## 國立臺灣大學理學院物理學系(所) 博士論文

### Department or Graduate Institute of Physics College of Science

### National Taiwan University Doctoral Dissertation

## R-R背景場下的D膜理論 D-brane in R-R field background

## 葉啟賢 Chi-Hsien Yeh

## 指導教授: 賀培銘 教授 Advisor: Professor Pei-Ming Ho

## 中華民國 101 年 1 月 January 2012

國立臺灣大學

物理學系

( 所 ) 博士論文

### R-R 背景場下的 D 膜理論 葉啟賢 撰

**101**

**1**

i

### Acknowledgment

I want to thank many people to help finish this thesis. Firstly, I want to thank my parents and my little brother. They give me many supports of spirit and life . Secondly, I need to thank my advisor professor Pei-Ming Ho. He always gives me kind suggestions and tells me how to correct my faults. Professor Ho lets me know how to become a theoretical physicist. Thirdly, I want to thank Hirotaka Irie, Chuan-Tsung Chan and Sheng-Yu Darren Shih. Doctor Irie kindly teaches me the main knowledge of matrix model theory.

Professor Chuan-Tsung Chan tells me the important things for doing research. Sheng-Yu Darren Shih teaches me much knowledge and many ideas in physics. We always discuss the matrix model together even on holidays. Finally, I want to thank many people who give me many helps during these years: Wen-Yu Wen, Hsien-Chung Kao, Jiunn-Wei Chen, Kazuyuki Furuuchi, Dan Tomino, Shoichi Kawamoto, Hiroshi Isono, Tomohisa Takimi, Shou-Huang Dai, Chen-Pin Yeh, Chien-Ho Chen, Xue-Yan Lin, Sheng-Lan Ko, Wei-Ming Chen, Kuo-Wei Huang, Yen-Ta Huang, and Chen-Te Ma.

### 論文摘要:

本篇論文主要研究在 Ramond-Ramond（R-R）背景場下 D 膜(D-brane)的有效場論。

由於 R-R 背景場的存在，這樣的理論在背景場的方向會有體積保持不變的對稱性

（volume-preserving diffeomorphism），這是這理論的主要特徵之一。之所以 會研究這樣的理論，起源於最近有關 M5 膜在大 C 背景場的有效場論的研究。高 一維度的理論可以透過丟掉場在這一維度的自由度，來得到低一維度在低能量極 限的有效場論。因此這樣的分析方法常常會有一些多餘的場殘留在低一維度的理 論中。要如何分辨哪些場是理論所必須的，而哪些場又是可以被積掉的，是這研 究的核心部分。在這篇論文中，我們發現原先所預期出現的規範場被隱藏在某些 場內，我們使用了對偶變換的方法來使這樣的規範場在理論中變的明顯。接著我 們討論了在這樣的變換下，要如何求出規範場的規範對稱變換以及超對稱變換。

我 們 研 究 了 在 規 範 對 稱 性 以 及 體 積 保 持 不 變 性 之 下 的 協 變 量 （ covariant variables）應是什麼樣子的，並利用它們來使理論易於推廣到不同的情形。最 後我們利用這理論所具有的超對稱去討論這理論的拓樸性質，即理論所允許的孤 立子解。

Abstract

In this paper, we try to understand the low energy effective theory of Dp-brane in large R-R (p-1)-form field background. To construct the effective theory, we start with the M5-brane theory in large C-field background [1, 2]. The C-field background defines the 3-dimensional volume form in M5-brane theory. Hence, the M5-brane theory can be described as a Nambu-Poisson-bracket gauge theory with volume-preserving diffeomor- phism symmetry (VPD). After doing double dimensional reduction, we obtain the effec- tive theory of D4-brane in large C-field background [3]. This theory has both the usual U(1) gauge symmetry and the new symmetry VPD. The VPD two-form gauge potential can be understood as the electric-magnetic dual of the one-form gauge field in the D4- brane theory. This theory is described by the one-form gauge field and the dual two-form gauge field at the same time. These results can be generalized to Dp-branes cases. In the last part of thesis, we study the supersymmetry (SUSY) algebra in this theory. We can calculate the central charges from the SUSY algebra in this theory, then we can know the possible topological quantities in this system. This interesting system may help us to understand M-theory, the models with volume-preserving diffeomorphism, the suit- able low energy description of Dp-branes in different field backgrounds, some new soliton solutions, and so on.

## Table of Contents

1 Introduction 1

1.1 Dp-Branes with Different Field Backgrounds . . . 2

1.1.1 Terminology Explanation . . . 2

1.1.2 Dirac-Born-Infeld Action and Yang-Mills Gauge Theory . . . 3

1.1.3 Dp-Branes with NS-NS and R-R Fields . . . 4

1.2 Large Field Background Effects . . . 5

1.2.1 Dp-Branes in constant NS-NS B-field Background . . . 6

1.2.2 Volume-Preserving Diffeomorphism and Nambu-Poisson Bracket . 7 1.3 A Review of M Theory . . . 8

2 M5 in Large C-Field Background 10 2.1 Nambu-Poisson M5 Theory . . . 10

2.2 Action of Nambu-Poisson M5 Theory . . . 11

2.3 Symmetry of Nambu-Poisson M5 Theory . . . 13

2.3.1 Gauge Symmetry and VPD . . . 13

2.3.2 Supersymmetry . . . 14

2.4 Double Dimensional Reduction . . . 15

2.4.1 Poisson D4 Description From Nambu-Poisson M5 Theory . . . 15

3 D4 in R-R Three Form Background 17 3.1 D4-Brane in C Field Background via DDR . . . 17

3.1.1 Gauge Transformation of Fields . . . 18

3.1.2 Action . . . 18

3.2 Dual Transformation . . . 19

3.2.1 Equivalent Dual Action and Dual One Form Field . . . 20

3.2.2 Action after Dual Transformation . . . 21

3.3 Covariant Variables . . . 22

3.3.1 Gauge Symmetry after Dual Transformation . . . 22

3.3.2 Covariant Variable with U(1) and VPD Symmetry . . . 23

3.3.3 Action with Covariant Variables . . . 24

3.4 Order Expansion Analysis . . . 24

3.4.1 Zeroth Order Expansion . . . 24

3.4.2 First Order Expansion . . . 26

3.4.3 Electric-Magnetic (EM) duality . . . 27

4 Extension and Application 29 4.1 Dp-Branes in R-R field Background . . . 29

4.1.1 Generalize VPD in R-R (p-1) Form Field Background . . . 30

4.1.2 Gauge Symmetry and Covariant Variables in Multiple Dp-Branes Theory . . . 30

4.1.3 Ansatz of Action . . . 31

4.2 Couple to Matter fields . . . 32

4.2.1 D4 in C Field Background with Matter Fields . . . 33

4.2.2 Order Expansion Analysis . . . 34

4.2.3 Rewrite Action with Covariant Variables . . . 35

4.3 Supersymmetry Transformation . . . 37

4.3.1 Supersymmetry Law of Dual Field . . . 37

4.3.2 Non-linear Fermion Symmetry of Dual Field . . . 38

4.3.3 Linear Supersymmetry Transformation . . . 38

4.3.4 Supersymmetry Transformation Law of ˆBα˙µ Field . . . 39

4.4 Topological Quantities of D4 in Large C Field Background . . . 40

4.4.1 Central Charges of Superalgebra . . . 40

4.4.2 Instanton Solutions . . . 41

5 Conclusion and Discussion 42 5.1 Summary . . . 42

5.2 Discussion . . . 43

A Conventions and Notations 44 B Some Useful Identities 46 C Suitable Scaling Limit in Different Cases 48 C.1 Scaling Limit of Dp-brane in B-field Background . . . 48

C.2 Scaling Limit of M5 in Large C-field Background . . . 49 C.3 Scaling Limit of D4 in Large C-field Background . . . 50

References 51

## Chapter 1 Introduction

In this chapter, we will review several relevant elements of effective theory in a certain background. First of all, we will talk about the effective action of D-brane theory. To understand the string theory, we can start with the calculation of perturbative string scattering amplitudes. On the other hand, the nonperturbative effect of string theory is described by soliton solutions in ten-dimensional supergravity theory. These solitons are the Dp-branes, which are the extended object with p-spatial dimensions. The open string ends on these Dp-branes, hence the low energy effective field theory of Dp-brane can be obtained from the calculation of the open string scattering amplitudes. The Dp-branes theories have two main descriptions. One is the Dirac-Born-Infeld action [4]. Another is the Yang-Mills gauge theory [5]. They share part of the original brane theory in different limits, which are the slowly varying limit or zero slope limit of string theory. There are several good reviews of D-brane theory. For example, the review articles [6–8] are useful.

On the other hand, we want to introduce the well-known case of Dp-branes in constant NS-NS field background [9–12]. We will show noncommutative Yang-Mills theory as the effective field theory of the Dp-branes theory in the low energy limit. The first- order expansion of noncommutative algebra is described by Poisson bracket, which is the generator of Area-Preserving Diffeomorphism (APD). The noncommutative effect depends on the inverse of NS-NS B-field. Hence, the field theory of Poisson-bracket is relevant to Dp-branes in large NS-NS B-field background.

When we want to study the effective field theory in large n-form field background, we can focus on the symmetry in this theory. While the n-form field defines the n-dimensional volume form in the theory, we expect that the effective theory may have n-dimensional volume-preserving diffeomorphism (VPD) symmetry. We will give more detail description of the VPD symmetry, where the symmetry generator is Nambu-Poisson bracket.

In the last part of this chapter, we will review M theory, where similar phenomena can be found. The M5-brane worldvolume theory has their own action with C field, which is called PST action [25]. Recently, people [1, 2] found another action for M5 in large C field background. It is similar to the story of Dp-branes in the NS-NS B-field background. We will give more details of this theory in next chapter.

### 1.1 Dp-Branes with Different Field Backgrounds

The low energy effective theories of Dp-branes are called Dirac-Born-Infeld (DBI) action.

People also study the modification of DBI action in NS-NS and R-R field backgrounds.

For the theory to be gauge invariant and anomaly free, we need to replace U(1) field strength F by B + F and add Wess-Zumino terms into the original DBI action. In fact, the NS-NS and R-R fields are the massless mode of the close string spectrum. They are the background fields of open string scattering amplitudes just like the gravitational background. When we use the open string scattering amplitudes to study the effective theories of D-brane in NS-NS and R-R field backgrounds, we may have different inter- pretations for these background fields. For example, the NS-NS background fields can be absorbed into the field strength of open-string oscillation mode or the open-string metric, then we get different effective theories of D-brane. For these effective theories, we call all of them to be D-brane in field backgrounds. However, this terminology is confusing in this thesis. The effective theory, which we want to talk in this thesis, is the effective theory without manifest background fields. In this case, the effects of background fields hide in the geometry and the symmetry algebra of effective theory. We try to distinguish them in next subsection.

### 1.1.1 Terminology Explanation

When we want to talk about a theory in some field backgrounds, we need to know what it really means.

Firstly, the meaning of background field is that we neglect the dynamic behavior of this background field. The simple case is that we study the matter field in electric-magnetic fields background, in this case, we neglect the dynamic contribution of EM gauge fields.

So the background fields what we means are constant fields. The terminology “theory in constant field background” is the same as the terminology “theory in field background”.

Secondly, when we talk about the effective field theory in field background, the ef- fective theory usually does not include the manifest background fields dependence. The

effect of background field can appear in effective mass, effective coupling or new geometry.

Hence we will have a new field theory, then we can study the equivalent phenomena in the two theories. We do not have a simple example in field theory. However, the phenomena appear frequently in string theory. For example, the effective description of Dp-brane is not unique, we have more than one effective description. The first example is the com- mutative and noncommutative gauge theory for Dp-brane in NS-NS B field background.

People [12] understand this phenomena as the result of different regularization method of open string scattering amplitude analysis. The effective field theory will be different in the different regularization method, they can be related by changing variables. In this case, this change of variables is called the Seiberg-Witten map [12]. However, the two different effective field theories are not really the same after Seiberg-Witten map, they are different by higher derivative terms and total derivative terms. Hence, they stand for the different parts of the full D-brane theory, while they can have overlap in the scal- ing limit (Appendix C). Hence, in order to distinguish these two situations from other cases, we use the terminology of Dp-brane “with” NS-NS and R-R fields for originally well known DBI action. We use the terminology of Dp-brane “in” NS-NS and R-R fields

“background” for the case what we want to talk in this thesis. The effective field theory

“in” fields background does not have manifest background fields dependence.

Finally, we study the theory in large field background in the most part of this thesis.

In this limit, the effective field theory becomes simpler and easier to analyze.

### 1.1.2 Dirac-Born-Infeld Action and Yang-Mills Gauge Theory

In this subsection, we want to write down the explicit action form of effective field theory
of D-brane. It is called the Dirac-Born-Infeld(DBI) action [4]. Roughly speaking, the
DBI action comes from the calculation of open string scattering amplitude. When we
calculate the β-function of open string scattering amplitude, because the theory has
conformal invariance, the β-function must vanish. From these constraints, we can find
the constraints of fields. These fields are the oscillation mode of open string. These
constraints of fields can be understood as the equations of motion which are derived
from corresponding effective field theory action. The effective action (DBI action) is
described by p+1 coordinates ξ^{a}, a = 0, 1, . . . , p. The DBI action is written as^{1} [4]:

SDBI = Tp

Z

d^{p+1}ξpdet(Gab+ 2πα^{′}Fab), (1.1)

1In this chapter, we use the review paper of Dp-brane [6]

here T_{p} is defined by ^{1}

(2π)^{p}gsℓ^{p+1}s , which is the tension of Dp-brane. It is the generalization
of the string tension T_{F 1} = _{2πα}^{1} _{′}. The p labels the number of spatial dimensions for
Dp-brane. The g_{s} is string coupling and ℓ_{s} =√

α^{′} is identified as string length. The G_{ab}
is the induced metric in Dp-brane, it is usually complex in the fermionic part. Here, we
give the bosonic part of the induce metric:

G_{ab}= η_{M N}∂_{a}X^{M}∂_{b}X^{N}, (1.2)
where M is from 0 to p. We can choose gauge to let X^{a} = ξ^{a}. So, the remaining scalars
in DBI action are the transverse coordinates in target spacetime, and we label them with
2πα^{′}X^{I} I = p + 1, . . . , 9. Here, we use the factor 2πα^{′} to make the mass dimension of
X^{I} equal to one. Hence, we can rewrite action as:

SDBI = Tp

Z

d^{p+1}ξpdet(ηab+ 2πα^{′}∂aX^{I}∂^{a}X^{I} + 2πα^{′}Fab). (1.3)
The F is the field strength of one form gauge potential A, that is F = dA in Maxwell
theory. We can regard the DBI action as the high energy version of Maxwell action. To
take the low energy limit α^{′} → 0 and omit the scalar terms, we can get:

S_{DBI} = T_{p}
Z

d^{p+1}ξpdet ηab(1−1

4F^{ab}F_{ab}+ O(α^{′})). (1.4)
The low energy limit makes the D-brane theory to become simpler.

### 1.1.3 Dp-Branes with NS-NS and R-R Fields

The dynamics of Dp-Brane will be affected by background fields, which come from the
closed string NS-NS and R-R sector. In NS-NS sector, we have graviton gM N which is
symmetry rank-2 field, and NS-NS B-field 2πα^{′}BM N which is antisymmetry two-form
field. We also have dilaton field Φ, which is a scalar. All of them will modify the form of
DBI action. For simplicity, here we only consider the effect of NS-NS B-field. The action
of Dp-brane in NS-NS B field background can be written as:

SDBI = Tp

Z

d^{p+1}ξpdet(ηab+ 2πα^{′}∂aX^{I}∂^{a}X^{I} + 2πα^{′}(Fab+ Bab)), (1.5)
which can be realized by modification of Gab, the induce metric, in following way:

Gab = (ηM N + 2πα^{′}BM N)∂aX^{M}∂bX^{N}, (1.6)

the mixed terms of B and X will vanish for the antisymmetry of B field. The action form can have the gauge symmetry of two form field B with additional shift of one form field A:

B → B + dΛ, A→ A − Λ, (1.7)

such that B + F term do not transform.

The R-R sectors of close string are some higher ranks form. For example, the Dp-
brane can have R-R (p+1)-form,(p-1)-form,. . .,1-form (or 0-form for odd p), we label
them by C_{p+1}, C_{p−1}, . . . , C_{1} (or C_{0} for odd p).

The action of Dp-brane in R-R field background can be written as [13, 14]:

SDBI = Tp

Z

d^{p+1}ξpdet(Gab+ 2πα^{′}Fab) + SW Z, (1.8)
here the new term is written by:

SW Z = µp

Z

(Ce^{2πα}^{′}^{F})p+1, C ≡

8

X

n=0

Cn. (1.9)

The notation (· · · )^{p+1} is to keep the p+1 form inside the parentheses. The µp is the
electric charge of Dp-brane. In fact, the calculation of open-string scattering amplitude
in R-R background is very difficult. People do not know how to quantize the nonlinear
sigma model in curved spacetime. However, we can know the field contents, the gauge
symmetry, and the supersymmetry from the flat space calculation. Hence, we can use
these informations to analyze the effective worldvolume theory of Dp-brane with R-R
fields. For example, the Wess-Zumino term (SW Z) is introduced to cancel the gauge
anomaly in superstring theory.

While the DBI-like action of multiple Dp-branes is incomplete and unclear (the rel- evant papers [15, 16]), we can still use non-abelian Yang-Mills action to describe them.

Yang-Mills action is the leading term of multiple Dp-branes action after taking zero slope
limit (α^{′} → 0).

### 1.2 Large Field Background Effects

From effective theory viewpoint, the high derivative terms can be omitted in the low energy limit. However, when the system is embedded in large field background, this approximation is not true. The large field background can couple to these high derivative terms, which are still leading in low energy limit. Large field background will be have

The well-known example is the Dp-branes in constant NS-NS B field background. In this case, the effective field theory is not conventional Yang-Mills field theory, we should use the noncommutative Yang-Mills field theory to suitably describe the effective field theory of Dp-branes in constant NS-NS B field background [9–12]. The noncommutative field theory is a better description of D-brane in NS-NS B field background than orig- inal DBI action or Yang-Mills field theory. The reason is the noncommutative theory includes nonlocal behavior, which encodes the information of the higher derivative terms in original theory. As what we mentioned before, the large NS-NS B field coupled to higher derivative terms will remain after taking low energy limit, and noncommutativity emerges.

### 1.2.1 Dp-Branes in constant NS-NS B-field Background

When we calculate the scattering amplitudes of open string in constant NS-NS B field background, we use another regularization processes called point splitting regularization.

The different regularization methods will modify the forms of β-function. After imposing the vanishing β-function, we get the effective field theory of Dp-branes in constant NS- NS B field background. After taking scaling limit, we get the similar Yang-Mills type effective action. For example, the leading terms of three point open string scattering amplitude can be effectively obtained from the action [12]:

(α^{′})^{3−p}^{2}
4(2π)^{p−2}Gs

Z √

detGG^{ab}G^{cd}Fˆ_{ac}∗ ˆF_{bd}. (1.10)
It is called noncommutative Yang-Mills field theory. The noncommutativity is defined
by the Moyal product “∗”, such that

f (x)∗ g(x) = e^{2}^{i}^{θ}^{ab}^{∂ξa}^{∂} ^{∂ζb}^{∂} f (x + ξ)g(x + ζ)

ξ=ζ=0. (1.11)

The field strength ˆF is defined by gauge potential ˆa and Moyal product:

Fˆab = ∂aˆab− ∂^{b}ˆaa− iˆa^{a}∗ ˆa^{b} + iˆab ∗ ˆa^{a}, (1.12)
while the gauge symmetry is :

δλˆaa = ∂aλ + iλ∗ ˆa^{a}− ia^{a}∗ λ. (1.13)

When the background B field is large, the noncommutative factor θ becomes small. We can expand the Moyal product to the first order. Hence we will get (in U(1) case) [12]:

f (x)∗ g(x) = fg + i

2θ^{ab}∂af ∂bg + O(θ^{2}), (1.14)
Fˆab = ∂aˆab− ∂bˆaa+ θ^{cd}∂cˆaa∂daˆb+ O(θ^{2}), (1.15)
δλˆaa = ∂aλ− θ^{cd}∂cλ∂dˆaa+ O(θ^{2}). (1.16)
In this case, the main characteristic in large B field is appearance of the Poisson bracket
structure:

{f, g}^{pb}= ǫ^{ab}∂af ∂bg. (1.17)
We will discuss it more in next subsection.

### 1.2.2 Volume-Preserving Diffeomorphism and Nambu-Poisson Bracket

From previous subsection, we can understand the noncommutative gauge fields theory can be described by Moyal product. If we focus on the large NS-NS B field case, theory is handled by Poisson bracket, which is the generator of Area-Preserving Diffeomorphism (APD). In general. for higher ranks field background, they need a general Poisson bracket, which is called Nambu-Poisson bracket [17–21], to be the generator of Volume-Preserving Diffeomorphism (VPD).

To understand the reason why VPD emerges, we can think in following way. The original worldvolume theory has diffeomorphism symmetry:

x^{a}→ ´x^{a} = ´x^{a}(x^{a}). (1.18)

When we consider the theory in large field background, the original diffeomorphism sym- metry will be broken by background field, the remaining symmetry is volume-preserving diffeomorphism. The n-dimensional volume-preserving diffeomorphism is the reduced symmetry of n-dimensional general coordinate diffeomorphism, which is described by (infinitesimal transformation):

x^{a}→ ´x^{a} = ´x^{a}(x^{a}) = x^{a}+ κ^{a}, ∂aκ^{a}= 0. (1.19)
where a = 0, 1, ..., n − 1. The n-dimensional volume-preserving diffeomorphism can be
understood as that this transformation parameter κ^{a} has additional constraint as shown

in (1.19). To see how this constraint gives rise to the volume-preserving, we can investi- gate the Jacobian of coordinate transformation. For example, we can find the Jacobian of coordinate transformation for n=2 reads

ǫ^{ab}∂_{a}x´^{0}∂_{b}x´^{1} ={´x^{0}, ´x^{1}}. (1.20)
We consider the coordinate transformation in (1.19), after simple calculation, we can
find:

{´x^{0}, ´x^{1}} = 1 + ∂aκ^{a}+ O(κ^{2}). (1.21)
Therefore we can see the constraint ∂aκ^{a} = 0 makes area-preserving. Moreover, higher
ranked volume-preserving transformation can be generated by generalize Nambu-Poisson
bracket, defined by

{f^{1}, f2,· · · , f^{n}} ≡ ǫ^{a}^{1}^{a}^{2}^{···a}^{n}∂a1f1∂a2f2· · · ∂^{a}^{n}fn. (1.22)
Hence, we can define the VPD transformation as follows

δΛ1,...,Λ_{n−1}x^{a} ={Λ1, ..., Λ_{n−1}, x^{a}} = ǫ^{a}^{1}^{···a}^{n−1}^{a}∂a1Λ1· · · ∂a_{n−1}Λ_{n−1} ≡ κ^{a}. (1.23)
In the special case of APD (n=2):

δΛx^{a}={Λ, x^{a}}pb = κ^{a}. (1.24)
This is the simplest case in this kind of symmetry transformation. We can see the
Nambu-Poisson bracket is the generator of VPD.

Now we can ask the next question; what is the field theory with VPD? In fact, we already saw the example of field theory with APD in previous subsection. We can find the symmetry transformation is generated by Poisson bracket. We will see more examples in next three chapters.

### 1.3 A Review of M Theory

In order to understand more nonperturbative effect of superstring theory, people start to study the M theory. M theory is the complete picture of string theory. The five different perturbative string theories and eleven dimensional supergravity theory can be under- stood as the different descriptions of M theory. For example, M theory can be understood as strong coupling limit of type IIA superstring theory in one higher dimension. Hence, the low energy effective theory of M theory is eleven dimensional supergravity theory.

From the eleven-dimensional superalgebra analysis, there are two kind high-dimensional central charges [22]. They are carried by M2-brane and M5-brane, which are the extended objects 2-brane and 5-brane in eleven dimensions. Following the analysis of 2-brane and 5-brane soliton solutions in eleven-dimensional supergravity theory, we can know the field contents of the effective worldvolume theory of M2-brane and M5-brane. The action of effective field theory for single M2-brane and M5-brane is well known. The M2-brane ef- fective action is the generalized Nambu-Goto action. The effective action of M5-brane is more difficult because it involves the self-dual two-form gauge potential [23–28]. Recently, there are several interesting papers about self-dual gauge theory [29, 30].

The theory also have a background form fields as string does, and it is the three form field background. The M2-brane couple electrically to the 3-form field. As the research of D-brane in NS-NS B field background, people try to generalize the research into M theory.

For example, people [31] tried to study the quantization of open membrane in large C-field background, and they found similar noncommutative behavior as in the open string case because quantization processes naturally adopts Poisson structure. Moreover, people [32]

calculated the scattering amplitudes of open membrane in large C-field background which can be described by Nambu-Poisson algebra (or VPD gauge symmetry). It gives the candidate of generalization of Poisson bracket and Moyal product. To study the M theory in large C-filed background helps us understand the way to generalize Moyal product, which gives the way to quantize string. Hence, people try to apply these researches to understand how to quantize membrane. To study M5 in large C-field background has more interesting physic phenomena. This kind theory includes the self-dual two form, the non-abelian gauge algebra and new action form which is different from PST M5 action [25]. We will discuss this topic in next chapter.

## Chapter 2

## M5 in Large C-Field Background

Recently, Bagger, Lambert and Gustavsson imposed the Lie-3 algebra into Basu-Harvey BPS system to construct the theory of multiple membranes [33–36]. It is called BLG model, which describes the multiple M2-branes system. In this articles, we will not give any more detail of BLG theory. Latter, people [1, 2] started to impose the Nambu- Poisson structure into the three internal dimensions of BLG model, then they found the new description of single M5-brane theory. We denote it as Nambu-Poisson (NP) M5 theory.

### 2.1 Nambu-Poisson M5 Theory

The Nambu-Poisson algebra is an infinite dimensional Lie-3 algebra, which is used to de- scribe the algebra in BLG model. People [1,2] consider the additional three internal space dimensions (N ) with 3-dimensional volume-preserving diffeomorphism, which define the space of Nambu-Poisson bracket. Moreover, the worldvolume of multiple M2-branes (M) and the 3 internal dimensions N together can be identified as the worldvolume of M5 theory (M × N ).

These processes will divide the worldvolumes dimensions of M5 into two parts:

{x^{µ}; y^{˙µ}} = {x^{0}, x^{1}, x^{2}; y^{˙1}, y^{˙2}, y^{˙3}}. (2.1)
Here, the coordinate x^{µ} label the direction on M, which are the longitudinal directions
of M2-branes. Another coordinates y^{˙µ} label the internal directions on N , which are
the space of the volume-preserving diffeomorphism. Roughly speaking, this effective M5
description have 3-dimensional VPD, which is the main characteristic of theory in large C

field background. Hence, the Nambu-Poisson structure in M5 theory is the first evidence of M5 in large C field background.

The fields contents of NP M5 theory are self-dual two form (b_{µ ˙µ}, b_{˙µ ˙ν}), five scalar fields
X^{I}, and chiral Majorana fermion Ψ. The two form gauge fields are self-dual, so the
degree of freedom (DOF) of two form fields are ^{6}_{2} = 3 and we do not need b_{µν} in this
theory. The five scalar fields are the DOF of M5-brane on the transverse directions.

The Majorana fermion is reduction from 11 dimension which satisfies the 6-dimension
chirality condition Γ^{7}Ψ = Ψ. Hence the DOF of fermion are ^{1}_{2}2^{[}^{11}^{2}^{]} = 16. The one half
of fermion DOF^{1} (^{16}_{2} ) are equal to the bosonic DOF (3 + 5) in NP M5 theory, which is a
result of supersymmetry.

In next section, we will give the full action to describe the dynamics of these fields.

We will not give all the details about how to derive the action from the BLG theory.

The main process of action calculation is to replace the Lie-3 bracket by Nambu-Poisson
bracket: [•, •, •] → g^{2}{•, •, •}. The other details can be found in the papers [1, 2].

### 2.2 Action of Nambu-Poisson M5 Theory

In this section, we want to summarize the main result of action of NP M5 theory. The NP M5 action is the effective description of M5 in large C field background, so the description is well-defined in some suitable scaling limit. We put the discussion of suitable scaling limit in Appendix.

Following the result of the papers [1, 2], the action of NP M5 theory is written as:

S = TM 5

g^{2} (SX + SΨ+ Sgauge, ) , Sgauge = S_{H}^{2} + SCS, (2.2)

1Only one half of fermionic DOF is really equivalent to bosonic DOF, because the EOM of fermion involves first derivative.

where ^{2}
SX =

Z

d^{3}xd^{3}y

−1

2(DµX^{I})^{2}− 1

2(D˙λX^{I})^{2}

− 1
2g^{2} − g^{4}

4 {X^{˙µ}, X^{I}, X^{J}}^{2}− g^{4}

12{X^{I}, X^{J}, X^{K}}^{2}

, (2.3)

SΨ = Z

d^{3}xd^{3}y i

2ΨΓ^{µ}D^{µ}Ψ + i

2ΨΓ^{ρ}^{˙}D^{ρ}^{˙}Ψ
+ig^{2}

2 ΨΓ˙µΓ^{I}{X^{˙µ}, X^{I}, Ψ} − ig^{2}

4 ΨΓ^{IJ}Γ_{˙1˙2˙3}{X^{I}, X^{J}, Ψ}

, (2.4)

S_{H}^{2} =
Z

d^{3}xd^{3}y

− 1

12H^{2}˙µ ˙ν ˙ρ− 1
4Hλ ˙µ ˙ν^{2}

, (2.5)

SCS = Z

d^{3}xd^{3}y ǫ^{µνλ}ǫ^{˙µ ˙ν ˙λ}

−1

2∂˙µbµ ˙ν∂νb_{λ ˙λ}+g

6∂˙µbν ˙νǫ^{ρ ˙σ ˙τ}^{˙} ∂˙σbλ ˙ρ(∂_{˙λ}bµ ˙τ − ∂˙τb_{µ ˙λ})

.(2.6) In the above we use the notation

X^{˙µ}(y) ≡ y^{˙µ}
g +1

2ǫ^{˙µ ˙κ ˙λ}b_{˙κ ˙λ}(y)≡ y^{˙µ}

g + b^{˙µ}(y), (2.7)
{A, B, C} ≡ ǫ^{˙µ ˙ν ˙}^{ρ}∂˙µA∂˙νB∂ρ˙C. (2.8)
Here, we can find the effective field theory on worldvolume of NP M5 theory are described
by Nambu-Poisson bracket.

The covariant derivative is defined by(Φ = X^{I} or Ψ):

DµΦ ≡ ∂µΦ− g{bµ ˙ν, y^{˙ν}, Φ}, (2.9)
D^{˙µ}Φ ≡ g^{2}

2ǫ˙µ ˙ν ˙ρ{X^{˙ν}, X^{ρ}^{˙}, Φ}. (2.10)

We can find the covariant derivative is defined by two gauge fields: bµ ˙µ and b˙µ ˙ν. The definition of the 3-form field strength reads

Hλ ˙µ ˙ν = ∂λb˙µ ˙ν − ∂^{˙µ}bλ ˙ν + ∂˙νbλ ˙µ, (2.11)

H_{˙λ ˙µ ˙ν} = ∂_{˙λ}b_{˙µ ˙ν} + ∂_{˙µ}b_{˙ν ˙λ}+ ∂_{˙ν}b_{˙λ ˙µ}, (2.12)

which is no longer covariant under the non-Abelian gauge transformations. The covariant

2In original paper [2], they meet the unusually kinetic term of fermions, which has added Γ1 ˙^{˙}2 ˙3factor.

In order to solve the problem, they used the similar unitary transformation:Ψ = √^{1}

2(1− Γ^{1 ˙}^{˙}^{2 ˙}^{3})Ψ^{′}. Here,
we use the symbol Ψ, which was denoted by Ψ^{′} in [2].

3-form field strengths H should be defined as
Hλ ˙µ ˙ν = ǫ_{˙µ ˙ν ˙λ}DλX^{˙λ}

= Hλ ˙µ ˙ν − gǫ^{˙σ ˙τ ˙}^{ρ}(∂˙σbλ ˙τ)∂ρ˙b˙µ ˙ν, (2.13)

H˙1˙2˙3 = g^{2}{X^{˙1}, X^{˙2}, X^{˙3}} − 1
g

= H_{˙1˙2˙3}+g

2(∂˙µb^{˙µ}∂˙νb^{˙ν} − ∂^{˙µ}b^{˙ν}∂˙νb^{˙µ}) + g^{2}{b^{˙1}, b^{˙2}, b^{˙3}}. (2.14)
In fact, the deformations of field strengths come from the VPD symmetry. It is similar
to the theory with APD symmetry. We will give the more details of VPD symmetry
transformations of fields in next section.

### 2.3 Symmetry of Nambu-Poisson M5 Theory

In this section, we will show the symmetry in the NP M5 theory. The gauge symmetry of NP M5 theory is the volume-preserving diffeomorphism. On the other hand, the theory has also supersymmetry, which can be used to calculate the BPS states and central charges.

### 2.3.1 Gauge Symmetry and VPD

The fundamental fields transform under the gauge transformation as

δΛΦ = gκ^{ρ}^{˙}∂ρ˙Φ (Φ = X^{I}, Ψ), (2.15)

δΛb_{˙κ ˙λ} = ∂˙κΛ_{˙λ}− ∂˙λΛ˙κ+ gκ^{ρ}^{˙}∂ρ˙b_{˙κ ˙λ}, (2.16)

δΛbλ ˙σ = ∂λΛ˙σ − ∂^{˙σ}Λλ+ gκ^{˙τ}∂˙τbλ ˙σ+ g(∂˙σκ^{˙τ})bλ ˙τ, (2.17)
where

κ^{˙λ} ≡ ǫ^{˙λ ˙µ ˙ν}∂˙µΛ˙ν(x, y). (2.18)

The field strengths H transform like Φ.

The gauge transformations can be more concisely expressed in terms of the new
variables b^{˙µ}, Bµ˙µ

b^{˙µ} ≡ 1

2ǫ^{˙µ ˙ν ˙λ}b_{˙ν ˙λ}, (2.19)

Bµ˙µ ≡ ǫ^{˙µ ˙ν ˙λ}∂˙νb_{µ ˙λ} (2.20)

for the gauge fields as

δ_{Λ}b^{˙µ} = κ^{˙µ}+ gκ^{˙ν}∂_{˙ν}b^{˙µ}, (2.21)
δΛBµ˙µ = ∂µκ^{˙µ}+ gκ^{˙ν}∂˙νBµ˙µ− g(∂˙νκ^{˙µ})Bµ˙ν. (2.22)
In terms of Bµ˙ν, the covariant derivative Dµ acts as

DµΦ = ∂µΦ− gBµ˙µ∂˙µΦ. (2.23)
Another feature of the gauge transformations is that, in terms of X^{I}, Ψ, b^{˙µ} and Bµ˙µ,
all gauge transformations can be expressed solely in terms of κ^{˙µ}, without referring to Λ˙µ,
as long as one keeps in mind the constraint

∂˙µκ^{˙µ} = 0. (2.24)

This gauge transformation can be naturally interpreted as volume-preserving diffeomor- phism (VPD)

δy^{˙µ} = gκ^{˙µ}, with ∂˙µκ^{˙µ} = 0. (2.25)
The field b^{˙µ} is then interpreted as the gauge potential for the VPD in the 3-dimensional
space picked by the C-field background.

### 2.3.2 Supersymmetry

The M5-brane theory is also invariant under the supersymmetry transformations δχ and δǫ. We have

δχΨ = χ, δχX^{I} = δχb˙µ ˙ν = δχbµ ˙ν = 0, (2.26)
and ^{3}

δǫX^{I} = iǫΓ^{I}Ψ, (2.27)

δ_{ǫ}Ψ = DµX^{I}Γ^{µ}Γ^{I}ǫ +D˙µX^{I}Γ^{˙µ}Γ^{I}ǫ

−1

2H^{µ ˙ν ˙}^{ρ}Γ^{µ}Γ^{˙ν ˙}^{ρ}ǫ− 1

g (1 + gH˙1˙2˙3) Γ_{˙1˙2˙3}ǫ

−g^{2}

2{X^{˙µ}, X^{I}, X^{J}}Γ^{˙µ}Γ^{IJ}ǫ +g^{2}

6{X^{I}, X^{J}, X^{K}}Γ^{IJK}Γ^{˙1˙2˙3}ǫ, (2.28)

δǫb˙µ ˙ν = −i(ǫΓ˙µ ˙νΨ), (2.29)

δǫbµ ˙ν = −i (1 + gH˙1˙2˙3) ǫΓµΓ˙νΨ + ig(ǫΓµΓ^{I}Γ_{˙1˙2˙3}Ψ)∂˙νX^{I}. (2.30)

3ǫhere was denoted by ǫ^{′} in [2].

The SUSY transformation parameters χ, ǫ can be conveniently denoted as an 11D Ma- jorana spinor satisfying the 6D chirality condition

Γ^{7}χ =−χ, Γ^{7}ǫ =−ǫ. (2.31)

They are both nonlinear SUSY transformations, but a superposition of the two,

δχ+ gδǫ and χ = Γ^{˙1˙2˙3}ǫ, (2.32)

defines a linear SUSY transformation.

### 2.4 Double Dimensional Reduction

In order to understand that the NP M5 theory describes the M5-brane in large C field background. One can study the relative superstring theory. This relation between M theory and superstring theory can be done by dimensional reduction. There are several ways of dimensional reductions. One way is just to compactify one target space dimension on circle, then M2-brane and M5-brane in eleven dimensions will relate to D2-brane and NS5-brane in ten dimensions. Another way is to compactify one target space dimension and one worldvolume space dimension on a circle at the same time. It is called Double Dimensional Reduction (DDR). After DDR, the M2-brane and M5-brane in eleven di- mensions will relate to F1 string and D4-brane in ten dimensions. These objects (F1, D2, D4, and NS5) are the main elements in IIA superstring theory in ten dimensions.

Similarly, if we compactify one target space dimension on S^{1}/Z_{2}, it will relate to the
E_{8}× E8 heterotic superstring theory. In this section, we will focus on the DDR method,
then we can study the relative D4-brane action of NP M5 theory.

### 2.4.1 Poisson D4 Description From Nambu-Poisson M5 Theory

In this subsection, we will re-derive the D4 in large NS-NS B field background from the
NP M5 theory. Firstly, we know theory D4-brane theory can be obtained from M5-brane
theory after double dimensional reduction on a circle. The double dimensional reduction
(DDR) means that we do the dimensional reduction on worldvolume and target space at
the same time. In original, people [1, 2] want to show the evidence of the NP M5 theory
is the effective description of M5-brane in large C field background. Hence, they expect
to get the D4 in large NS-NS B field background after compactification the circle, which
live in the direction y^{˙3} and has radius R. There are several reasons for this choice. The

first thing is the C field background field C_{˙1˙2˙3} will be explained as B_{˙1˙2} after DDR on y^{˙3}.
The relation between C_{˙1˙2˙3} and B_{˙1˙2} is written by:

Z y^{3}^{˙}=2πR
y^{3}^{˙}=0

C_{˙1˙2˙3}dy^{˙1}dy^{˙2}dy^{˙3} ≡ B˙1˙2dy^{˙1}dy^{˙2}. (2.33)

The second thing is the Nambu-Poisson bracket will relate to Poisson bracket by this
way: {f, g, y^{˙3}} = ǫ^{α ˙}^{˙}^{β ˙3}∂˙µf ∂˙νg ≡ {f, g}p.b.. Here the indices ˙α are {˙1, ˙2}.

After integrating out the auxiliary field (bµ ˙α) and renaming some fields, we get^{4}:

S_{D4inB} =

Z

d^{3}xd^{2}y

−1

2( ˆD_{a}X^{I})^{2}− 1

4( ˆF_{ab})^{2}− g^{2}

4{X^{I}, X^{J}}^{2}− 1
2g^{2}
+i

2

Ψ^{′}Γ^{a}Dˆ_{a}Ψ^{′}+ gΨ^{′}Γ^{I}{X^{I}, Ψ^{′}}

, (2.34)

where we use the unitary transformation of fermion Ψ = ^{√}^{1}_{2}(Γ_{˙3} + Γ^{7})Ψ^{′} to keep the
chirality condition of gaugino (Ψ^{′}) on D4-brane: Γ_{˙3}Ψ^{′} = Ψ^{′}. The gauge field b_{µ ˙3} and b_{α ˙3}_{˙}
are understood as the one form gauge field in D4-brane theory after DDR. The gauge
field ˆaa:= b_{a ˙3} can be used to define the covariant derivative and field strength:

δΛˆaa = ∂aΛ− g{Λ, ˆaa}p.b., Λ≡ Λ˙3, (2.35)
Fˆab = ∂aˆab − ∂^{b}ˆaa+ g{ˆa^{a}, ˆab}^{p.b.}, (2.36)
DˆaΦ = ∂aΦ + g{ˆaa, Φ}p.b.. (2.37)
This theory describes the D4-brane in large NS-NS B field background(B_{˙1˙2}).

In this chapter, we show the main characters of NP M5 theory. We give several evi-
dences of the M5-brane in large C field background. For example, the constant term exists
in action, the supersymmetry law is nonlinear, the two form gauge field has non-abelian
structure, and it reproduces D4-brane in NS-NS B field background, etc. However, we
find another possible D4-brane formalism, which can describe the D4-brane in large C
field background. It can be achieved by DDR on another circle x^{2}. We will deal with it
in next chapter.

4Here the indices ‘a’ are{µ; ˙α}.

## Chapter 3

## D4 in R-R Three Form Background

In this chapter, we will start to consider the effective action of D4-brane in the large three-form background. This is motivated from NP M5 theory, which describes the single M5-brane in large C field background. If we do double dimensional reduction along the codimension of C field. We will get the effective description of D4-brane in large C field background.

### 3.1 D4-Brane in C Field Background via DDR

To carry out the double dimensional reduction (DDR) for the M5-brane along the x^{2}-
direction, we set

x^{2} ∼ x^{2}+ 2πR, (3.1)

and let all other fields to be independent of x^{2}. As a result we can set ∂_{2} to zero when
it acts on any field. Here R is the radius of the circle of compactification and we should
take R ≪ 1 such that the 6 dimensional field theory on M5 reduces to a 5 dimensional
field theory for D4. To keep zero mode of fields in x^{2} direction, we need to explain the
meaning of field with component 2. For example, the b_{µ ˙µ} → {b2 ˙µ, b_{α ˙µ}}, where α = 0, 1
and the field b_{2 ˙µ} is understood by one form field on D4-brane theory. Hence, we define

b_{˙µ2} ≡ a˙µ. (3.2)

On the other hand, the Gamma matrix Γ^{2} is understood by ten dimensional chirality
matrix. It is used to define the chirality condition of fermion (gaugino) in D4-brane
theory.

### 3.1.1 Gauge Transformation of Fields

As what we have mentioned, the fields after DDR are X^{I}, Ψ, bα ˙µ, b2 ˙µ, and b^{˙µ}. After DDR,
the gauge transformation of fields are:

δΛΦ = gκ^{ρ}^{˙}∂ρ˙Φ (Φ = X^{I}, Ψ), (3.3)
δ_{Λ}b_{α ˙σ} = ∂_{α}Λ_{˙σ}− ∂˙σΛ_{α}+ gκ^{˙τ}∂_{˙τ}b_{α ˙σ}+ g(∂_{˙σ}κ^{˙τ})b_{α ˙τ}, (3.4)
δΛb2 ˙σ = −∂^{˙σ}Λ2+ gκ^{˙τ}∂˙τb2 ˙σ+ g(∂˙σκ^{˙τ})b2 ˙τ, (3.5)

δ_{Λ}b^{˙µ} = κ^{˙µ}+ gκ^{˙ν}∂_{˙ν}b^{˙µ}. (3.6)

We expect that the U (1) gauge symmetry on the D4-brane has its origin in the gauge
transformations (2.16), (2.17) on the M5-brane. The gauge transformation parameter
Λ_{2} shall be identified with the U (1) gauge transformation parameter. This is consistent
with the identification of a_{˙µ} with b_{˙µ2}. The gauge symmetry parametrized by Λ_{˙µ}, i.e., the
VPD, is also still present on the D4-brane. Hence, we can have the gauge transformation
of a_{˙µ}:

δ_{Λ}a_{˙µ} = ∂_{˙µ}λ + g(κ^{˙ν}∂_{˙ν}a_{˙µ}+ a_{˙ν}∂_{˙µ}κ^{˙ν}). (3.7)
The gauge symmetry combines U(1) gauge symmetry and volume-preserving diffeomor-
phism symmetry. This is the first new character of the new D4 theory. The 3-dimensional
volume-preserving diffeomorphism is the evidence of D4 in large C-field background. We
want to ask how to find the other DOF of one form fields (aα), and we also want to know
how to find the gauge transformation law of aα. We will deal with it in next section.

### 3.1.2 Action

After keeping the zero mode of fields in x^{2} direction, we get the effective description of
five dimensions worldvolume theory. The action is what we expect for the new D4-brane
action, which describe the effective action of D4-brane in large C-field background. The
complete action form can be represented in different parts. The result of DDR on Sgauge

is

S_{gauge}^{(1)} =
Z

d^{2}xd^{3}y

−1

2H^{2}_{˙1˙2˙3}− 1

4H^{2}2 ˙µ ˙ν − 1
4Hα ˙µ ˙ν^{2}

+ǫ^{αβ}ǫ^{˙µ ˙ν ˙}^{ρ}∂βaρ˙∂˙µbα ˙ν +g

2ǫ^{αβ}ǫ˙µ ˙ν ˙ρǫ^{˙µ ˙δ ˙τ}ǫ^{˙ν ˙σ ˙λ}ǫ^{ρ ˙η ˙}^{˙} ^{ξ}∂_{˙δ}bα ˙τ∂˙σb_{β ˙λ}∂˙ηaξ˙

o, (3.8)

where we use the definition of ǫ^{αβ2} ≡ ǫ^{αβ}. The result of DDR on S_{X} is
S_{X}^{(1)} =

Z

d^{2}xd^{3}y

−1

2D˙µX^{I}D^{˙µ}X^{I} −1

2∂αX^{I}∂^{α}X^{I} + gB_{α}^{˙µ}∂˙µX^{I}∂^{α}X^{I}

−g^{2}

2B_{α}^{˙µ}B^{α}_{˙ν}∂˙µX^{I}∂^{˙ν}X^{I} − g^{2}

8ǫ^{˙µ ˙}^{ρ ˙τ}ǫ_{˙ν ˙σ ˙δ}Fρ ˙τ˙ F^{˙σ ˙δ}∂˙µX^{I}∂^{˙ν}X^{I}

− 1
2g^{2} − g^{4}

4{X^{˙µ}, X^{I}, X^{J}}^{2}− g^{4}

12{X^{I}, X^{J}, X^{K}}^{2}

. (3.9)

The result of DDR on SΨ is
S_{Ψ}^{(1)} =

Z

d^{2}xd^{3}y i

2ΨΓ¯ ^{α}∂αΨ + i

2ΨΓ¯ ^{ρ}^{˙}Dρ˙Ψ + gi

4ΨΓ¯ ^{2}ǫ^{˙µ ˙ν ˙}^{ρ}F˙ν ˙ρ∂˙µΨ− gi

2ΨΓ¯ ^{α}B_{α}^{˙µ}∂˙µΨ
+g^{2}i

2ΨΓ¯ ˙µΓ^{I}{X^{˙µ}, X^{I}, Ψ} − g^{2}i

4ΨΓ¯ ^{IJ}Γ_{˙1˙2˙3}{X^{I}, X^{J}, Ψ}

. (3.10)

In this chapter, we will focus on the gauge field part. To understand if the gauge part has a well description of D4 in large C-field background will teach us how to deal with matter fields part. After turning off the mater fields, we only need to consider the equation (3.8).

Focus on the action of gauge fields after DDR, we identify a˙µ as components of the one- form potential on the D4-brane. In terms of the field strength

F_{˙µ ˙ν} ≡ ∂˙µa_{˙ν} − ∂˙νa_{˙µ}, (3.11)

we can rewrite H2 ˙µ ˙ν as

H2 ˙µ ˙ν = F_{˙µ ˙ν} +g

2ǫ_{˙µ ˙ν ˙λ}ǫ^{˙σ ˙}^{ρ ˙τ}∂_{˙σ}b^{˙λ}F_{ρ ˙τ}_{˙} . (3.12)

In the above we see that part of the two-form potential b on the M5-brane transforms into part of the one-form potential a on D4. However, in order to interpret this action as a D4-brane action, we still need to identify the rest of the components aα of the one- form gauge potential, and to re-interpret bα ˙µ and b˙µ ˙ν from the D4-brane viewpoint. We also need to find all components of field strength or find all covariant variables in this theory. On the other hand, we also need to understand the new D4 action in usually D4 viewpoint. We will deal with these problems in different sections.

### 3.2 Dual Transformation

In this section, we use the method which is called dual transformation to find the other
components of one form fields (a_{α}). This one form component is not suddenly adding

form field a_{α}. It also can be understood as the electric-magnetic duality in 3-dimensional
spaces (y^{˙µ}). This is the dual description between the one form (b_{α})_{˙µ} and the zero form
(a_{α}). We will see this fact in this section.

### 3.2.1 Equivalent Dual Action and Dual One Form Field

In order to understand the physical meaning of the action (3.8), we try to simplify the action by integrating out the remaining components of the 2-form gauge field b as much as possible, since there is no 2-form gauge potential in the usual description of a D4-brane.

First we note that the action (3.8) depends on b_{α ˙µ} only through the variable B_{α}^{˙µ}
(2.20). In terms of B_{α}^{˙µ}, we have

Hα ˙µ ˙ν = ǫ_{˙µ ˙ν ˙λ}(∂_{α}b^{˙λ} − V˙σ˙λB_{α}^{˙σ}), (3.13)

where

V_{˙ν} ^{˙µ} ≡ δ˙ν^{˙µ}+ g∂˙νb^{˙µ}. (3.14)
Hence we can rewrite the action (3.8) as

S^{(2)}[b^{˙µ}, a˙µ, B_{α}^{˙µ}] =
Z

d^{2}xd^{3}y

−1

2H^{2}_{˙1˙2˙3}− 1
4H^{2}2 ˙µ ˙ν

−1

2(∂αb^{˙µ}− V˙σ^{˙µ}B_{α}^{˙σ})^{2} +ǫ^{αβ}∂βa˙µB_{α}^{˙µ}+ g

2ǫ^{αβ}F˙µ ˙νB_{α}^{˙µ}B_{β}^{˙ν}o

.(3.15)
It turns out that it is possible to extract the components aα on the D4-brane by
dualizing the field B_{α}^{˙µ}. We can introduce the Lagrange multiplier fα ˙µ to rewrite the
action (3.15) as

S^{(3)}[b^{˙µ}, a˙µ, bα ˙µ, ˘B_{α}^{˙µ}, fβ ˙µ] =
Z

d^{2}xd^{3}y

−1

2H^{2}_{˙1˙2˙3}− 1

4H^{2}2 ˙µ ˙ν− 1

2(∂αb^{˙µ}− V˙σ^{˙µ}B˘_{α}^{˙σ})^{2}
+ǫ^{αβ}∂βa˙µB˘_{α}^{˙µ}+ g

2ǫ^{αβ}F˙µ ˙νB˘_{α}^{˙µ}B˘_{β}^{˙ν}

−ǫ^{αβ}fβ ˙µ[ ˘B_{α}^{˙µ}− ǫ^{˙µ ˙ν ˙}^{ρ}∂˙νbα ˙ρ]o

, (3.16)

where we used the notation ˘B for a new variable independent of bα ˙µ. If we integrate out
the Lagrange multiplier fβ ˙µ, we will get ˘B_{α}^{˙µ} = B_{α}^{˙µ}, and the action above reduces back
to (3.15).

Instead, we can integrate out ˘B_{α}^{˙µ} and bα ˙µ to dualize the field B_{α}^{˙µ}. First we integrate
out bα ˙µ, and find the constraint on fα ˙µ

ǫ^{˙µ ˙ν ˙λ}∂˙µfα ˙ν = 0. (3.17)

It implies that, locally

f_{α ˙µ} = ∂_{˙µ}a_{α} (3.18)

for some potential aα. Hence, after integrating out bα ˙µ, we get
S^{(4)}[b^{˙µ}, a˙µ, aα, ˘B_{α}^{˙µ}] =

Z

d^{2}xd^{3}y

−1

2H^{2}_{˙1˙2˙3}− 1

4H^{2}2 ˙µ ˙ν − 1

2(∂αb^{˙µ}− V˙σ^{˙µ}B˘_{α}^{˙σ})^{2}
+ǫ^{αβ}∂βa˙µB˘_{α}^{˙µ}+g

2ǫ^{αβ}F˙µ ˙νB˘_{α}^{˙µ}B˘_{β}^{˙ν} −ǫ^{αβ}∂˙µaβB˘_{α}^{˙µ}o

. (3.19) In order to find the final form of dual action, we should also need to integrate out the ˘B.

We will get the complete form in next subsection.

### 3.2.2 Action after Dual Transformation

Since the action is at most quadratic in ˘B_{α}^{˙µ}, the result of integrating out ˘B_{α}^{˙µ} is the same
as replacing ˘B_{α}^{˙µ} by the solution to its equation of motion, which is a constraint

V_{˙µ}^{˙ν}(∂^{α}b˙ν − V^{ρ}^{˙}˙νB˘^{α}_{ρ}_{˙}) + ǫ^{αβ}Fβ ˙µ+ gǫ^{αβ}F˙µ ˙νB˘_{β}^{˙ν} = 0. (3.20)
The solution of ˘B_{α}^{˙µ}, denoted as ˆB_{α}^{˙µ}, is given by

Bˆ_{α}^{˙µ} ≡ (M^{−1})αβ˙µ ˙ν(V_{˙ν}^{˙σ}∂^{β}b˙σ+ ǫ^{βγ}Fγ ˙ν), (3.21)

where

M˙µ ˙ναβ ≡ V˙µ ˙ρV˙νρ˙δ^{αβ} − gǫ^{αβ}F˙µ ˙ν, (3.22)

and M^{−1} is defined by

(M^{−1})γα˙λ ˙µM˙µ ˙ναβ = δ^{˙λ}˙νδ_{γ}^{β}. (3.23)

After integrating out ˘B_{α}^{˙µ}, we get
S^{(5)}[b^{˙µ}, a˙µ, aα] =

Z

d^{2}xd^{3}y

−1

2H^{2}_{˙1˙2˙3}− 1

4(F˙ν ˙ρ+g

2ǫ˙µ ˙ν ˙ρǫ^{˙σ ˙δ ˙τ}∂˙σb^{˙µ}F_{˙δ ˙τ})^{2}− 1

2∂αb^{˙µ}∂^{α}b˙µ

+1

2(ǫ^{αγ}Fγ ˙µ+ V_{˙µ}^{˙σ}∂^{α}b˙σ)(M^{−1})αβ˙µ ˙ν(ǫ^{βδ}Fδ ˙ν + V˙ν˙λ∂^{β}b_{˙λ})

. (3.24) At the quantum level, there is a one-loop contribution to the action when we integrate out ˘Bα˙µ. It is

∆S_{1−loop} =−~

2T r(Log(M_{˙µ ˙ν}^{αβ})). (3.25)
The action (3.24) is only remotely resembling the familiar Maxwell action for a U (1)