1

Superstring vertex operators in type IIB matrix model

Satoshi Nagaoka (KEK)

with Yoshihisa Kitazawa (KEK & Sokendai)

String Theory and Quantum Field Theory Kinki university/August 7 2007

2

Introduction

Type IIB matrix model: [Ishibashi-Kawai-Kitazawa-Tsuchiya `96]

Aμ: N×N Hermitian matricesψ: Ten dimensional Majorana-Weyl spinor, N×N matrices

• Dimensional reduction of 10-dim SYM theory to 0-dim• Matrix regularization of type IIB Green-Schwarz superstrin

g

3

Introduction

(Green-Schwarz light-cone closed) superstring:

• Supersymmetry transformation determines the interaction vertex for GS light-cone superstring.

• T-duality transformation

radius R α’/R

winding number w momentum p+

4

• In this talk, Green-Schwarz light-cone closed superstring theory is derived from type IIB matrix model.

• Low energy excitations of closed string are described by the matrices.

Introduction

5

Plan of Talk

IIB matrix model SUSY trf. vertex operators

Light-cone GS string action SUSY trf. vertex op.

1. Introduction

2. GS light-cone superstring action from IIB matrix model

3. Supersymmetry transformation

4. Superstring vertex operators in IIB matrix model

5. Summary

6

GS superstring from IIB matrix model

Background is determined by the classical solutions.

Expanding this action around 2D background :

2D N=8 U(n) noncommutative Yang-Mills theory

8 scalar fields: φi 8v representation of SO(8)

16 spinor fields: 8c, 8s rep. of SO(8)

7

GS superstring from IIB matrix model

1. Mapping the coordinate system from R2 into R1×S1 as

The origin of x coordinate is the point where vertex operators are inserted.

2. In the low energy limit, • Noncommutative product → commutative product :• Diagonal components are favored rather than off-diagonal

components.• Gauge fields on 2-dim decouple to other fields.

8

GS superstring from IIB matrix model

We obtain the action:

• w : a winding number along σ direction

• Multiple strings are obtained in general.

n=∑i wi

3. By identifying , we obtain GS light-cone

superstring action.

9

GS superstring from IIB matrix model

Duality relation:

IIA D0 F1: fundamental string

T-duality D1: D-string

IIB D1 (2D-YM) D1 D0: D-particle

S-duality S-duality

IIB F1 F1

T-duality T-duality

IIA F1 F1

DVV’s matrix string string in IKKT model[Dijkgraaf-Verlinde-Verlinde `97]

10

Supersymmetry transformation

N=2 SUSY transformation in IIB matrix model :

In 2D background and low energy limit, this transformation reduces to

This transformation leaves the GS light-cone string action invariant.

11

Construction of vertex operator in GS superstring

16 supercahrges :

left mover

right mover

For the vertex operators,

the coefficients B and F are determined by the SUSY trf.

12

Construction of vertex operator in GS superstring

Light-cone open superstring vertex operators are composed of bosonic (vector) and fermionic (spinor) operators:

where

k: external momentum, (ki)2=0 , k+=0

ζi, ua : polarization vectors (spinors) which represent the wave functions for vector (spinor) states

13

Construction of vertex operator in GS superstring

There are 256 massless states in type IIA superstring theory. Field contents are composed of

NS-NS sector: 8v×8v=[0]+[2]+(2)=1+28+35v

R-R sector: 8c×8s=[1]+[3]=8v+56v

NS-R sector: 8v×8s=[1]+[3]=8c+56c

R-NS sector: 8c×8v=[1]+[3]=8s+56s

Vertex operators for closed string modes are constructed by the product of left-mover and right-mover (~open string):

14

Vertex operator in type IIB matrix model

Vertex operators in type IIB SUGRA multiplet are constructed for type IIB matrix model. [Kitazawa (2002), Iso-Terachi-Umetsu (2004), Kitazawa-Mizoguchi-Saito (2006)]

(1) Linear couplings to the background fields

Sint= ∑i Vi (A,ψ) fi

(2) Related with each other by the SUSY transf.

∑i Vi (δA, δψ) fi = ∑i Vi (A,ψ) δfi

15

Vertex operator in type IIB matrix model

Type IIB supergravity multiplet

Dilaton vertex operator:

Dilatino vertex operator:

:

Graviton vertex operator:

:

where

16

Superstring vertex operator from type IIB matrix model

1) 2D background 8 scalar fields, 16 spinor fields

2) We construct corresponding states for IIB matrix model as closed string states (~(left-mover) × (right-mover)).

3) Since left and right mover of the fermion have an opposite chirality in 2D, we factorize operators by their chiralities.

4) Type IIA supergravity modes are derived by the compactification along S1 in matrix model. (~T-duality transformation).

5) We consider the kinematics (ki)2=0, k+=0

17

Superstring vertex operator from type IIB matrix model

Example: graviton Graviton vertex operators include the contribution such as

In the second term, although there are four right-handed spinors, there is no left-handed spinor.

Since we have defined the Fock space of this theory as that of closed string states, level matching condition forbids the contribution from these terms.

18

Superstring vertex operator from type IIB matrix model

Many terms vanish by the level matching condition.

The remaining terms are

This operator is equivalent with the vertex operator for graviton in GS light-cone superstring.

The interaction of multi-graviton gives the same amplitude.

Gravitino, Cμνρ 　　　　　　　　 the same vertex operators

19

Summary

IIB matrix model SUSY trf. vertex operators

Light-cone GS string action SUSY trf. vertex op.

We have derived Green-Schwarz light-cone superstring theory from type IIB matrix model.

In the low energy limit, 2-dim NC background in type IIB matrix model reduces to GS superstring action.

We have derived supersymmetry transformation for GS light-cone string from IIB matrix model.

20

Summary

IIB matrix model SUSY trf. vertex operators

Light-cone GS string action SUSY trf. vertex op.

We have identified superstring vertex operators with those in type IIB matrix model.

Multi-string interaction

• Recombination of (D-)string?

• Closed superstring field theory?