Wilson loops and Vertex operators in matrix models

Wilson loops and Vertex operators in matrix models

＠NTU 04/10/29

KEK

Satoshi （暁） Iso（磯）

based on collaborations

with H. Terachi and H. Umetsu

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[1] Introduction

Various matrix models

・ II B(IKKT) D(-1) ; constituents of space-time and matter

・ II A(BFSS) D0 ; time is introduced as a parameter

Both models have been proposed as a nonperturbative formulation of string theory.

Dimensional reduction of U(N) super Yang-Mills with 16 supercharges.

BFSS : reduction to d=1

IKKT : reduction to d=0 Large N (AdS/CFT : reduction to d=4)

keio.ppt

ten-dim Majorana Weyl fermion

Eventually we should take large N limit.

Questions

• Where is background space-time?

• How can we see gravity in this model?

• What is the meaning of SO(10) symmetry of the action?

• Does this model have diffeomorphism invariance?

Space-time = distribution of eigenvalues

N-discrete points

links between i and j.

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Branched polymers

・ ・

dynamics of eigevalues = dynamics of space-time

Aoki Iso Kawai Kitazawa Tada (1998)

SO(10)

IKKT : Schild action of strings in flat d=10 space-time or in the D(-1) picture, D-instantons in .

Embed matirces in larger matrices.

N

>> N

←Mean field

heat (matrix) bath

Configuration of modifies the action for

Analogies

(1) Ising model

symmetry breaking B B→0

B

(2) themodynamics

Heat bath → temperature, chemical pot. etc

In IKKT matrix model

SO(10) = background generated by the other D(-1)

modify matrix bath = modify sugra bgd

background independent matrix model ? string

string in flat space

condensation of graviton string in curved space

string field theory

cubic action background ind. string action

In the following we will consider the effect of weak background fields in matrix models.

(1) How can we describe backgrounds in matrix models ? (2) equation of motion for the background

(3) Can we modify the backgound by condensation of D(-1)?

String theory

(1) vertex operators

(2) conformal invariance

・beta function =0 → Einstein equation

・or to impose that vertex operators have dim. (1,1).

→ graviton

・calculate effective action Æ Einstein Hilbert action (3) Æ string field theory

What plays the role of conformal invariance in matrix models?

Large N renormalization group in the space of coupling constants

N N D(-1) in the background of M

Integrate n D(-1)s N-n

M+n

Effective action conserved EM tensor

integrate over effective action

Integration is very difficult

→ investigate N dependence for the coefficients c.

Induced gravity

graviton bound state?

divergence of the coefficients bosonic Æ infra div.

super Æ ultra div.

[2] Wilson loops and vertex operators

It is important to obtain the correct form of vertex op.

Kitazawa 01

conservation law

susy multiplet under N=2

N

M=1 : mean field D(-1) wave function for mean field D(-1)

= conjugate variable to supergravity background field Æ background for N D(-1)s.

N =2 supersymmetry

N 1 Integrate over the off-diagonal block fields

and obtain effective action for the diagonal blocks Supersymmetry for (N+1) matrix can be written as

In the leading order of perturbation, eff.action is invariant under

susy generators for N D(-1)

susy generators for mean field D(-1)

on

Supersymmetric Wilson loop

massive and massless vertex operators

Since we are now interested in massless multiplet, we consider the Wilson loop with one segment:

This is invariant under

if we use equation of motion.

note

Massless vertex operators

on shell

16 components 8 components

256 independent wave func.

= massless type II B sugra.

complex

Charge conjugation for wave functions

normalization of the measure

under charge conjugation

two supersymmetries are interchanged.

Sugra multiplet of wave functions

9 8 45-9-8=28

All the wave func satisfy e.o.m.

etc

Susy transformations up to gauge transf.

Vertex operators

expand the Wilson loop as a power series of

symmetrized trace

・ Vertex op. for dilaton and dilatino

・ Antisymmetric tensor field

It satisfies the conservation law

・Gravitino

By using e.o.m , it satisfies

・Graviton

It satisfies by using eom.

Self dual 4-th rank tensor

It satisfies

Conservation law

supersymmetry

Invariance under susy tr. holds

if we use e.o.m and drop gauge tr. for wave.fun.

[3] Condensation of massless fields

N

M=1 : mean field D(-1) in flat background

Condensation of mean field D(-1) with certain wave func.

Æ background for N D(-1)s.

flat background Æ curved background condensation

By integrating over off-diagonal blocks,

we can obtain effective action for N D(-1) and mean field D(-1).

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expand in powers of 1/x^2.

Lower orders are cancelled due to supersymmetries.

Finally we will have interactions between

N D(-1) instantons and coordinates of the mean field D(-1).

N D(-1) ● mean field D(-1) interaction

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For example, we have such a term as

Then mulply an appropriate wave function for mean field D(-1) and integrate over the coordinates.

No infra div but ultra div.

Chern Simons term or gravity term is induced.

IKKT matrix model SO(10)

Myers effect condensation of mean field D(-1) Chern Simons term is induced

Fuzzy sphere is a solution

[4] Summary

(1) Susy Wilson loop Æ massless w.f and vertex op.

in type II B matrix model

They satisfy conservation laws and transform covariantly under susy tr.

(a) massive case ?

Wilson loop with more than one segment (b) conservation law susy

Can we understand it as a part of larger symmetry?

origin of diffeo ?

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(2) Matrix model in flat bg. Æ by integrating mean field D(-1) with certain wave functions,

we can generate backgrounds.

Matrix Myers effect (a) on shell and off shell

(b) perform the calculation more systematically (3) Large N RG in the coupling constant sp.

integrate MF D(-1)

N D(-1) (N+1) D(-1)

RG flow

RG flow can be interpreted as RG flow of the coefficients of the effective action.

RG flow of