Page 1

arXiv:hep-th/0503101v2 23 Aug 2005

KEK-TH-1006

OIQP-05-01

Fermionic Backgrounds and Condensation of

Supergravity Fields in IIB Matrix Model

Satoshi Iso∗1, Fumihiko Sugino†2, Hidenori Terachi∗3and Hiroshi Umetsu†4

∗Institute of Particle and Nuclear Studies

High Energy Accelerator Research Organization (KEK)

Oho 1-1, Tsukuba, Ibaraki 305-0801, Japan

and

Department of Particle and Nuclear Physics,

the Graduate University for Advanced Studies (Sokendai)

Oho 1-1, Tsukuba, Ibaraki 305-0801, Japan

†Okayama Institute for Quantum Physics

1-9-1 Kyoyama, Okayama 700-0015, Japan

Abstract

In a previous paper [1] we constructed wave functions of a D-instanton and vertex

operators in type IIB matrix model by expanding supersymmetric Wilson line operators.

They describe couplings of a D-instanton and type IIB matrix model to the massless

closed string fields respectively and form a multiplet of D = 10 N = 2 supersymmetries.

In this paper we consider fermionic backgrounds and condensation of supergravity fields

in IIB matrix model by using these wave functions. We start from the type IIB matrix

model in a flat background whose matrix size is (N + 1) × (N + 1), or equivalently the

effective action for (N + 1) D-instantons. We then calculate an effective action for N D-

instantons by integrating out one D-instanton (which we call a mean-field D-instanton) with

an appropriate wave function and show that various terms can be induced corresponding

to the choice of the wave functions. In particular, a Chern-Simons-like term is induced

when the mean-field D-instanton has a wave function of the antisymmetric tensor field. A

fuzzy sphere becomes a classical solution to the equation of motion for the effective action.

1satoshi.iso@kek.jp

2fumihiko sugino@pref.okayama.jp

3terachi@post.kek.jp

4hiroshi umetsu@pref.okayama.jp

Page 2

We also give an interpretation of the above wave functions in the superstring theory

side as overlaps of the D-instanton boundary state with the closed string massless states

in the Green-Schwarz formalism.

2

Page 3

1Introduction

Type IIB (IKKT) matrix model has been proposed as a nonperturbative formulation of

superstring theory of type IIB [2]. As an evidence for the nonperturbative formulation, the

Schwinger-Dyson equation of Wilson lines is shown to describe the string field equation of

motion of type IIB superstring in the light-cone gauge under some plausible assumptions

about the continuum limit [3]. Although there are still many issues to be resolved, the

model has an advantage to other formulations of superstrings that we can discuss dynamics

of space-time more directly [4, 5, 6, 7]. The action of the model is given by

SIKKT= −1

4tr [Aµ,Aν]2−1

2tr¯ψΓµ[Aµ,ψ], (1.1)

where Aµ(µ = 0,··· ,9) and ten-dimensional Majorana-Weyl fermion ψ are N×N bosonic

and fermionic hermitian matrices. The action was originally derived from the Schild action

for the type IIB superstring by regularizing the world sheet coordinates by matrices. It

is interesting that the same action describes the effective action for N D-instantons [8].

This suggests a possibility that D-instantons can be considered as fundamental objects

to generate the space-time itself as well as the dynamical fields (or strings) on the space-

time. The bosonic matrices represent noncommutative coordinates of D-instantons and

the distribution of eigenvalues of Aµ is interpreted to form space-time. The fermionic

coordinates ψ are collective coordinates associated with broken supersymmetries of D-

branes but in the matrix model interpretation they describe internal structures of our

space-time.

If we take the above interpretation that the space-time is constructed by distribution of

D-instantons, how can we interpret the SO(9,1) rotational symmetry of the matrix model

action? This symmetry can be interpreted in the sense of mean-field. Namely we can

consider that the system of N D-instantons is embedded in larger size (N +M)×(N +M)

matrices as

?ND(−1)

and consider the action (1.1) as an effective action in the background where the rest, M

eigenvalues, distribute uniformly in 10 dimensions. If the M eigenvalues distribute inho-

mogeneously, we may expect that the effective action for N D-instantons is modified so

that they live in a curved space-time. This is analogous to a thermodynamic system. In a

canonical ensemble, a subsystem in a heat bath is characterized by several thermodynamic

quantities like temperature and pressure. Similarly a subsystem of N D-instantons in a

“matrix bath” can be considered to be characterized by several “thermodynamic quanti-

ties” in a certain large N limit.

Since the matrix model has the N = 2 type IIB supersymmetry

?δAµ= i¯ ǫ1Γµψ,

2[Aµ,Aν]Γµνǫ1+ ǫ21N,

MD(−1) as background for ND(−1)

?

, (1.2)

δψ = −i

(1.3)

1

Page 4

we can expect that the configuration of the M background D-instantons describes conden-

sation of massless fields of the type IIB supergravity and the thermodynamic quantities of

the matrix bath are characterized by the values of the condensation.

In the following, we consider the simplest case that the background is represented by

a wave function of one D-instanton (namely M = 1 with an appropriate wave function

introduced). This simplification can be considered as a mean-field approximation that the

configuration of M D-instantons is represented by a mean-field wave function described by

a single D-instanton. We call this extra D-instanton a mean-field D-instanton. This

kind of idea was first discussed by Yoneya in [9]. In the previous paper [1], we constructed

a set of wave functions for the mean-field D-instanton. This wave functions have a stringy

interpretation, namely, as we see in this paper, they can be interpreted as overlaps of

D-instanton boundary states with closed string massless states.

In this paper, we calculate the effect of the mean-field D-instanton on the N D-

instantons. We first start from the IIB matrix model with a size (N + 1) × (N + 1),

or equivalently a system of (N +1) D-instantons, and integrate the mean-field D-instanton

with an appropriate wave function. This corresponds to condensation of supergravity fields

and the effective action for the N D-instantons is modified. We particularly consider two

types of wave functions, namely those describing an antisymmetric tensor field or a gravi-

ton field. In the former case, a Chern-Simons like term is induced in the leading order of

the perturbation. (This term vanishes if we assume that the N D-instantons satisfy the

equation of motion for the original IIB action.) With this term, a fuzzy sphere becomes a

solution to the equation of motion. This phenomenon is similar to the Myers effect [10].

In both cases for the antisymmetric tensor field and the graviton, if we assume that the

configuration satisfies the classical equation of motion, the modification of the effective

action is given by a vertex operator for each supergravity field.

The content of the paper is as follows. In the next section, we review the results of the

previous paper [1] on the wave functions of a D-instanton and the vertex operators for closed

string massless states in IIB matrix model. In section 3, we give a stringy interpretation of

the D-instanton wave functions as overlaps of D-instanton boundary states with massless

states of the closed string. In section 4, we calculate the one-loop effective action in general

fermionic backgrounds. In section 5, we apply this calculation to a system of (N + 1) D-

instantons and integrate the mean-field D-instanton to obtain an effective action for the

rest N D-instantons. We particularly consider the wave functions of the antisymmetric

tensor field and the graviton field. We summarize our results in section 6. In appendix,

we review the boundary states in the Green-Schwarz formalism.

2 Wave functions and Vertex operators

In this section we summarize our previous results on wave functions for a D-instanton and

vertex operators in type IIB matrix model. Wave functions are functions of a d = 10

coordinate (or its conjugate momentum) and d = 10 Majorana Weyl spinor and contain

information on the couplings of a D-instanton to the closed string massless states. Their

2

Page 5

physical meaning in superstring theories will be clarified in the next section.

other hand, in the matrix model, interactions corresponding to the supergravity modes are

induced as quantum effects and their couplings to these modes are described through the

vertex operators.

On the

2.1 Supersymmetric Wilson line operator

The degrees of freedom of a D-instanton are described by its coordinate, a ten-dimensional

vector yµand a Majorana-Weyl fermion λ, and thus information of its state is encoded in

functions of yµand λ, that is, wave functions. Here we give wave functions corresponding

to the supergravity modes in the form of fA(λ)e−ik·ywith a momentum k, where the index

A specifies each mode. Wave functions are defined to form a multiplet of the following

d = 10 N = 2 supersymmetry transformations

δ(1)f(λ) = [¯ ǫ1q1,f(λ)] = ǫ1

∂

∂λf(λ), (2.1)

δ(2)f(λ) = [¯ ǫ2q2,f(λ)] = (¯ ǫ2/ kλ)f(λ), (2.2)

where ǫi(i = 1,2) are the Majorana-Weyl spinors.

The Majorana-Weyl fermion λ contains 16 degrees of freedom and there are 216inde-

pendent wave functions for λ. To reduce the number, we impose the massless condition

for the momentum k; k2= 0. Then, since / kλ in (2.2) has only 8 independent degrees of

freedom, the supersymmetry can generate only 28= 256 independent wave functions for

λ. They form a massless type IIB supergravity multiplet containing a complex dilaton Φ,

a complex dilatino˜Φ, a complex antisymmetric tensor Bµν, a complex gravitino Ψµ, a real

graviton hµνand a real 4th-rank self-dual antisymmetric tensor Aµνρσ. A physical meaning

of the wave functions in string theories is given by using boundary states of a D-instanton

in section 3.

Vertex operators VA(Aµ,ψ;k) covariantly transform under the following N = 2 super-

symmetry of the IIB matrix model,

?δ(1)Aµ= i¯ ǫ1Γµψ,

δ(1)ψ = −i

2[Aµ,Aν]Γµνǫ1,

?

δ(2)Aµ= 0,

δ(2)ψ = ǫ21N.

(2.3)

We denote the generator of δ(i)(i = 1,2) as Qi(i = 1,2), respectively. Since the N = 2

supersymmetry algebra closes only on shell, in this section we assume that the N × N

matrices Aµand ψ satisfy the equations of motion for the IKKT action (1.1),

[Aν,[Aµ,Aν]] −1

Γµ[Aµ,ψ] = 0.

2(Γ0Γµ)αβ{ψα,ψβ} = 0,(2.4)

(2.5)

In order to construct vertex operators systematically, we start from a supersymmetric

Wilson line operator first introduced in [11] for the IIB matrix model;

ω(C) = tr

?

j

e¯λjQ1e−iǫkµ

jAµe−¯λjQ1.(2.6)

3