two fundamental puzzles and lattice susy s.arianos, a.d’adda, a.feo, i.kanamori, k.nagata, j.saito...
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![Page 1: 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,](https://reader034.vdocuments.site/reader034/viewer/2022051216/5697c0021a28abf838cc3369/html5/thumbnails/1.jpg)
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,
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Majorana fermion
fermion + gravity
Motivations
Boulatov &Kazakov
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Fractal Structure of 2D Quantum Gravity
N.K. & Yotsuji
N.K. & Watabiki
Q state Potts model on random surface
(c: central charge matter )
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success of lattice QCD
success of 2-dim. lattice quantum gravity
gauge theory + matter fermion + gravity on random lattice
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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
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Dirac Kaehler Fermion
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staggered phase
species doublers
Puzzle 1
Is the staggered phase or species doublers or the “flavour” degrees of freedom physical ?
dual
Dirac-Kaehler fermion
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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
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9
N=2 SUSY in two dimensions
Dirac-Kaehler Twist (N=2)
Cont:
Latt:
Gauged Latt:
Twisted N=2 SUSY
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Compatibility of Shifts
We need a modified Leibniz rule for too !
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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.
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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
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Jacobi Identities
…
Define fermionic link components
…
Auxiliary Field
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Twisted N=2 Super Yang-Mills Action
Action has twisted SUSY exact form. Off-shell SUSY invariancefor all twisted super charges.
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Bosonic part of the Action
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Fermionic part of the Action
…
…
(1)
(2)
(1) (2)
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Higer dimensional extension is possible:
3-dim. N=4 super Yang-Mills
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“inconsistency”When
BruckmannKok
but if we introduce the following “mild non-commutativity”:
then
In general
Two Problems
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Modified Leibniz rule +Mild non-commutativity
Hopf algebraicField Theory
Concrete representation of this non-commutativity
Lattice version of Moyal product
Orbifold condition
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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
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Gauge Theory on the Random Lattice
0
1
2
3
・・
・・ ・
・
Form Simplex
1 3 0 2
1
Gauge Theory + Gravity ?
SUSY ?
Boson Fermion ?
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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
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Puzzle 2
What is the role of “quaternion” in generalized gauge theory ?
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Single lattice translation as SUSY transformation
Super parameter
SUSY algebra
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Matrix Representation
are diagonal.
Two step translation as SUSY transformation
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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.