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Lattice Formulation of N=4 D=3 Lattice Formulation of N=4 D=3 Twisted Super Yang-MillsTwisted Super Yang-Mills
Kazuhiro NAGATA Dept. of Phys., Indiana Univ.
A. D’Adda INFN Torino, ItalyI. Kanamori RIKEN, JapanN. Kawamoto Hokkaido Univ., Japan
Based on the collaboration with
Phys. Lett. B 633 (2006) 645-652Nucl. Phys. B 707 (2005) 100-144
arXiv:0707.3533 [hep-lat]
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IntroductionSUSY 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 ObstacleMain 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
MotivationMotivationss
a developing field…
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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. ActionLattice SUSY inv. Action
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Implementation of Lattice SUSY Algebra
forfor
Leibniz rule conditions
..
N=2 D=2 4 SuperchargesN=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
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N=4 D=3 Twisted SUSY Algebra
Dirac-Kahler expansions..
Twisted Superchargesof N=4 D=3
On the Lattice ?
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Leibniz rule conditions
N=4 D=3 Twisted Lattice SUSY AlgebraSymm. Choice
Asymm. Choice
Each represents3D Simplicial element
0-form 1-form 2-form 3-form
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Lattice N=4 D=3 SYM formulationBosonic & FermioincGauge Link Variables
Lattice SYM Multiplet via Jacobi identities
3-Gauge fields + 3-Scalars
Auxiliary fields
N=4 D=3 Twisted Fermions
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Lattice N=4 D=3 Twisted SUSY trans. laws
Resulting SUSY Algebra closes off-shell
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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
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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)
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Boson part of ActionPlaquettes Zero-area loops :
Fermion part of Action
etc..
Contribution of Scalar fields
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Naïve Continuum limit
Agreement with Cont. Twisted N=4 D=3 SYM
Contribution of
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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 addressedDim. 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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