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. Majorana fermion fermion + gravity Motivations Boulatov &Kazakov

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

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

5. Lattice Fermions Free Dirac Naïve Staggered Kogut-Susskind Dirac-Kaehler (N.K. & J.Smit) (N.K. & I.Kanamori) (Kluberg-Stern et.al. ＆ Gliozzi) y2x Ivanenko&Landau ‘28 i : flavour ? Staggered phase

6. Dirac Kaehler Fermion

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

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

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

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

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

12. N=D=2 SUSY Dirac-Kaehler Twist Dirac-Kaehler fermion i : flavour ? Extended SUSY suffix y2x 2-dim. N=2 3-dim. N=4 4-dim. N=4 #boson = #fermion super charges in d-dim. Dirac-Kaehler twisting Answer to the Puzzle 1

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

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

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

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

20. 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 ! operation makes link holes and thus loses gauge invariance. gets coordinate dependence super gravity

21. Gauge Theory on the Random Lattice ０１２３０１２３ ・・・・ ・・・・ ・・・・ Form Simplex １ ３０２ １ Gauge Theory + Gravity ？ SUSY ? Boson Fermion ?

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

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.