Two Fundamental Puzzles And Lattice SUSY S.Arianos, A.D’Adda, A.Feo, I.Kanamori, K.Nagata, J.Saito...
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Transcript of Two Fundamental Puzzles And Lattice SUSY S.Arianos, A.D’Adda, A.Feo, I.Kanamori, K.Nagata, J.Saito...
Two Fundamental Puzzles And Lattice SUSY
S.Arianos, A.D’Adda, A.Feo, I.Kanamori, K.Nagata, J.Saito
J.Kato, A.Miyake, T.Tsukioka, Y.Uchida,
Majorana fermion
fermion + gravity
Motivations
Boulatov &Kazakov
Fractal Structure of 2D Quantum Gravity
N.K. & Yotsuji
N.K. & Watabiki
Q state Potts model on random surface
(c: central charge matter )
success of lattice QCD
success of 2-dim. lattice quantum gravity
gauge theory + matter fermion + gravity on random lattice
Lattice FermionsFree Dirac
Naïve Staggered
Kogut-Susskind
Dirac-Kaehler
(N.K. & J.Smit)
(N.K. & I.Kanamori)
(Kluberg-Stern et.al.& Gliozzi)
y 2x
Ivanenko&Landau ‘28
i : flavour ?
Staggered phase
Dirac Kaehler Fermion
staggered phase
species doublers
Puzzle 1
Is the staggered phase or species doublers or the “flavour” degrees of freedom physical ?
dual
Dirac-Kaehler fermion
Quantization and Twisted SUSY
(Two dimensional Abelian BF)
Nilpotency of BRS charge s
Auxiliary fieldOff-shell invariance
Kato,N.K.&Uchida
Continuum
N=D=2 Twisted SUSY
Tsukioka, N.K., Kato, Miyake, Uchida
9
N=2 SUSY in two dimensions
Dirac-Kaehler Twist (N=2)
Cont:
Latt:
Gauged Latt:
Twisted N=2 SUSY
Compatibility of Shifts
We need a modified Leibniz rule for too !
Symm. Choice
Asymm. Choice
Twisted N=D=2 Lattice SUSY Algebra
Cond. for Twisted N=D=2
Solutions
Equivalent to orbifold construction: by Kaplan et.al.
N=D=2 SUSY
Dirac-Kaehler Twist
Dirac-Kaehler fermion
i : flavour ? Extended SUSY suffix
y 2x
2-dim. N=2 3-dim. N=44-dim. N=4
#boson = #fermion
super charges in d-dim.
Dirac-Kaehler twisting
Answer to the Puzzle 1
Jacobi Identities
…
Define fermionic link components
…
Auxiliary Field
Twisted N=2 Super Yang-Mills Action
Action has twisted SUSY exact form. Off-shell SUSY invariancefor all twisted super charges.
Bosonic part of the Action
Fermionic part of the Action
…
…
(1)
(2)
(1) (2)
Higer dimensional extension is possible:
3-dim. N=4 super Yang-Mills
“inconsistency”When
BruckmannKok
but if we introduce the following “mild non-commutativity”:
then
In general
Two Problems
Modified Leibniz rule +Mild non-commutativity
Hopf algebraicField Theory
Concrete representation of this non-commutativity
Lattice version of Moyal product
Orbifold condition
A possible solution
We claim: if there is covariantly constant super parameter which has opposite shift of and commutes with all the super covariant derivatives:
compensates the link holes.
lattice SUSY and gauge invariant !
operation makes link holes and thus loses gauge invariance.
gets coordinate dependence super gravity
Gauge Theory on the Random Lattice
0
1
2
3
・・
・・ ・
・
Form Simplex
1 3 0 2
1
Gauge Theory + Gravity ?
SUSY ?
Boson Fermion ?
Generalized Gauge Theories in arbitrary dimensions
gauge field
gauge parameter
derivative
curvature
gauge trans.
Chern-Simons
Topological Yang-Mills
Yang-Mills
N.K. & Watabiki ‘91
Puzzle 2
What is the role of “quaternion” in generalized gauge theory ?
Single lattice translation as SUSY transformation
Super parameter
SUSY algebra
Matrix Representation
are diagonal.
Two step translation as SUSY transformation
Partial answer to Puzzle 2
Quaternion may be fundamentally related to the lattice SUSY transformation. Chirality may play an important role in the transformation.
Differential form structure for Dirac-Kaeher mechanism should be essentially introduced to accommodate super gravity nature.