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.

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)

Jun Nishimura (KEK) 10.5.14 Univ. of Maryland

Large-N reduction and supersymmetry

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Large-N reduction

U(N) gauge theory in D-dim. torus

large-N reduced model

reduce to a point

Eguchi-Kawai (’82)

is NOT spontaneouslybrokenBhanot-Heller-Neuberger (’82)



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A continuum version of the large-N reduced modelGross-Kitazawa (’82)

Gonzalez-Arroyo & Korthals-Altes (’83)

is NOT spontaneouslybroken

zero volume limit

Revival of this type of model in two different contexts,where supersymmetry plays a crucial role.



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Plan of the talk0. 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



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1. A novel large-N reductionas a supersymmetric regulator

zero volume limit

many classical vacua preserving SUSY

all degenerate

Ishiki-Ishii-Shimasaki-Tsuchiya (’08)



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A novel large N reduction as a supersymmetric regulator (cont’d)



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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 VEV2) 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)



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Important application: First principle test of AdS/CFT

CFT

conformal mapping

1d SYM with 9 adjoint scalars (16 SUSY)



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Monte Carlo results (preliminary)

all order

weak coupling

strong coupling

work in progressHonda-Ishiki-J.N.-Tsuchiya

1

2



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Non-renormalization theorem from a computer

3pt function of chiral primary operators

Strong coupling resultsagree with free theoryup to an overall const.

work in progressHonda-Ishiki-Kim-J.N.-Tsuchiya

consistent with the AdS/CFT duality!

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



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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



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Dynamical compactification from 10d to 4d

Eigenvalues :

in the limit

The order parameter forthe SSB of SO(10)

e.g.) SO(10) → SO(4)

(“moment of inertia” tensor)



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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



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Remarkable properties of the phase J.N.-Vernizzi (’00)

Stationarityof the phaseincreasesfor lower d

This effect compensates the entropy loss for lower d !



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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),…



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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)



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Results for the 4d toy model

J.N.-Okubo-Sugino (’04)



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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!



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Monte Carlo simulation omitting the phase

0.6

no SSB of SO(6) symmetry without the phase.

Anagnostopoulos-Azuma-J.N.



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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



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Reconsiderations of previous GEM results for the IKKT model

from GEM

from MC

consistent with GEM results

free energy is lowerfor SO(4) than SO(7)

Aoyama-J.N.-Okubo, in prep.

4. Summary



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Summary and future prospectsLarge-N reduction 　 (Eguchi-Kawai ’82)

supersymmetric regularization ofplanar gauge theories

nonperturbative formulation ofsuperstring 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)



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What does the IKKT model describe?

The extents in the extended directions and shrunken directions are BOTH finite.

comparison of free energy for d=3,4,5,6based 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 BangThen, how did the time appear?

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.

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