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.