1. Large N reduction and supersymmetry MCFP Workshop on Large N Gauge Theories, May 13-15, 2010, University of Maryland, College Park Jun Nishimura (KEK Theory Center)

2. Jun Nishimura (KEK) 10.5.14 Univ. of Maryland Large-N reduction and supersymmetry 2 Large-N reduction U(N) gauge theory in D-dim. torus large-N reduced model reduce to a point Eguchi-Kawai (’82) is NOT spontaneously broken Bhanot-Heller-Neuberger (’82)

3. Jun Nishimura (KEK) 10.5.14 Univ. of Maryland Large-N reduction and supersymmetry 3 A continuum version of the large-N reduced model Gross-Kitazawa (’82) Gonzalez-Arroyo & Korthals-Altes (’83) is NOT spontaneously broken zero volume limit Revival of this type of model in two different contexts, where supersymmetry plays a crucial role.

4. Jun Nishimura (KEK) 10.5.14 Univ. of Maryland Large-N reduction and supersymmetry 4 Plan of the talk 0. Introduction 1. A novel large-N reduction as a supersymmetric regulator first-principle test of the AdS/CFT correspondence 2. A large-N reduced model as non-perturbative superstring theory dynamical compactification 3. Summary

5. Jun Nishimura (KEK) 10.5.14 Univ. of Maryland Large-N reduction and supersymmetry 5 1.A novel large-N reduction as a supersymmetric regulator zero volume limit many classical vacua preserving SUSY all degenerate Ishiki-Ishii-Shimasaki-Tsuchiya (’08)

6. Jun Nishimura (KEK) 10.5.14 Univ. of Maryland Large-N reduction and supersymmetry 6 A novel large N reduction as a supersymmetric regulator (cont ’ d)

7. Jun Nishimura (KEK) 10.5.14 Univ. of Maryland Large-N reduction and supersymmetry 7 Comments Well, this does not harm anything… Here we are interested in the SUSY case anyway. needed for 2 purposes. 1)equivalence spoiled by radiative corrections to the VEV 2)the background becomes unstable above critical coupling 1) to remove non-planar diagrams, which disagree with their field theoretic counterparts 2) to suppress transitions to other vacua The equivalence does not hold for the bosonic case. c.f.) Azuma-Bal-Nagao-J.N.(’04)

8. Jun Nishimura (KEK) 10.5.14 Univ. of Maryland Large-N reduction and supersymmetry 8 Important application: First principle test of AdS/CFT CFT conformal mapping 1d SYM with 9 adjoint scalars (16 SUSY)

9. Jun Nishimura (KEK) 10.5.14 Univ. of Maryland Large-N reduction and supersymmetry 9 Monte Carlo results (preliminary) all order weak coupling strong coupling work in progress Honda-Ishiki-J.N.-Tsuchiya 1 2

10. Jun Nishimura (KEK) 10.5.14 Univ. of Maryland Large-N reduction and supersymmetry 10 Non-renormalization theorem from a computer 3pt function of chiral primary operators Strong coupling results agree with free theory up to an overall const. work in progress Honda-Ishiki-Kim-J.N.-Tsuchiya consistent with the AdS/CFT duality!

11. 2. A large-N reduced model as nonperturbative superstring theory

12. Jun Nishimura (KEK) 10.5.14 Univ. of Maryland Large-N reduction and supersymmetry 12 A large-N reduced model as superstrings a non-perturbative formulation of type IIB superstring theory in 10 dim. (conjecture) Ishibashi-Kawai-Kitazawa-Tsuchiya ’96 zero volume limit

13. Jun Nishimura (KEK) 10.5.14 Univ. of Maryland Large-N reduction and supersymmetry 13 Dynamical compactification from 10d to 4d Eigenvalues : in the limit The order parameter for the SSB of SO(10) e.g.) SO(10) → SO(4) (“moment of inertia” tensor)

14. Jun Nishimura (KEK) 10.5.14 Univ. of Maryland Large-N reduction and supersymmetry 14 Complex fermion determinant fermion determinant reweighting method simulate the phase quenched model cannot be treated as the Boltzmann weight complex in general suppressed as effective sampling becomes difficult

15. Jun Nishimura (KEK) 10.5.14 Univ. of Maryland Large-N reduction and supersymmetry 15 Remarkable properties of the phase J.N.-Vernizzi (’00) Stationarity of the phase increases for lower d This effect compensates the entropy loss for lower d !

16. Jun Nishimura (KEK) 10.5.14 Univ. of Maryland Large-N reduction and supersymmetry 16 This is a dilemma ! Phase of the fermion determinant  important for the possible SSB of SO(10)  difficult to include in Monte Carlo simulation Gaussian expansion method Sugino-J.N. (’00), Kawai et al. (’01),… New Monte Carlo technique (factorization method) Anagnostopoulos-J.N. (’01),…

17. Jun Nishimura (KEK) 10.5.14 Univ. of Maryland Large-N reduction and supersymmetry 17 Models with similar properties 6d IKKT model 4d toy model (non SUSY) (SSB of SO(D) expected due to complex fermion det.) 10d IKKT model J.N. (’01)

18. Jun Nishimura (KEK) 10.5.14 Univ. of Maryland Large-N reduction and supersymmetry 18 Results for the 4d toy model J.N.-Okubo-Sugino (’04)

19. Jun Nishimura (KEK) 10.5.14 Univ. of Maryland Large-N reduction and supersymmetry 19 Results for the 6d IKKT model Aoyama-J.N.-Okubo, in prep. In fact, there are also solutions with larger free energy. universal! for all solutions!

20. Jun Nishimura (KEK) 10.5.14 Univ. of Maryland Large-N reduction and supersymmetry 20 Monte Carlo simulation omitting the phase 0.6 no SSB of SO(6) symmetry without the phase. Anagnostopoulos-Azuma-J.N.

21. Jun Nishimura (KEK) 10.5.14 Univ. of Maryland Large-N reduction and supersymmetry 21 Understanding based on LEET treat them as small fluctuations and keep only quadratic terms Aoki-Iso-Kawai-Kitazawa-Tada(’98) Ambjorn-Anagnostopoulos-Bietenholz-Hotta-J.N.(’00) branched-polymer-like structure

22. Jun Nishimura (KEK) 10.5.14 Univ. of Maryland Large-N reduction and supersymmetry 22 Reconsiderations of previous GEM results for the IKKT model from GEM from MC consistent with GEM results free energy is lower for SO(4) than SO(7) Aoyama-J.N.-Okubo, in prep.

23. 4. Summary

24. Jun Nishimura (KEK) 10.5.14 Univ. of Maryland Large-N reduction and supersymmetry 24 Summary and future prospects Large-N reduction (Eguchi-Kawai ’82) supersymmetric regularization of planar gauge theories nonperturbative formulation of superstring theory first principle test of AdS/CFT dynamical compactification to 4d twisted Eguchi-Kawai model (Gonzalez-Arroyo & Okawa ’83) as field theories on noncommutative torus (Aoki-Ishibashi-Iso-Kawai-Kitazawa-Tada ’99, Ambjorn-Makeenko-J.N.-Szabo ’99)

25. Jun Nishimura (KEK) 10.5.14 Univ. of Maryland Large-N reduction and supersymmetry 25 What does the IKKT model describe? The extents in the extended directions and shrunken directions are BOTH finite. comparison of free energy for d=3,4,5,6 based on Monte Carlo simulation Is SO(4) sym. solution the true vacuum ? The space-time is assumed to have Euclidean signature. Possible interpretation : Early universe before Big Bang Then, how did the time appear?