Large N reduction and supersymmetry

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

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

4. Plan of the talk • 0. Introduction • 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 Large-N reduction and supersymmetry

5. A novel large-N reductionas a supersymmetric regulator Ishiki-Ishii-Shimasaki-Tsuchiya (’08) zero volume limit many classical vacua preserving SUSY all degenerate Large-N reduction and supersymmetry

6. A novel large N reduction as a supersymmetric regulator (cont’d) Large-N reduction and supersymmetry

7. Comments • needed for 2 purposes. 1) to remove non-planar diagrams, which disagree with their field theoretic counterparts2) to suppress transitions to other vacua • The equivalence does not hold for the bosonic case. Well, this does not harm anything… Here we are interested in the SUSY case anyway. • equivalence spoiled by radiative corrections to the VEV • the background becomes unstable above critical coupling c.f.) Azuma-Bal-Nagao-J.N.(’04) 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) Large-N reduction and supersymmetry

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

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

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

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

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

14. Complex fermion determinant • fermion determinant • reweighting method simulate the phase quenched model complex in general cannot be treated as the Boltzmann weight suppressed as effective sampling becomes difficult 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 ! 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),… Large-N reduction and supersymmetry

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

18. Results for the 4d toy model J.N.-Okubo-Sugino (’04) 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! Large-N reduction and supersymmetry

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

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

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

23. 4. Summary

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

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