









+ (119872) oplus Ω2minus (119872) (7)



119877 = (119882+ + 112119904 119861119861푇 119882minus + 112119904) (8)




The gauge theory formulation of six-dimensional spinmanifolds also leads to a valuable perspective for the dou-bling The first useful access is to identify a gauge theoryobject corresponding to a CY 3-fold in the same sense thata gravitational instanton (or a hyper-Kahler manifold) canbe identified with an 119878119880(2) Yang-Mills instanton in fourdimensions [11ndash13] An obvious guess goes toward a six-dimensional generalization of the four-dimensional Yang-Mills instantons known as Hermitian Yang-Mills (HYM)instantons Indeed this relationship has been well-knownto string theorists and mathematicians under the name ofthe Donaldson-Uhlenbeck-Yau (DUY) theorem [17 18] Wequote a paragraph in [19] (21) to clearly summarize thispicture





+ (119872) oplus Ω2minus (119872) (9)





















) (13)


)= 12 (120596(+)


) (14)








T푐 (16)


T푎 (17)






) (19)





it is given by














E = E퐴Γ퐴 = ( 0 119890(+)퐴120574퐴119890(minus)퐴120574퐴 0 ) (28)






A (119890(+)퐴 120596(+)퐴퐵)

B (119890(minus)퐴 120596(minus)퐴퐵) (31)


푀푁 (119909) 119889119909푀 otimes 119889119909푁(32)


or ( 120597120597119904)2

plusmn





퐵 and 119890(plusmn)퐵 = 0 (34)


퐴퐵 and 119890(plusmn)퐵 = 0 (35)








0 휔(+)119860119861퐽119860119861+




DR = (119863(+)119877(+) 00 119863(minus)119877(minus)) = 0 (38)










C119897 (6) cong Λlowast119872 = 6⨁푘=0

Ω푘 (119872) (41)



119881 = 3⨁푘=0


푘=0





푘 even













퐴퐵 = (119865(+)푎퐴퐵 119879푎


2 ) = F퐴퐵 (49)









plusmn (119872) (51)
















A 12119877(+)퐴퐵퐶퐷119869퐶퐷+ equiv 119865(+)푎

퐴퐵 (119879푎 oplus 0) B 12119877(minus)



A 119877(+)퐴퐵퐶퐷 = minus119865(+)푎

















A 119865(+)푎퐴퐵 = 119891푎푏

(++)120578푏A퐵 (62)

B 119865(minus)푎퐴퐵 = 119891푎푏




A 119877(+)퐴퐵퐶퐷 = 119891푎푏

(++)120578푎퐴퐵120578푏퐶퐷 (64)










퐷퐸퐹퐺 = 0 (66)






15 otimes 15 = (1 + 15 + 20 + 84)푆 oplus (15 + 45 + 45)퐴푆

(67)






(minusminus) (68)



퐶퐷퐸퐹 = 0 (69)














∓ = 0 (73)






∓ = minus18119868퐴퐵119868퐶퐷 (75)


(76)






minus 119898(+) 푎퐶퐷 = 0119875퐴퐵퐶퐷

+ 119898(+) 푎퐶퐷 = 119898(+) 푎


(79)



















+ (119872) = Λ21 oplus Λ2

6 oplus Λ28 (86)



and Λ28 by Λ2

8 = 1198696+ 1198697+ 11986911+ 11986912+ 11986913+ 11986914+ 119870+ 119871+ (88)












푎=0

120579푎120582푎) equiv 119890Θ (93)





푖 = 119864훼

푖= 0 (95)


훽119889119911훼 and 119889119911훽= 119894119892(plusmn)

훼훽119889119911훼 and 119889119911훽 (96)




푖푗= 0 (97)





(98)


(99)








(++)120578푏퐴퐵119890(+)퐴 and 119890(+)퐵 = 0 (102)





8 oplus Λ21 (103)













(++) + 1radic31198918푎(++) + 1radic611989115푎

(++) = 0 (105)


(++) + 1radic31198918푎(++) + 1radic611989115푎

(++)) 120578푎퐴퐵= 119865(+)3퐴퐵 + 1radic3119865(+)8

퐴퐵 + 1radic6119865(+)15퐴퐵 = 0 (106)
















































푟=0

(minus)푟 119887푟 (120)






푀119888푛 (119864) (122)



int푀




119877푀5푀6퐴5퐴6 (124)




and 119877(+)퐴3퐴4

and 119877(+)퐴5퐴6= minus 1210 sdot 31205873

int푀



119889푎푏푐119865(+)푎 and 119865(+)푏 and 119865(+)푐(125)



int푀



and 119877(minus)퐴5퐴6= 1961205873

int푀







int푀


(++)119891푐푓

(++) (127)





(minusminus) (128)
















푌푀

int푀





(136)





int푀


푌푀

int푀

1198896119909Tr119865 and 119865 and Ω (138)


Using the fact



Tr119865 and 119865 and Ω = 119889 (119870 and Ω) (140)


































(++) + 1radic31198918푎(++) + 1radic611989115푎

(++) = 0119868퐴퐵119865(minus)푎퐴퐵 = 119891푎푏



(151)








int푀plusmn




int푀plusmn




int푀Tr119865(+) and 119865(+) and 119865(+)


119889푎푏푐119865(+)푎 and 119865(+)푏 and 119865(+)푐

= minus 11921205873int푀


(++)119891푐푓

(++)120594minus (119865) = minus 119894241205873


= 1961205873int푀


= 11921205873int푀



(minusminus)

(157)





6 Discussion






















Appendix


Γ퐴 = ( 0 120574퐴120574퐴 0 ) 119860 = 1 6 (A1)


1205741 = ( 0 0 0 minus10 0 1 00 1 0 0minus1 0 0 0 )


(A2)










1205821 = (0 1 0 01 0 0 00 0 0 00 0 0 0)

1205822 = (((

0 minus119894 0 0119894 0 0 00 0 0 00 0 0 0)))



(A8)








1205781퐴퐵 = 12 (((((

0 0 0 1 0 00 0 1 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= 11989421205822 otimes 1205901

1205782퐴퐵 = minus12 (((((

0 0 1 0 0 00 0 0 minus1 0 0minus1 0 0 0 0 00 1 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= minus 11989421205822 otimes 1205903

1205783퐴퐵 = 12 (((((


1205784퐴퐵 = 12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 1 00 0 0 minus1 0 00 0 minus1 0 0 0)))))

= 11989421205827 otimes 1205901

1205785퐴퐵 = minus12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 minus10 0 minus1 0 0 00 0 0 1 0 0)))))

= minus 11989421205827 otimes 1205903

1205786퐴퐵 = 12 (((((

0 0 0 0 0 10 0 0 0 minus1 00 0 0 0 0 00 0 0 0 0 00 1 0 0 0 0minus1 0 0 0 0 0)))))

= 11989421205824 otimes 1205902


1205787퐴퐵 = minus12 ((((((

0 0 0 0 1 00 0 0 0 0 10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 minus1 0 0 0 0))))))

= minus 11989421205825 otimes I2

1205788퐴퐵 = minus 12radic3 ((((((


1205789퐴퐵 = minus12 ((((((

0 0 0 0 0 10 0 0 0 1 00 0 0 0 0 00 0 0 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 0))))))

= minus 11989421205825 otimes 1205901

12057810퐴퐵 = 12 ((((((

0 0 0 0 1 00 0 0 0 0 minus10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 1 0 0 0 0))))))

= 11989421205825 otimes 1205903

12057811퐴퐵 = 12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 minus1 00 0 0 1 0 00 0 minus1 0 0 0))))))

= 11989421205826 otimes 1205902

12057812퐴퐵 = minus12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 10 0 minus1 0 0 00 0 0 minus1 0 0))))))

= minus 11989421205827 otimes I2

12057813퐴퐵 = 12 ((((((

0 0 0 1 0 00 0 minus1 0 0 00 1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205821 otimes 1205902

12057814퐴퐵 = 12 ((((((

0 0 1 0 0 00 0 0 1 0 0minus1 0 0 0 0 00 minus1 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205822 otimes I2

12057815퐴퐵 = 1radic6 ((((((


(B1)









퐴퐵=1



plusmn (B5)








푧120572푧120573120578푎

훼훽= 120578푎



= 1198944 120578322

= 1198944 120578613

= 1198944 120578713

= minus14 120578811

= minus 1198944radic3 120578822

= 1198944radic3 120578833

= 1198942radic3 1205781123

= 1198944 1205781223

= minus14 1205781312

= 1198944 1205781412

= 14


1205781511

= 1198942radic6 1205781522

= minus 1198942radic6 1205781533

= 1198942radic6 (B17)





Acknowledgments


References










































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




M Spin(6) SU(4)

CY3 HYM

CY3 HYM

Spin(6) SU(4)

lArrrArr

lArrrArr

M

９3

９3A rarr

B rarr A rarr

B rarr （９－

（９－









+ (119872) oplus Ω2minus (119872) (7)



119877 = (119882+ + 112119904 119861119861푇 119882minus + 112119904) (8)









+ (119872) oplus Ω2minus (119872) (9)





















) (13)


)= 12 (120596(+)


) (14)








T푐 (16)


T푎 (17)






) (19)





it is given by














E = E퐴Γ퐴 = ( 0 119890(+)퐴120574퐴119890(minus)퐴120574퐴 0 ) (28)






A (119890(+)퐴 120596(+)퐴퐵)

B (119890(minus)퐴 120596(minus)퐴퐵) (31)


푀푁 (119909) 119889119909푀 otimes 119889119909푁(32)


or ( 120597120597119904)2

plusmn





퐵 and 119890(plusmn)퐵 = 0 (34)


퐴퐵 and 119890(plusmn)퐵 = 0 (35)








0 휔(+)119860119861퐽119860119861+




DR = (119863(+)119877(+) 00 119863(minus)119877(minus)) = 0 (38)










C119897 (6) cong Λlowast119872 = 6⨁푘=0

Ω푘 (119872) (41)



119881 = 3⨁푘=0


푘=0





푘 even













퐴퐵 = (119865(+)푎퐴퐵 119879푎


2 ) = F퐴퐵 (49)









plusmn (119872) (51)
















A 12119877(+)퐴퐵퐶퐷119869퐶퐷+ equiv 119865(+)푎

퐴퐵 (119879푎 oplus 0) B 12119877(minus)



A 119877(+)퐴퐵퐶퐷 = minus119865(+)푎

















A 119865(+)푎퐴퐵 = 119891푎푏

(++)120578푏A퐵 (62)

B 119865(minus)푎퐴퐵 = 119891푎푏




A 119877(+)퐴퐵퐶퐷 = 119891푎푏

(++)120578푎퐴퐵120578푏퐶퐷 (64)










퐷퐸퐹퐺 = 0 (66)






15 otimes 15 = (1 + 15 + 20 + 84)푆 oplus (15 + 45 + 45)퐴푆

(67)






(minusminus) (68)



퐶퐷퐸퐹 = 0 (69)














∓ = 0 (73)






∓ = minus18119868퐴퐵119868퐶퐷 (75)


(76)






minus 119898(+) 푎퐶퐷 = 0119875퐴퐵퐶퐷

+ 119898(+) 푎퐶퐷 = 119898(+) 푎


(79)



















+ (119872) = Λ21 oplus Λ2

6 oplus Λ28 (86)



and Λ28 by Λ2

8 = 1198696+ 1198697+ 11986911+ 11986912+ 11986913+ 11986914+ 119870+ 119871+ (88)












푎=0

120579푎120582푎) equiv 119890Θ (93)





푖 = 119864훼

푖= 0 (95)


훽119889119911훼 and 119889119911훽= 119894119892(plusmn)

훼훽119889119911훼 and 119889119911훽 (96)




푖푗= 0 (97)





(98)


(99)








(++)120578푏퐴퐵119890(+)퐴 and 119890(+)퐵 = 0 (102)





8 oplus Λ21 (103)













(++) + 1radic31198918푎(++) + 1radic611989115푎

(++) = 0 (105)


(++) + 1radic31198918푎(++) + 1radic611989115푎

(++)) 120578푎퐴퐵= 119865(+)3퐴퐵 + 1radic3119865(+)8

퐴퐵 + 1radic6119865(+)15퐴퐵 = 0 (106)
















































푟=0

(minus)푟 119887푟 (120)






푀119888푛 (119864) (122)



int푀




119877푀5푀6퐴5퐴6 (124)




and 119877(+)퐴3퐴4

and 119877(+)퐴5퐴6= minus 1210 sdot 31205873

int푀



119889푎푏푐119865(+)푎 and 119865(+)푏 and 119865(+)푐(125)



int푀



and 119877(minus)퐴5퐴6= 1961205873

int푀







int푀


(++)119891푐푓

(++) (127)





(minusminus) (128)
















푌푀

int푀





(136)





int푀


푌푀

int푀

1198896119909Tr119865 and 119865 and Ω (138)


Using the fact



Tr119865 and 119865 and Ω = 119889 (119870 and Ω) (140)


































(++) + 1radic31198918푎(++) + 1radic611989115푎

(++) = 0119868퐴퐵119865(minus)푎퐴퐵 = 119891푎푏



(151)








int푀plusmn




int푀plusmn




int푀Tr119865(+) and 119865(+) and 119865(+)


119889푎푏푐119865(+)푎 and 119865(+)푏 and 119865(+)푐

= minus 11921205873int푀


(++)119891푐푓

(++)120594minus (119865) = minus 119894241205873


= 1961205873int푀


= 11921205873int푀



(minusminus)

(157)





6 Discussion






















Appendix


Γ퐴 = ( 0 120574퐴120574퐴 0 ) 119860 = 1 6 (A1)


1205741 = ( 0 0 0 minus10 0 1 00 1 0 0minus1 0 0 0 )


(A2)










1205821 = (0 1 0 01 0 0 00 0 0 00 0 0 0)

1205822 = (((

0 minus119894 0 0119894 0 0 00 0 0 00 0 0 0)))



(A8)








1205781퐴퐵 = 12 (((((

0 0 0 1 0 00 0 1 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= 11989421205822 otimes 1205901

1205782퐴퐵 = minus12 (((((

0 0 1 0 0 00 0 0 minus1 0 0minus1 0 0 0 0 00 1 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= minus 11989421205822 otimes 1205903

1205783퐴퐵 = 12 (((((


1205784퐴퐵 = 12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 1 00 0 0 minus1 0 00 0 minus1 0 0 0)))))

= 11989421205827 otimes 1205901

1205785퐴퐵 = minus12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 minus10 0 minus1 0 0 00 0 0 1 0 0)))))

= minus 11989421205827 otimes 1205903

1205786퐴퐵 = 12 (((((

0 0 0 0 0 10 0 0 0 minus1 00 0 0 0 0 00 0 0 0 0 00 1 0 0 0 0minus1 0 0 0 0 0)))))

= 11989421205824 otimes 1205902


1205787퐴퐵 = minus12 ((((((

0 0 0 0 1 00 0 0 0 0 10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 minus1 0 0 0 0))))))

= minus 11989421205825 otimes I2

1205788퐴퐵 = minus 12radic3 ((((((


1205789퐴퐵 = minus12 ((((((

0 0 0 0 0 10 0 0 0 1 00 0 0 0 0 00 0 0 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 0))))))

= minus 11989421205825 otimes 1205901

12057810퐴퐵 = 12 ((((((

0 0 0 0 1 00 0 0 0 0 minus10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 1 0 0 0 0))))))

= 11989421205825 otimes 1205903

12057811퐴퐵 = 12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 minus1 00 0 0 1 0 00 0 minus1 0 0 0))))))

= 11989421205826 otimes 1205902

12057812퐴퐵 = minus12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 10 0 minus1 0 0 00 0 0 minus1 0 0))))))

= minus 11989421205827 otimes I2

12057813퐴퐵 = 12 ((((((

0 0 0 1 0 00 0 minus1 0 0 00 1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205821 otimes 1205902

12057814퐴퐵 = 12 ((((((

0 0 1 0 0 00 0 0 1 0 0minus1 0 0 0 0 00 minus1 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205822 otimes I2

12057815퐴퐵 = 1radic6 ((((((


(B1)









퐴퐵=1



plusmn (B5)








푧120572푧120573120578푎

훼훽= 120578푎



= 1198944 120578322

= 1198944 120578613

= 1198944 120578713

= minus14 120578811

= minus 1198944radic3 120578822

= 1198944radic3 120578833

= 1198942radic3 1205781123

= 1198944 1205781223

= minus14 1205781312

= 1198944 1205781412

= 14


1205781511

= 1198942radic6 1205781522

= minus 1198942radic6 1205781533

= 1198942radic6 (B17)





Acknowledgments


References










































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics










+ (119872) oplus Ω2minus (119872) (9)





















) (13)


)= 12 (120596(+)


) (14)








T푐 (16)


T푎 (17)






) (19)





it is given by














E = E퐴Γ퐴 = ( 0 119890(+)퐴120574퐴119890(minus)퐴120574퐴 0 ) (28)






A (119890(+)퐴 120596(+)퐴퐵)

B (119890(minus)퐴 120596(minus)퐴퐵) (31)


푀푁 (119909) 119889119909푀 otimes 119889119909푁(32)


or ( 120597120597119904)2

plusmn





퐵 and 119890(plusmn)퐵 = 0 (34)


퐴퐵 and 119890(plusmn)퐵 = 0 (35)








0 휔(+)119860119861퐽119860119861+




DR = (119863(+)119877(+) 00 119863(minus)119877(minus)) = 0 (38)










C119897 (6) cong Λlowast119872 = 6⨁푘=0

Ω푘 (119872) (41)



119881 = 3⨁푘=0


푘=0





푘 even













퐴퐵 = (119865(+)푎퐴퐵 119879푎


2 ) = F퐴퐵 (49)









plusmn (119872) (51)
















A 12119877(+)퐴퐵퐶퐷119869퐶퐷+ equiv 119865(+)푎

퐴퐵 (119879푎 oplus 0) B 12119877(minus)



A 119877(+)퐴퐵퐶퐷 = minus119865(+)푎

















A 119865(+)푎퐴퐵 = 119891푎푏

(++)120578푏A퐵 (62)

B 119865(minus)푎퐴퐵 = 119891푎푏




A 119877(+)퐴퐵퐶퐷 = 119891푎푏

(++)120578푎퐴퐵120578푏퐶퐷 (64)










퐷퐸퐹퐺 = 0 (66)






15 otimes 15 = (1 + 15 + 20 + 84)푆 oplus (15 + 45 + 45)퐴푆

(67)






(minusminus) (68)



퐶퐷퐸퐹 = 0 (69)














∓ = 0 (73)






∓ = minus18119868퐴퐵119868퐶퐷 (75)


(76)






minus 119898(+) 푎퐶퐷 = 0119875퐴퐵퐶퐷

+ 119898(+) 푎퐶퐷 = 119898(+) 푎


(79)



















+ (119872) = Λ21 oplus Λ2

6 oplus Λ28 (86)



and Λ28 by Λ2

8 = 1198696+ 1198697+ 11986911+ 11986912+ 11986913+ 11986914+ 119870+ 119871+ (88)












푎=0

120579푎120582푎) equiv 119890Θ (93)





푖 = 119864훼

푖= 0 (95)


훽119889119911훼 and 119889119911훽= 119894119892(plusmn)

훼훽119889119911훼 and 119889119911훽 (96)




푖푗= 0 (97)





(98)


(99)








(++)120578푏퐴퐵119890(+)퐴 and 119890(+)퐵 = 0 (102)





8 oplus Λ21 (103)













(++) + 1radic31198918푎(++) + 1radic611989115푎

(++) = 0 (105)


(++) + 1radic31198918푎(++) + 1radic611989115푎

(++)) 120578푎퐴퐵= 119865(+)3퐴퐵 + 1radic3119865(+)8

퐴퐵 + 1radic6119865(+)15퐴퐵 = 0 (106)
















































푟=0

(minus)푟 119887푟 (120)






푀119888푛 (119864) (122)



int푀




119877푀5푀6퐴5퐴6 (124)




and 119877(+)퐴3퐴4

and 119877(+)퐴5퐴6= minus 1210 sdot 31205873

int푀



119889푎푏푐119865(+)푎 and 119865(+)푏 and 119865(+)푐(125)



int푀



and 119877(minus)퐴5퐴6= 1961205873

int푀







int푀


(++)119891푐푓

(++) (127)





(minusminus) (128)
















푌푀

int푀





(136)





int푀


푌푀

int푀

1198896119909Tr119865 and 119865 and Ω (138)


Using the fact



Tr119865 and 119865 and Ω = 119889 (119870 and Ω) (140)


































(++) + 1radic31198918푎(++) + 1radic611989115푎

(++) = 0119868퐴퐵119865(minus)푎퐴퐵 = 119891푎푏



(151)








int푀plusmn




int푀plusmn




int푀Tr119865(+) and 119865(+) and 119865(+)


119889푎푏푐119865(+)푎 and 119865(+)푏 and 119865(+)푐

= minus 11921205873int푀


(++)119891푐푓

(++)120594minus (119865) = minus 119894241205873


= 1961205873int푀


= 11921205873int푀



(minusminus)

(157)





6 Discussion






















Appendix


Γ퐴 = ( 0 120574퐴120574퐴 0 ) 119860 = 1 6 (A1)


1205741 = ( 0 0 0 minus10 0 1 00 1 0 0minus1 0 0 0 )


(A2)










1205821 = (0 1 0 01 0 0 00 0 0 00 0 0 0)

1205822 = (((

0 minus119894 0 0119894 0 0 00 0 0 00 0 0 0)))



(A8)








1205781퐴퐵 = 12 (((((

0 0 0 1 0 00 0 1 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= 11989421205822 otimes 1205901

1205782퐴퐵 = minus12 (((((

0 0 1 0 0 00 0 0 minus1 0 0minus1 0 0 0 0 00 1 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= minus 11989421205822 otimes 1205903

1205783퐴퐵 = 12 (((((


1205784퐴퐵 = 12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 1 00 0 0 minus1 0 00 0 minus1 0 0 0)))))

= 11989421205827 otimes 1205901

1205785퐴퐵 = minus12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 minus10 0 minus1 0 0 00 0 0 1 0 0)))))

= minus 11989421205827 otimes 1205903

1205786퐴퐵 = 12 (((((

0 0 0 0 0 10 0 0 0 minus1 00 0 0 0 0 00 0 0 0 0 00 1 0 0 0 0minus1 0 0 0 0 0)))))

= 11989421205824 otimes 1205902


1205787퐴퐵 = minus12 ((((((

0 0 0 0 1 00 0 0 0 0 10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 minus1 0 0 0 0))))))

= minus 11989421205825 otimes I2

1205788퐴퐵 = minus 12radic3 ((((((


1205789퐴퐵 = minus12 ((((((

0 0 0 0 0 10 0 0 0 1 00 0 0 0 0 00 0 0 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 0))))))

= minus 11989421205825 otimes 1205901

12057810퐴퐵 = 12 ((((((

0 0 0 0 1 00 0 0 0 0 minus10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 1 0 0 0 0))))))

= 11989421205825 otimes 1205903

12057811퐴퐵 = 12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 minus1 00 0 0 1 0 00 0 minus1 0 0 0))))))

= 11989421205826 otimes 1205902

12057812퐴퐵 = minus12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 10 0 minus1 0 0 00 0 0 minus1 0 0))))))

= minus 11989421205827 otimes I2

12057813퐴퐵 = 12 ((((((

0 0 0 1 0 00 0 minus1 0 0 00 1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205821 otimes 1205902

12057814퐴퐵 = 12 ((((((

0 0 1 0 0 00 0 0 1 0 0minus1 0 0 0 0 00 minus1 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205822 otimes I2

12057815퐴퐵 = 1radic6 ((((((


(B1)









퐴퐵=1



plusmn (B5)








푧120572푧120573120578푎

훼훽= 120578푎



= 1198944 120578322

= 1198944 120578613

= 1198944 120578713

= minus14 120578811

= minus 1198944radic3 120578822

= 1198944radic3 120578833

= 1198942radic3 1205781123

= 1198944 1205781223

= minus14 1205781312

= 1198944 1205781412

= 14


1205781511

= 1198942radic6 1205781522

= minus 1198942radic6 1205781533

= 1198942radic6 (B17)





Acknowledgments


References










































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics
















) (13)


)= 12 (120596(+)


) (14)








T푐 (16)


T푎 (17)






) (19)





it is given by














E = E퐴Γ퐴 = ( 0 119890(+)퐴120574퐴119890(minus)퐴120574퐴 0 ) (28)






A (119890(+)퐴 120596(+)퐴퐵)

B (119890(minus)퐴 120596(minus)퐴퐵) (31)


푀푁 (119909) 119889119909푀 otimes 119889119909푁(32)


or ( 120597120597119904)2

plusmn





퐵 and 119890(plusmn)퐵 = 0 (34)


퐴퐵 and 119890(plusmn)퐵 = 0 (35)








0 휔(+)119860119861퐽119860119861+




DR = (119863(+)119877(+) 00 119863(minus)119877(minus)) = 0 (38)










C119897 (6) cong Λlowast119872 = 6⨁푘=0

Ω푘 (119872) (41)



119881 = 3⨁푘=0


푘=0





푘 even













퐴퐵 = (119865(+)푎퐴퐵 119879푎


2 ) = F퐴퐵 (49)









plusmn (119872) (51)
















A 12119877(+)퐴퐵퐶퐷119869퐶퐷+ equiv 119865(+)푎

퐴퐵 (119879푎 oplus 0) B 12119877(minus)



A 119877(+)퐴퐵퐶퐷 = minus119865(+)푎

















A 119865(+)푎퐴퐵 = 119891푎푏

(++)120578푏A퐵 (62)

B 119865(minus)푎퐴퐵 = 119891푎푏




A 119877(+)퐴퐵퐶퐷 = 119891푎푏

(++)120578푎퐴퐵120578푏퐶퐷 (64)










퐷퐸퐹퐺 = 0 (66)






15 otimes 15 = (1 + 15 + 20 + 84)푆 oplus (15 + 45 + 45)퐴푆

(67)






(minusminus) (68)



퐶퐷퐸퐹 = 0 (69)














∓ = 0 (73)






∓ = minus18119868퐴퐵119868퐶퐷 (75)


(76)






minus 119898(+) 푎퐶퐷 = 0119875퐴퐵퐶퐷

+ 119898(+) 푎퐶퐷 = 119898(+) 푎


(79)



















+ (119872) = Λ21 oplus Λ2

6 oplus Λ28 (86)



and Λ28 by Λ2

8 = 1198696+ 1198697+ 11986911+ 11986912+ 11986913+ 11986914+ 119870+ 119871+ (88)












푎=0

120579푎120582푎) equiv 119890Θ (93)





푖 = 119864훼

푖= 0 (95)


훽119889119911훼 and 119889119911훽= 119894119892(plusmn)

훼훽119889119911훼 and 119889119911훽 (96)




푖푗= 0 (97)





(98)


(99)








(++)120578푏퐴퐵119890(+)퐴 and 119890(+)퐵 = 0 (102)





8 oplus Λ21 (103)













(++) + 1radic31198918푎(++) + 1radic611989115푎

(++) = 0 (105)


(++) + 1radic31198918푎(++) + 1radic611989115푎

(++)) 120578푎퐴퐵= 119865(+)3퐴퐵 + 1radic3119865(+)8

퐴퐵 + 1radic6119865(+)15퐴퐵 = 0 (106)
















































푟=0

(minus)푟 119887푟 (120)






푀119888푛 (119864) (122)



int푀




119877푀5푀6퐴5퐴6 (124)




and 119877(+)퐴3퐴4

and 119877(+)퐴5퐴6= minus 1210 sdot 31205873

int푀



119889푎푏푐119865(+)푎 and 119865(+)푏 and 119865(+)푐(125)



int푀



and 119877(minus)퐴5퐴6= 1961205873

int푀







int푀


(++)119891푐푓

(++) (127)





(minusminus) (128)
















푌푀

int푀





(136)





int푀


푌푀

int푀

1198896119909Tr119865 and 119865 and Ω (138)


Using the fact



Tr119865 and 119865 and Ω = 119889 (119870 and Ω) (140)


































(++) + 1radic31198918푎(++) + 1radic611989115푎

(++) = 0119868퐴퐵119865(minus)푎퐴퐵 = 119891푎푏



(151)








int푀plusmn




int푀plusmn




int푀Tr119865(+) and 119865(+) and 119865(+)


119889푎푏푐119865(+)푎 and 119865(+)푏 and 119865(+)푐

= minus 11921205873int푀


(++)119891푐푓

(++)120594minus (119865) = minus 119894241205873


= 1961205873int푀


= 11921205873int푀



(minusminus)

(157)





6 Discussion






















Appendix


Γ퐴 = ( 0 120574퐴120574퐴 0 ) 119860 = 1 6 (A1)


1205741 = ( 0 0 0 minus10 0 1 00 1 0 0minus1 0 0 0 )


(A2)










1205821 = (0 1 0 01 0 0 00 0 0 00 0 0 0)

1205822 = (((

0 minus119894 0 0119894 0 0 00 0 0 00 0 0 0)))



(A8)








1205781퐴퐵 = 12 (((((

0 0 0 1 0 00 0 1 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= 11989421205822 otimes 1205901

1205782퐴퐵 = minus12 (((((

0 0 1 0 0 00 0 0 minus1 0 0minus1 0 0 0 0 00 1 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= minus 11989421205822 otimes 1205903

1205783퐴퐵 = 12 (((((


1205784퐴퐵 = 12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 1 00 0 0 minus1 0 00 0 minus1 0 0 0)))))

= 11989421205827 otimes 1205901

1205785퐴퐵 = minus12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 minus10 0 minus1 0 0 00 0 0 1 0 0)))))

= minus 11989421205827 otimes 1205903

1205786퐴퐵 = 12 (((((

0 0 0 0 0 10 0 0 0 minus1 00 0 0 0 0 00 0 0 0 0 00 1 0 0 0 0minus1 0 0 0 0 0)))))

= 11989421205824 otimes 1205902


1205787퐴퐵 = minus12 ((((((

0 0 0 0 1 00 0 0 0 0 10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 minus1 0 0 0 0))))))

= minus 11989421205825 otimes I2

1205788퐴퐵 = minus 12radic3 ((((((


1205789퐴퐵 = minus12 ((((((

0 0 0 0 0 10 0 0 0 1 00 0 0 0 0 00 0 0 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 0))))))

= minus 11989421205825 otimes 1205901

12057810퐴퐵 = 12 ((((((

0 0 0 0 1 00 0 0 0 0 minus10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 1 0 0 0 0))))))

= 11989421205825 otimes 1205903

12057811퐴퐵 = 12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 minus1 00 0 0 1 0 00 0 minus1 0 0 0))))))

= 11989421205826 otimes 1205902

12057812퐴퐵 = minus12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 10 0 minus1 0 0 00 0 0 minus1 0 0))))))

= minus 11989421205827 otimes I2

12057813퐴퐵 = 12 ((((((

0 0 0 1 0 00 0 minus1 0 0 00 1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205821 otimes 1205902

12057814퐴퐵 = 12 ((((((

0 0 1 0 0 00 0 0 1 0 0minus1 0 0 0 0 00 minus1 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205822 otimes I2

12057815퐴퐵 = 1radic6 ((((((


(B1)









퐴퐵=1



plusmn (B5)








푧120572푧120573120578푎

훼훽= 120578푎



= 1198944 120578322

= 1198944 120578613

= 1198944 120578713

= minus14 120578811

= minus 1198944radic3 120578822

= 1198944radic3 120578833

= 1198942radic3 1205781123

= 1198944 1205781223

= minus14 1205781312

= 1198944 1205781412

= 14


1205781511

= 1198942radic6 1205781522

= minus 1198942radic6 1205781533

= 1198942radic6 (B17)





Acknowledgments


References










































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






T푐 (16)


T푎 (17)






) (19)





it is given by














E = E퐴Γ퐴 = ( 0 119890(+)퐴120574퐴119890(minus)퐴120574퐴 0 ) (28)






A (119890(+)퐴 120596(+)퐴퐵)

B (119890(minus)퐴 120596(minus)퐴퐵) (31)


푀푁 (119909) 119889119909푀 otimes 119889119909푁(32)


or ( 120597120597119904)2

plusmn





퐵 and 119890(plusmn)퐵 = 0 (34)


퐴퐵 and 119890(plusmn)퐵 = 0 (35)








0 휔(+)119860119861퐽119860119861+




DR = (119863(+)119877(+) 00 119863(minus)119877(minus)) = 0 (38)










C119897 (6) cong Λlowast119872 = 6⨁푘=0

Ω푘 (119872) (41)



119881 = 3⨁푘=0


푘=0





푘 even













퐴퐵 = (119865(+)푎퐴퐵 119879푎


2 ) = F퐴퐵 (49)









plusmn (119872) (51)
















A 12119877(+)퐴퐵퐶퐷119869퐶퐷+ equiv 119865(+)푎

퐴퐵 (119879푎 oplus 0) B 12119877(minus)



A 119877(+)퐴퐵퐶퐷 = minus119865(+)푎

















A 119865(+)푎퐴퐵 = 119891푎푏

(++)120578푏A퐵 (62)

B 119865(minus)푎퐴퐵 = 119891푎푏




A 119877(+)퐴퐵퐶퐷 = 119891푎푏

(++)120578푎퐴퐵120578푏퐶퐷 (64)










퐷퐸퐹퐺 = 0 (66)






15 otimes 15 = (1 + 15 + 20 + 84)푆 oplus (15 + 45 + 45)퐴푆

(67)






(minusminus) (68)



퐶퐷퐸퐹 = 0 (69)














∓ = 0 (73)






∓ = minus18119868퐴퐵119868퐶퐷 (75)


(76)






minus 119898(+) 푎퐶퐷 = 0119875퐴퐵퐶퐷

+ 119898(+) 푎퐶퐷 = 119898(+) 푎


(79)



















+ (119872) = Λ21 oplus Λ2

6 oplus Λ28 (86)



and Λ28 by Λ2

8 = 1198696+ 1198697+ 11986911+ 11986912+ 11986913+ 11986914+ 119870+ 119871+ (88)












푎=0

120579푎120582푎) equiv 119890Θ (93)





푖 = 119864훼

푖= 0 (95)


훽119889119911훼 and 119889119911훽= 119894119892(plusmn)

훼훽119889119911훼 and 119889119911훽 (96)




푖푗= 0 (97)





(98)


(99)








(++)120578푏퐴퐵119890(+)퐴 and 119890(+)퐵 = 0 (102)





8 oplus Λ21 (103)













(++) + 1radic31198918푎(++) + 1radic611989115푎

(++) = 0 (105)


(++) + 1radic31198918푎(++) + 1radic611989115푎

(++)) 120578푎퐴퐵= 119865(+)3퐴퐵 + 1radic3119865(+)8

퐴퐵 + 1radic6119865(+)15퐴퐵 = 0 (106)
















































푟=0

(minus)푟 119887푟 (120)






푀119888푛 (119864) (122)



int푀




119877푀5푀6퐴5퐴6 (124)




and 119877(+)퐴3퐴4

and 119877(+)퐴5퐴6= minus 1210 sdot 31205873

int푀



119889푎푏푐119865(+)푎 and 119865(+)푏 and 119865(+)푐(125)



int푀



and 119877(minus)퐴5퐴6= 1961205873

int푀







int푀


(++)119891푐푓

(++) (127)





(minusminus) (128)
















푌푀

int푀





(136)





int푀


푌푀

int푀

1198896119909Tr119865 and 119865 and Ω (138)


Using the fact



Tr119865 and 119865 and Ω = 119889 (119870 and Ω) (140)


































(++) + 1radic31198918푎(++) + 1radic611989115푎

(++) = 0119868퐴퐵119865(minus)푎퐴퐵 = 119891푎푏



(151)








int푀plusmn




int푀plusmn




int푀Tr119865(+) and 119865(+) and 119865(+)


119889푎푏푐119865(+)푎 and 119865(+)푏 and 119865(+)푐

= minus 11921205873int푀


(++)119891푐푓

(++)120594minus (119865) = minus 119894241205873


= 1961205873int푀


= 11921205873int푀



(minusminus)

(157)





6 Discussion






















Appendix


Γ퐴 = ( 0 120574퐴120574퐴 0 ) 119860 = 1 6 (A1)


1205741 = ( 0 0 0 minus10 0 1 00 1 0 0minus1 0 0 0 )


(A2)










1205821 = (0 1 0 01 0 0 00 0 0 00 0 0 0)

1205822 = (((

0 minus119894 0 0119894 0 0 00 0 0 00 0 0 0)))



(A8)








1205781퐴퐵 = 12 (((((

0 0 0 1 0 00 0 1 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= 11989421205822 otimes 1205901

1205782퐴퐵 = minus12 (((((

0 0 1 0 0 00 0 0 minus1 0 0minus1 0 0 0 0 00 1 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= minus 11989421205822 otimes 1205903

1205783퐴퐵 = 12 (((((


1205784퐴퐵 = 12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 1 00 0 0 minus1 0 00 0 minus1 0 0 0)))))

= 11989421205827 otimes 1205901

1205785퐴퐵 = minus12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 minus10 0 minus1 0 0 00 0 0 1 0 0)))))

= minus 11989421205827 otimes 1205903

1205786퐴퐵 = 12 (((((

0 0 0 0 0 10 0 0 0 minus1 00 0 0 0 0 00 0 0 0 0 00 1 0 0 0 0minus1 0 0 0 0 0)))))

= 11989421205824 otimes 1205902


1205787퐴퐵 = minus12 ((((((

0 0 0 0 1 00 0 0 0 0 10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 minus1 0 0 0 0))))))

= minus 11989421205825 otimes I2

1205788퐴퐵 = minus 12radic3 ((((((


1205789퐴퐵 = minus12 ((((((

0 0 0 0 0 10 0 0 0 1 00 0 0 0 0 00 0 0 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 0))))))

= minus 11989421205825 otimes 1205901

12057810퐴퐵 = 12 ((((((

0 0 0 0 1 00 0 0 0 0 minus10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 1 0 0 0 0))))))

= 11989421205825 otimes 1205903

12057811퐴퐵 = 12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 minus1 00 0 0 1 0 00 0 minus1 0 0 0))))))

= 11989421205826 otimes 1205902

12057812퐴퐵 = minus12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 10 0 minus1 0 0 00 0 0 minus1 0 0))))))

= minus 11989421205827 otimes I2

12057813퐴퐵 = 12 ((((((

0 0 0 1 0 00 0 minus1 0 0 00 1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205821 otimes 1205902

12057814퐴퐵 = 12 ((((((

0 0 1 0 0 00 0 0 1 0 0minus1 0 0 0 0 00 minus1 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205822 otimes I2

12057815퐴퐵 = 1radic6 ((((((


(B1)









퐴퐵=1



plusmn (B5)








푧120572푧120573120578푎

훼훽= 120578푎



= 1198944 120578322

= 1198944 120578613

= 1198944 120578713

= minus14 120578811

= minus 1198944radic3 120578822

= 1198944radic3 120578833

= 1198942radic3 1205781123

= 1198944 1205781223

= minus14 1205781312

= 1198944 1205781412

= 14


1205781511

= 1198942radic6 1205781522

= minus 1198942radic6 1205781533

= 1198942radic6 (B17)





Acknowledgments


References










































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




or ( 120597120597119904)2

plusmn





퐵 and 119890(plusmn)퐵 = 0 (34)


퐴퐵 and 119890(plusmn)퐵 = 0 (35)








0 휔(+)119860119861퐽119860119861+




DR = (119863(+)119877(+) 00 119863(minus)119877(minus)) = 0 (38)










C119897 (6) cong Λlowast119872 = 6⨁푘=0

Ω푘 (119872) (41)



119881 = 3⨁푘=0


푘=0





푘 even













퐴퐵 = (119865(+)푎퐴퐵 119879푎


2 ) = F퐴퐵 (49)









plusmn (119872) (51)
















A 12119877(+)퐴퐵퐶퐷119869퐶퐷+ equiv 119865(+)푎

퐴퐵 (119879푎 oplus 0) B 12119877(minus)



A 119877(+)퐴퐵퐶퐷 = minus119865(+)푎

















A 119865(+)푎퐴퐵 = 119891푎푏

(++)120578푏A퐵 (62)

B 119865(minus)푎퐴퐵 = 119891푎푏




A 119877(+)퐴퐵퐶퐷 = 119891푎푏

(++)120578푎퐴퐵120578푏퐶퐷 (64)










퐷퐸퐹퐺 = 0 (66)






15 otimes 15 = (1 + 15 + 20 + 84)푆 oplus (15 + 45 + 45)퐴푆

(67)






(minusminus) (68)



퐶퐷퐸퐹 = 0 (69)














∓ = 0 (73)






∓ = minus18119868퐴퐵119868퐶퐷 (75)


(76)






minus 119898(+) 푎퐶퐷 = 0119875퐴퐵퐶퐷

+ 119898(+) 푎퐶퐷 = 119898(+) 푎


(79)



















+ (119872) = Λ21 oplus Λ2

6 oplus Λ28 (86)



and Λ28 by Λ2

8 = 1198696+ 1198697+ 11986911+ 11986912+ 11986913+ 11986914+ 119870+ 119871+ (88)












푎=0

120579푎120582푎) equiv 119890Θ (93)





푖 = 119864훼

푖= 0 (95)


훽119889119911훼 and 119889119911훽= 119894119892(plusmn)

훼훽119889119911훼 and 119889119911훽 (96)




푖푗= 0 (97)





(98)


(99)








(++)120578푏퐴퐵119890(+)퐴 and 119890(+)퐵 = 0 (102)





8 oplus Λ21 (103)













(++) + 1radic31198918푎(++) + 1radic611989115푎

(++) = 0 (105)


(++) + 1radic31198918푎(++) + 1radic611989115푎

(++)) 120578푎퐴퐵= 119865(+)3퐴퐵 + 1radic3119865(+)8

퐴퐵 + 1radic6119865(+)15퐴퐵 = 0 (106)
















































푟=0

(minus)푟 119887푟 (120)






푀119888푛 (119864) (122)



int푀




119877푀5푀6퐴5퐴6 (124)




and 119877(+)퐴3퐴4

and 119877(+)퐴5퐴6= minus 1210 sdot 31205873

int푀



119889푎푏푐119865(+)푎 and 119865(+)푏 and 119865(+)푐(125)



int푀



and 119877(minus)퐴5퐴6= 1961205873

int푀







int푀


(++)119891푐푓

(++) (127)





(minusminus) (128)
















푌푀

int푀





(136)





int푀


푌푀

int푀

1198896119909Tr119865 and 119865 and Ω (138)


Using the fact



Tr119865 and 119865 and Ω = 119889 (119870 and Ω) (140)


































(++) + 1radic31198918푎(++) + 1radic611989115푎

(++) = 0119868퐴퐵119865(minus)푎퐴퐵 = 119891푎푏



(151)








int푀plusmn




int푀plusmn




int푀Tr119865(+) and 119865(+) and 119865(+)


119889푎푏푐119865(+)푎 and 119865(+)푏 and 119865(+)푐

= minus 11921205873int푀


(++)119891푐푓

(++)120594minus (119865) = minus 119894241205873


= 1961205873int푀


= 11921205873int푀



(minusminus)

(157)





6 Discussion






















Appendix


Γ퐴 = ( 0 120574퐴120574퐴 0 ) 119860 = 1 6 (A1)


1205741 = ( 0 0 0 minus10 0 1 00 1 0 0minus1 0 0 0 )


(A2)










1205821 = (0 1 0 01 0 0 00 0 0 00 0 0 0)

1205822 = (((

0 minus119894 0 0119894 0 0 00 0 0 00 0 0 0)))



(A8)








1205781퐴퐵 = 12 (((((

0 0 0 1 0 00 0 1 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= 11989421205822 otimes 1205901

1205782퐴퐵 = minus12 (((((

0 0 1 0 0 00 0 0 minus1 0 0minus1 0 0 0 0 00 1 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= minus 11989421205822 otimes 1205903

1205783퐴퐵 = 12 (((((


1205784퐴퐵 = 12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 1 00 0 0 minus1 0 00 0 minus1 0 0 0)))))

= 11989421205827 otimes 1205901

1205785퐴퐵 = minus12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 minus10 0 minus1 0 0 00 0 0 1 0 0)))))

= minus 11989421205827 otimes 1205903

1205786퐴퐵 = 12 (((((

0 0 0 0 0 10 0 0 0 minus1 00 0 0 0 0 00 0 0 0 0 00 1 0 0 0 0minus1 0 0 0 0 0)))))

= 11989421205824 otimes 1205902


1205787퐴퐵 = minus12 ((((((

0 0 0 0 1 00 0 0 0 0 10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 minus1 0 0 0 0))))))

= minus 11989421205825 otimes I2

1205788퐴퐵 = minus 12radic3 ((((((


1205789퐴퐵 = minus12 ((((((

0 0 0 0 0 10 0 0 0 1 00 0 0 0 0 00 0 0 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 0))))))

= minus 11989421205825 otimes 1205901

12057810퐴퐵 = 12 ((((((

0 0 0 0 1 00 0 0 0 0 minus10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 1 0 0 0 0))))))

= 11989421205825 otimes 1205903

12057811퐴퐵 = 12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 minus1 00 0 0 1 0 00 0 minus1 0 0 0))))))

= 11989421205826 otimes 1205902

12057812퐴퐵 = minus12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 10 0 minus1 0 0 00 0 0 minus1 0 0))))))

= minus 11989421205827 otimes I2

12057813퐴퐵 = 12 ((((((

0 0 0 1 0 00 0 minus1 0 0 00 1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205821 otimes 1205902

12057814퐴퐵 = 12 ((((((

0 0 1 0 0 00 0 0 1 0 0minus1 0 0 0 0 00 minus1 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205822 otimes I2

12057815퐴퐵 = 1radic6 ((((((


(B1)









퐴퐵=1



plusmn (B5)








푧120572푧120573120578푎

훼훽= 120578푎



= 1198944 120578322

= 1198944 120578613

= 1198944 120578713

= minus14 120578811

= minus 1198944radic3 120578822

= 1198944radic3 120578833

= 1198942radic3 1205781123

= 1198944 1205781223

= minus14 1205781312

= 1198944 1205781412

= 14


1205781511

= 1198942radic6 1205781522

= minus 1198942radic6 1205781533

= 1198942radic6 (B17)





Acknowledgments


References










































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





푘 even













퐴퐵 = (119865(+)푎퐴퐵 119879푎


2 ) = F퐴퐵 (49)









plusmn (119872) (51)
















A 12119877(+)퐴퐵퐶퐷119869퐶퐷+ equiv 119865(+)푎

퐴퐵 (119879푎 oplus 0) B 12119877(minus)



A 119877(+)퐴퐵퐶퐷 = minus119865(+)푎

















A 119865(+)푎퐴퐵 = 119891푎푏

(++)120578푏A퐵 (62)

B 119865(minus)푎퐴퐵 = 119891푎푏




A 119877(+)퐴퐵퐶퐷 = 119891푎푏

(++)120578푎퐴퐵120578푏퐶퐷 (64)










퐷퐸퐹퐺 = 0 (66)






15 otimes 15 = (1 + 15 + 20 + 84)푆 oplus (15 + 45 + 45)퐴푆

(67)






(minusminus) (68)



퐶퐷퐸퐹 = 0 (69)














∓ = 0 (73)






∓ = minus18119868퐴퐵119868퐶퐷 (75)


(76)






minus 119898(+) 푎퐶퐷 = 0119875퐴퐵퐶퐷

+ 119898(+) 푎퐶퐷 = 119898(+) 푎


(79)



















+ (119872) = Λ21 oplus Λ2

6 oplus Λ28 (86)



and Λ28 by Λ2

8 = 1198696+ 1198697+ 11986911+ 11986912+ 11986913+ 11986914+ 119870+ 119871+ (88)












푎=0

120579푎120582푎) equiv 119890Θ (93)





푖 = 119864훼

푖= 0 (95)


훽119889119911훼 and 119889119911훽= 119894119892(plusmn)

훼훽119889119911훼 and 119889119911훽 (96)




푖푗= 0 (97)





(98)


(99)








(++)120578푏퐴퐵119890(+)퐴 and 119890(+)퐵 = 0 (102)





8 oplus Λ21 (103)













(++) + 1radic31198918푎(++) + 1radic611989115푎

(++) = 0 (105)


(++) + 1radic31198918푎(++) + 1radic611989115푎

(++)) 120578푎퐴퐵= 119865(+)3퐴퐵 + 1radic3119865(+)8

퐴퐵 + 1radic6119865(+)15퐴퐵 = 0 (106)
















































푟=0

(minus)푟 119887푟 (120)






푀119888푛 (119864) (122)



int푀




119877푀5푀6퐴5퐴6 (124)




and 119877(+)퐴3퐴4

and 119877(+)퐴5퐴6= minus 1210 sdot 31205873

int푀



119889푎푏푐119865(+)푎 and 119865(+)푏 and 119865(+)푐(125)



int푀



and 119877(minus)퐴5퐴6= 1961205873

int푀







int푀


(++)119891푐푓

(++) (127)





(minusminus) (128)
















푌푀

int푀





(136)





int푀


푌푀

int푀

1198896119909Tr119865 and 119865 and Ω (138)


Using the fact



Tr119865 and 119865 and Ω = 119889 (119870 and Ω) (140)


































(++) + 1radic31198918푎(++) + 1radic611989115푎

(++) = 0119868퐴퐵119865(minus)푎퐴퐵 = 119891푎푏



(151)








int푀plusmn




int푀plusmn




int푀Tr119865(+) and 119865(+) and 119865(+)


119889푎푏푐119865(+)푎 and 119865(+)푏 and 119865(+)푐

= minus 11921205873int푀


(++)119891푐푓

(++)120594minus (119865) = minus 119894241205873


= 1961205873int푀


= 11921205873int푀



(minusminus)

(157)





6 Discussion






















Appendix


Γ퐴 = ( 0 120574퐴120574퐴 0 ) 119860 = 1 6 (A1)


1205741 = ( 0 0 0 minus10 0 1 00 1 0 0minus1 0 0 0 )


(A2)










1205821 = (0 1 0 01 0 0 00 0 0 00 0 0 0)

1205822 = (((

0 minus119894 0 0119894 0 0 00 0 0 00 0 0 0)))



(A8)








1205781퐴퐵 = 12 (((((

0 0 0 1 0 00 0 1 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= 11989421205822 otimes 1205901

1205782퐴퐵 = minus12 (((((

0 0 1 0 0 00 0 0 minus1 0 0minus1 0 0 0 0 00 1 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= minus 11989421205822 otimes 1205903

1205783퐴퐵 = 12 (((((


1205784퐴퐵 = 12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 1 00 0 0 minus1 0 00 0 minus1 0 0 0)))))

= 11989421205827 otimes 1205901

1205785퐴퐵 = minus12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 minus10 0 minus1 0 0 00 0 0 1 0 0)))))

= minus 11989421205827 otimes 1205903

1205786퐴퐵 = 12 (((((

0 0 0 0 0 10 0 0 0 minus1 00 0 0 0 0 00 0 0 0 0 00 1 0 0 0 0minus1 0 0 0 0 0)))))

= 11989421205824 otimes 1205902


1205787퐴퐵 = minus12 ((((((

0 0 0 0 1 00 0 0 0 0 10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 minus1 0 0 0 0))))))

= minus 11989421205825 otimes I2

1205788퐴퐵 = minus 12radic3 ((((((


1205789퐴퐵 = minus12 ((((((

0 0 0 0 0 10 0 0 0 1 00 0 0 0 0 00 0 0 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 0))))))

= minus 11989421205825 otimes 1205901

12057810퐴퐵 = 12 ((((((

0 0 0 0 1 00 0 0 0 0 minus10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 1 0 0 0 0))))))

= 11989421205825 otimes 1205903

12057811퐴퐵 = 12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 minus1 00 0 0 1 0 00 0 minus1 0 0 0))))))

= 11989421205826 otimes 1205902

12057812퐴퐵 = minus12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 10 0 minus1 0 0 00 0 0 minus1 0 0))))))

= minus 11989421205827 otimes I2

12057813퐴퐵 = 12 ((((((

0 0 0 1 0 00 0 minus1 0 0 00 1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205821 otimes 1205902

12057814퐴퐵 = 12 ((((((

0 0 1 0 0 00 0 0 1 0 0minus1 0 0 0 0 00 minus1 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205822 otimes I2

12057815퐴퐵 = 1radic6 ((((((


(B1)









퐴퐵=1



plusmn (B5)








푧120572푧120573120578푎

훼훽= 120578푎



= 1198944 120578322

= 1198944 120578613

= 1198944 120578713

= minus14 120578811

= minus 1198944radic3 120578822

= 1198944radic3 120578833

= 1198942radic3 1205781123

= 1198944 1205781223

= minus14 1205781312

= 1198944 1205781412

= 14


1205781511

= 1198942radic6 1205781522

= minus 1198942radic6 1205781533

= 1198942radic6 (B17)





Acknowledgments


References










































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








A 12119877(+)퐴퐵퐶퐷119869퐶퐷+ equiv 119865(+)푎

퐴퐵 (119879푎 oplus 0) B 12119877(minus)



A 119877(+)퐴퐵퐶퐷 = minus119865(+)푎

















A 119865(+)푎퐴퐵 = 119891푎푏

(++)120578푏A퐵 (62)

B 119865(minus)푎퐴퐵 = 119891푎푏




A 119877(+)퐴퐵퐶퐷 = 119891푎푏

(++)120578푎퐴퐵120578푏퐶퐷 (64)










퐷퐸퐹퐺 = 0 (66)






15 otimes 15 = (1 + 15 + 20 + 84)푆 oplus (15 + 45 + 45)퐴푆

(67)






(minusminus) (68)



퐶퐷퐸퐹 = 0 (69)














∓ = 0 (73)






∓ = minus18119868퐴퐵119868퐶퐷 (75)


(76)






minus 119898(+) 푎퐶퐷 = 0119875퐴퐵퐶퐷

+ 119898(+) 푎퐶퐷 = 119898(+) 푎


(79)



















+ (119872) = Λ21 oplus Λ2

6 oplus Λ28 (86)



and Λ28 by Λ2

8 = 1198696+ 1198697+ 11986911+ 11986912+ 11986913+ 11986914+ 119870+ 119871+ (88)












푎=0

120579푎120582푎) equiv 119890Θ (93)





푖 = 119864훼

푖= 0 (95)


훽119889119911훼 and 119889119911훽= 119894119892(plusmn)

훼훽119889119911훼 and 119889119911훽 (96)




푖푗= 0 (97)





(98)


(99)








(++)120578푏퐴퐵119890(+)퐴 and 119890(+)퐵 = 0 (102)





8 oplus Λ21 (103)













(++) + 1radic31198918푎(++) + 1radic611989115푎

(++) = 0 (105)


(++) + 1radic31198918푎(++) + 1radic611989115푎

(++)) 120578푎퐴퐵= 119865(+)3퐴퐵 + 1radic3119865(+)8

퐴퐵 + 1radic6119865(+)15퐴퐵 = 0 (106)
















































푟=0

(minus)푟 119887푟 (120)






푀119888푛 (119864) (122)



int푀




119877푀5푀6퐴5퐴6 (124)




and 119877(+)퐴3퐴4

and 119877(+)퐴5퐴6= minus 1210 sdot 31205873

int푀



119889푎푏푐119865(+)푎 and 119865(+)푏 and 119865(+)푐(125)



int푀



and 119877(minus)퐴5퐴6= 1961205873

int푀







int푀


(++)119891푐푓

(++) (127)





(minusminus) (128)
















푌푀

int푀





(136)





int푀


푌푀

int푀

1198896119909Tr119865 and 119865 and Ω (138)


Using the fact



Tr119865 and 119865 and Ω = 119889 (119870 and Ω) (140)


































(++) + 1radic31198918푎(++) + 1radic611989115푎

(++) = 0119868퐴퐵119865(minus)푎퐴퐵 = 119891푎푏



(151)








int푀plusmn




int푀plusmn




int푀Tr119865(+) and 119865(+) and 119865(+)


119889푎푏푐119865(+)푎 and 119865(+)푏 and 119865(+)푐

= minus 11921205873int푀


(++)119891푐푓

(++)120594minus (119865) = minus 119894241205873


= 1961205873int푀


= 11921205873int푀



(minusminus)

(157)





6 Discussion






















Appendix


Γ퐴 = ( 0 120574퐴120574퐴 0 ) 119860 = 1 6 (A1)


1205741 = ( 0 0 0 minus10 0 1 00 1 0 0minus1 0 0 0 )


(A2)










1205821 = (0 1 0 01 0 0 00 0 0 00 0 0 0)

1205822 = (((

0 minus119894 0 0119894 0 0 00 0 0 00 0 0 0)))



(A8)








1205781퐴퐵 = 12 (((((

0 0 0 1 0 00 0 1 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= 11989421205822 otimes 1205901

1205782퐴퐵 = minus12 (((((

0 0 1 0 0 00 0 0 minus1 0 0minus1 0 0 0 0 00 1 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= minus 11989421205822 otimes 1205903

1205783퐴퐵 = 12 (((((


1205784퐴퐵 = 12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 1 00 0 0 minus1 0 00 0 minus1 0 0 0)))))

= 11989421205827 otimes 1205901

1205785퐴퐵 = minus12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 minus10 0 minus1 0 0 00 0 0 1 0 0)))))

= minus 11989421205827 otimes 1205903

1205786퐴퐵 = 12 (((((

0 0 0 0 0 10 0 0 0 minus1 00 0 0 0 0 00 0 0 0 0 00 1 0 0 0 0minus1 0 0 0 0 0)))))

= 11989421205824 otimes 1205902


1205787퐴퐵 = minus12 ((((((

0 0 0 0 1 00 0 0 0 0 10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 minus1 0 0 0 0))))))

= minus 11989421205825 otimes I2

1205788퐴퐵 = minus 12radic3 ((((((


1205789퐴퐵 = minus12 ((((((

0 0 0 0 0 10 0 0 0 1 00 0 0 0 0 00 0 0 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 0))))))

= minus 11989421205825 otimes 1205901

12057810퐴퐵 = 12 ((((((

0 0 0 0 1 00 0 0 0 0 minus10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 1 0 0 0 0))))))

= 11989421205825 otimes 1205903

12057811퐴퐵 = 12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 minus1 00 0 0 1 0 00 0 minus1 0 0 0))))))

= 11989421205826 otimes 1205902

12057812퐴퐵 = minus12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 10 0 minus1 0 0 00 0 0 minus1 0 0))))))

= minus 11989421205827 otimes I2

12057813퐴퐵 = 12 ((((((

0 0 0 1 0 00 0 minus1 0 0 00 1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205821 otimes 1205902

12057814퐴퐵 = 12 ((((((

0 0 1 0 0 00 0 0 1 0 0minus1 0 0 0 0 00 minus1 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205822 otimes I2

12057815퐴퐵 = 1radic6 ((((((


(B1)









퐴퐵=1



plusmn (B5)








푧120572푧120573120578푎

훼훽= 120578푎



= 1198944 120578322

= 1198944 120578613

= 1198944 120578713

= minus14 120578811

= minus 1198944radic3 120578822

= 1198944radic3 120578833

= 1198942radic3 1205781123

= 1198944 1205781223

= minus14 1205781312

= 1198944 1205781412

= 14


1205781511

= 1198942radic6 1205781522

= minus 1198942radic6 1205781533

= 1198942radic6 (B17)





Acknowledgments


References










































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





퐶퐷퐸퐹 = 0 (69)














∓ = 0 (73)






∓ = minus18119868퐴퐵119868퐶퐷 (75)


(76)






minus 119898(+) 푎퐶퐷 = 0119875퐴퐵퐶퐷

+ 119898(+) 푎퐶퐷 = 119898(+) 푎


(79)



















+ (119872) = Λ21 oplus Λ2

6 oplus Λ28 (86)



and Λ28 by Λ2

8 = 1198696+ 1198697+ 11986911+ 11986912+ 11986913+ 11986914+ 119870+ 119871+ (88)












푎=0

120579푎120582푎) equiv 119890Θ (93)





푖 = 119864훼

푖= 0 (95)


훽119889119911훼 and 119889119911훽= 119894119892(plusmn)

훼훽119889119911훼 and 119889119911훽 (96)




푖푗= 0 (97)





(98)


(99)








(++)120578푏퐴퐵119890(+)퐴 and 119890(+)퐵 = 0 (102)





8 oplus Λ21 (103)













(++) + 1radic31198918푎(++) + 1radic611989115푎

(++) = 0 (105)


(++) + 1radic31198918푎(++) + 1radic611989115푎

(++)) 120578푎퐴퐵= 119865(+)3퐴퐵 + 1radic3119865(+)8

퐴퐵 + 1radic6119865(+)15퐴퐵 = 0 (106)
















































푟=0

(minus)푟 119887푟 (120)






푀119888푛 (119864) (122)



int푀




119877푀5푀6퐴5퐴6 (124)




and 119877(+)퐴3퐴4

and 119877(+)퐴5퐴6= minus 1210 sdot 31205873

int푀



119889푎푏푐119865(+)푎 and 119865(+)푏 and 119865(+)푐(125)



int푀



and 119877(minus)퐴5퐴6= 1961205873

int푀







int푀


(++)119891푐푓

(++) (127)





(minusminus) (128)
















푌푀

int푀





(136)





int푀


푌푀

int푀

1198896119909Tr119865 and 119865 and Ω (138)


Using the fact



Tr119865 and 119865 and Ω = 119889 (119870 and Ω) (140)


































(++) + 1radic31198918푎(++) + 1radic611989115푎

(++) = 0119868퐴퐵119865(minus)푎퐴퐵 = 119891푎푏



(151)








int푀plusmn




int푀plusmn




int푀Tr119865(+) and 119865(+) and 119865(+)


119889푎푏푐119865(+)푎 and 119865(+)푏 and 119865(+)푐

= minus 11921205873int푀


(++)119891푐푓

(++)120594minus (119865) = minus 119894241205873


= 1961205873int푀


= 11921205873int푀



(minusminus)

(157)





6 Discussion






















Appendix


Γ퐴 = ( 0 120574퐴120574퐴 0 ) 119860 = 1 6 (A1)


1205741 = ( 0 0 0 minus10 0 1 00 1 0 0minus1 0 0 0 )


(A2)










1205821 = (0 1 0 01 0 0 00 0 0 00 0 0 0)

1205822 = (((

0 minus119894 0 0119894 0 0 00 0 0 00 0 0 0)))



(A8)








1205781퐴퐵 = 12 (((((

0 0 0 1 0 00 0 1 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= 11989421205822 otimes 1205901

1205782퐴퐵 = minus12 (((((

0 0 1 0 0 00 0 0 minus1 0 0minus1 0 0 0 0 00 1 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= minus 11989421205822 otimes 1205903

1205783퐴퐵 = 12 (((((


1205784퐴퐵 = 12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 1 00 0 0 minus1 0 00 0 minus1 0 0 0)))))

= 11989421205827 otimes 1205901

1205785퐴퐵 = minus12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 minus10 0 minus1 0 0 00 0 0 1 0 0)))))

= minus 11989421205827 otimes 1205903

1205786퐴퐵 = 12 (((((

0 0 0 0 0 10 0 0 0 minus1 00 0 0 0 0 00 0 0 0 0 00 1 0 0 0 0minus1 0 0 0 0 0)))))

= 11989421205824 otimes 1205902


1205787퐴퐵 = minus12 ((((((

0 0 0 0 1 00 0 0 0 0 10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 minus1 0 0 0 0))))))

= minus 11989421205825 otimes I2

1205788퐴퐵 = minus 12radic3 ((((((


1205789퐴퐵 = minus12 ((((((

0 0 0 0 0 10 0 0 0 1 00 0 0 0 0 00 0 0 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 0))))))

= minus 11989421205825 otimes 1205901

12057810퐴퐵 = 12 ((((((

0 0 0 0 1 00 0 0 0 0 minus10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 1 0 0 0 0))))))

= 11989421205825 otimes 1205903

12057811퐴퐵 = 12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 minus1 00 0 0 1 0 00 0 minus1 0 0 0))))))

= 11989421205826 otimes 1205902

12057812퐴퐵 = minus12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 10 0 minus1 0 0 00 0 0 minus1 0 0))))))

= minus 11989421205827 otimes I2

12057813퐴퐵 = 12 ((((((

0 0 0 1 0 00 0 minus1 0 0 00 1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205821 otimes 1205902

12057814퐴퐵 = 12 ((((((

0 0 1 0 0 00 0 0 1 0 0minus1 0 0 0 0 00 minus1 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205822 otimes I2

12057815퐴퐵 = 1radic6 ((((((


(B1)









퐴퐵=1



plusmn (B5)








푧120572푧120573120578푎

훼훽= 120578푎



= 1198944 120578322

= 1198944 120578613

= 1198944 120578713

= minus14 120578811

= minus 1198944radic3 120578822

= 1198944radic3 120578833

= 1198942radic3 1205781123

= 1198944 1205781223

= minus14 1205781312

= 1198944 1205781412

= 14


1205781511

= 1198942radic6 1205781522

= minus 1198942radic6 1205781533

= 1198942radic6 (B17)





Acknowledgments


References










































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics


















+ (119872) = Λ21 oplus Λ2

6 oplus Λ28 (86)



and Λ28 by Λ2

8 = 1198696+ 1198697+ 11986911+ 11986912+ 11986913+ 11986914+ 119870+ 119871+ (88)












푎=0

120579푎120582푎) equiv 119890Θ (93)





푖 = 119864훼

푖= 0 (95)


훽119889119911훼 and 119889119911훽= 119894119892(plusmn)

훼훽119889119911훼 and 119889119911훽 (96)




푖푗= 0 (97)





(98)


(99)








(++)120578푏퐴퐵119890(+)퐴 and 119890(+)퐵 = 0 (102)





8 oplus Λ21 (103)













(++) + 1radic31198918푎(++) + 1radic611989115푎

(++) = 0 (105)


(++) + 1radic31198918푎(++) + 1radic611989115푎

(++)) 120578푎퐴퐵= 119865(+)3퐴퐵 + 1radic3119865(+)8

퐴퐵 + 1radic6119865(+)15퐴퐵 = 0 (106)
















































푟=0

(minus)푟 119887푟 (120)






푀119888푛 (119864) (122)



int푀




119877푀5푀6퐴5퐴6 (124)




and 119877(+)퐴3퐴4

and 119877(+)퐴5퐴6= minus 1210 sdot 31205873

int푀



119889푎푏푐119865(+)푎 and 119865(+)푏 and 119865(+)푐(125)



int푀



and 119877(minus)퐴5퐴6= 1961205873

int푀







int푀


(++)119891푐푓

(++) (127)





(minusminus) (128)
















푌푀

int푀





(136)





int푀


푌푀

int푀

1198896119909Tr119865 and 119865 and Ω (138)


Using the fact



Tr119865 and 119865 and Ω = 119889 (119870 and Ω) (140)


































(++) + 1radic31198918푎(++) + 1radic611989115푎

(++) = 0119868퐴퐵119865(minus)푎퐴퐵 = 119891푎푏



(151)








int푀plusmn




int푀plusmn




int푀Tr119865(+) and 119865(+) and 119865(+)


119889푎푏푐119865(+)푎 and 119865(+)푏 and 119865(+)푐

= minus 11921205873int푀


(++)119891푐푓

(++)120594minus (119865) = minus 119894241205873


= 1961205873int푀


= 11921205873int푀



(minusminus)

(157)





6 Discussion






















Appendix


Γ퐴 = ( 0 120574퐴120574퐴 0 ) 119860 = 1 6 (A1)


1205741 = ( 0 0 0 minus10 0 1 00 1 0 0minus1 0 0 0 )


(A2)










1205821 = (0 1 0 01 0 0 00 0 0 00 0 0 0)

1205822 = (((

0 minus119894 0 0119894 0 0 00 0 0 00 0 0 0)))



(A8)








1205781퐴퐵 = 12 (((((

0 0 0 1 0 00 0 1 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= 11989421205822 otimes 1205901

1205782퐴퐵 = minus12 (((((

0 0 1 0 0 00 0 0 minus1 0 0minus1 0 0 0 0 00 1 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= minus 11989421205822 otimes 1205903

1205783퐴퐵 = 12 (((((


1205784퐴퐵 = 12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 1 00 0 0 minus1 0 00 0 minus1 0 0 0)))))

= 11989421205827 otimes 1205901

1205785퐴퐵 = minus12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 minus10 0 minus1 0 0 00 0 0 1 0 0)))))

= minus 11989421205827 otimes 1205903

1205786퐴퐵 = 12 (((((

0 0 0 0 0 10 0 0 0 minus1 00 0 0 0 0 00 0 0 0 0 00 1 0 0 0 0minus1 0 0 0 0 0)))))

= 11989421205824 otimes 1205902


1205787퐴퐵 = minus12 ((((((

0 0 0 0 1 00 0 0 0 0 10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 minus1 0 0 0 0))))))

= minus 11989421205825 otimes I2

1205788퐴퐵 = minus 12radic3 ((((((


1205789퐴퐵 = minus12 ((((((

0 0 0 0 0 10 0 0 0 1 00 0 0 0 0 00 0 0 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 0))))))

= minus 11989421205825 otimes 1205901

12057810퐴퐵 = 12 ((((((

0 0 0 0 1 00 0 0 0 0 minus10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 1 0 0 0 0))))))

= 11989421205825 otimes 1205903

12057811퐴퐵 = 12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 minus1 00 0 0 1 0 00 0 minus1 0 0 0))))))

= 11989421205826 otimes 1205902

12057812퐴퐵 = minus12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 10 0 minus1 0 0 00 0 0 minus1 0 0))))))

= minus 11989421205827 otimes I2

12057813퐴퐵 = 12 ((((((

0 0 0 1 0 00 0 minus1 0 0 00 1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205821 otimes 1205902

12057814퐴퐵 = 12 ((((((

0 0 1 0 0 00 0 0 1 0 0minus1 0 0 0 0 00 minus1 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205822 otimes I2

12057815퐴퐵 = 1radic6 ((((((


(B1)









퐴퐵=1



plusmn (B5)








푧120572푧120573120578푎

훼훽= 120578푎



= 1198944 120578322

= 1198944 120578613

= 1198944 120578713

= minus14 120578811

= minus 1198944radic3 120578822

= 1198944radic3 120578833

= 1198942radic3 1205781123

= 1198944 1205781223

= minus14 1205781312

= 1198944 1205781412

= 14


1205781511

= 1198942radic6 1205781522

= minus 1198942radic6 1205781533

= 1198942radic6 (B17)





Acknowledgments


References










































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







푖 = 119864훼

푖= 0 (95)


훽119889119911훼 and 119889119911훽= 119894119892(plusmn)

훼훽119889119911훼 and 119889119911훽 (96)




푖푗= 0 (97)





(98)


(99)








(++)120578푏퐴퐵119890(+)퐴 and 119890(+)퐵 = 0 (102)





8 oplus Λ21 (103)













(++) + 1radic31198918푎(++) + 1radic611989115푎

(++) = 0 (105)


(++) + 1radic31198918푎(++) + 1radic611989115푎

(++)) 120578푎퐴퐵= 119865(+)3퐴퐵 + 1radic3119865(+)8

퐴퐵 + 1radic6119865(+)15퐴퐵 = 0 (106)
















































푟=0

(minus)푟 119887푟 (120)






푀119888푛 (119864) (122)



int푀




119877푀5푀6퐴5퐴6 (124)




and 119877(+)퐴3퐴4

and 119877(+)퐴5퐴6= minus 1210 sdot 31205873

int푀



119889푎푏푐119865(+)푎 and 119865(+)푏 and 119865(+)푐(125)



int푀



and 119877(minus)퐴5퐴6= 1961205873

int푀







int푀


(++)119891푐푓

(++) (127)





(minusminus) (128)
















푌푀

int푀





(136)





int푀


푌푀

int푀

1198896119909Tr119865 and 119865 and Ω (138)


Using the fact



Tr119865 and 119865 and Ω = 119889 (119870 and Ω) (140)


































(++) + 1radic31198918푎(++) + 1radic611989115푎

(++) = 0119868퐴퐵119865(minus)푎퐴퐵 = 119891푎푏



(151)








int푀plusmn




int푀plusmn




int푀Tr119865(+) and 119865(+) and 119865(+)


119889푎푏푐119865(+)푎 and 119865(+)푏 and 119865(+)푐

= minus 11921205873int푀


(++)119891푐푓

(++)120594minus (119865) = minus 119894241205873


= 1961205873int푀


= 11921205873int푀



(minusminus)

(157)





6 Discussion






















Appendix


Γ퐴 = ( 0 120574퐴120574퐴 0 ) 119860 = 1 6 (A1)


1205741 = ( 0 0 0 minus10 0 1 00 1 0 0minus1 0 0 0 )


(A2)










1205821 = (0 1 0 01 0 0 00 0 0 00 0 0 0)

1205822 = (((

0 minus119894 0 0119894 0 0 00 0 0 00 0 0 0)))



(A8)








1205781퐴퐵 = 12 (((((

0 0 0 1 0 00 0 1 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= 11989421205822 otimes 1205901

1205782퐴퐵 = minus12 (((((

0 0 1 0 0 00 0 0 minus1 0 0minus1 0 0 0 0 00 1 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= minus 11989421205822 otimes 1205903

1205783퐴퐵 = 12 (((((


1205784퐴퐵 = 12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 1 00 0 0 minus1 0 00 0 minus1 0 0 0)))))

= 11989421205827 otimes 1205901

1205785퐴퐵 = minus12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 minus10 0 minus1 0 0 00 0 0 1 0 0)))))

= minus 11989421205827 otimes 1205903

1205786퐴퐵 = 12 (((((

0 0 0 0 0 10 0 0 0 minus1 00 0 0 0 0 00 0 0 0 0 00 1 0 0 0 0minus1 0 0 0 0 0)))))

= 11989421205824 otimes 1205902


1205787퐴퐵 = minus12 ((((((

0 0 0 0 1 00 0 0 0 0 10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 minus1 0 0 0 0))))))

= minus 11989421205825 otimes I2

1205788퐴퐵 = minus 12radic3 ((((((


1205789퐴퐵 = minus12 ((((((

0 0 0 0 0 10 0 0 0 1 00 0 0 0 0 00 0 0 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 0))))))

= minus 11989421205825 otimes 1205901

12057810퐴퐵 = 12 ((((((

0 0 0 0 1 00 0 0 0 0 minus10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 1 0 0 0 0))))))

= 11989421205825 otimes 1205903

12057811퐴퐵 = 12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 minus1 00 0 0 1 0 00 0 minus1 0 0 0))))))

= 11989421205826 otimes 1205902

12057812퐴퐵 = minus12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 10 0 minus1 0 0 00 0 0 minus1 0 0))))))

= minus 11989421205827 otimes I2

12057813퐴퐵 = 12 ((((((

0 0 0 1 0 00 0 minus1 0 0 00 1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205821 otimes 1205902

12057814퐴퐵 = 12 ((((((

0 0 1 0 0 00 0 0 1 0 0minus1 0 0 0 0 00 minus1 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205822 otimes I2

12057815퐴퐵 = 1radic6 ((((((


(B1)









퐴퐵=1



plusmn (B5)








푧120572푧120573120578푎

훼훽= 120578푎



= 1198944 120578322

= 1198944 120578613

= 1198944 120578713

= minus14 120578811

= minus 1198944radic3 120578822

= 1198944radic3 120578833

= 1198942radic3 1205781123

= 1198944 1205781223

= minus14 1205781312

= 1198944 1205781412

= 14


1205781511

= 1198942radic6 1205781522

= minus 1198942radic6 1205781533

= 1198942radic6 (B17)





Acknowledgments


References










































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







8 oplus Λ21 (103)













(++) + 1radic31198918푎(++) + 1radic611989115푎

(++) = 0 (105)


(++) + 1radic31198918푎(++) + 1radic611989115푎

(++)) 120578푎퐴퐵= 119865(+)3퐴퐵 + 1radic3119865(+)8

퐴퐵 + 1radic6119865(+)15퐴퐵 = 0 (106)
















































푟=0

(minus)푟 119887푟 (120)






푀119888푛 (119864) (122)



int푀




119877푀5푀6퐴5퐴6 (124)




and 119877(+)퐴3퐴4

and 119877(+)퐴5퐴6= minus 1210 sdot 31205873

int푀



119889푎푏푐119865(+)푎 and 119865(+)푏 and 119865(+)푐(125)



int푀



and 119877(minus)퐴5퐴6= 1961205873

int푀







int푀


(++)119891푐푓

(++) (127)





(minusminus) (128)
















푌푀

int푀





(136)





int푀


푌푀

int푀

1198896119909Tr119865 and 119865 and Ω (138)


Using the fact



Tr119865 and 119865 and Ω = 119889 (119870 and Ω) (140)


































(++) + 1radic31198918푎(++) + 1radic611989115푎

(++) = 0119868퐴퐵119865(minus)푎퐴퐵 = 119891푎푏



(151)








int푀plusmn




int푀plusmn




int푀Tr119865(+) and 119865(+) and 119865(+)


119889푎푏푐119865(+)푎 and 119865(+)푏 and 119865(+)푐

= minus 11921205873int푀


(++)119891푐푓

(++)120594minus (119865) = minus 119894241205873


= 1961205873int푀


= 11921205873int푀



(minusminus)

(157)





6 Discussion






















Appendix


Γ퐴 = ( 0 120574퐴120574퐴 0 ) 119860 = 1 6 (A1)


1205741 = ( 0 0 0 minus10 0 1 00 1 0 0minus1 0 0 0 )


(A2)










1205821 = (0 1 0 01 0 0 00 0 0 00 0 0 0)

1205822 = (((

0 minus119894 0 0119894 0 0 00 0 0 00 0 0 0)))



(A8)








1205781퐴퐵 = 12 (((((

0 0 0 1 0 00 0 1 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= 11989421205822 otimes 1205901

1205782퐴퐵 = minus12 (((((

0 0 1 0 0 00 0 0 minus1 0 0minus1 0 0 0 0 00 1 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= minus 11989421205822 otimes 1205903

1205783퐴퐵 = 12 (((((


1205784퐴퐵 = 12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 1 00 0 0 minus1 0 00 0 minus1 0 0 0)))))

= 11989421205827 otimes 1205901

1205785퐴퐵 = minus12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 minus10 0 minus1 0 0 00 0 0 1 0 0)))))

= minus 11989421205827 otimes 1205903

1205786퐴퐵 = 12 (((((

0 0 0 0 0 10 0 0 0 minus1 00 0 0 0 0 00 0 0 0 0 00 1 0 0 0 0minus1 0 0 0 0 0)))))

= 11989421205824 otimes 1205902


1205787퐴퐵 = minus12 ((((((

0 0 0 0 1 00 0 0 0 0 10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 minus1 0 0 0 0))))))

= minus 11989421205825 otimes I2

1205788퐴퐵 = minus 12radic3 ((((((


1205789퐴퐵 = minus12 ((((((

0 0 0 0 0 10 0 0 0 1 00 0 0 0 0 00 0 0 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 0))))))

= minus 11989421205825 otimes 1205901

12057810퐴퐵 = 12 ((((((

0 0 0 0 1 00 0 0 0 0 minus10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 1 0 0 0 0))))))

= 11989421205825 otimes 1205903

12057811퐴퐵 = 12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 minus1 00 0 0 1 0 00 0 minus1 0 0 0))))))

= 11989421205826 otimes 1205902

12057812퐴퐵 = minus12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 10 0 minus1 0 0 00 0 0 minus1 0 0))))))

= minus 11989421205827 otimes I2

12057813퐴퐵 = 12 ((((((

0 0 0 1 0 00 0 minus1 0 0 00 1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205821 otimes 1205902

12057814퐴퐵 = 12 ((((((

0 0 1 0 0 00 0 0 1 0 0minus1 0 0 0 0 00 minus1 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205822 otimes I2

12057815퐴퐵 = 1radic6 ((((((


(B1)









퐴퐵=1



plusmn (B5)








푧120572푧120573120578푎

훼훽= 120578푎



= 1198944 120578322

= 1198944 120578613

= 1198944 120578713

= minus14 120578811

= minus 1198944radic3 120578822

= 1198944radic3 120578833

= 1198942radic3 1205781123

= 1198944 1205781223

= minus14 1205781312

= 1198944 1205781412

= 14


1205781511

= 1198942radic6 1205781522

= minus 1198942radic6 1205781533

= 1198942radic6 (B17)





Acknowledgments


References










































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






































푟=0

(minus)푟 119887푟 (120)






푀119888푛 (119864) (122)



int푀




119877푀5푀6퐴5퐴6 (124)




and 119877(+)퐴3퐴4

and 119877(+)퐴5퐴6= minus 1210 sdot 31205873

int푀



119889푎푏푐119865(+)푎 and 119865(+)푏 and 119865(+)푐(125)



int푀



and 119877(minus)퐴5퐴6= 1961205873

int푀







int푀


(++)119891푐푓

(++) (127)





(minusminus) (128)
















푌푀

int푀





(136)





int푀


푌푀

int푀

1198896119909Tr119865 and 119865 and Ω (138)


Using the fact



Tr119865 and 119865 and Ω = 119889 (119870 and Ω) (140)


































(++) + 1radic31198918푎(++) + 1radic611989115푎

(++) = 0119868퐴퐵119865(minus)푎퐴퐵 = 119891푎푏



(151)








int푀plusmn




int푀plusmn




int푀Tr119865(+) and 119865(+) and 119865(+)


119889푎푏푐119865(+)푎 and 119865(+)푏 and 119865(+)푐

= minus 11921205873int푀


(++)119891푐푓

(++)120594minus (119865) = minus 119894241205873


= 1961205873int푀


= 11921205873int푀



(minusminus)

(157)





6 Discussion






















Appendix


Γ퐴 = ( 0 120574퐴120574퐴 0 ) 119860 = 1 6 (A1)


1205741 = ( 0 0 0 minus10 0 1 00 1 0 0minus1 0 0 0 )


(A2)










1205821 = (0 1 0 01 0 0 00 0 0 00 0 0 0)

1205822 = (((

0 minus119894 0 0119894 0 0 00 0 0 00 0 0 0)))



(A8)








1205781퐴퐵 = 12 (((((

0 0 0 1 0 00 0 1 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= 11989421205822 otimes 1205901

1205782퐴퐵 = minus12 (((((

0 0 1 0 0 00 0 0 minus1 0 0minus1 0 0 0 0 00 1 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= minus 11989421205822 otimes 1205903

1205783퐴퐵 = 12 (((((


1205784퐴퐵 = 12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 1 00 0 0 minus1 0 00 0 minus1 0 0 0)))))

= 11989421205827 otimes 1205901

1205785퐴퐵 = minus12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 minus10 0 minus1 0 0 00 0 0 1 0 0)))))

= minus 11989421205827 otimes 1205903

1205786퐴퐵 = 12 (((((

0 0 0 0 0 10 0 0 0 minus1 00 0 0 0 0 00 0 0 0 0 00 1 0 0 0 0minus1 0 0 0 0 0)))))

= 11989421205824 otimes 1205902


1205787퐴퐵 = minus12 ((((((

0 0 0 0 1 00 0 0 0 0 10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 minus1 0 0 0 0))))))

= minus 11989421205825 otimes I2

1205788퐴퐵 = minus 12radic3 ((((((


1205789퐴퐵 = minus12 ((((((

0 0 0 0 0 10 0 0 0 1 00 0 0 0 0 00 0 0 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 0))))))

= minus 11989421205825 otimes 1205901

12057810퐴퐵 = 12 ((((((

0 0 0 0 1 00 0 0 0 0 minus10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 1 0 0 0 0))))))

= 11989421205825 otimes 1205903

12057811퐴퐵 = 12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 minus1 00 0 0 1 0 00 0 minus1 0 0 0))))))

= 11989421205826 otimes 1205902

12057812퐴퐵 = minus12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 10 0 minus1 0 0 00 0 0 minus1 0 0))))))

= minus 11989421205827 otimes I2

12057813퐴퐵 = 12 ((((((

0 0 0 1 0 00 0 minus1 0 0 00 1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205821 otimes 1205902

12057814퐴퐵 = 12 ((((((

0 0 1 0 0 00 0 0 1 0 0minus1 0 0 0 0 00 minus1 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205822 otimes I2

12057815퐴퐵 = 1radic6 ((((((


(B1)









퐴퐵=1



plusmn (B5)








푧120572푧120573120578푎

훼훽= 120578푎



= 1198944 120578322

= 1198944 120578613

= 1198944 120578713

= minus14 120578811

= minus 1198944radic3 120578822

= 1198944radic3 120578833

= 1198942radic3 1205781123

= 1198944 1205781223

= minus14 1205781312

= 1198944 1205781412

= 14


1205781511

= 1198942radic6 1205781522

= minus 1198942radic6 1205781533

= 1198942radic6 (B17)





Acknowledgments


References










































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








int푀


(++)119891푐푓

(++) (127)





(minusminus) (128)
















푌푀

int푀





(136)





int푀


푌푀

int푀

1198896119909Tr119865 and 119865 and Ω (138)


Using the fact



Tr119865 and 119865 and Ω = 119889 (119870 and Ω) (140)


































(++) + 1radic31198918푎(++) + 1radic611989115푎

(++) = 0119868퐴퐵119865(minus)푎퐴퐵 = 119891푎푏



(151)








int푀plusmn




int푀plusmn




int푀Tr119865(+) and 119865(+) and 119865(+)


119889푎푏푐119865(+)푎 and 119865(+)푏 and 119865(+)푐

= minus 11921205873int푀


(++)119891푐푓

(++)120594minus (119865) = minus 119894241205873


= 1961205873int푀


= 11921205873int푀



(minusminus)

(157)





6 Discussion






















Appendix


Γ퐴 = ( 0 120574퐴120574퐴 0 ) 119860 = 1 6 (A1)


1205741 = ( 0 0 0 minus10 0 1 00 1 0 0minus1 0 0 0 )


(A2)










1205821 = (0 1 0 01 0 0 00 0 0 00 0 0 0)

1205822 = (((

0 minus119894 0 0119894 0 0 00 0 0 00 0 0 0)))



(A8)








1205781퐴퐵 = 12 (((((

0 0 0 1 0 00 0 1 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= 11989421205822 otimes 1205901

1205782퐴퐵 = minus12 (((((

0 0 1 0 0 00 0 0 minus1 0 0minus1 0 0 0 0 00 1 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= minus 11989421205822 otimes 1205903

1205783퐴퐵 = 12 (((((


1205784퐴퐵 = 12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 1 00 0 0 minus1 0 00 0 minus1 0 0 0)))))

= 11989421205827 otimes 1205901

1205785퐴퐵 = minus12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 minus10 0 minus1 0 0 00 0 0 1 0 0)))))

= minus 11989421205827 otimes 1205903

1205786퐴퐵 = 12 (((((

0 0 0 0 0 10 0 0 0 minus1 00 0 0 0 0 00 0 0 0 0 00 1 0 0 0 0minus1 0 0 0 0 0)))))

= 11989421205824 otimes 1205902


1205787퐴퐵 = minus12 ((((((

0 0 0 0 1 00 0 0 0 0 10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 minus1 0 0 0 0))))))

= minus 11989421205825 otimes I2

1205788퐴퐵 = minus 12radic3 ((((((


1205789퐴퐵 = minus12 ((((((

0 0 0 0 0 10 0 0 0 1 00 0 0 0 0 00 0 0 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 0))))))

= minus 11989421205825 otimes 1205901

12057810퐴퐵 = 12 ((((((

0 0 0 0 1 00 0 0 0 0 minus10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 1 0 0 0 0))))))

= 11989421205825 otimes 1205903

12057811퐴퐵 = 12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 minus1 00 0 0 1 0 00 0 minus1 0 0 0))))))

= 11989421205826 otimes 1205902

12057812퐴퐵 = minus12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 10 0 minus1 0 0 00 0 0 minus1 0 0))))))

= minus 11989421205827 otimes I2

12057813퐴퐵 = 12 ((((((

0 0 0 1 0 00 0 minus1 0 0 00 1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205821 otimes 1205902

12057814퐴퐵 = 12 ((((((

0 0 1 0 0 00 0 0 1 0 0minus1 0 0 0 0 00 minus1 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205822 otimes I2

12057815퐴퐵 = 1radic6 ((((((


(B1)









퐴퐵=1



plusmn (B5)








푧120572푧120573120578푎

훼훽= 120578푎



= 1198944 120578322

= 1198944 120578613

= 1198944 120578713

= minus14 120578811

= minus 1198944radic3 120578822

= 1198944radic3 120578833

= 1198942radic3 1205781123

= 1198944 1205781223

= minus14 1205781312

= 1198944 1205781412

= 14


1205781511

= 1198942radic6 1205781522

= minus 1198942radic6 1205781533

= 1198942radic6 (B17)





Acknowledgments


References










































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






푌푀

int푀





(136)





int푀


푌푀

int푀

1198896119909Tr119865 and 119865 and Ω (138)


Using the fact



Tr119865 and 119865 and Ω = 119889 (119870 and Ω) (140)


































(++) + 1radic31198918푎(++) + 1radic611989115푎

(++) = 0119868퐴퐵119865(minus)푎퐴퐵 = 119891푎푏



(151)








int푀plusmn




int푀plusmn




int푀Tr119865(+) and 119865(+) and 119865(+)


119889푎푏푐119865(+)푎 and 119865(+)푏 and 119865(+)푐

= minus 11921205873int푀


(++)119891푐푓

(++)120594minus (119865) = minus 119894241205873


= 1961205873int푀


= 11921205873int푀



(minusminus)

(157)





6 Discussion






















Appendix


Γ퐴 = ( 0 120574퐴120574퐴 0 ) 119860 = 1 6 (A1)


1205741 = ( 0 0 0 minus10 0 1 00 1 0 0minus1 0 0 0 )


(A2)










1205821 = (0 1 0 01 0 0 00 0 0 00 0 0 0)

1205822 = (((

0 minus119894 0 0119894 0 0 00 0 0 00 0 0 0)))



(A8)








1205781퐴퐵 = 12 (((((

0 0 0 1 0 00 0 1 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= 11989421205822 otimes 1205901

1205782퐴퐵 = minus12 (((((

0 0 1 0 0 00 0 0 minus1 0 0minus1 0 0 0 0 00 1 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= minus 11989421205822 otimes 1205903

1205783퐴퐵 = 12 (((((


1205784퐴퐵 = 12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 1 00 0 0 minus1 0 00 0 minus1 0 0 0)))))

= 11989421205827 otimes 1205901

1205785퐴퐵 = minus12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 minus10 0 minus1 0 0 00 0 0 1 0 0)))))

= minus 11989421205827 otimes 1205903

1205786퐴퐵 = 12 (((((

0 0 0 0 0 10 0 0 0 minus1 00 0 0 0 0 00 0 0 0 0 00 1 0 0 0 0minus1 0 0 0 0 0)))))

= 11989421205824 otimes 1205902


1205787퐴퐵 = minus12 ((((((

0 0 0 0 1 00 0 0 0 0 10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 minus1 0 0 0 0))))))

= minus 11989421205825 otimes I2

1205788퐴퐵 = minus 12radic3 ((((((


1205789퐴퐵 = minus12 ((((((

0 0 0 0 0 10 0 0 0 1 00 0 0 0 0 00 0 0 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 0))))))

= minus 11989421205825 otimes 1205901

12057810퐴퐵 = 12 ((((((

0 0 0 0 1 00 0 0 0 0 minus10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 1 0 0 0 0))))))

= 11989421205825 otimes 1205903

12057811퐴퐵 = 12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 minus1 00 0 0 1 0 00 0 minus1 0 0 0))))))

= 11989421205826 otimes 1205902

12057812퐴퐵 = minus12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 10 0 minus1 0 0 00 0 0 minus1 0 0))))))

= minus 11989421205827 otimes I2

12057813퐴퐵 = 12 ((((((

0 0 0 1 0 00 0 minus1 0 0 00 1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205821 otimes 1205902

12057814퐴퐵 = 12 ((((((

0 0 1 0 0 00 0 0 1 0 0minus1 0 0 0 0 00 minus1 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205822 otimes I2

12057815퐴퐵 = 1radic6 ((((((


(B1)









퐴퐵=1



plusmn (B5)








푧120572푧120573120578푎

훼훽= 120578푎



= 1198944 120578322

= 1198944 120578613

= 1198944 120578713

= minus14 120578811

= minus 1198944radic3 120578822

= 1198944radic3 120578833

= 1198942radic3 1205781123

= 1198944 1205781223

= minus14 1205781312

= 1198944 1205781412

= 14


1205781511

= 1198942radic6 1205781522

= minus 1198942radic6 1205781533

= 1198942radic6 (B17)





Acknowledgments


References










































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics

























(++) + 1radic31198918푎(++) + 1radic611989115푎

(++) = 0119868퐴퐵119865(minus)푎퐴퐵 = 119891푎푏



(151)








int푀plusmn




int푀plusmn




int푀Tr119865(+) and 119865(+) and 119865(+)


119889푎푏푐119865(+)푎 and 119865(+)푏 and 119865(+)푐

= minus 11921205873int푀


(++)119891푐푓

(++)120594minus (119865) = minus 119894241205873


= 1961205873int푀


= 11921205873int푀



(minusminus)

(157)





6 Discussion






















Appendix


Γ퐴 = ( 0 120574퐴120574퐴 0 ) 119860 = 1 6 (A1)


1205741 = ( 0 0 0 minus10 0 1 00 1 0 0minus1 0 0 0 )


(A2)










1205821 = (0 1 0 01 0 0 00 0 0 00 0 0 0)

1205822 = (((

0 minus119894 0 0119894 0 0 00 0 0 00 0 0 0)))



(A8)








1205781퐴퐵 = 12 (((((

0 0 0 1 0 00 0 1 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= 11989421205822 otimes 1205901

1205782퐴퐵 = minus12 (((((

0 0 1 0 0 00 0 0 minus1 0 0minus1 0 0 0 0 00 1 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= minus 11989421205822 otimes 1205903

1205783퐴퐵 = 12 (((((


1205784퐴퐵 = 12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 1 00 0 0 minus1 0 00 0 minus1 0 0 0)))))

= 11989421205827 otimes 1205901

1205785퐴퐵 = minus12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 minus10 0 minus1 0 0 00 0 0 1 0 0)))))

= minus 11989421205827 otimes 1205903

1205786퐴퐵 = 12 (((((

0 0 0 0 0 10 0 0 0 minus1 00 0 0 0 0 00 0 0 0 0 00 1 0 0 0 0minus1 0 0 0 0 0)))))

= 11989421205824 otimes 1205902


1205787퐴퐵 = minus12 ((((((

0 0 0 0 1 00 0 0 0 0 10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 minus1 0 0 0 0))))))

= minus 11989421205825 otimes I2

1205788퐴퐵 = minus 12radic3 ((((((


1205789퐴퐵 = minus12 ((((((

0 0 0 0 0 10 0 0 0 1 00 0 0 0 0 00 0 0 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 0))))))

= minus 11989421205825 otimes 1205901

12057810퐴퐵 = 12 ((((((

0 0 0 0 1 00 0 0 0 0 minus10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 1 0 0 0 0))))))

= 11989421205825 otimes 1205903

12057811퐴퐵 = 12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 minus1 00 0 0 1 0 00 0 minus1 0 0 0))))))

= 11989421205826 otimes 1205902

12057812퐴퐵 = minus12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 10 0 minus1 0 0 00 0 0 minus1 0 0))))))

= minus 11989421205827 otimes I2

12057813퐴퐵 = 12 ((((((

0 0 0 1 0 00 0 minus1 0 0 00 1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205821 otimes 1205902

12057814퐴퐵 = 12 ((((((

0 0 1 0 0 00 0 0 1 0 0minus1 0 0 0 0 00 minus1 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205822 otimes I2

12057815퐴퐵 = 1radic6 ((((((


(B1)









퐴퐵=1



plusmn (B5)








푧120572푧120573120578푎

훼훽= 120578푎



= 1198944 120578322

= 1198944 120578613

= 1198944 120578713

= minus14 120578811

= minus 1198944radic3 120578822

= 1198944radic3 120578833

= 1198942radic3 1205781123

= 1198944 1205781223

= minus14 1205781312

= 1198944 1205781412

= 14


1205781511

= 1198942radic6 1205781522

= minus 1198942radic6 1205781533

= 1198942radic6 (B17)





Acknowledgments


References










































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






int푀plusmn




int푀plusmn




int푀Tr119865(+) and 119865(+) and 119865(+)


119889푎푏푐119865(+)푎 and 119865(+)푏 and 119865(+)푐

= minus 11921205873int푀


(++)119891푐푓

(++)120594minus (119865) = minus 119894241205873


= 1961205873int푀


= 11921205873int푀



(minusminus)

(157)





6 Discussion






















Appendix


Γ퐴 = ( 0 120574퐴120574퐴 0 ) 119860 = 1 6 (A1)


1205741 = ( 0 0 0 minus10 0 1 00 1 0 0minus1 0 0 0 )


(A2)










1205821 = (0 1 0 01 0 0 00 0 0 00 0 0 0)

1205822 = (((

0 minus119894 0 0119894 0 0 00 0 0 00 0 0 0)))



(A8)








1205781퐴퐵 = 12 (((((

0 0 0 1 0 00 0 1 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= 11989421205822 otimes 1205901

1205782퐴퐵 = minus12 (((((

0 0 1 0 0 00 0 0 minus1 0 0minus1 0 0 0 0 00 1 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= minus 11989421205822 otimes 1205903

1205783퐴퐵 = 12 (((((


1205784퐴퐵 = 12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 1 00 0 0 minus1 0 00 0 minus1 0 0 0)))))

= 11989421205827 otimes 1205901

1205785퐴퐵 = minus12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 minus10 0 minus1 0 0 00 0 0 1 0 0)))))

= minus 11989421205827 otimes 1205903

1205786퐴퐵 = 12 (((((

0 0 0 0 0 10 0 0 0 minus1 00 0 0 0 0 00 0 0 0 0 00 1 0 0 0 0minus1 0 0 0 0 0)))))

= 11989421205824 otimes 1205902


1205787퐴퐵 = minus12 ((((((

0 0 0 0 1 00 0 0 0 0 10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 minus1 0 0 0 0))))))

= minus 11989421205825 otimes I2

1205788퐴퐵 = minus 12radic3 ((((((


1205789퐴퐵 = minus12 ((((((

0 0 0 0 0 10 0 0 0 1 00 0 0 0 0 00 0 0 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 0))))))

= minus 11989421205825 otimes 1205901

12057810퐴퐵 = 12 ((((((

0 0 0 0 1 00 0 0 0 0 minus10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 1 0 0 0 0))))))

= 11989421205825 otimes 1205903

12057811퐴퐵 = 12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 minus1 00 0 0 1 0 00 0 minus1 0 0 0))))))

= 11989421205826 otimes 1205902

12057812퐴퐵 = minus12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 10 0 minus1 0 0 00 0 0 minus1 0 0))))))

= minus 11989421205827 otimes I2

12057813퐴퐵 = 12 ((((((

0 0 0 1 0 00 0 minus1 0 0 00 1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205821 otimes 1205902

12057814퐴퐵 = 12 ((((((

0 0 1 0 0 00 0 0 1 0 0minus1 0 0 0 0 00 minus1 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205822 otimes I2

12057815퐴퐵 = 1radic6 ((((((


(B1)









퐴퐵=1



plusmn (B5)








푧120572푧120573120578푎

훼훽= 120578푎



= 1198944 120578322

= 1198944 120578613

= 1198944 120578713

= minus14 120578811

= minus 1198944radic3 120578822

= 1198944radic3 120578833

= 1198942radic3 1205781123

= 1198944 1205781223

= minus14 1205781312

= 1198944 1205781412

= 14


1205781511

= 1198942radic6 1205781522

= minus 1198942radic6 1205781533

= 1198942radic6 (B17)





Acknowledgments


References










































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




















Appendix


Γ퐴 = ( 0 120574퐴120574퐴 0 ) 119860 = 1 6 (A1)


1205741 = ( 0 0 0 minus10 0 1 00 1 0 0minus1 0 0 0 )


(A2)










1205821 = (0 1 0 01 0 0 00 0 0 00 0 0 0)

1205822 = (((

0 minus119894 0 0119894 0 0 00 0 0 00 0 0 0)))



(A8)








1205781퐴퐵 = 12 (((((

0 0 0 1 0 00 0 1 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= 11989421205822 otimes 1205901

1205782퐴퐵 = minus12 (((((

0 0 1 0 0 00 0 0 minus1 0 0minus1 0 0 0 0 00 1 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= minus 11989421205822 otimes 1205903

1205783퐴퐵 = 12 (((((


1205784퐴퐵 = 12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 1 00 0 0 minus1 0 00 0 minus1 0 0 0)))))

= 11989421205827 otimes 1205901

1205785퐴퐵 = minus12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 minus10 0 minus1 0 0 00 0 0 1 0 0)))))

= minus 11989421205827 otimes 1205903

1205786퐴퐵 = 12 (((((

0 0 0 0 0 10 0 0 0 minus1 00 0 0 0 0 00 0 0 0 0 00 1 0 0 0 0minus1 0 0 0 0 0)))))

= 11989421205824 otimes 1205902


1205787퐴퐵 = minus12 ((((((

0 0 0 0 1 00 0 0 0 0 10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 minus1 0 0 0 0))))))

= minus 11989421205825 otimes I2

1205788퐴퐵 = minus 12radic3 ((((((


1205789퐴퐵 = minus12 ((((((

0 0 0 0 0 10 0 0 0 1 00 0 0 0 0 00 0 0 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 0))))))

= minus 11989421205825 otimes 1205901

12057810퐴퐵 = 12 ((((((

0 0 0 0 1 00 0 0 0 0 minus10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 1 0 0 0 0))))))

= 11989421205825 otimes 1205903

12057811퐴퐵 = 12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 minus1 00 0 0 1 0 00 0 minus1 0 0 0))))))

= 11989421205826 otimes 1205902

12057812퐴퐵 = minus12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 10 0 minus1 0 0 00 0 0 minus1 0 0))))))

= minus 11989421205827 otimes I2

12057813퐴퐵 = 12 ((((((

0 0 0 1 0 00 0 minus1 0 0 00 1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205821 otimes 1205902

12057814퐴퐵 = 12 ((((((

0 0 1 0 0 00 0 0 1 0 0minus1 0 0 0 0 00 minus1 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205822 otimes I2

12057815퐴퐵 = 1radic6 ((((((


(B1)









퐴퐵=1



plusmn (B5)








푧120572푧120573120578푎

훼훽= 120578푎



= 1198944 120578322

= 1198944 120578613

= 1198944 120578713

= minus14 120578811

= minus 1198944radic3 120578822

= 1198944radic3 120578833

= 1198942radic3 1205781123

= 1198944 1205781223

= minus14 1205781312

= 1198944 1205781412

= 14


1205781511

= 1198942radic6 1205781522

= minus 1198942radic6 1205781533

= 1198942radic6 (B17)





Acknowledgments


References










































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







1205821 = (0 1 0 01 0 0 00 0 0 00 0 0 0)

1205822 = (((

0 minus119894 0 0119894 0 0 00 0 0 00 0 0 0)))



(A8)








1205781퐴퐵 = 12 (((((

0 0 0 1 0 00 0 1 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= 11989421205822 otimes 1205901

1205782퐴퐵 = minus12 (((((

0 0 1 0 0 00 0 0 minus1 0 0minus1 0 0 0 0 00 1 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= minus 11989421205822 otimes 1205903

1205783퐴퐵 = 12 (((((


1205784퐴퐵 = 12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 1 00 0 0 minus1 0 00 0 minus1 0 0 0)))))

= 11989421205827 otimes 1205901

1205785퐴퐵 = minus12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 minus10 0 minus1 0 0 00 0 0 1 0 0)))))

= minus 11989421205827 otimes 1205903

1205786퐴퐵 = 12 (((((

0 0 0 0 0 10 0 0 0 minus1 00 0 0 0 0 00 0 0 0 0 00 1 0 0 0 0minus1 0 0 0 0 0)))))

= 11989421205824 otimes 1205902


1205787퐴퐵 = minus12 ((((((

0 0 0 0 1 00 0 0 0 0 10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 minus1 0 0 0 0))))))

= minus 11989421205825 otimes I2

1205788퐴퐵 = minus 12radic3 ((((((


1205789퐴퐵 = minus12 ((((((

0 0 0 0 0 10 0 0 0 1 00 0 0 0 0 00 0 0 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 0))))))

= minus 11989421205825 otimes 1205901

12057810퐴퐵 = 12 ((((((

0 0 0 0 1 00 0 0 0 0 minus10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 1 0 0 0 0))))))

= 11989421205825 otimes 1205903

12057811퐴퐵 = 12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 minus1 00 0 0 1 0 00 0 minus1 0 0 0))))))

= 11989421205826 otimes 1205902

12057812퐴퐵 = minus12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 10 0 minus1 0 0 00 0 0 minus1 0 0))))))

= minus 11989421205827 otimes I2

12057813퐴퐵 = 12 ((((((

0 0 0 1 0 00 0 minus1 0 0 00 1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205821 otimes 1205902

12057814퐴퐵 = 12 ((((((

0 0 1 0 0 00 0 0 1 0 0minus1 0 0 0 0 00 minus1 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205822 otimes I2

12057815퐴퐵 = 1radic6 ((((((


(B1)









퐴퐵=1



plusmn (B5)








푧120572푧120573120578푎

훼훽= 120578푎



= 1198944 120578322

= 1198944 120578613

= 1198944 120578713

= minus14 120578811

= minus 1198944radic3 120578822

= 1198944radic3 120578833

= 1198942radic3 1205781123

= 1198944 1205781223

= minus14 1205781312

= 1198944 1205781412

= 14


1205781511

= 1198942radic6 1205781522

= minus 1198942radic6 1205781533

= 1198942radic6 (B17)





Acknowledgments


References










































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








1205781퐴퐵 = 12 (((((

0 0 0 1 0 00 0 1 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= 11989421205822 otimes 1205901

1205782퐴퐵 = minus12 (((((

0 0 1 0 0 00 0 0 minus1 0 0minus1 0 0 0 0 00 1 0 0 0 00 0 0 0 0 00 0 0 0 0 0)))))

= minus 11989421205822 otimes 1205903

1205783퐴퐵 = 12 (((((


1205784퐴퐵 = 12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 1 00 0 0 minus1 0 00 0 minus1 0 0 0)))))

= 11989421205827 otimes 1205901

1205785퐴퐵 = minus12 (((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 minus10 0 minus1 0 0 00 0 0 1 0 0)))))

= minus 11989421205827 otimes 1205903

1205786퐴퐵 = 12 (((((

0 0 0 0 0 10 0 0 0 minus1 00 0 0 0 0 00 0 0 0 0 00 1 0 0 0 0minus1 0 0 0 0 0)))))

= 11989421205824 otimes 1205902


1205787퐴퐵 = minus12 ((((((

0 0 0 0 1 00 0 0 0 0 10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 minus1 0 0 0 0))))))

= minus 11989421205825 otimes I2

1205788퐴퐵 = minus 12radic3 ((((((


1205789퐴퐵 = minus12 ((((((

0 0 0 0 0 10 0 0 0 1 00 0 0 0 0 00 0 0 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 0))))))

= minus 11989421205825 otimes 1205901

12057810퐴퐵 = 12 ((((((

0 0 0 0 1 00 0 0 0 0 minus10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 1 0 0 0 0))))))

= 11989421205825 otimes 1205903

12057811퐴퐵 = 12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 minus1 00 0 0 1 0 00 0 minus1 0 0 0))))))

= 11989421205826 otimes 1205902

12057812퐴퐵 = minus12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 10 0 minus1 0 0 00 0 0 minus1 0 0))))))

= minus 11989421205827 otimes I2

12057813퐴퐵 = 12 ((((((

0 0 0 1 0 00 0 minus1 0 0 00 1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205821 otimes 1205902

12057814퐴퐵 = 12 ((((((

0 0 1 0 0 00 0 0 1 0 0minus1 0 0 0 0 00 minus1 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205822 otimes I2

12057815퐴퐵 = 1radic6 ((((((


(B1)









퐴퐵=1



plusmn (B5)








푧120572푧120573120578푎

훼훽= 120578푎



= 1198944 120578322

= 1198944 120578613

= 1198944 120578713

= minus14 120578811

= minus 1198944radic3 120578822

= 1198944radic3 120578833

= 1198942radic3 1205781123

= 1198944 1205781223

= minus14 1205781312

= 1198944 1205781412

= 14


1205781511

= 1198942radic6 1205781522

= minus 1198942radic6 1205781533

= 1198942radic6 (B17)





Acknowledgments


References










































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




1205787퐴퐵 = minus12 ((((((

0 0 0 0 1 00 0 0 0 0 10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 minus1 0 0 0 0))))))

= minus 11989421205825 otimes I2

1205788퐴퐵 = minus 12radic3 ((((((


1205789퐴퐵 = minus12 ((((((

0 0 0 0 0 10 0 0 0 1 00 0 0 0 0 00 0 0 0 0 00 minus1 0 0 0 0minus1 0 0 0 0 0))))))

= minus 11989421205825 otimes 1205901

12057810퐴퐵 = 12 ((((((

0 0 0 0 1 00 0 0 0 0 minus10 0 0 0 0 00 0 0 0 0 0minus1 0 0 0 0 00 1 0 0 0 0))))))

= 11989421205825 otimes 1205903

12057811퐴퐵 = 12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 minus1 00 0 0 1 0 00 0 minus1 0 0 0))))))

= 11989421205826 otimes 1205902

12057812퐴퐵 = minus12 ((((((

0 0 0 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 10 0 minus1 0 0 00 0 0 minus1 0 0))))))

= minus 11989421205827 otimes I2

12057813퐴퐵 = 12 ((((((

0 0 0 1 0 00 0 minus1 0 0 00 1 0 0 0 0minus1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205821 otimes 1205902

12057814퐴퐵 = 12 ((((((

0 0 1 0 0 00 0 0 1 0 0minus1 0 0 0 0 00 minus1 0 0 0 00 0 0 0 0 00 0 0 0 0 0))))))

= 11989421205822 otimes I2

12057815퐴퐵 = 1radic6 ((((((


(B1)









퐴퐵=1



plusmn (B5)








푧120572푧120573120578푎

훼훽= 120578푎



= 1198944 120578322

= 1198944 120578613

= 1198944 120578713

= minus14 120578811

= minus 1198944radic3 120578822

= 1198944radic3 120578833

= 1198942radic3 1205781123

= 1198944 1205781223

= minus14 1205781312

= 1198944 1205781412

= 14


1205781511

= 1198942radic6 1205781522

= minus 1198942radic6 1205781533

= 1198942radic6 (B17)





Acknowledgments


References










































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics









퐴퐵=1



plusmn (B5)








푧120572푧120573120578푎

훼훽= 120578푎



= 1198944 120578322

= 1198944 120578613

= 1198944 120578713

= minus14 120578811

= minus 1198944radic3 120578822

= 1198944radic3 120578833

= 1198942radic3 1205781123

= 1198944 1205781223

= minus14 1205781312

= 1198944 1205781412

= 14


1205781511

= 1198942radic6 1205781522

= minus 1198942radic6 1205781533

= 1198942radic6 (B17)





Acknowledgments


References










































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




1205781511

= 1198942radic6 1205781522

= minus 1198942radic6 1205781533

= 1198942radic6 (B17)





Acknowledgments


References










































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics














FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Calabi-Yau Manifolds, Hermitian Yang-Mills Instantons, and...

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