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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.