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

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

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  1. 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

  2. So far, limited # of Supercharges has been realized on the Lattice. Ex. • 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 Introduction 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

  3. 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

  4. Leibniz rule conditions for for .. 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

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

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

  7. 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

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

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

  10. 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

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

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

  13. 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-tastes) (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

  14. LATTICE 2007 @ Regensburg

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