Lattice Formulation of N4 D3 Twisted Super YangMills

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 ar. Xiv: 0707. 3533 [hep-lat] LATTICE 2007 @ Regensburg 1

Introduction SUSY on a Lattice a developing field… • S. Catterall, T. Wiseman, [ar. Xiv: 0706. 3518] • P. H. Damgaad, S. Matsuura, [ar. Xiv: 0706. 3007, 0704. 2696] • H. Suzuki, [ar. Xiv: 0706. 1392] • T. Takimi, [ar. Xiv: 0705. 3831] • J. Giedt, Po. S LAT 2006: 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 … l. Motivations (1) Fermionic structure of regularized spacetime (2) Constructive formulation of SUSY models (3) Non-perturbative dynamics … l. Main Obstacle Leibniz rule (Cont. ) Leibniz rule (Lattice) So far, limited # of Supercharges has been realized on the Lattice. Ex. LATTICE 2007 @ Regensburg 2

Our Strategy of Lattice SUSY Formulation Lattice SUSY Algebra Difference op. Lattice SUSY Multiplet Cont. SUSY Algebra SUSY Multiplet SUSY inv. Action … Lattice SUSY inv. Action … Pursuing All Supercharges to be exactly realized on the Lattice…. . LATTICE 2007 @ Regensburg 3

Implementation of Lattice SUSY Algebra Leibniz rule conditions for. . Satisfied for Dirac-Kahler Twisted Algebra of : Nucl. Phys. , B 707 (2005) 100 -144) N=2 D=2 4 Supercharges (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) … LATTICE 2007 @ Regensburg 4

N=4 D=3 Twisted SUSY Algebra . . Dirac-Kahler expansions On the Lattice ? Twisted Supercharges of N=4 D=3 LATTICE 2007 @ Regensburg 5

N=4 D=3 Twisted Lattice SUSY Algebra Symm. Choice Leibniz rule conditions Asymm. Choice Each represents 3 D Simplicial element 0 -form 1 -form 2 -form 3 -form LATTICE 2007 @ Regensburg 6

Lattice N=4 D=3 SYM formulation Bosonic & Fermioinc Gauge Link Variables Lattice SYM Multiplet via Jacobi identities N=4 D=3 Twisted Fermions Auxiliary fields 3 -Gauge fields + 3 -Scalars LATTICE 2007 @ Regensburg 7

Lattice N=4 D=3 Twisted SUSY trans. laws Resulting SUSY Algebra closes off-shell LATTICE 2007 @ Regensburg 8

Lattice N=4 D=3 Twisted SYM Action Exact form w. r. t. all the supercharges Manifest SUSY Inv. for Each term forms closed loop. Manifest Gauge Inv. with help of cyclic property under LATTICE 2007 @ Regensburg 9

Remarks 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 10

Boson part of Action Contribution of Scalar fields Zero-area loops : Plaquettes Fermion part of Action etc. . LATTICE 2007 @ Regensburg 11

Naïve Continuum limit Contribution of Agreement with Cont. Twisted N=4 D=3 SYM LATTICE 2007 @ Regensburg 12

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 Twisted Fermions (N-extended SUSY) Staggered Fermions (N-tastes) 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 13

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