Lattice Formulation of N=4 D=3 Twisted Super Yang-Mills
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Lattice Formulation of N=4 D=3 Twisted Super Yang-Mills. Kazuhiro NAGATA Dept. of Phys., Indiana Univ. Based on the collaboration with. A. D’Adda INFN Torino, Italy I. Kanamori RIKEN, Japan N. Kawamoto Hokkaido Univ., Japan. Nucl. Phys. B 707 (2005) 100-144.

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

Kazuhiro NAGATA

Dept. of Phys., Indiana Univ.

Based on the collaboration with

A. D’Adda INFN Torino, Italy

I. Kanamori RIKEN, Japan

N. Kawamoto Hokkaido Univ., Japan

Nucl. Phys. B 707 (2005) 100-144

Phys. Lett. B 633 (2006) 645-652

arXiv:0707.3533 [hep-lat]

LATTICE 2007 @ Regensburg

So far, limited # of Supercharges has been

realized on the Lattice.


  • Main Obstacle

    Leibniz rule (Cont.) Leibniz rule (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


SUSY on a Lattice

a developing field…

  • Motivations

(1) Fermionic structure of regularized spacetime

(2) Constructive formulation of SUSY models

(3) Non-perturbative dynamics

LATTICE 2007 @ Regensburg

Difference op.

Our Strategyof Lattice SUSY Formulation

Cont. SUSY Algebra

Lattice SUSY Algebra

SUSY Multiplet

Lattice SUSY Multiplet

SUSY inv. Action

Lattice SUSY inv. Action

Pursuing All Supercharges to

be exactly realized on the Lattice…..

LATTICE 2007 @ Regensburg

Leibniz rule conditions




Satisfied for Dirac-Kahler Twisted Algebra of

N=2 D=2 4 Supercharges

(DKKN : Nucl. Phys., B 707 (2005) 100-144)

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

N=4 D=3 8 Supercharges

Main Topic of this Talk

N=4 D=4 16 Supercharges

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

Implementation of Lattice SUSY Algebra

LATTICE 2007 @ Regensburg

On the Lattice ?

N=4 D=3 Twisted SUSY Algebra


Dirac-Kahler expansions

Twisted Supercharges

of N=4 D=3

LATTICE 2007 @ Regensburg

Each represents

3D Simplicial element





N=4 D=3 Twisted Lattice SUSY Algebra

Symm. Choice

Leibniz rule conditions

Asymm. Choice

LATTICE 2007 @ Regensburg

via Jacobi identities

Lattice SYM Multiplet

3-Gauge fields + 3-Scalars

N=4 D=3 Twisted Fermions

Auxiliary fields

Lattice N=4 D=3 SYM formulation

Bosonic & Fermioinc

Gauge Link Variables

LATTICE 2007 @ Regensburg

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

Resulting SUSY Algebra closes off-shell

LATTICE 2007 @ Regensburg

Exact form w.r.t.

all the supercharges

Manifest SUSY Inv.


Each term forms

closed loop.


Gauge Inv.

Lattice N=4 D=3 Twisted SYM Action

with help of cyclic property under

LATTICE 2007 @ Regensburg


F. Bruckmann, M. de Kok, Phys.Rev.D 73, 074511 (2006)

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

Since we are dealing with link objects throughout the formulation,

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

Gauge Covariant SUSY variations for comp. fields

: Covariantly constant fermionic parameter

: SUSY inv of Action

LATTICE 2007 @ Regensburg

Boson part of Action

Contribution of Scalar fields

Zero-area loops :


Fermion part of Action


LATTICE 2007 @ Regensburg

Contribution of

Naïve Continuum limit

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

LATTICE 2007 @ Regensburg

Summary & Discussions

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

Lattice SUSY Algebra for All Supercharges

Construction of manifest SUSY inv.Action

with covariantly constant fermionic parameter

Why Twisted SUSY ?


Dirac-Kahler (Simplicial) Structure of Fermions

Staggered Fermions

Twisted Fermions


(N-extended SUSY)

To be addressed

Dim. Red. to N=4 D=2 v.s. Hermiticity on the lattice

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

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

LATTICE 2007 @ Regensburg