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

Page 6

Since we are interested in the massless multiplet, we here consider the simplest straight

Wilson line operator with a global momentum k;

ω(λ,k) = e¯λQ1treik·Ae−¯λQ1. (2.7)

Here, though the Majorana-Weyl spinor λ is a parameter of the supersymmetry transfor-

mation, it is eventually interpreted as a fermionic collective coordinate of a D-instanton.

This supersymmetric Wilson line operator ω(λ,k) is invariant under simultaneous su-

persymmetry transformations for N × N matrices Aµ,ψ and the parameters (λ,k) as

[¯ ǫ1Q1,ω(λ,k)] − [¯ ǫ1q1,ω(λ,k)] = 0,

[¯ ǫ2Q2,ω(λ,k)] − [¯ ǫ2q2,ω(λ,k)] = 0.

By expanding ω(λ,k) in terms of the wave functions for λ, which are constructed in the

manner stated above, as

(2.8)

(2.9)

ω(λ,k) =

?

A

fA(λ) VA(Aµ,ψ; k), (2.10)

it is understood from eqs.(2.8) and (2.9) that VA(Aµ,ψ; k) correctly transform under the

N = 2 supersymmetry. Therefore VA(Aµ,ψ; k) can be regarded as candidates for the

vertex operators. Indeed it will be shown explicitly in section 5 that a system of N D-

instantons couples to the supergravity modes through these vertex operators.

2.2Wave functions of a D-instanton

We here summarize our results of the wave functions for the massless multiplet and their su-

persymmetry transformations. In constructing wave functions which transform covariantly

under the supersymmetries, we first assume that the dilaton wave function is proportional

to exp(−ik ·y), namely fA(λ) = 1. It is annihilated by the supersymmetry transformation

q1. Then the other wave functions can be determined by supersymmetry transformations.

For more details, see [1].

By defining a fermion bilinear as bµν= kσ¯λΓµνσλ, the supersymmetry multiplet of the

wave functions is given as follows;

• dilaton

Φ(λ,k) = 1,(2.11)

• dilatino

˜Φ(λ,k) = / kλ,(2.12)

• antisymmetric tensor field

Bµν(λ,k) = −1

2bµν(λ),(2.13)

4

Page 7

• gravitino

Ψµ(λ,k) = −i

24(kσΓνσλ)bµν(λ),(2.14)

• graviton

hµν(λ,k) =

1

96bρ

µbρν(λ),(2.15)

• 4-th rank self-dual antisymmetric tensor field

Aµνρσ(λ,k) = −

i

32(4!)2b[µνbρσ](λ), (2.16)

• gravitino (charge conjugation of (2.14))

Ψc

µ(λ,k) = −

i

4 · 5!kρΓρλλbλσbσµ(λ), (2.17)

• antisymmetric tensor field (charge conjugation of (2.13))

Bc

µν(λ,k) = −1

6!bµρbρσbσν(λ),(2.18)

• dilatino (charge conjugation of (2.12))

˜Φc(λ,k) =1

8!kαΓµναλbνρbρσbσµ(λ), (2.19)

• dilaton (charge conjugation of (2.11))

Φc(λ,k) =

1

8 · 8!bν

µbλ

νbσ

λbµ

σ(λ).(2.20)

In these expressions we have chosen a specific gauge for each wave function. These wave

functions can be interpreted as overlaps of D-instanton boundary states and closed string

massless states as we will see in the next section. In the usual convention of superstrings,

the first dilaton (2.11) corresponds to a wave function of (dilaton +i axion) and the second

one (2.20) corresponds to (dilaton −i axion). The other complex fields also have the same

structure.

The supersymmetry transformations (2.1, 2.2) lead to the following transformations

between these wave functions;

5

Page 8

δΦ = ¯ ǫ2˜Φ,

δ˜Φ = / kǫ1Φ −

δBµν

= −¯ ǫ1Γµν˜Φ + 2i(¯ ǫ2Γ[µΨν]+ k[µΛν]),

δΨµ =

i

24Γµνρǫ2Hµνρ,

1

24 · 4[9Γνρǫ1Hµνρ− Γµνρσǫ1Hνρσ] +i

+

4 · 5!Γρ1···ρ5Γµǫ2Fρ1···ρ5+ kµξ,

= −i

1

(4!)2¯ ǫ1Γ[µνρΨσ]−

i

2Γνρkρhµνǫ1+

1

24 · 4

= 2i(¯ ǫ1Γ[µΨc

= −i

= ¯ ǫ1˜Φc,

2Γνρkρhµνǫ2

i

δhµν

2¯ ǫ1Γ(µΨν)−i

2¯ ǫ2Γ(µΨc

ν)+ k(µξν),

δAµνρσ

= −

1

(4!)2¯ ǫ2Γ[µνρΨc

σ]+ k[µξνρσ],

δΨc

µ

=

i

4 · 5!Γρ1···ρ5Γµǫ1Fρ1···ρ5

?9Γνρǫ2Hc

+

µνρ− Γµνρσǫ2Hc

ν]+ k[µΛc

νρσ

?+ kµξc,

δBc

µνν]) − ¯ ǫ2Γµν˜Φc,

µνρ+ / kǫ2Φc,

δ˜Φc

24Γµνρǫ1Hc

δΦc

(2.21)

where ξ,ξc,ξµ,ξµνρ,Λµand Λc

strengths of Bµν,Bc

same as that in [12] up to normalizations.

µare gauge parameters. Hµνρ,Hc

µνand Aµνρσ, respectively. This supersymmetry transformation is the

µνρand Fρ1···ρ5are the field

2.3Vertex operators

Construction of the vertex operators can be done systematically by expanding the super-

symmetric Wilson line operator in terms of the wave functions fA(λ) given in the previous

subsection. In section 5, we will show that these vertex operators indeed describe couplings

of type IIB matrix model to the supergravity modes. The derivation itself is systematic

but the complete calculation is cumbersome. Partial results were obtained in [13]. More

complete analysis was given in [1]1. The results are as follows.

• dilaton

VΦ= tr eik·A,(2.22)

• dilatino

V

˜Φ= tr eik·A¯ψ,(2.23)

1Similar calculations were performed in the BFSS matrix model in [14] and [15].

6

Page 9

• antisymmetric tensor field

VB

µν

= Str eik·A

?1

16kρ(¯ψ · Γµνρψ) −i

2[Aµ,Aν]

?

,(2.24)

• gravitino

VΨ

µ

= Str eik·A

?

−i

12kρ(¯ψ · Γµνρψ) − 2[Aµ,Aν]

?

·¯ψΓν, (2.25)

• graviton

Vh

µν

= 2 Str eik·A

?

−

[Aµ,Aρ] · [Aν,Aρ] +1

1

8 · 4!kλkτ(¯ψ · Γ

4

¯ψ · Γ(µ[Aν),ψ] −i

8kρ¯ψ · Γρσ(µψ · [Aν),Aσ]

σ

µλψ) · (¯ψ · Γντσψ)

?

,(2.26)

• 4-th rank self-dual antisymmetric tensor field

VA

µνρσ

= −i Str eik·A

?

F[µν· Fρσ]+ c¯ψ · Γ[µνρ[Aσ],ψ] −3i

1

8 · 4!kλkτ(¯ψ · Γλ[µνψ) · (¯ψ · Γρσ]τψ)

4ckλ¯ψ · Γλ[µνψ · Fρσ]

−

?

,(2.27)

where c = −1/3. We fixed the value of c by another calculation (See Section IV-E in

[1]).

Hereafter we write down only the leading order terms of vertex operators.

• charge conjugation of gravitino

VΨc

µ

= Str eik·A

?

[Aµ,Aν] · [Aρ,Aσ] · ΓρσΓνψ +2

3

¯ψ · Γν[Aµ,ψ] · Γνψ

?

,(2.28)

• charge conjugation of antisymmetric tensor field

?

Str eik·A

[Aµ,Aρ] · [Aρ,Aσ] · [Aσ,Aν] −1

4[Aµ,Aν] · [Aρ,Aσ] · [Aσ,Aρ]

?

,(2.29)

• charge conjugation of dilatino

V

˜Φc

= Str eik·A

??

[Aµ,Aρ] · [Aρ,Aσ] · [Aσ,Aν] −1

4[Aµ,Aν] · [Aρ,Aσ] · [Aσ,Aρ]

?

?

· Γµνψ

+1

24[Aµ,Aν] · [Aρ,Aσ] · [Aλ,Aτ] · Γµνρσλτψ,(2.30)

7

Page 10

• charge conjugation of dilaton

VΦc

= Str eik·A

?

[Aµ,Aν] · [Aν,Aρ] · [Aρ,Aσ] · [Aσ,Aµ]

−1

+[Aσ,Aµ] · [Aν,Aρ] ·¯ψΓµΓνρ· [Aσ,ψ]

4[Aµ,Aν] · [Aν,Aµ] · [Aρ,Aσ] · [Aσ,Aρ]

?

.(2.31)

Str means a symmetrized trace which is defined by

Str eik·AB1· B2···Bn =

?1

×tr eik·At1B1eik·A(t2−t1)B2···eik·A(tn−1−tn−2)Bn−1eik·A(1−tn−1)Bn

+( permutations of Bi’s (i = 2,3,··· ,n) ).

The center-dot on the left hand side means that the operators Bi are symmetrized. In

the first term in (2.26), for example, B1and B2correspond to [Aµ,Aρ] and [Aν,Aρ], re-

spectively. See the appendix of [1] for various properties of the symmetrized trace. For

notational simplicity we sometimes use Str also for a single operator like Str (eik·AB) which

is equivalent to the ordinary trace.

0

dt1

?1

t1

dt2···

?1

tn−2

dtn−1

(2.32)

3Stringy Interpretation of Wave Functions

In this section we show that the wave functions obtained in the previous section can be

interpreted as overlaps of D-instanton boundary states and closed string massless states in

the Green-Schwarz formalism of type IIB superstring. The ordinary D-instanton is known

to be coupled only with the dilaton and the axion states [16] and becomes a source for

these closed string modes only. But the D-instanton is a half-BPS state and breaks a

half of the supersymmetries and we can construct a supersymmetry multiplet by acting

broken supersymmetry generators successively on the simplest D-instanton boundary state.

Namely the D-instanton has an internal structure and these multiplet states are coupled

also to the other closed string massless states such as gravitons or antisymmetric tensor

fields. Hence they become a source for these fields, although the couplings contain higher

derivatives. Such internal structures of D-branes were discussed in various papers [17, 18,

19, 20, 21, 14, 22, 23]. In the following, we show that the wave functions in the previous

section are nothing but overlaps of such D-instanton boundary states with the closed string

massless states.

We adopt the Green-Schwarz formalism of type IIB superstring and take the light-cone

gauge. Our notations and brief summaries of a construction of boundary states in the

Green-Schwarz formalism are given in the appendix. Definitions of the supercharges and

a boundary state for the D-instanton are obtained by setting

η = +1,Mij= δij,Mab= δab,M˙ a˙b= δ˙ a˙b,(3.1)

8

Page 11

in the corresponding equations in the appendix ((A.45)-(A.65)).

The type IIB superstring has N = 2 supersymmetries with 32 supercharges. A bound-

ary state for the D-instanton is defined by the boundary conditions in eqs.(A.45)-(A.47)

with (3.1),

∂σXi|B? = 0,

Q+a|B? = 0,

Q+˙ a|B? = 0,(3.2)

and a solution of these conditions is given in eq.(A.60) as

|B? = e

?

n>0(1

nαi

−n˜ αi

−n−iSa

−n˜Sa

−n)|B0?, (3.3)

where Sa

superstring. From eq.(A.61), the zero-mode part becomes

nand˜Sa

nare fermionic modes and αi

nand ˜ αi

nare bosonic modes of the type IIB

|B0? = C (|i?|i? − i|˙ a?|˙ a?),(3.4)

where C is a normalization constant. The D-instanton boundary state preserves a half of

supersymmetries Q+aand Q+˙ a, and breaks the other half Q−aand Q−˙ awhich are defined

in (A.64) and (A.65). The broken and unbroken supercharges satisfy the algebra

{Q+a,Q−b} = 4p+δab,

{Q+a,Q−˙b} = 2√2γi

{Q+˙ a,Q−b} = 2√2γi

{Q+˙ a,Q−˙b} = 2(P−+˜P−)δ˙ a˙b.

a˙bpi,

˙ abpi,

(3.5)

The other anticommutators vanish.

3.1Coupling of D-instanton boundary states to supergravity

modes

States obtained by acting the broken generators Q−a,Q−˙ aon the D-instanton boundary

states couple to the supergravity modes. Here we concentrate on massless modes and

ignore massive excitations.

The zero-mode part of the boundary state of the D-instanton is given by

|D(−1)? =

1

√2(|i?|i? − i|˙ a?|˙ a?),(3.6)

where we set the normalization constant C = 1/√2 for simplicity. This state couples to a

linear combination of the dilaton and axion,

|Φ? ≡

1

√2(|i?|i? − i|˙ a?|˙ a?).(3.7)

9

Page 12

The coupling is given by

?Φ|D(−1)? = 1.(3.8)

Acting the broken charge λaQ−aon |D(−1)?, we obtain the fermionic state

λaQ−a|D(−1)? =

This couples to the following linear combination of dilatino states

?2p+γi

a˙ aλa(|˙ a?|i? − i|i?|˙ a?).(3.9)

|˜Φa? ∼

?

p+γi

a˙ a(|˙ a?|i? − i|i?|˙ a?),(3.10)

and the coupling is given by

?˜Φa|λbQ−b|D(−1)? ∼ p+λa.(3.11)

The normalizations of states for the supergravity modes are fixed so that the supersym-

metry transformations of them satisfy eqs.(2.21).

By further acting the broken supersymmetry charges, we can construct the following

state

λa1λa2Q−a1Q−a2|D(−1)?

= 2√2p+γij

a1a2λa1λa2|i?|j? −

√2ip+?

γi

a1˙ aγi

a2˙b− γi

a2˙ aγi

a1˙b

?

λa1λa2|˙ a?|˙b?. (3.12)

This state couples to the antisymmetric tensor field Bµν,

|Bij? ∼ |i?|j? − |j?|i? −i

2γij

˙ a˙b|˙ a?|˙b?.(3.13)

The coupling between these states is given by

?Bij|λa1λa2Q−a1Q−a2|D(−1)? ∼ p+?γij

a1a2λa1λa2?. (3.14)

Since the coupling contains momentum p+, the boundary state (3.12) has a derivative-

coupling to the antisymmetric tensor field.

The state multiplied by three broken charges is given by

λa1λa2λa3Q−a1Q−a2Q−a3|D(−1)?

=?2p+?3

2

?

γj

a1˙ aγji

a2a3+1

2γi

a1˙b

?

γj

a2˙ aγj

a3˙b− γj

a3˙ aγj

a2˙b

??

λa1λa2λa3(|˙ a?|i? + i|i?|˙ a?).

(3.15)

This state couples to a linear combination of gravitino states

|Ψ˙ a

i? ∼

?

p+

?

|˙ a?|i? + i|i?|˙ a? −1

8γi

˙ abγj

b˙b

?

|˙b?|j? + i|j?|˙b?

??

.(3.16)

Hence the coupling between the boundary state (3.15) and the gravitino state (3.16) be-

comes

?Ψ˙ a

i|λa1λa2λa3Q−a1Q−a2Q−a3|D(−1)? ∼?p+?2γj

˙ aa1λa1?γji

a2a3λa2λa3?.(3.17)

10

Page 13

A boundary state which is obtained by acting four broken generators on |D(−1)? be-

comes

λa1···λa4Q−a1···Q−a4|D(−1)?

= 8√2(p+)2??γik

a1a2λa1λa2??γkj

a3a4λa3λa4?|i?|j? − i?γij

a3a4λa3λa4??

γi

a1˙ aγj

a2˙bλa1λa2?

|˙ a?|˙b?

(3.18)

?

.

This state couples to the graviton state

|hij? ∼ |i?|j? + |i?|j? −1

4δij|k?|k?, (3.19)

and its coupling is given by

?hij|λa1···λa4Q−a1···Q−a4|D(−1)? ∼?p+?2?γik

The coupling contains two derivatives.

We can similarly construct states by acting more broken supersymmetry generators.

They couple to the other massless states of type IIB closed string through derivative

couplings2.

a1a2λa1λa2??γkj

a3a4λa3λa4?. (3.20)

3.2Wave functions with light-cone momentum

In order to compare the wave functions of the mean-field D-instanton in section 2.2 to the

results in the previous subsection, we take the light-cone momentum and rewrite the wave

functions in section 2.2.

Let us take the frame where the momentum is represented as

kµ= (E,0,··· ,0,E),(3.21)

namely, only the k+component is non-vanishing. Then the following relations hold:

/ k = E (Γ0+ Γ9) = −E?Γ0− Γ9?= −√2EΓ−

bij= 4E?γij

bi+= 0.

2Note that both of Q−aand Q−˙ ahave the same structure Sa

long as the massless closed string states are concerned, it is sufficient to consider only one of those two

generators, namely λaQ−a.

/ kλ = 2iE (λa,0,−λa,0)T,

abλaλb?,bi−= 4√2E?γi

0− i˜Sa

˙ aaλ˙ aλa?,

0in the zero-mode part. Hence as

11

Page 14

By using these relations, transverse components of the wave functions in section 2.2 become

Φ = 1,

˜Φ = 2iE (λa,0,−λa,0)T,

Bij

= −2E?γij

Ψi = 4iE2?γij

hij

=

6E2?γik

?

Γ11˜Φ = +˜Φ

?

abλaλb?,

bcλbλc??0,−γj

˙ aaλa,0,γj

˙ aaλa?T,

1

abλaλb??

γkj

cdλcλd?

,

(3.22)

They are the same as the overlaps in the previous subsection with the identification p+=

√2E, up to normalizations. Hence we have shown that the wave functions of the mean-field

D-instanton represent couplings of a supersymmetry multiplet of a D-instanton to closed

string massless states.

3.3Fermionic coherent state of D-instanton

So far we have constructed boundary states by acting a fixed number of broken super-

symmetry generators on |D(−1)? so that they form an ordinary set of a supersymmetry

multiplet. In order to see the above interpretation more systematically, we construct a

fermionic coherent state by acting the unitary operator exp(−λaQ−a) on |D(−1)?;

|λ? = exp(−λaQ−a)|D(−1)?.

Due to the commutation relations (3.5), this state satisfies modified boundary conditions

(3.23)

Q+a|λ? = 4p+λa|λ?

Q+˙ a|λ? = 2√2piγi

(3.24)

(3.25)

˙ aaλa|λ?.

In the IIB matrix model, the bosonic coordinates are interpreted as the coordinates of

space-time. From the consideration here, the fermionic coordinates can be interpreted as

the fermionic parameters which bestow an internal structure on the space-time constructed

from the bosonic coordinates.

The wave functions for the mean-field D-instanton can be written as

fA(λ) = ?A|λ?,(3.26)

for each supergravity state A. In the previous subsection, this relation has been shown

separately for each state |A? up to normalization. It can be understood more directly

as follows. When the momentum k is taken as (3.21), the supercharges q1and q2for a

D-instanton, (2.1) and (2.2), have the following forms,

qa

1= −i

∂

∂λa,qa

2= 2iEλa,(3.27)

12

Page 15

and satisfy the algebra

{qa

1, qb

2} = 2Eδab,others = 0.(3.28)

On the other hand, as far as massless states are concerned, this algebra is equivalent to

the ones among the supercharges Q±aand Q±˙ awith pi= P−=˜P−= 0, eq.(3.5). Actually

actions of qa

which act on the massless state of the supergravity modes |A? as follows,

1fA(λ) = −i∂

qa

ion the wave functions (3.26) can be regarded as insertions of Q−aand Q+a

qa

∂λfA(λ) = i??A|Q−a?|λ?,

2fA(λ) = 2iEλafA(λ) =

i

2√2

??A|Q+a?|λ?,

where we have used eqs.(3.24) and (3.25). Hence a construction of the supergravity multi-

plet by acting Q±aon the closed string massless state ?A| corresponds to the one by acting

qa

couplings between a D-instanton and various supergravity modes.

ion wave functions fA(λ) and the wave functions we constructed describe the (derivative)

4One-loop Effective Action

In the latter half of the paper, we discuss condensation of massless supergravity fields in

type IIB matrix model. We consider a matrix model of size (N +1)×(N +1) and integrate

over one D-instanton with the wave functions given in section 2. In this way, we can obtain

a modified effective action in a weak supergravity background of N D-instantons.

In this section we first give a systematic evaluation of the one-loop effective action with

general fermionic backgrounds. The results were partly given in [24] and [25]. Similar

calculations were performed in the BFSS matrix model in [14]. Since we are interested in

condensation, we do not use the matrix model equation of motion in this section.

We start from the type IIB matrix model with a size (N + 1) × (N + 1) and write

(N + 1) × (N + 1) bosonic and fermionic hermitian matrices as A′

We then decompose them into backgrounds (Xµ,Φ) and fluctuations (aµ,ϕ) around them

as

µ(µ = 0,··· ,9) and ψ′.

A′

ψ′

µ

= Xµ+ aµ,

= Φ + ϕ. (4.1)

In order to perform perturbative calculations, we fix a gauge and add the following terms

to the action (1.1),

Sg.f.+ghost= −tr

?1

2[Xµ,aµ][Xν,aν] + [Xµ,b][A′µ,c]

?

,(4.2)

where c and b are ghost and anti-ghost fields respectively. Substituting the decompositions

(4.1) into the action (1.1) and (4.2), we obtain the following expression up to the 2nd order

13

Page 16

of the fluctuations,

SIKKT+ Sg.f.+ghost = SIKKT(X,Φ) −1

2tr(aµ[Xν,[Xν,aµ]] + 2aµ[[Xµ,Xν],aν])

2tr ¯ ϕΓµ[Xµ,ϕ] − tr¯ΦΓµ[aµ,ϕ] + tr b[Xµ,[Xµ,c]]

= SIKKT(X,Φ) +1

2tr aµ

−1

Γ ·˜ X

+tr b˜X2c + higher orders,

−1

?

δµν˜ X2+ 2˜Fµν+˜ ¯ΦΓµ

1

Γ ·˜ XΓν˜Φ

1

Γ ·˜ XΓν[aν,Φ]

?

aν

2tr

?

¯ ϕ +?¯Φ,aµ

?Γµ

1

??

Γ ·˜ X

??

ϕ +

?

(4.3)

where we defined Fµν= [Xµ,Xν], Γ·X ≡ ΓµXµand denoted the adjoint action of a general

operator O as˜OS ≡ [O,S]. Then the one-loop partition function of the IIB matrix model

becomes

?

∼ e−SIKKT(X,Φ)det−1

??

Thus the free energy is given by

Z(X,Φ) =daµdϕdbdc e−(SIKKT+Sg.f.+ghost)

2

?

δµν˜ X2+ 2˜Fµν+˜ ¯ΦΓµ

1

Γ ·˜ XΓν˜Φ

?˜ X2?

?

×det

1

4

˜ X2+1

2Γµν˜Fµν

?1 + Γ11

2

?

det.(4.4)

F(X,Φ) = −lnZ(X,Φ) = SIKKT(X,Φ) + Fb+ Ff,

Fb =

2T r ln

−T rln˜ X2,

1

2T r ln

(4.5)

1

?

δµν˜ X2+ 2˜Fµν

?

−1

4T r ln

?

˜ X2+1

2Γµν˜Fµν1 + Γ11

2

?

(4.6)

Ff

=

?

δµν+

?

1

˜ X2+ 2˜F

?

µρ

˜ ¯ΦΓρ

1

Γ ·˜ XΓν˜Φ

?

,(4.7)

where T r is the trace of the adjoint operators.

We first expand Ff formally with respect to the inverse powers of˜ X. To this end we

use the following formulas,

1

˜X2+ 2˜F

=

1

˜ X2˜F

1

2˜ X2Γ ·˜F

1

1 +

1 +

2

1

˜X2,(4.8)

1

Γ ·˜ X

=

1 +

1

2

1

1

˜ X2Γ ·˜ X

1

˜ X2Γ ·˜ X +1

(4.9)

=

1

2˜ X2Γ ·˜F

2Γ ·˜ X

1

1 +

1

2˜ X2Γ ·˜F

1

˜ X2,(4.10)

where

?

Γ ·˜X

?2

=˜X2+1

2Γ ·˜F,(4.11)

14

Page 17

and Γ·˜F ≡ Γµν˜Fµν. In the following we expand the free energy with respect to 1/˜ X. Since

the leading part of˜ X is a distance between N D-instantons and a single D-instanton,

this expansion is valid when the single D-instanton is far separated from the other N

D-instantons.

4.1Second order terms of Φ

First let us focus on the terms with two fermions. As is seen in section 5, these terms

are relevant for condensation of the antisymmetric tensor field. After using eqs.(4.8) and

(4.10), the second order terms of the fermionic background Φ are given by

Ff

???

Φ2

=

1

4T r

?

1

˜ X2˜F

1 +

2

?

µν

1

˜ X2

˜ ¯ΦΓν

1

1 +

1

2˜ X2Γ ·˜F

1

˜ X2

?

Γ ·˜ X

?

Γµ˜Φ

+

?

1

˜ X2˜F

1 +

2

?

µν

1

˜ X2

˜ ¯ΦΓν

?

Γ ·˜ X

?

1

1

1 +

2˜ X2Γ ·˜F

1

˜ X2Γµ˜Φ

. (4.12)

We now expand the effective action with two Φ’s (4.12) with respect to 1/˜ X.

4.1.1

˜ X−3

The leading order starts from 1/˜X3and is given by

1

4T r

?1

˜ X2

˜ ¯Φ1

˜ X2Γµ

1

˜ X2

?

Γ ·˜ X

?˜Xµ,˜Φ

?

Γµ˜Φ +

1

˜ X2

˜ ¯ΦΓµ

?

Γ ·˜ X

?

Γµ

1

˜ X2˜Φ

?

= −2T r

˜ ¯Φ1

˜ X2Γµ

?

.(4.13)

This is proportional to the equation of motion for the fermion.

4.1.2

˜ X−5

The next-to-leading order is proportional to 1/˜ X5. At this order we have the following

terms,

1

4T r

??

?

−2

−2

˜ X2˜Fµν

?

1

˜ X2

1

˜X2

˜ ¯ΦΓν

1

˜ X2

?

Γ ·˜ X

?1

?

Γµ˜Φ +

1

˜ X2

1

˜X2

˜ ¯ΦΓµ

?

Γ ·˜ X

−

1

2˜ X2Γ ·˜F

??

?

1

1

˜ X2

?

Γ ·˜ X

?

?

Γµ˜Φ

+

˜X2˜Fµν

?

˜ ¯ΦΓν

?

Γ ·˜ X

˜X2Γµ˜Φ +

˜ ¯ΦΓµ

?

−

2˜X2Γ ·˜F

1

˜X2Γµ˜Φ

?

.

(4.14)

15

Page 18

After some calculations, these terms are rewritten as

−1

2T r

1

˜ X2˜Fµν

1

˜ X2˜Fµν

1

˜ X2˜Fµν

1

˜ X2

1

˜ X2

1

˜ X2˜ Xµ˜ ¯Φ1

˜ ¯Φ1

˜ X2Γµν· Γρ

˜ ¯Φ1

˜ X2Γν˜ Xµ˜Φ − T r

˜ X2Γν˜Φ + T r

?

˜Xρ,˜Φ

?

−1

2T r

1

˜ X2

1

˜ X2

1

˜ X2˜Fµν

˜ ¯Φ1

˜ X2Γν˜Φ˜ Xµ

˜ ¯Φ˜ Xµ

˜ X2Γν˜Φ.

1

˜ X2

?˜ ¯Φ,˜Xρ

?

Γρ· Γµν

1

˜ X2˜Φ

−T r

1

˜ X2˜Fµν

1

˜ X2˜Fµν

+T r

1

(4.15)

The first two terms are proportional to the equation of motion. It is noted that the terms

in the second and the third lines vanish if˜ Xµis replaced with dµ. Here dµis a vector

directed to the center of the N D-instantons from the single D-instanton. Therefore these

terms are actually O(d−6) in the 1/d expansions.

4.1.3

˜ X−7

The terms of the order˜ X−7are given by

1

4T r

??2

+1

˜ X2

?2

+

˜ X2˜Fµν

??2

?

?

??2

?

?

˜ X2˜Fνρ

1

2˜ X2Γ ·˜F

1

˜ X2

?

1

˜ X2

??

?

?

1

2˜X2Γ ·˜F

˜ ¯ΦΓρ

1

˜ X2

?

Γ ·˜ X

?

1

˜ X2

?

?

Γ ·˜X

?1

Γµ˜Φ

˜ ¯ΦΓµ

1

2˜ X2Γ ·˜F

1

2˜X2Γ ·˜F

1

˜ X2

1

˜ X2

Γ ·˜ X

?

Γµ˜Φ

Γµ˜Φ

+

˜ X2˜Fµν

?2

+1

˜X2

?2

˜ ¯ΦΓν

?

?

??

˜ X2˜Fµν

˜ X2˜Fνρ

??

1

˜ X2

˜ ¯ΦΓρ

Γ ·˜ X

1

2˜X2Γ ·˜F

1

2˜ X2Γ ·˜F

˜ X2Γµ˜Φ

?

?

˜ ¯ΦΓµ

Γ ·˜ X

??

??

1

˜X2Γµ˜Φ

1

˜ X2Γµ˜Φ

+

˜ X2˜Fµν

˜ ¯ΦΓν

?

Γ ·˜X

?

.(4.16)

16

Page 19

These are rewritten as

1

4T r

1

˜X2˜Fµν

1

˜ X2˜Fµν

1

˜ X2˜Fµν

1

˜ X2˜Fµν

1

˜ X2˜Fµν

1

˜ X2˜Fµν

1

˜X2˜Fµν

1

˜X2˜Fµν

1

˜ X2˜Fµν

1

˜ X2˜Fµν

1

˜ X2˜Fµν

1

˜X2

1

˜ X2˜Fνρ

1

˜ X2˜Fνµ

1

˜ X2

1

˜ X2˜Fρσ

1

˜ X2˜Fρσ

1

˜X2

1

˜X2˜Fνρ

1

˜ X2˜Fνρ

1

˜ X2˜Fνρ

1

˜ X2

˜ ¯Φ1

˜X2˜Fρσ

1

˜X2ΓµνρσΓλ

˜ ¯Φ1

˜ X2ΓµρΓσ

˜ ¯Φ1

˜ X2Γρ

˜ X2˜Fνµ

1

˜ X2˜ Xν˜ ¯ΦΓµρσ

1

˜ X2

˜ ¯Φ1

˜X2˜Fρσ

1

˜X2

1

˜ X2

1

˜ X2˜Xµ˜ ¯ΦΓρ

˜ ¯Φ1

˜ X2˜Fνρ

?˜ Xλ,˜Φ

?˜ Xσ,˜Φ

?˜Xρ,˜Φ

˜ X2Γρ

˜ X2˜Φ −1

˜ ¯ΦΓµνσ

?

+ T r

˜ X2˜Fµν

+T r

1

˜ X2

1

˜ X2

˜ ¯Φ1

?

1

˜ X2˜Fµν

1

˜ X2˜Fνρ

˜ X2˜Fνµ

1

˜ X2

?˜ ¯Φ,˜Xρ

?˜ ¯Φ,˜ Xσ

?

˜ X2˜Φ

ΓσΓµρ

1

˜ X2˜Φ

−T r

−1

−1

−1

−1

?

− T r

?

2T r

1

11

˜ X2

?

Γρ

1

2T r

1

?

˜ Xρ,˜Φ

2T r

1

1

˜ X2˜Fµν

1

˜ X2˜Fµν

1

˜X2˜Fµν

˜X2˜Fµν

˜ X2˜Fµν

1

˜ X2˜Fµν

1

˜ X2˜Fµν

1

˜ X2˜Fρσ

1

˜ X2˜Fρσ

1

˜X2

˜X2˜Fνρ

˜ X2˜Fνρ

1

˜ X2˜Fνρ

1

˜ X2

1

˜ X2˜ Xσ˜ ¯ΦΓµνρ

1

˜ X2

˜ ¯Φ˜ Xν

˜X2˜Fρσ

1

˜X2

1

˜ X2

1

˜ X2˜Xρ˜ ¯ΦΓµ

˜ ¯Φ˜ Xµ

˜ X2˜Fνρ

1

˜ X2˜Φ

2T r

1

˜ X2˜Φ˜ Xρ−1

˜X2˜ XνΓµρσ˜Φ +1

˜ ¯Φ1

˜X2Γρ˜ Xµ˜Φ + 2T r

˜ ¯ΦΓµ

˜ X2˜Φ˜ Xρ+ T r

1

˜ X2˜Φ + T r

1

˜ X2˜ XµΓρ˜Φ + T r

2T r

˜ ¯ΦΓνρσ

1

˜ X2˜Φ˜ Xµ

1

˜X2Γµρσ˜Φ

˜X2Γµ˜Φ

˜ X2˜Φ˜ Xµ

1

˜ X2˜Φ

1

˜ X2Γρ˜Φ.

2T r

1

2T r

1

+2T r

1

1

˜ ¯Φ˜ Xρ

1

−T r

1

1

1

˜ ¯ΦΓρ

1

−T r

+T r

1

(4.17)

The first six terms vanish if the fermionic background satisfies the equation of motion.

17

Page 20

4.1.4

˜ X−9

The terms of the order˜ X−9are given by

T r

?

−21

˜ X2˜Fµν

−1

32

˜ X2

−1

2

−1

8

−21

−1

32

˜ X2

−1

2

−1

8

1

˜ X2˜Fνρ

1

˜ X2˜Fρσ

1

˜ X2

?1

˜ ¯ΦΓρ

˜ ¯ΦΓσ

1

˜ X2

?

Γ ·˜ X

?

Γ ·˜F

?

?1

?1

Γ ·˜F

?

?1

Γµ˜Φ

1

˜ ¯ΦΓµ

1

˜ X2

1

˜X2˜Fνρ

1

˜ X2

1

˜ X2˜Fνρ

˜ ¯ΦΓµ

?

Γ ·˜F

1

˜X2

1

˜ X2

1

˜ X2˜Fρσ

Γ ·˜X

˜ X2˜Fνρ

1

˜ X2

˜ X2

?

?

?1

˜ ¯ΦΓσ

Γ ·˜F

Γ ·˜F

?1

?1

?

Γ ·˜X

?1

?1

?

˜ X2

??1

?

?

˜ X2

Γµ˜Φ

?

Γ ·˜ X

?

Γµ˜Φ

1

˜X2˜Fµν

1

˜ X2˜Fµν

˜ X2˜Fµν

1

1

˜X2

˜X2

Γ ·˜ X

˜ ¯ΦΓν

?

Γ ·˜F

1

˜ X2

?

?

?1

˜ X2

Γ ·˜F

˜ X2

Γ ·˜ X

?

Γµ˜Φ

?

˜ X2Γµ˜Φ

?1

Γ ·˜F

??1

1

˜ X2

˜ X2

˜ ¯ΦΓρ

Γ ·˜F

Γ ·˜ X

˜ X2

?

˜ X2

?1

?

?

Γ ·˜F

?1

˜ X2Γµ˜Φ

1

˜ X2˜Fµν

1

˜ X2˜Fµν

1

˜ X2

˜ X2Γµ˜Φ

?1

˜ ¯ΦΓν

?

Γ ·˜ X

˜ X2

Γ ·˜F

˜ X2

Γ ·˜F

˜ X2Γµ˜Φ

?

. (4.18)

4.2Fourth order terms of Φ

Now let us consider four-fermion terms. These terms are relevant for condensation of

gravitons. The fourth order terms of Φ are given by

Ff

???

Φ4

= −1

4T r

?

1

˜ X2˜F1 +

2

?

µν

?

1

˜ X2

˜ ¯ΦΓν

1

1

1 +

2˜ X2Γ ·˜F

1

1 +

1

˜ X2

?

Γ ·˜ X

?

Γρ˜Φ

×

?

1

˜ X2˜F1 +

2

ρσ

1

˜ X2

˜ ¯ΦΓσ

1

2˜ X2Γ ·˜F

1

˜ X2

?

Γ ·˜ X

?

Γµ˜Φ.(4.19)

4.2.1

˜ X−6

The leading order term is proportional to 1/˜X6. The term of this order is given by

−1

4T r

1

˜X2

˜ ¯ΦΓµ

1

˜X2

?

Γ ·˜ X

?

Γν˜Φ1

˜X2

˜ ¯ΦΓν

1

˜X2

?

Γ ·˜ X

?

Γµ˜Φ.(4.20)

18

Page 21

4.2.2

˜ X−8

The next order terms are proportional to 1/˜ X8and given by

−1

2T r

??

+1

˜ X2

−2

˜ X2˜Fµν

?

2˜ X2Γ ·˜F

1

˜ X2

1

˜ ¯ΦΓν

1

˜ X2

?

?

1

Γ ·˜ X

?

Γρ˜Φ1

˜ X2

Γν˜Φ1

˜ ¯ΦΓρ

1

˜ X2

?

1

Γ ·˜ X

?

Γµ˜Φ

˜ ¯ΦΓµ

?

−

˜ X2

?

Γ ·˜ X

?

˜ X2

˜ ¯ΦΓν

˜ X2

?

Γ ·˜ X

?

Γµ˜Φ

?

.(4.21)

4.2.3

˜ X−10

The terms of the order 1/˜ X10become

−T r

?

+1

21

˜ X2˜Fµν

1

˜ X2

+1

2

+1

2

+1

˜ X2Fµν

+1

16

1

˜ X2˜Fνρ

˜ ¯ΦΓµ

˜ X2Γλρ˜Fλρ

1

˜X2

1

˜ X2

1

˜ X2

˜ ¯ΦΓµ

˜ X2Γλρ˜Fλρ

1

˜ X2

˜ ¯ΦΓρ

1

˜ X2

1

˜ X2Γστ˜Fστ

˜X2Γστ˜Fστ

1

˜ X2

1

˜ X2

1

˜ X2

?

Γ ·˜X

?

Γσ˜Φ1

˜ X2

˜ ¯ΦΓσ

1

˜ X2

?

˜ ¯ΦΓν

Γ ·˜X

?

Γµ˜Φ

8

1

1

˜ X2

?

Γρ˜Φ1

?

Γ ·˜ X

?

˜ ¯ΦΓρ

?

Γν˜Φ1

˜ X2

˜ ¯ΦΓρ

1

˜ X2

?

?

?

Γ ·˜ X

?

?

Γµ˜Φ

?

Γµ˜Φ

1

˜X2Fµν

1

˜ X2Fµν

˜ ¯ΦΓν

1

1

˜X2

?

Γρ˜Φ1

Γ ·˜ XΓρ˜Φ1

˜X2

1

˜X2

1

˜ X2

?

Γ ·˜ X

Γ ·˜ X

?

?

Γµ˜Φ

˜ ¯ΦΓν

?

Γ ·˜X

?

Γ ·˜ X

˜ X2

1

˜ X2Γστ˜Fστ

1

˜ X2

˜ ¯ΦΓν

˜ X2Γστ˜Fστ

Γµ˜Φ

˜ ¯ΦΓν

??

˜ X2Fρσ

Γν˜Φ1

˜ X2

˜ ¯ΦΓσ

1

˜ X2

Γ ·˜X

1

˜ X2

1

Γ ·˜X

?

1

Γ ·˜X

?

Γµ˜Φ

?

.(4.22)

5Condensation of the Supergravity Modes

In this section, we discuss modifications of the effective actions for the IIB matrix model

by condensation of D-instantons with appropriate wave functions. They correspond to

condensation of massless type IIB supergravity fields. We here consider backgrounds pro-

duced by a mean-field D-instanton. As we saw in sections 2 and 3, a D-instanton forms a

supersymmetry multiplet by acting broken supersymmetry generators on the ordinary D-

instanton boundary state. These states couple to the closed string massless states through

derivative couplings and become a source for these fields.

Now we write (N + 1) × (N + 1) matrices in the decomposition (4.1) as follows,

?

aµ=

α†

µ

0

Xµ=

xµ1N+ Aµ

0

0

yµ

?

ϕ =

,Φ =

?

?

ψ 0

0ξ

?

,(5.1)

?

0αµ

?

,

?

0

φ†

φ

0

. (5.2)

Aµand ψ are N × N traceless matrices. yµis a bosonic coordinate of a (mean-field) D-

instanton. ξ is a fermionic coordinate and a Majorana-Weyl spinor. They represent degrees

19

Page 22

of freedom of the mean-field D-instanton. Off-diagonal components αµand φ are N-vectors

corresponding to interactions between the diagonal blocks, which we have integrated out at

the one-loop order in the previous section. Hence the free energy (4.5) is a function of these

diagonal components, F(X,Φ) = F(A,x,ψ;y,ξ). By choosing wave functions fk(y,ξ) for

the mean-field D-instanton and integrating over y,ξ, we can obtain a modified effective

action Seff(A,x,ψ;fk) by condensation of the massless modes;

e−Seff(A,x,ψ;fk)=

?

dydξ e−F(A,x,ψ;y,ξ)fk(y,ξ). (5.3)

In what follows, we mainly look at terms without fermionic matrices ψ and replace all

fermionic variables by the D-instanton fermionic coordinate ξ.

5.1 Condensation of the dilaton

In order to express condensation of dilaton in terms of the wave function of the mean-field

D-instanton, we put fk(y,ξ) as

fD(y,ξ) =

?

d10k eik·y ˜fD(k,ξ) =

?

d10k eik·yf(k)

?16

γ=1

?

ξγ

?

.(5.4)

Then the ξ integration is already saturated by the wave function. The leading contribution

to the effective action is easily shown to be proportional to (the charge conjugation of )

the dilaton vertex operator (2.31).

5.2Condensation of the antisymmetric tensor Bµν

We now calculate the effective action with an insertion of a wave function describing the

antisymmetric tensor field Bµν. In the present calculation, the supersymmetry multiplet

starts from the dilaton wave function (5.4) and the other functions in the multiplet can be

constructed by acting a derivative operator ∂/∂ξ.

By replacing λ in eq.(2.13) with ∂/∂ξ and applying the differential operator on

we obtain the wave function for the antisymmetric tensor field;

??16

γ=1ξγ

?

,

fB(y,ξ) =

?

?

d10k eik·y ˜fB(k,ξ)(5.5)

=d10k eik·y(ζµν(k)kρ+ ζνρ(k)kµ+ ζρµ(k)kν)

×(ΓµνρΓ0)αβ

∂

∂ξα

∂

∂ξβ

?16

γ=1

?

ξγ

?

,

where ζµν(k) is a polarization tensor, ζµν(k) = −ζνµ(k). Since our SUSY algebra closes

only on shell,˜fD(k,ξ) and˜fB(k,ξ) fall into the supergravity multiplet for the case k2= 0.

Hereafter we, however, formally extend the wave functions to the off-shell and integrate

over the whole momentum region.

20

Page 23

5.2.1Contribution at O(1/d8) and Myers-like effect

Let us first look at contributions from the second order terms of Φ. In these terms we can

simply replace Φ with ξ and thus the terms of the order˜ X−3and˜ X−5vanish. The terms

of the order˜ X−7, eq. (4.17), become

1

2

?¯ξΓµρσξ?T r

?

−˜Fµν

1

˜ X2˜Fρσ

1

˜ X2˜ Xν

?2

?2

?1

1

˜ X2˜Fρσ

˜ X2

?2

−˜Fµν

1

˜ X2˜ Xν

?1

?1

1

˜ X2˜Fρσ

˜ X2

?2

˜Fρσ

1

˜ X2

+˜Fµν

?1

?1

˜ X2

˜ Xν

1

˜ X2+˜Fµν

1

˜ X2˜Fρσ

˜ X2

?2

˜ Xν

1

˜ X2

?2?

−˜Fµν

˜ X2

˜Fρσ

1

˜ X2˜ Xν

1

˜ X2+˜Fµν

1

˜ X2˜ Xν

?1

˜ X2

. (5.6)

We then expand these terms with respect to the inverse powers of dµ ≡ xµ− yµ. For

example, 1/˜ X2is expanded as follows,

1

˜ X2=1

d2

?

1 − 2d · A

d2

?

+ O

?1

d4

?

.(5.7)

It is easily realized that the leading terms with 1/d7vanish. The 1/d8term has the following

simple form,

−1

2d8

The 1/d8dependence of the term indicates that the interaction is induced by an exchange

of massless antisymmetric field.

We then integrate over yµand ξ with the wave function (5.5) in order to derive the

effective action under condensation of the antisymmetric tensor field. In this calculation,

we take our wave function (5.5) such that it damps at the infrared region where |y −x| →

∞. Such a choice of wave function is natural from the view point of the dynamics of

the eigenvalues in the matrix model. It was indeed shown that the distributions of the

eigenvalues of Aµare bounded in a finite region dynamically [4]. It is therefore natural to

consider that the wave function damps far from the D-instantons. The size of the eigenvalue

distribution is a function of N. If the eigenvalues are distributed on d-dim hypersurface

uniformly, it is proportional to N1/d. The natural scale of the infrared cutoff of the wave

function depends on the dynamics of the matrix models, which we do not discuss in the

present paper.

The integration over ξ and y can be easily performed as

?¯ξΓµρσξ?tr[Aν,Fµν]Fρσ. (5.8)

?

d10y d16ξ fB(y,ξ)

−1

2(x − y)8¯ξΓµρσξ tr[Aν,Fµν]Fρσ

=

?

d10y d10k eik·y(ζµν(k)kρ+ ζνρ(k)kµ+ ζρµ(k)kν)

−1

2(x − y)8tr[Aσ,Fµσ]Fρν

= −π5

3

?

d10keik·x

k2(ζµν(k)kρ+ ζνρ(k)kµ+ ζρµ(k)kν) tr[Aσ,Fµσ]Fρν. (5.9)

21

Page 24

Because of our choice of the wave function, ζµν(k) damps at small k.

We therefore obtain the following effective action

Seff(A,x,ψ;fB) = SIKKT− i

?

d10kfµνρ(k)eik·xtr[Aσ,Fµσ]Fνρ,(5.10)

where fµνρ(k) = −iπ5

This effective action shows that the Chern-Simons-like term is induced by an effect

of condensation of the antisymmetric tensor. This phenomenon is similar to the Myers

effect [10], but there is a difference. In the case of the Myers effect for D0-branes, a cubic

term of bosonic matrices is induced in the RR three-form background. This term can be

interpreted as a vertex operator for the RR potential. In our case, however, the leading

order of the induced term in eq.(5.10) is different from the expected vertex operator for

the charge conjugation of the antisymmetric tensor field (2.29). Such a term appears at

the next order in the 1/d expansion as shown in the next subsection. The reason can

be understood as follows. If we also calculate the fermionic term containing ψ, we would

expect to obtain a term like tr?¯ψΓµψ?Fνρand the leading order term in (5.10) with this

IKKT action. This kind of terms can not be seen in the vertex operators since we have

assumed the equation of motion (2.4) and (2.5) in their construction. Here, since we are

interested in investigating the effective actions under condensation of the antisymmetric

tensor fields, we do not want to use the equations of motion of the original IKKT action

and the term in (5.10) should not be omitted.

Let us see an effect of the induced term in (5.10) for a particular form of the polarization

tensor. Assuming that the coefficient?d10kfµνρ(k)eik·xis proportional to ǫijkwith a specific

modified matrix model action becomes

3(kµζνρ+ kνζρµ+ kρζµν)/k2.

fermionic term would be cancelled by using the equation of motion (2.4) of the original

direction (i,j,k) = (1,2,3) and that the region k ∼ 0 is dominant in the k-integration, the

Seff(A,x,ψ;fB) = SIKKT− iαǫijktr[Aν,Fiν]Fjk,

with a constant coefficient α. This action has a fuzzy sphere classical solution;

1

10αLi,

Aa = 0,

ψ = 0.

(5.11)

Ai =

(i = 1,2,3)

(for the other directions)

(5.12)

The radius of the fuzzy sphere is in inverse proportion to the coefficient α and in the

α → 0 limit the fuzzy sphere is expanded and becomes a flat plane. It contrasts with

matrix models with the ordinary cubic Chern-Simons term (see, for example [26]) where

the radius of the fuzzy sphere is proportional to the coefficient of the Chern-Simons term.

In addition to the fuzzy sphere solution, flat D-branes

[Aµ, Aν] = iθµν1N

(θµν= −θνµ).(5.13)

with a constant θµνare also classical solutions of the effective action (for an infinite N). It

will be interesting to compare stabilities of these solutions to the fuzzy sphere solution by

calculating loop corrections around them.

22

Page 25

5.2.2Contribution at O(1/d9) and Bµν vertex operator

The induced term in the previous subsection vanishes if we use the equation of motion

for the configuration Aµ. Then the next order O(1/d9) term becomes the leading order.

From the dimensional analysis, it is expected that the vertex operator corresponding to

the charge conjugation of the antisymmetric tensor field (2.29) would appear at the order

of 1/d9.

Expanding the O(˜X−7) term (5.6) with respect to 1/d, we obtain O(1/d9) terms

2dλ

d10

+2

d10

?¯ξΓµρσξ?tr(FµνFρσFνλ+ FµνFνλFρσ)

?¯ξΓµρσξ?tr([Aν,Fµν](d · A)Fρσ+ [Aν,Fµν]Fρσ(d · A)).

The same order terms with O(1/d9) can be obtained also from eq. (4.18) as

−12dλ

d10

−2dλ

d10

Therefore the interaction terms between the mean-field D-instanton and the N × N block

are given at this order by

(5.14)

?¯ξΓµνλξ?tr

?¯ξΓµρσξ?tr(FµνFρσFνλ+ FµνFνλFρσ).

?

FµρFρσFσν−1

4FµνFρσFσρ

?

(5.15)

−12dλ

+2

d10

d10

?¯ξΓµνλξ?tr

?¯ξΓµρσξ?tr([Aν,Fµν](d · A)Fρσ+ [Aν,Fµν]Fρσ(d · A)).

The first line represents an interaction through the vertex operator for (the charge conju-

gation of) the antisymmetric tensor field (2.29). The second term is similar to the eq.(5.8)

except for the insertion of d·A. By integrating over yµand ξ with the wave function (5.5),

the following terms are added to the effective action,

?

FµρFρσFσν−1

4FµνFρσFσρ

?

(5.16)

−i

?

d10kfµνρ(k)eik·x

?

− 3ikρtr

?

FµσFσλFλν−1

4FµνFσλFλσ

?

+1

2tr([Aσ,Fµσ](ik · A)Fνρ+ [Aσ,Fµσ]Fνρ(ik · A))

?

. (5.17)

The first term represents a derivative coupling of D-instantons to the vertex operator of

the antisymmetric tensor field. The second term can be combined with eq.(5.10) into a

form

− i

?

d10kfµνρ(k)eik·xStr eik·A[Aσ,Fµσ] · Fνρ.(5.18)

If we calculate higher order terms in the 1/d expansion, we would expect to obtain higher

order terms of (5.18) with respect to the number of bosonic fields Aµ.

23

Page 26

5.3Condensation of the graviton

Effects of the condensation of gravitons can be seen from the fourth order terms of ξ. The

term eq.(4.20) vanishes by substituting ξ for Φ because of the identity for the Majorana-

Weyl spinor,?¯ξΓµνρξ?Γνρξ = 0. Therefore the leading contribution in the 1/d expansion

comes from the˜ X−8terms, eq.(4.21) by replacing Φ with ξ as,

?¯ξΓνλρξ??¯ξΓµρσξ?T r

Contribution at O(1/d8) and O(1/d9)

Order O(1/d8) terms vanish

1

˜ X2˜Fµν

?1

˜ X2

?2

˜ Xλ

?1

˜ X2

?2

˜ Xσ.(5.19)

5.3.1

dλdσ

d10

?¯ξΓνλρξ??¯ξΓµρσξ?tr Fµν= 0,(5.20)

since dλdσ

Similarly order O(1/d9) terms also vanish

1

d10

?¯ξΓνλρξ??¯ξΓµρσξ?is symmetric under an exchange of (µ,ν).

?¯ξΓνλρξ??¯ξΓµρσξ?tr(dλFµνAσ+ dσFµνAλ) = 0.

Contribution at O(1/d10)

Hence the leading order terms start from O(1/d10) terms. Contributions from the above

˜ X−8term (5.19) are given by

(5.21)

5.3.2

1

2d10

?¯ξΓµρλξ??¯ξΓνσλξ?tr FµνFρσ+

where cµν(ξ) ≡ dρ

(4.22) as

4

d12dλ

?¯ξΓµρσξ?cµν(ξ)tr FνρFσλ,(5.22)

?¯ξΓµνρξ?. The same order terms are also obtained from the˜ X−10terms

?¯ξΓµνλξ??¯ξΓρσλξ?tr FµνFρσ−

2d12cµρcνσtr FµνFρσ+

−

1

8d10

9

1

2d12cµρcρνtr FµσFσν+

?¯ξΓµνρξ?tr FµνFσλ.

3

2d12cµνcρσtr FµνFρσ

−

3

2d12dλcρσ

(5.23)

By using the following Fierz identity,

cµνcρσ

=

1

3(cµνcρσ+ cµρcσν+ cµσcνρ)

−1

+1

6

+d2

6

6(gµρcνλcλσ− gµσcνλcλρ− gνρcµλcλσ+ gνσcµλcλρ)

?¯ξΓρσλξ?− dνcλµ

?¯ξΓµνλξ??¯ξΓρσλξ?.

?dµcλν

?¯ξΓρσλξ?+ dρcλσ

?¯ξΓµνλξ?− dσcλρ

?¯ξΓµνλξ??

(5.24)

24

Page 27

the sum of these two terms, eqs.(5.22) and (5.23), can be simplified and depends on ξ only

in the form of cµν(ξ) as

(5.22) + (5.23) = −1

d12cµρcρνtr FµσFσν.(5.25)

It represents a derivative coupling of a single D-instanton to the graviton vertex operator

constructed from the N D-instantons. If we insert the graviton wave function and integrate

over the single D-instanton coordinates, we can obtain the graviton vertex operator as an

induced term in the effective action.

Similarly interactions mediated by the 4-th rank self-dual antisymmetric tensor field

would appear, but such terms vanish in the leading order because of the cyclic property of

the trace and the Jacobi identity,

cµνcρσtr(FµνFρσ+ FµρFσν+ FµσFνρ)

= cµνcρσtrAµ([Aν,[Aρ,Aσ]] + [Aρ,[Aσ,Aν]] + [Aσ,[Aν,Aρ]]) = 0.(5.26)

If we calculate higher order terms, we would expect to obtain the terms which can be

produced by expanding the exponential in

1

d12cµνcρσStr eik·A(Fµν· Fρσ+ Fµρ· Fσν+ Fµσ· Fνρ). (5.27)

6Conclusion

In this paper, we have considered fermionic backgrounds and condensation of supergravity

fields in the IIB matrix model. We start from the type IIB matrix model in a flat back-

ground with the size (N + 1) × (N + 1), namely a system of (N + 1) D-instantons. We

then integrate one D-instanton (which we call a mean-field D-instanton) and obtain an

effective action for N D-instantons by assuming particular forms of wave functions of the

mean-field D-instanton. If we assume that the configurations of N D-instantons satisfy the

equation of motion, we show that vertex operators obtained in our previous paper [1] are

induced in the effective action as leading contributions. If we do not assume it, extra terms

also appear. In particular if we take the wave function as that of the antisymmetric tensor

field, a Chern-Simons like term is induced in the leading order of perturbations. Though

this term is quintic with respect to the field Aµ, a fuzzy sphere becomes a solution to the

equation of motion. In this sense, this is a similar mechanism to the Myers effect.

We have also given a stringy interpretation of the wave functions of the mean-field

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

states. The ordinary D-instanton only couples with the dilaton and the axion states.

But since a D-instanton is a half-BPS state and breaks one half of the supersymmetries,

we can obtain other states by acting broken supersymmetry generators on the ordinary

D-instanton state. They couple to other supergravity fields through derivative couplings

and form a supersymmetry multiplet in type IIB supergravity. We showed that the wave

25

Page 28

functions are nothing but the overlaps of these D-instanton boundary states with massless

closed string states.

It is interesting to investigate effective actions under condensation of every massless

closed string mode systematically, besides the charge conjugation of the antisymmetric

tensor field and graviton we studied in this paper. Though it is expected from the analysis

of the string theory side that each mode couples to the vertex operator of the N D-instanton

system through an appropriate derivative coupling, other types of couplings like the quintic

term derived here can also appear. We think that such studies clarify how the IIB matrix

model contains dynamics of closed strings.

Acknowledgements

We would like to thank H. Aoki, K. Hamada, Y. Kimura, Y. Kitazawa, J. Nishimura,

T. Suyama, D. Tomino, A. Tsuchiya and K. Yoshida for useful discussions. We would

especially thank K. Yoshida for discussions on section 3.

Appendix

In this appendix, we briefly review the boundary states in the Green-Schwarz formalism

of type IIB superstrings in the light-cone gauge.

We first summarize our notations:

• Space-time quantities (in (9 + 1)-dimensions)

Metric:

ηµν= diag(−1,+1,··· ,+1),(A.1)

Gamma matrices (in Majorana representation):

{Γµ,Γν} = −2ηµν,

Γ0= σ2⊗ 116,

Γi= iσ1⊗ γi, (i = 1,2,··· ,8)

Γ9= iσ3⊗ 116,

Γ11= Γ0Γ1···Γ9= −σ1⊗

?

γi

γi

(A.2)

(A.3)

(A.4)

(A.5)

?18

a˙ a= γi

0

0

−18

?

, (A.6)

γi=

0γi

a˙ a

0γi

˙ aa

?

,γi

˙ aa,(A.7)

a˙ aγj

a˙ aγi

˙ ab+ γj

b˙b+ γi

a˙ aγi

b˙ aγi

˙ ab= 2δijδab,

a˙b= 2δabδ˙ a˙b,

(A.8)

(A.9)

Spinors:

θ =?θa

1,θ˙ a

1,θa

2,θ˙ a

2

?T,(A.10)

26

Page 29

Weyl spinors:

Γ11θ = θ =⇒ θ =?θa,θ˙ a,−θa,θ˙ a?T, (A.11)

Γ11θ = −θ =⇒ θ =?θa,θ˙ a,θa,−θ˙ a?T, (A.12)

• World-sheet quantities

Metric:

ηαβ= diag(−1,+1)(A.13)

Gamma matrices:

{ρα,ρβ} = −2ηαβ,

ρ0= σ2,

(A.14)

(A.15)ρ1= iσ1,

Antisymmetric tensor ǫαβ:

ǫ01= +1,(A.16)

In the Green-Schwarz formalism, the IIB superstring theory is described by ten real

bosons Xµ(µ = 0,1,··· ,9) and two Majorana-Weyl fermions θA(A = 1,2) with the same

chirality Γ11θA= −θA. Here we take the light-cone gauge;

Γ+θA

= 0

−→

X+

= x++ p+τ.

θA=?θAa,0,θAa,0?, (A.17)

(A.18)

The light-cone components are defined as

Γ±

=

1

√2

1

√2

?Γ0± Γ9?,

?X0± X9?.

(A.19)

X±

=

(A.20)

The explicit forms of Γ±are

Γ+

=

i

√2

i

√2

?

?

1 −1

1 −1

−1 −1

1

?

⊗ 116,

?

(A.21)

Γ−

=

1

⊗ 116.(A.22)

The world-sheet action in the light-cone gauge is given by

Sl.c. = −1

= −1

4π

?

?

d2σ?∂αXi∂αXi− i¯Saρα∂αSa?

d2σ

?

4π

−?∂τXi?2+?∂σXi?2− iS1a(∂τ+ ∂σ)S1a− iS2a(∂τ− ∂σ)S2a?

,

(A.23)

27

Page 30

where SAaare proportional to θAa; SAa∝√p+θAa. The coordinates are expanded with

respect to the Fourier modes as

Xi

= xi+ piτ +

i

√2

?

n?=0

1

n

?αi

ne−in(τ−σ)+ ˜ αi

ne−in(τ+σ)?,(A.24)

S1a

=

?

?

n

Sa

ne−in(τ−σ),(A.25)

S2a

=

n

˜Sa

ne−in(τ+σ).(A.26)

Under the quantization, the mode operators satisfy the hermiticity conditions

α†

n= α−n,˜ α†

n= ˜ α−n,(Sa

n)†= Sa

−n,

?˜Sa

n

?†

=˜Sa

−n. (A.27)

Also the commutation relations among them are given by

[xi,pj] = iδij,

{Sa

[αi

m,αj

n] = mδijδm+n,0,

{˜Sa

[˜ αi

m, ˜ αj

n] = mδijδm+n,0,(A.28)

(A.29)

m,Sb

n} = δabδm+n,0,

m,˜Sb

n} = δabδm+n,0.

The action (A.23) has the N = 2 supersymmetry consisting of the kinematical SUSY

δSAa

=

?

= 0,

2p+ǫAa,(A.30)

(A.31)δXi

and the dynamical SUSY

δS1a

=

1

√p+(∂τ− ∂σ)Xiγi

1

√p+(∂τ+ ∂σ)Xiγi

i

√p+ǫA˙ aγi

a˙ aǫ1˙ a,(A.32)

δS2a

=

a˙ aǫ2˙ a,(A.33)

δXi

= −

˙ aaSAa. (A.34)

These transformations are generated by the following supercharges

Q1a

=

?2π

?2π

?2π

?2π

0

dσ

2π

dσ

2π

?

?

√p+(∂τ− ∂σ)Xiγi

2p+S1a=

?

?

2p+Sa

0,(A.35)

Q2a

=

0

2p+S2a=2p+ ˜Sa

0,(A.36)

Q1˙ a

=

0

dσ

2π

1

˙ aaS1a=

1

√p+γi

˙ aa

?

?

piSa

0+

√2

?

?

n?=0

αi

nSa

−n

?

?

,(A.37)

Q2˙ a

=

0

dσ

2π

1

√p+(∂τ+ ∂σ)Xiγi

˙ aaS2a=

1

√p+γi

˙ aa

pi˜Sa

0+

√2

n?=0

˜ αi

n˜Sa

−n

,(A.38)

28

Page 31

which satisfy the algebra

{QAa,QBb} = 2p+δABδab,

{Q1˙ a,Q1˙b} = 2P−δ˙ a˙b,

{Q2˙ a,Q2˙b} = 2˜P−δ˙ a˙b,

{QAa,QB˙ a} =

(A.39)

(A.40)

(A.41)

√2γi

a˙ apiδAB,(A.42)

with

P−

=

1

p+

?

?

pipi

2

+

?

?

n?=0

?nSa

?

−nSa

n+ αi

−nαi

n

?

??

?

, (A.43)

˜P−

=

1

p+

pipi

2

+

n?=0

n˜Sa

−n˜Sa

n+ ˜ αi

−n˜ αi

n

. (A.44)

A boundary state is usually defined by a set of the boundary conditions on a constant

τ surface:

?(∂τ− ∂σ)Xi− Mij(∂τ+ ∂σ)Xj?|B,η? = 0,

Q+˙ a

?

(A.45)

Q+a

η|B,η? ≡?Q1a+ iηMabQ2b?|B,η? = 0,

Q1˙ a+ iηM˙ a˙bQ2˙b?

(A.46)

η|B,η? ≡|B,η? = 0,(A.47)

where η is a parameter (η2= 1), and Mij is an element of SO(8). For the Neumann

directions Mij= −δijand for the Dirichlet directions Mij= δij. Maband M˙ a˙bare deter-

mined by consistency requirements as follows. Taking the surface τ = 0, the conditions

(A.45)-(A.47) are written in terms of the mode operators as

?pi− Mijpj?|B,η? = 0,

?αi

?

?

(A.48)

n− Mij˜ αj

Sa

−n

?|B,η? = 0,

0

|B,η? = 0,

(A.49)

0+ iηMab˜Sb

?

(A.50)

γi

˙ aapiSa

0+ iηM˙ a˙bγi

˙ba˜Sa

0+

√2

?

n?=0

?

γi

˙ aaαi

nSa

−n+ iηM˙ a˙bγi

˙ba˜ αi

n˜Sa

−n

??

|B,η? = 0.

(A.51)

Let us determine Maband M˙ a˙b. From {Q+a,Q+b}|B,η? = 0, we find

MacMbc= δab, (A.52)

meaning that Mabis an orthogonal matrix. Next, {Q+a,Q+˙ a}|B,η? = 0 leads to

?γi

a˙ api− MabM˙ a˙bγi

˙bbpi?|B,η? = 0.(A.53)

29

Page 32

Comparing this with (A.48), we have

γi

a˙ aMij− MabM˙ a˙bγj

˙bb= 0.(A.54)

The consistency between eq.(A.45) and eq.(A.47) requires

?

γi

˙ aaSa

n+ iηMijM˙ a˙bγj

˙bb˜Sb

−n

?

|B,η? = 0for n ?= 0,(A.55)

by using (A.54), which are rewritten as

γi

˙ aa

?

Sa

n+ iηMab˜Sb

−n

?

|B,η? = 0for n ?= 0.(A.56)

Since Mijis an element of SO(8), it can be written as Mij=

?

eΩklΣkl?

ijwith?Σkl?

ij=

δk

iδl

j− δl

iδk

jbeing generators of SO(8). Eq.(A.54) can be solved in terms of Ωijas

Mab =

?

?

e

1

2Ωijγij?

1

ab,(A.57)

M˙ a˙b

=e

2Ωij˜ γij?

˙ a˙b=1

˙ a˙b,(A.58)

where

γij

ab=1

2

?γi

a˙ aγj

˙ ab− γj

a˙ aγi

˙ ab

?,˜ γij

2

?

γi

˙ aaγj

a˙b− γj

˙ aaγi

a˙b

?

. (A.59)

The boundary state |B,η? can be expressed in the form

|B,η? = e

with the zero-mode part

?

n>0(1

nMijαi

−n˜ αj

−n−iηMabSa

−n˜Sb

−n)|B0,η?

(A.60)

|B0,η? = C

?

Mij|i?|j? − iηM˙ a˙b|˙ a?|˙b?

?

.(A.61)

C is a normalization constant, and the ground states |i? and |˙ a? are defined by

αj

0|i? =γi

n|i? = Sa

Sa

n|i? = αi

a˙ a

√2|˙ a?,

n|˙ a? = Sa

Sa

n|˙ a? = 0,

a˙ a

√2|i?.

(for n > 0),(A.62)

0|˙ a? =γi

(A.63)

Broken supercharges are given by

Q−a

η

Q−˙ a

η

≡ Q1a− iηMabQ2b,

≡ Q1˙ a− iηM˙ a˙bQ2˙b,

(A.64)

(A.65)

and the algebra of broken and unbroken supercharges becomes

{Q+a

{Q+a

{Q+˙ a

{Q+˙ a

η,Q−b

η,Q−˙b

η} = 4p+δab,

η} =

η,Q−b

η,Q−˙b

(A.66)

√2γi

√2γi

a˙b

?pi+ Mijpj?,

?pi+ Mijpj?,

(A.67)

η} =

η} = 2(P−+˜P−)δ˙ a˙b= 2P−

˙ ab

(A.68)

clδ˙ a˙b,(A.69)

and the other anticommutators vanish.

30

Page 33

References

[1] S. Iso, H. Terachi and H. Umetsu, “Wilson loops and vertex operators in matrix

model,” Phys. Rev. D 70, 125005 (2004) [arXiv:hep-th/0410182].

[2] N. Ishibashi, H. Kawai, Y. Kitazawa and A. Tsuchiya, “A large-N reduced model as

superstring,” Nucl. Phys. B 498, 467 (1997) [arXiv:hep-th/9612115].

[3] M. Fukuma, H. Kawai, Y. Kitazawa and A. Tsuchiya, “String field theory from IIB

matrix model,” Nucl. Phys. B 510, 158 (1998) [arXiv:hep-th/9705128].

[4] H. Aoki, S. Iso, H. Kawai, Y. Kitazawa and T. Tada, “Space-time structures from IIB

matrix model,” Prog. Theor. Phys. 99, 713 (1998) [arXiv:hep-th/9802085].

[5] J. Nishimura and F. Sugino, “Dynamical generation of four-dimensional space-time in

the IIB matrix model,” JHEP 0205, 001 (2002) [arXiv:hep-th/0111102].

[6] H. Kawai, S. Kawamoto, T. Kuroki, T. Matsuo and S. Shinohara, “Mean field approx-

imation of IIB matrix model and emergence of four dimensional space-time,” Nucl.

Phys. B 647, 153 (2002) [arXiv:hep-th/0204240].

[7] H. Kawai, S. Kawamoto, T. Kuroki and S. Shinohara, “Improved perturbation theory

and four-dimensional space-time in IIB matrix model,” Prog. Theor. Phys. 109, 115

(2003) [arXiv:hep-th/0211272].

[8] E. Witten, “Bound states of strings and p-branes,” Nucl. Phys. B 460, 335 (1996)

[arXiv:hep-th/9510135].

[9] T. Yoneya, “Supergravity, AdS/CFT correspondence, and matrix models,” Prog.

Theor. Phys. Suppl. 134, 182 (1999) [arXiv:hep-th/9902200].

[10] R. C. Myers, “Dielectric-branes,” JHEP 9912, 022 (1999) [arXiv:hep-th/9910053].

[11] K. J. Hamada, “Supersymmetric Wilson loops in IIB matrix model,” Phys. Rev. D

56, 7503 (1997) [arXiv:hep-th/9706187].

[12] J. H. Schwarz and P. C. West, “Symmetries And Transformations Of Chiral N=2 D

= 10 Supergravity,” Phys. Lett. B 126, 301 (1983).

[13] Y. Kitazawa, “Vertex operators in IIB matrix model,” JHEP 0204, 004 (2002)

[arXiv:hep-th/0201218].

[14] W. I. Taylor and M. Van Raamsdonk, “Supergravity currents and linearized interac-

tions for matrix theory configurations with fermionic backgrounds,” JHEP 9904, 013

(1999) [arXiv:hep-th/9812239].

[15] A. Dasgupta, H. Nicolai and J. Plefka, “Vertex operators for the supermembrane,”

JHEP 0005, 007 (2000) [arXiv:hep-th/0003280].

31

Page 34

[16] M. B. Green, “Point - like states for type 2b superstrings,” Phys. Lett. B 329, 435

(1994) [arXiv:hep-th/9403040].

[17] M. B. Green and M. Gutperle, “Light-cone supersymmetry and D-branes,” Nucl. Phys.

B 476, 484 (1996) [arXiv:hep-th/9604091].

[18] M. B. Green and M. Gutperle, “Effects of D-instantons,” Nucl. Phys. B 498, 195

(1997) [arXiv:hep-th/9701093].

[19] J. F. Morales, C. Scrucca and M. Serone, “A note on supersymmetric D-brane dy-

namics,” Phys. Lett. B 417, 233 (1998) [arXiv:hep-th/9709063].

[20] M. J. Duff, J. T. Liu and J. Rahmfeld, “g = 1 for Dirichlet 0-branes,” Nucl. Phys. B

524, 129 (1998) [arXiv:hep-th/9801072].

[21] J. F. Morales, C. A. Scrucca and M. Serone, “Scale independent spin effects in D-brane

dynamics,” Nucl. Phys. B 534, 223 (1998) [arXiv:hep-th/9801183].

[22] W. I. Taylor and M. Van Raamsdonk, “Multiple D0-branes in weakly curved back-

grounds,” Nucl. Phys. B 558, 63 (1999) [arXiv:hep-th/9904095].

[23] K. Millar, W. Taylor and M. Van Raamsdonk, “D-particle polarizations with multipole

moments of higher-dimensional branes,” arXiv:hep-th/0007157.

[24] T. Suyama and A. Tsuchiya (unpublished), talk at JPS meeting 1999 and private

communication

[25] Y. Kimura and Y. Kitazawa, “Supercurrent interactions in noncommutative Yang-

Mills and IIB matrix model,” Nucl. Phys. B 598, 73 (2001) [arXiv:hep-th/0011038].

[26] S. Iso, Y. Kimura, K. Tanaka and K. Wakatsuki, “Noncommutative gauge the-

ory on fuzzy sphere from matrix model,” Nucl. Phys. B 604, 121 (2001)

[arXiv:hep-th/0101102].

32