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Lattice Formulation of N=4 D=3 Lattice Formulation of N=4 D=3 Twisted Super Yang-MillsTwisted Super Yang-Mills

Kazuhiro NAGATA Dept. of Phys., Indiana Univ.

A. D’Adda INFN Torino, Italy

I. Kanamori RIKEN, Japan

N. Kawamoto Hokkaido Univ., Japan

Based on the collaboration with

Phys. Lett. B 633 (2006) 645-652Nucl. Phys. B 707 (2005) 100-144

arXiv:0707.3533 [hep-lat]

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Introduction

SUSY on a Lattice

• S. Catterall, T. Wiseman, [arXiv:0706.3518]• P.H. Damgaad, S. Matsuura, [arXiv:0706.3007, 0704.2696]• H. Suzuki, [arXiv:0706.1392]• T. Takimi, [arXiv:0705.3831]• J. Giedt, PoS LAT2006:008, 2006• S. Catterall, JHEP 0704:015, 2007• K. Ohta, T. Takimi, Prog. Theor. Phys. 117, 317-345, 2007• F. Bruckmann, S. Catterall, M. de Kok, Phys.Rev.D 75, 045016,2007 • S. Catterall, G.Ghadab, JHEP 0610:063, 2006 …

Main ObstacleMain Obstacle Leibniz rule (Cont.) Leibniz rule (Lattice)

So far, limited # of Supercharges has been realized on the Lattice.

Ex.

…(3) Non-perturbative dynamics

(1) Fermionic structure of regularized spacetime

(2) Constructive formulation of SUSY models

MotivationMotivationss

a developing field…

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Our Strategy of Lattice SUSY Formulation

Cont. SUSY Algebra

SUSY Multiplet

…

Lattice SUSY Algebra

Difference op.

Lattice SUSY Multiplet

…

Pursuing All Supercharges to be exactly realized on the Lattice…..

SUSY inv. Action

Lattice SUSY inv. Action

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Implementation of Lattice SUSY Algebra

for

for

Leibniz rule conditions

..

N=2 D=2 4 Supercharges

N=4 D=3 8 Supercharges N=4 D=4 16 Supercharges

(DKKN : Phys. Lett. B 633 (2006) 645-652) (DKKN : Nucl. Phys., B 707 (2005) 100-144)

Satisfied for Dirac-Kahler Twisted Algebra of

(DKKN : Phys. Lett. B 633 (2006) 645-652) …

Main Topic of this Talk

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N=4 D=3 Twisted SUSY Algebra

Dirac-Kahler expansions..

Twisted Superchargesof N=4 D=3

On the Lattice ?

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Leibniz rule conditions

N=4 D=3 Twisted Lattice SUSY Algebra

Symm. Choice

Asymm. Choice

Each represents3D Simplicial element

0-form 1-form 2-form 3-form

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Lattice N=4 D=3 SYM formulation

Bosonic & FermioincGauge Link Variables

Lattice SYM Multiplet via Jacobi identities

3-Gauge fields + 3-Scalars

Auxiliary fields

N=4 D=3 Twisted Fermions

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Lattice N=4 D=3 Twisted SUSY trans. laws

Resulting SUSY Algebra closes off-shell

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Each term forms closed loop.

Manifest Gauge Inv.

Exact form w.r.t. all the supercharges

Manifest SUSY Inv. for

Lattice N=4 D=3 Twisted SYM Action

with help of cyclic property under

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Remarks

Gauge Covariant SUSY variations for comp. fields

: Covariantly constant fermionic parameter

Since we are dealing with link objects throughout the formulation,

special attentions are needed to Gauge cov. vs SUSY inv..

: SUSY inv of Action

F. Bruckmann, M. de Kok, Phys.Rev.D 73, 074511 (2006)F. Bruckmann, S. Catterall,M. de Kok, Phys.Rev.D 75, 045016 (2007)

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Boson part of ActionPlaquettes Zero-area loops :

Fermion part of Action

etc..

Contribution of Scalar fields

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Naïve Continuum limit

Agreement with Cont. Twisted N=4 D=3 SYM

Contribution of

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Summary & Discussions

N=D=4 Lattice SYM (Leibniz rule is O.K.)

…

N=4 D=3 Twisted SYM on a Lattice (8 supercharges)

Construction of manifest SUSY inv. Action

Lattice SUSY Algebra for All Supercharges

Why Twisted SUSY ?

Twisted Fermions Staggered Fermions (N-extended SUSY) (N-tastes)

Matrix Formulation (Talk by A. D’Adda & Poster by S. Arianos)

To be addressedDim. Red. to N=4 D=2 v.s. Hermiticity on the lattice

Dirac-Kahler (Simplicial) Structure of Fermions..

with covariantly constant fermionic parameter

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