1. 10/31/2006Joint Meeting of Pacific Region Particle Physics Communities 1 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 hep-lat/06 *****

2. 10/31/2006Joint Meeting of Pacific Region Particle Physics Communities 2 Introduction SUSY on a Lattice K. Fujikawa, Nucl.Phys. Phys.Rev. D66 (2002) 074510 D.B. Kaplan, M. Unsal, JHEP 0509:042,2005 etc. F. Sugino, JHEP 0501 (2005) 016, tec. S. Catterall, JHEP 0506:027,2005, etc. Y. Kikukawa, H. Suzuki, JHEP 0502 (2005) 012 K. Itoh, M. Kato, H. Sawanaka, H. So,N.Ukita JHEP 0302 (2003) 033,etc. M. Bonini, A. Feo, Phys.Rev.D71:114512,2005 T. Onogi, T. Takimi, Phys.Rev.D72:074504,2005 H. Suzuki, Y. Taniguchi, JHEP 0510:082,2005 … Main Obstacle Main Obstacle Leibniz rule (Cont.) Leibniz rule (Lattice) So far, limited # of Supercharges has been realized on the Lattice. Ex. … Non-perturbative dynamics Fermionic structure of regularized spacetime Constructive formulation of SUSY models Motivations Motivations

3. 10/31/2006Joint Meeting of Pacific Region Particle Physics Communities 3 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. 10/31/2006Joint Meeting of Pacific Region Particle Physics Communities 4 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 only for Dirac-Kahler Twisted Algebra of (DKKN : Phys. Lett. B 633 (2006) 645-652) …

5. 10/31/2006Joint Meeting of Pacific Region Particle Physics Communities 5 N=4 D=3 Twisted SUSY Algebra Dirac-Kahler expansions.. Twisted Supercharges of N=4 D=3 On the Lattice ?

6. 10/31/2006Joint Meeting of Pacific Region Particle Physics Communities 6 Leibniz rule conditions N=4 D=3 Twisted Lattice SUSY Algebra Symm. Choice Asymm. Choice Each represents Simplicial element of 3D

7. 10/31/2006Joint Meeting of Pacific Region Particle Physics Communities 7 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. 10/31/2006Joint Meeting of Pacific Region Particle Physics Communities 8 Lattice N=4 D=3 Twisted SUSY trans. laws Resulting SUSY Algebra closes off-shell

9. 10/31/2006Joint Meeting of Pacific Region Particle Physics Communities 9 Lattice N=4 D=3 Twisted SYM Action Each term forms closed loop. Manifest Gauge Inv. Exact form w.r.t. all the supercharges Manifest SUSY Inv. for

10. 10/31/200610 Boson part Plaquettes Zero-area loops : Fermion part etc.. Contribution of Scalar fields

11. 10/31/2006Joint Meeting of Pacific Region Particle Physics Communities 11 Naïve Continuum limit Agreement with Cont. Twisted N=4 D=3 SYM Contribution of

12. 10/31/2006Joint Meeting of Pacific Region Particle Physics Communities 12 Summary & Discussions N=D=4 Lattice SYM (Leibniz rule is O.K.) … N=4 D=3 Twisted SYM on a Lattice Construction of manifest SUSY inv. Action Lattice SUSY Algebra for All Supercharges Why Twisted SUSY ? Twisted Fermions Staggered Fermions (N-extended SUSY) (N-flavors) Matrix Formulation (Rigid construction) To be addressed Dim. Red. to N=4 D=2 v.s. Hermiticity on the lattice Dirac-Kahler (Simplicial) Structure of Fermions.. Moduli problem c.f. Kanamori’s talk c.f. Saito’s talk 16:30 today @ Hilo

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