1. LATTICE 2007 @ Regensburg1 Lattice Formulation of N=4 D=3 Twisted 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) 645-652 Nucl. Phys. B 707 (2005) 100-144 arXiv:0707.3533 [hep-lat]

2. LATTICE 2007 @ Regensburg2 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 Obstacle Main 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 Motivations Motivations a developing field…

3. LATTICE 2007 @ Regensburg3 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

4. LATTICE 2007 @ Regensburg4 Implementation of Lattice SUSY Algebra 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

5. LATTICE 2007 @ Regensburg5 N=4 D=3 Twisted SUSY Algebra Dirac-Kahler expansions.. Twisted Supercharges of N=4 D=3 On the Lattice ?

6. LATTICE 2007 @ Regensburg6 Leibniz rule conditions N=4 D=3 Twisted Lattice SUSY Algebra Symm. Choice Asymm. Choice Each represents 3D Simplicial element 0-form1-form 2-form3-form

7. LATTICE 2007 @ Regensburg7 Lattice N=4 D=3 SYM formulation Bosonic & Fermioinc Gauge Link Variables Lattice SYM Multiplet via Jacobi identities 3-Gauge fields + 3-Scalars Auxiliary fields N=4 D=3 Twisted Fermions

8. LATTICE 2007 @ Regensburg8 Lattice N=4 D=3 Twisted SUSY trans. laws Resulting SUSY Algebra closes off-shell

9. LATTICE 2007 @ Regensburg9 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

10. LATTICE 2007 @ Regensburg10 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)

11. LATTICE 2007 @ Regensburg11 Boson part of Action Plaquettes Zero-area loops : Fermion part of Action etc.. Contribution of Scalar fields

12. LATTICE 2007 @ Regensburg12 Naïve Continuum limit Agreement with Cont. Twisted N=4 D=3 SYM Contribution of

13. LATTICE 2007 @ Regensburg13 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 addressed Dim. 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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