Stringy Differential Geometry, beyond Riemann€¦ · Stringy differential geometry We propose a...
Transcript of Stringy Differential Geometry, beyond Riemann€¦ · Stringy differential geometry We propose a...
Stringy Differential Geometry, beyond Riemann
Imtak Jeon
Sogang University, Seoul
12 July. 2011
Talk is based on works
in collaboration with Jeong-Hyuck Park and Kanghoon Lee
• Differential geometry with a projection:
Application to double field theory
JHEP 1104:014 (2011), arXiv:1011.1324
• Double field formulation of Yang-Mills theory
Phys.Lett.B701:260-264 (2011) , arXiv:1102.0419
• Stringy differential geometry, beyond Riemann
arXiv: 1105.6294
Introduction
• In Riemannian geometry, the fundamental object is the metric, gµν .
• String theory puts gµν , Bµν and φ on an equal footing.
• This may suggests the existence of a veiled unifying description of them,beyond Riemann.
Introduction
Symmetry
• guides the structure of Lagrangians and organizes the physical laws into simpleforms.
• for example, in Maxwell theory,
• U(1) gauge symmetry forbids m2AµAµ
• Lorentz symmetry unifies the original 4 eqs into 2.
Introduction
• Essence of Riemannian geometry: Diffeomorphism
• ∂µ −→ ∇µ = ∂µ + Γµ
• ∇λgµν = 0 −→ Γλµν = 1
2 gλρ(∂µgνρ + ∂νgµρ − ∂ρgµν)
• Curvature, [∇µ,∇ν ], −→ Rg.
• Main purpose : Generalization to a geometry for stringy theory.
• Key property in string theory : T-duality
Outline
• T-duality in string theory
• String effective action and Double Field Theory
• Stringy differential geometry as underlying geometry of DFT.
• Local inertial frame
• Non-Abelian YM gauge field in DFT.
Closed string on a Toroidal background
• For closed string wraping around a circle, the modes expansion is
X = XL + XR ,
XL(σ+) = 1
2 (x + x) + 12 (p + w)σ+ + · · · ,
XR(σ−) = 1
2 (x− x) + 12 (p− w)σ− + · · · .
where
σ± := τ ± σ, p : momentum mode , w : winding mode ,x : center of mass , x is introduced .
• Physical condition (Virasoro constraint) implies the level matching and theon-shell condition,
L0 − L0 = 0L0 + L0 = 2 ⇐⇒
N − N − p · w = 0M
2 = p2 + w2 + 2(N + N − 2)
T-duality
• T-duality transformation is
XµL + Xµ
R −→ XµL − Xµ
R ,
such that the two pairs, (x, p) and (x, w), are exchanged by each other,
(x, x, p, w) −→ (x, x, w, p) ,
Then, the level matching condition and the mass spectrum is invariant.
• On d-dimensional toroidal background, the T-duality is realized as O(d, d, Z)
transformation preserving J =
„0 11 0
«
2d×2d.
• xi : Conjugate to the momenta pi
xi : Conjugate to the winding number wi
It is natural to introduce the additional coordinate xi
• Closed string field theory treats momenta and winding modes symmetrically.So, target spacetime fields are also depend on both x and x.: Φ(xi, xi)[Kugo,Zwiebach 1992]
Effective theory
• Describes a gravity : gµν , Bµν , φ are on an equal footing completing themassless sector.
• Low energy effective action of them:
Seff. =
ZdxD√
−ge−2φ `Rg + 4∂µφ∂µφ− 1
12 HλµνHλµν´
• Diffeomorphism and one-form gauge symmetry are manifest
xµ→ xµ + δxµ , Bµν → Bµν + ∂µΛν − ∂νΛµ .
Effective theory
• Describes a gravity : gµν , Bµν , φ are on an equal footing completing themassless sector.
• Low energy effective action of them:
Seff. =
ZdxD√
−ge−2φ `Rg + 4∂µφ∂µφ− 1
12 HλµνHλµν´
• Though not manifest, if there is isometry, this enjoys T-duality which mixesgµν , Bµν , φ (Buscher )
• Include the x dependence. −→ double field theory
Double field theory (DFT)• Hull and Zwiebach , later with Hohm constructed action which has explicit
T-duality,
SDFT =
Zdy2D e−2d
hH
AB `4∂A∂Bd − 4∂Ad∂Bd + 1
8 ∂AHCD∂BHCD
−12 ∂AH
CD∂CHBD´
+ 4∂AHAB∂Bd − ∂A∂BH
ABi.
• d is double field theory ’dilaton’ given by
e−2d =√−ge−2φ
and HAB is 2D× 2D matrix, ‘generalized metric’
HAB =
„gµν
−gµκBκσ
Bρκgκν gρσ − BρκgκλBλσ
«.
• Spacetime dimension is ‘formally’ doubled from D to 2D,
xµ→ yA = (xµ, xν) , ∂µ → ∂A = (∂µ, ∂ν) .
Double field theory (DFT)
• Indices, A, B, C, · · · , are 2D-dimensional vector indices, which can be loweredor raised by O(D, D) metric JAB.
JAB :=
„0 11 0
«.
• Under global O(D, D) rotation, L ∈ O(D, D),
HAB(y) −→ LACLB
DHCD(y) , d(y) −→ d(y) ,
manifest that DFT action is invariant.
• Note that O(D, D) T-duality is background independent.
Double field theory (DFT)
• By ’Level matching condition,’
DFT action ≡ D-dimensional closed string effective action
• It possesses a gauge symmetry, say double gauge symmetry, through’generalized Lie derivative’;
double gauge symmetry = diffeomorphism + 1-form gauge symmety
Level matching constraint
• Level matching condition for the massless sector,
p · w ≡ 0 ⇐⇒∂2
∂xµ∂xµ= ∂A∂A
≡ 0 .
• Strong constraint : all the fields and the gauge parameters as well as all of theirproducts should be annihilated by ∂2 = ∂A∂A
∂2Φ ≡ 0 , ∂AΦ1∂AΦ2 ≡ 0 ⇐⇒ ∂ ≡ 0 .
• DFT is restricted in the strong constraint.
• Actually meaning that theory is not truly doubled: we can impose ∂ ≡ 0 byO(D, D) rotation.
• DFT is reorganization of effective action: Upon the level matching constraint,
SDFT =⇒ Seff. =
ZdxD√
−ge−2φ“
Rg + 4(∂φ)2−
112 H2
”.
Double gauge symmetry
• Introducing an unifying doubled gauge parameter,
XA = (Λµ, δxν)
• diffeomorphism and 1-form gauge transformation is expressed in unifiedfashion, upon the level matching constraint,
δXHAB ≡ XC∂CHAB + 2∂[AXC]HC
B + 2∂[BXC]HAC ,
δX`e−2d´
≡ ∂A`XAe−2d´
.
• In fact, these coincide with the generalized Lie derivative,
δXHAB = LXHAB , δX(e−2d) = LX(e−2d) = −2(LXd)e−2d.
Generalized Lie derivative
• Definition, Siegel, Courant, Grana ...
LXTA1···An := XB∂BTA1···An + ω∂BXBTA1···An +nX
i=1
2∂[Ai XB]TA1···Ai−1B
Ai+1···An .
• cf. ordinary one,
LXTA1···An := XB∂BTA1···An + ω∂BXBTA1···An +nX
i=1
∂Ai XBTA1···Ai−1B
Ai+1···An .
• Definition of tensor(density) in DFT : ’double gauge covariant’ quantity
δXTA1A2···An = LXTA1A2···An .
• HAB: rank 2 generalized tensor, e−2d: weight 1 generalized scalar,
Algebra of generalized Lie derivative
• Commutator of generalized Lie derivatives is closed, up to the level matchingcondition, by using c-bracket,
[LX, LY ] ≡ L[X,Y]C ,
where [X, Y]C denotes C-bracket
[X, Y]AC := XB∂BYA− YB∂BXA + 1
2 YB∂AXB −12 XB∂AYB ,
known to be euqivalent the Courant bracket if dropping ∂.
Diffeomorphism & one-form gauge symmetry
• Direct computation shows that
LXHAB ≡ XC∂CHAB + 2∂[AXC]HC
B + 2∂[BXC]HAC ,
LX`e−2d´
≡ ∂A`XAe−2d´
,
are symmetry of DFT action by Hull, Zwiebach and Hohm,
SDFT =
Zdy2D e−2d
R(H, d) ,
where ’generalized curvature’ is
R(H, d) = HAB `
4∂A∂Bd − 4∂Ad∂Bd + 18 ∂AH
CD∂BHCD −12 ∂AH
CD∂CHBD´
+4∂AHAB∂Bd − ∂A∂BH
AB .
• This expression may be analogous to the case of writing the scalar curvature,Rg, in terms of the metric and its derivative.
Diffeomorphism & one-form gauge symmetry
• Direct computation shows that
LXHAB ≡ XC∂CHAB + 2∂[AXC]HC
B + 2∂[BXC]HAC ,
LX`e−2d´
≡ ∂A`XAe−2d´
,
are symmetry of the action by Hull, Zwiebach and Hohm,
SDFT =
Zdy2D e−2d
R(H, d) ,
where
R(H, d) = HAB `
4∂A∂Bd − 4∂Ad∂Bd + 18 ∂AH
CD∂BHCD −12 ∂AH
CD∂CHBD´
+4∂AHAB∂Bd − ∂A∂BH
AB .
• What is the underlying geometry?
Diffeomorphism & one-form gauge symmetry
• Direct computation shows that
LXHAB ≡ XC∂CHAB + 2∂[AXC]HC
B + 2∂[BXC]HAC ,
LX`e−2d´
≡ ∂A`XAe−2d´
,
are symmetry of the action by Hull, Zwiebach and Hohm,
SDFT =
Zdy2D e−2d
R(H, d) ,
where
R(H, d) = HAB `
4∂A∂Bd − 4∂Ad∂Bd + 18 ∂AH
CD∂BHCD −12 ∂AH
CD∂CHBD´
+4∂AHAB∂Bd − ∂A∂BH
AB .
• connection and covariant derivative of the geometry?
Stringy differential geometry
We propose a novel differential geometry which
• enables us to rewrite the low energy effective action as a single term in a
geometrical manner,
Seff. =
Zdx2D e−2d
HABSAB ,
• treats the three objects of the massless sector in a unified way,
• manifests not only diffeomorphism and one-form gauge symmetry but also
O(D, D) T-duality,
Stringy differential geometry• Motivated by the observation that,
H =
„g−1
−g−1BBg−1 g− Bg−1B
«
is of the most general form to satisfy
HACHC
B = δAB , HAB = HBA ,
and the upper left D× D block of H is non-degenerate,
• we focus on a symmetric projection,
PABPB
C = PAC PAB = PBA ,
which is related to H by
PAB = 12 (JAB +HAB) .
• Three basic objects:JAB , PAB , d .
Stringy differential geometry
• We postulate a “semi-covariant" derivative,∇A,
∇CTA1A2···An= ∂CTA1A2···An−ωΓBBCTA1A2···An +
nX
i=1
ΓCAiBTA1···Ai−1BAi+1···An .
• In particular,
∇C(e−2d) = ∂Ce−2d− ΓB
BCe−2d = −2(∇Cd)e−2d
=⇒ ∇Cd := ∂Cd + 12 Γ
BBC
Stringy differential geometry
• We demand the following compatibility conditions,
∇AJBC = 0 , ∇APBC = 0 , ∇Ad = 0 ,
as for the unifying description of the massless modes(cf. ∇λgµν = 0 in Riemannian geometry).
• Further we require,
ΓCAB + ΓCBA = 0 , ΓABC + ΓCAB + ΓBCA = 0 .
Stringy differential geometry
• Then, we may replace ∂A by∇A in LX and also in [X, Y]AC,
LXTA1···An = XB∇BTA1···An + ω∇BXBTA1···An +
Pni=1 2∇[Ai XB]TA1···Ai−1
BAi+1···An ,
[X, Y]AC =XB∇BYA
− YB∇BXA + 1
2 YB∇
AXB −12 XB∇
AYB .
• cf. In Riemannian geometry, torsion free condition implies
LXTµ1···µn = Xν∇νTµ1···µn + ω∇νXνTµ1···µn +
Pni=1∇µi X
νTµ1···µi−1νµi+1···µn ,
[X, Y]µ = Xν∇νYµ
− Yν∇νXµ .
Stringy differential geometry
• Explicitly, the connection is
ΓCAB = 2(P∂CPP)[AB] + 2`P[A
DPB]E− P[A
DPB]E´
∂DPEC
−4
D−1
`PC[APB]
D + PC[APB]D´`
∂Dd + (P∂EPP)[ED]
´.
where P is the complementary projection,
P = 12 (1−H) .
and P and P are orthogonal,
PP = PP = 0 .
• Is this derviativ∇A double gauge covariant?
Stringy differential geometry
• For usefulness, we set
PCABDEF := PC
DP[A[EPB]
F] + 2D−1 PC[APB]
[EPF]D ,
PCABDEF := PC
DP[A[EPB]
F] + 2D−1 PC[APB]
[EPF]D ,
which satisfy
PCABDEF = PDEFCAB = PC[AB]D[EF] ,
PCABDEFPDEF
GHI = PCABGHI ,
PA
ABDEF = 0 , PABPABCDEF = 0 , etc.
• The connection belongs to the kernel of these rank six-projectors,
PCABDEFΓDEF = 0 , PCAB
DEFΓDEF = 0 .
Stringy differential geometry
• Under double-gauge transform, δXPAB = LXPAB and δXd = LXd, thediffeomorphism and the one-form gague tranform,we obtain
(δX−LX)ΓCAB ≡ 2ˆ(P+P)CAB
FDE− δ F
C δ DA δ E
B˜∂F∂[DXE] ,
and
(δX−LX)∇CTA1···An≡
X
i
2(P+P)CAiBFDE∂F∂[DXE]TA1···B···An .
• Hence, these are not double-gauge covariant,
δX = LX .
Stringy differential geometry
• However, the characteristic property of our derivative,∇A, is that, combinedwith the projections, it can generate various O(D, D) and double-gaugecovariant quantities:
PCDPA1
B1 PA2B2 · · · PAn
Bn∇DTB1B2···Bn ,
PCDPA1
B1 PA2B2 · · ·PAn
Bn∇DTB1B2···Bn ,
PAB∇ATB , PAB
∇ATB ,
PABPC1D1 · · · PCn
Dn∇A∇BTD1···Dn ,
PABPC1D1 · · ·PCn
Dn∇A∇BTD1···Dn .
• This suggests us to call∇A as semi-covariant derivative .
Curvature
• The usual curvature,
RCDAB = ∂AΓBCD − ∂BΓACD + ΓACEΓBED − ΓBC
EΓAED ,
satisfying
[∇A,∇B]TC1C2···Cn = −ΓDAB∇DTC1C2···Cn +
nX
i=1
RCiDAB TC1···Ci−1D
Ci+1···Cn ,
is NOT double-gauge covariant,
δXRABCD = LXRABCD .
It satisfy RABCD = R[AB][CD], but
RABCD = RCDAB .
Generalized curvature
• Instead, we define, as for a key quantity in our formalism,
SABCD := 12
`RABCD +RCDAB − ΓE
ABΓECD´
.
• This can be read off from the commutaor,
PIAPJ
B[∇A,∇B]TC ≡ 2PIAPJ
BSCDABTD .
• It can be shown, by brute force computation, to satisfy
• just like the Riemann curvature,
SABCD = 12 (S[AB][CD] + S[CD][AB]) ≡ SABCD , SA[BCD] = 0 ,
• and further
P AI P B
J P CK P D
L SABCD ≡ 0 , P AI P B
J P CK P D
L SABCD ≡ 0 , etc.
Generalized curvature
• SABCD is not double-gauge covariant.
• Under the double-gauge transformations, we get
(δX − LX)SABCD ≡ 4∇A
h(P+P)BCD
EFG∂E∂[FXG] .i
.
• Nevertheless, contracting indices we can obtain covariant quantities.
Covariant curvature
By settingSAB= SBA:= SC
ACB ,
which turns out to be traceless,SA
A ≡ 0 ,
and contracting with projection operators, we get
• Double-gauge covariant rank two-tensor,
PIAPJ
BSAB : generalized curvature tensor
• Double-gauge covariant scalar,
HABSAB : generalized curvature scalar
Reproduction of DFT
• Natural DFT action is proposed by
SDFT =
Zdy2D e−2d
HABSAB ,
• In fact, the covariant scalar constitutes the effective action as
HABSAB ≡ Rg + 4φ− 4∂µφ∂µφ− 1
12 HλµνHλµν .
• It also agrees with Hull, Zwiebach and Hohm,
HABSAB ≡ H
AB `4∂A∂Bd − 4∂Ad∂Bd + 1
8 ∂AHCD∂BHCD −
12 ∂AH
CD∂CHBD´
+4∂AHAB∂Bd − ∂A∂BH
AB .
Deriving Equations of motion
• It is easy to rederive the equation of motion.
• Under arbitrary infinitesimal transformations of the dilaton and the projection,we get
δSeff. ≡
Zdy2D 2e−2d
“δPABSAB − δdHABSAB + δSAB
”.
• The third therm is total derivative as
δSABCD = ∇[AδΓB]CD +∇[CδΓD]AB , ∇Ad = 0 ,
where explicitly
δΓCAB = 2P D[A P E
B]∇CδPDE + 2(P D[A P E
B] − P D[A P E
B])∇DδPEC
−4
D−1 (PC[AP DB] + PC[AP D
B] )(∂Dδd + PE[G∇GδPE
D])
−ΓFDE δ(P + P)CABFDE ,
Deriving Equations of motion
• It is easy to rederive the equation of motion.
• Under arbitrary infinitesimal transformations of the dilaton and the projection,we get
δSeff. ≡
Zdy2D 2e−2d
“δPABSAB − δdHABSAB
”.
• from the relationδP = PδPP + PδPP ,
the equations of motions are easily obtained:
P(IAPJ)
BSAB = 0 , HABSAB = 0 .
Local inertial frame and Double-vielbein
• JAB and HAB can be simultaneously diagonalized,
J =`
V V´ “
η−1 00 −η
” `V V
´t,
H =`
V V´ “
η−1 00 η
” `V V
´t.
Here η and η are two copies of the D-dimensional Minkowskian metric. Both Vand V are 2D×D matrices which we name ‘double-vielbein’.
• They must satisfy
V = PV , Vη−1Vt = P , VtJ V = η , Vt
J V = 0 ,V = PV , V η V t = −P , V t
J V = −η−1 .
• There are two copies of independent vielbein(Siegel, Tseytlin )
Local inertial frame and Double-vielbein
• Our dobule-vielbein is of the general form,
VAm = 1√2
0
@e m
µ
(B + e)νm
1
A, VAn = 1√
2
0
@e nµ
(B− e)νn
1
A ,
where Bνm = Bνλeλm and Bν
n = Bνλe λn.
• Here, eµm and eν
n are two copies of the D-dimensional vielbeincorresponding to the same spacetime metric,
eµmeνm = eµ
neνn = gµν .
• We may identify (B + e)µm and (B− e)ν
n as two copies of the vielbeinfor the winding mode coordinate, xµ, since
(B + e)µm(B + e)νm = (B− e)µ
n(B− e)νn = (g− Bg−1B)µν .
Local inertial frame and Double vielbein
• Internal symmetry group is
SO(1, D−1)× SO(D−1, 1) ,
• Taking single diagonal local Lorentz group SO(1, D−1) or SO(D−1, 1) bygauge fixing corresponds to
VAm = 1√2
„e m
µ
(B + e)νm
«, UA
m := 1√2
„(e−1)mµ
(B− e)νm
«,
or
UAm = 1√2
„e m
µ
(B + e)νm
«, VA
m := 1√2
„(e−1)mµ
(B− e)νm
«.
where we define “twins" of the double-vielbein, by exchanging eµm and eµm inV and V .
• While V and V are O(D, D) covariant, the twins are not O(D, D) covariant.
Double spin connection
• Gauging each diagonal local Lorentz symmetry, we get doubled spinconnection
ΩAmn = PABVCm∇BVC
n − PABUCm∇BUC
n .
orΩAmn = PA
BUCm∇BUCn − PA
BVCm∇BVCn .
• Upon the level matching constraint, they are expressed in terms ofD-dimensional notation,
ΩA ≡
0
@−
12 Hµ
ων −12 BνρHρ
1
A , ΩA ≡
0
@−
12 Hµ
ων −12 BνρHρ
1
A ,
Pull back to D-dimensional theory
• Double-vielbein can pull back the chiral and the anti-chiral 2D indices to themore familiar D-dimensional ones
• We pull back the double-gauge covariant rank two-tensor to obtain,
SABVAmVB
n ≡ Rmn + 2DmDnφ− 14 HmµνHn
µν + (∂λφ)Hλmn −12∇
λHλmn .
• As expected, its symmetric and the anti-symmetric parts correspond to theequations of motion of the effective action for gµν and Bµν respectively.
• Pullback of various covariant quantities
VAlDATk1 k2···kn ≡
1√2
DlTk1 k2···kn ,
VAlDATk1k2···kn ≡
1√2
DlTk1k2···kn ,
PABDATBk1···kn ≡
1√2
DlTlk1···kn −√
2 Dlφ Tlk1···kn ,
PABDATBk1···kn ≡ −
1√2
DlTlk1···kn +√
2 Dlφ Tlk1···kn ,
PABDADBTk1···kn≡
12 DµDµTk1···kn − Dµφ DµTk1···kn ,
PABDADBTk1···kn≡ −
12 DµDµTk1···kn + Dµφ DµTk1···kn ,
where we put, as for D-dimensional tensors,
Tk1k2···kn = TA1A2···An VA1 k1 VA2 k2 · · ·VAn
kn ,Tk1 k2···kn = TA1A2···An VA1
k1VA2
k2· · · VAn
kn ,Tlk1···kn = TBA1···An VB
lVA1 k1 · · ·V
Ankn ,
Tlk1···kn = TBA1···An VBlVA1
k1· · · VAn
kn ,
which are O(D, D) singlets, and we set, for Dm = (e−1)mµDµ and
Dn = (e−1)nµDµ,
Dµ := µ + (ωµ + 12 Hµ) + (ωµ −
12 Hµ) .
Symmetry structure
• Symmetry structure for the double field theory
• O(D, D) ’T-duality’
• Double-gauge symmetry
DiffeomorphismOne-form gauge symmetry for Bµν
• Commutator between so(D, D) and generalized Lie derivative does notgenerate any symmetry of double field theory.
[δh, LX] = LY , ∂CYA∂CTB1B2···Bn = 0 .
• O(D, D) transformation rotates the entire hyperplane on which DFT lives.
Application to Yang-Mills
• Symmetry structure for the double field theory
• O(D, D) ’T-duality’
• Double-gauge symmetry
DiffeomorphismOne-form gauge symmetry for Bµν
• two copies of local Lorentz symmetry
Application to Yang-Mills
• Symmetry structure for the double field theory
• O(D, D) ’T-duality’
• Double-gauge symmetry
DiffeomorphismOne-form gauge symmetry for Bµν
• Yang-Mills gauge symmetry
• Apply the formalism to couple the non-Abelian Yang-Mills gauge field to theDFT action.
Application to Yang-Mills
• We postulate a vector potential, VA, which
• is O(D, D) and double-gauge covariant,• and transforms under non-Abelian gauge group, g ∈ G,
VA −→ gVAg−1− i(∂Ag)g
−1 .
Application to Yang-Mills
• The usual field strength,
FAB = ∂AVB − ∂BVA − i [VA,VB] ,
is YM gauge covariant, but it is NOT double-gauge covariant,
δXFAB = LXFAB .
Application to Yang-Mills
• Instead, we consider with the semi-covariant derivative,
FAB := ∇AVB −∇BVA − i [VA,VB] = FAB − ΓCABVC .
• While this is neither YM gauge nor double-gauge covariant,
FAB −→ gFABg−1 + iΓC
AB(∂Cg)g−1 ,
δXFAB = LXFAB ,
• if projected properly, it can be covariant up to level matching condtion,
PACPB
DFCD −→ PA
CPBD
gFCDg−1 ,
δX(PACPB
DFCD) = LX(PA
CPBDFCD) .
Application to Yang-Mills
PACPB
DFCD is DFT field strength which is fully covariant with respect to
• O(D, D) T-duality
• Gauge symmetry
• Double gauge = Diffeomorphism + one form gauge symmetry• Yang-Mills gauge
Yang-Mills action
• Our double field formulation of Yang-Mills action is
SYM = g−2YM
Zdy2D e−2d Tr
“PABPCD
FACFBD
”,
• Manifestly this action is invariant under O(D, D) T-duality, double-gauge andYang-Mills gauge transformation.
• Corresponding D-dimensional action of the Double field YM action?
Yang-Mills in components
• Decompose the vector potential into chiral and anti-chiral ones,
VA = V+A + V−A ,
V+A = PA
BVB , V−A = PA
BVB .
• Their general forms are
V+A = 1
2
0
@A+λ
(g+B)µνA+ν
1
A , V−A = 12
0
@−A−λ
(g−B)µνA−ν
1
A .
Yang-Mills in components
• With the field redefinition,
Aµ :=12(A+
µ + A−µ ) , φµ :=12(A+
µ − A−µ ) ,
we get a general form of double guage field
VA =
„φλ
Aµ + Bµνφν
«.
• Aµ and φν will be YM gauge connection and YM gauge covariant one-formrespectively.
Yang-Mills in components
• Turning off the x-dependence reduces the action to
SYM ≡ g−2YM
ZdxD √
−ge−2φ Tr“−
14 f µν fµν
”,
wherefµν := fµν − Dµφν − Dνφµ + i [φµ, φν ] + Hµνλφλ ,
and
Tr“
fµν f µν”
= Tr“
fµν f µν + 2DµφνDµφν + 2DµφνDνφµ− [φµ, φν ][φµ, φν ]
+2i fµν [φµ, φν ] + 2 (f µν + i[φµ, φν ]) Hµνσφσ + HµνσHµντφσφτ
”.
• For T-duality, we need YM gauge covariant 1-form field.
• Similar to topologically twisted Yang-Mills, but differs in detail.
• Curved D-branes are known to convert adjoint scalars into one-form,φa→ φµ, Bershadsky
More on fully covariant quantities
• Even power of the field strength, FAB := PACPB
DFCD,
Tr“F
A1B1 FA2B1 FA2B2 FA3B2 · · · F
AnBnFA1Bn
”.
• For the Abelian group, DBI type action
det“ηAB + κ FACFB
C”
= det“ηAB + κ FCAF
CB
”,
κ is a constant.No square root is necessary since this is a scalar, not a density.
Concluding remarks
• O(D, D) T-duality, diffeomorphism, one-form gauge symmetry fixes the lowenergy effective action,
Seff. =
ZdxD e−2d
HABSAB .
• Non-Abelian Yang-Mills field is incorporated in DFT formulation.
Concluding remarks
• Yet, string theory interpretation of the YM theory is not clear.
• D-brane in DFT Albertsson, Dai, Kao, Lin
• Fermion in DFT Coimbra, Strickland-Constable, Waldram
• Extension to RR fields Hull, Kwak, Zwiebach
• Application to ‘doubled sigma model’ and generalization to M-theory are ofinterest Hull, Berman, Perry, ... ... etc...
• Concluding:Perhaps, our formalism may provide some clue to a new framework for stringtheory, beyond Riemann.
Concluding remarks
• Yet, string theory interpretation of the YM theory is not clear.
• D-brane in DFT Albertsson, Dai, Kao, Lin
• Fermion in DFT Coimbra, Strickland-Constable, Waldram
• Extension to RR fields Hull, Kwak, Zwiebach
• Application to ‘doubled sigma model’ and generalization to M-theory are ofinterest Hull, Berman, Perry, ... ... etc...
• Concluding:Perhaps, our formalism may provide some clue to a new framework for stringtheory, beyond Riemann.
Thank you.