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Superstring vertex operators in type IIB matrix model

arXiv:0708.1077[hep-th], 0710.0709[hep-th]

Satoshi Nagaoka (KEK)

with Yoshihisa Kitazawa (KEK & Sokendai)

KEK workshop/March 5 2008

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Motivation

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

Aμ: N×N Hermitian matrices

ψ: Ten dimensional Majorana-Weyl spinor, N×N matrices

Perturbative picture of fundamental strings

Vertex operators?

Interactions?

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Approach

Noncommutative (NC) gauge theory

• Worldsheets are obtained from two dimensional NC background.

• Supersymmetry transformation for IIB matrix model (superstring) completely determines the vertex operators for IIB matrix model (superstring).

IIB matrix model SUSY trf. vertex op.

Light-cone string action SUSY trf. vertex op.

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Green-Schwarz superstring from IIB matrix model

Expanding this action around 2-dim background:

2-dim N=8 U(n) noncommutative Yang-Mills theory i , j= 2,…,9 [Aoki-Ishibashi-Iso-Kawai-Kitazawa-Tada `99]

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Green-Schwarz superstring from IIB matrix model

(i) IR limit: the NCYM action becomes commutative N=8 SYM:

Perturbative vacua:

8 scalar fields: φi 8v representation of SO(8) 16 spinor fields: 8c, 8s reps. of SO(8)

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Green-Schwarz superstring from IIB matrix model

• We assume that all the eigenvalues of matrices do not coincide with each other at any points on the worldsheet.

(ii) all the excitations of off-diagonal modes become massive. In the low energy limit,

Gauge fields on 2-dim decouple to other fields.

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Green-Schwarz superstring from IIB matrix model

(iii) Mapping the coordinate system from R2 into R1×S1:

We identify the eigenvalue

• w : a winding number along σ direction

Then, the integral is described as

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Green-Schwarz superstring from IIB matrix model

By the rescaling

we obtain the action:

Multiple strings are obtained in general.

n=∑a wa

By identifying , we obtain Green-Schwarz lig

ht-cone superstring action.

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Supersymmetry transformation

N=2 supersymmetry transformation in IIB matrix model :

On the 2-dim NC background, it reduces in the low energy limit to

This transformation leaves the Green-Schwarz light-cone superstring action invariant.

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Vertex operators in IIB matrix model

Vertex operators (type IIB supergravity multiplet) [Kitazawa (2002), Iso-Terachi-Umetsu (2004), Kitazawa-Mizoguchi-Saito (2006)]

Dilaton vertex operator:

Dilatino vertex operator:

:

Graviton vertex operator:

:

where

Graviton, gravitino, Bμν and C4 are completely determined.

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Superstring vertex operators in IIB matrix model

1) Trace

2-dim background 8 scalar fields, 16 spinor fields

By the scaling,

and the field redefinition,

we obtain

where dimension (1,1) operators in (…) are proportional to 1.

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Superstring vertex operators in IIB matrix model

2) Physical states

In IIB matrix model, we construct closed string states in a radial quantization as the product of separate states corresponding to the left and right movers.

Open string physical states |Λ, k > are obtained by inserting the vertex operators in the far past

There are (8+8) massless ground states in each sector:

| i > , | à >

They are mapped to each other

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Superstring vertex operators in IIB matrix model

3) Chirality

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

4) Type IIB → type IIA

Type IIA supergravity modes are derived by singling out the light-cone direction. For R-R 4-form tensor,

: 3-form

: 2-form

…

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Superstring vertex operators in IIB matrix model

5) Consistency with the light-cone formalism

In our formalism, correlation functions between gauge fields in the exponential factor are calculated as

where

On the other hand, in the light-cone gauge formalism,

with (ki)2=0, k+=0 .

After τ→ - i τ, exponential factor gives

to the scattering amplitude.

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Superstring vertex operators in IIB matrix model

Graviton vertex operator (8v×8v)

where

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Superstring vertex operators in IIB matrix model

Bμν vertex operator (8v×8v)

The operator in IIB matrix model which reduces to Bij type (1,1) operator should be either of the following four forms:

Thus, we obtain

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Superstring vertex operators in IIB matrix model

R-R 3-form (8c×8s)In the type IIB supergravity multiplet, RR 4-form vertex op.

From this vertex operator, we obtain 4,3 and 2-form fields:

4-form :

3-form :

3-form :

2-form :

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Superstring vertex operators in IIB matrix model

The correspondence between IIB matrix model / superstring vertex operators

hij, Bij, Cijl , Ci and Ψ: completely the same vertex operators

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Future directions

weak coupling ← gYM→ strong coupling

strong coupling ← gs → weak coupling

UV IR → 0

Perturbative (NC)YM/ holographic description →Free string

(Green-Schwarz string)

Multi-string interaction

• Light-cone superstring field theory

• Twist field formalism