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Two Fundamental Puzzles And Lattice SUSY

Two Fundamental Puzzles And Lattice SUSY. S.Arianos, A.D’Adda, A.Feo, I.Kanamori, K.Nagata, J.Saito. J.Kato, A.Miyake, T.Tsukioka, Y.Uchida,. Motivations. Majorana fermion. fermion + gravity. Boulatov &Kazakov. Fractal Structure of 2D Quantum Gravity.

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Two Fundamental Puzzles And Lattice SUSY

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  1. Two Fundamental Puzzles And Lattice SUSY S.Arianos, A.D’Adda, A.Feo, I.Kanamori, K.Nagata, J.Saito J.Kato, A.Miyake, T.Tsukioka, Y.Uchida,

  2. Motivations Majoranafermion fermion + gravity Boulatov &Kazakov

  3. Fractal Structure of 2D Quantum Gravity (c: central charge matter ) N.K. & Watabiki Q state Potts model on random surface N.K. & Yotsuji

  4. success of lattice QCD success of 2-dim. lattice quantum gravity gauge theory + matter fermion + gravity on random lattice

  5. y 2x Lattice Fermions Free Dirac Staggered phase Naïve Staggered (Kluberg-Stern et.al. ＆Gliozzi) (N.K. & J.Smit) Kogut-Susskind (N.K. & I.Kanamori) Dirac-Kaehler Ivanenko&Landau ‘28 i: flavour ?

  6. Dirac Kaehler Fermion

  7. staggered phase species doublers dual Dirac-Kaehler fermion Puzzle 1 Is the staggered phase or species doublers or the “flavour” degrees of freedom physical ?

  8. Quantization and Twisted SUSY Tsukioka, N.K., Kato, Miyake, Uchida Continuum (Two dimensional Abelian BF) Auxiliary field Off-shell invariance Nilpotency of BRScharge s N=D=2 Twisted SUSY Kato,N.K.&Uchida

  9. N=2 SUSY in two dimensions Dirac-Kaehler Twist (N=2) Twisted N=2 SUSY Cont: Latt: Gauged Latt:

  10. We need a modified Leibniz rule for too ! Compatibility of Shifts

  11. Symm. Choice Asymm. Choice Cond. for Twisted N=D=2 Solutions Twisted N=D=2 Lattice SUSY Algebra Equivalent to orbifold construction: by Kaplan et.al.

  12. i: flavour ? Extended SUSY suffix y 2x Answer to the Puzzle 1 N=D=2 SUSY Dirac-Kaehler Twist Dirac-Kaehler fermion super charges in d-dim. 2-dim. N=2 3-dim. N=4 4-dim. N=4 Dirac-Kaehler twisting #boson = #fermion

  13. Jacobi Identities … Define fermionic link components … Auxiliary Field

  14. Twisted N=2 Super Yang-Mills Action Off-shell SUSY invariance for all twisted super charges. Action has twisted SUSY exact form.

  15. Bosonic part of the Action

  16. Fermionic part of the Action … (1) … (2) (1) (2)

  17. Higer dimensional extension is possible: 3-dim. N=4 super Yang-Mills

  18. Two Problems Bruckmann Kok “inconsistency” When but if we introduce the following “mild non-commutativity”: then In general

  19. Modified Leibniz rule + Mild non-commutativity Hopf algebraic Field Theory Concrete representation of this non-commutativity Orbifold condition Lattice version of Moyal product

  20. operation makes link holes and thus loses gauge invariance. A possible solution We claim: if there is covariantly constant super parameter which has opposite shift of and commutes with all the super covariant derivatives: compensates the link holes. lattice SUSY and gauge invariant ! gets coordinate dependence super gravity

  21. ３ ０ ２ １ １ Gauge Theory on the Random Lattice Simplex Gauge Theory + Gravity ？ Form ０ １ ２ ３ BosonFermion ? SUSY ? ・ ・ ・ ・ ・ ・

  22. Generalized Gauge Theories in arbitrary dimensions N.K.＆Watabiki　‘９１ gauge field gauge parameter derivative curvature gauge trans. Chern-Simons Topological Yang-Mills Yang-Mills

  23. Ｐｕｚｚｌｅ　２ What is the role of “quaternion” in generalized gauge theory ?

  24. Single lattice translation as SUSY transformation Super parameter SUSY algebra

  25. Matrix Representation are diagonal. Two step translation as SUSY transformation

  26. Partial answer to Puzzle 2 Quaternion may be fundamentally related to the lattice SUSY transformation. Chirality may play an important role in the transformation. Differential form structure for Dirac-Kaeher mechanism should be essentially introduced to accommodate super gravity nature.

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