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.