Strings in Background Fields

M. de Leeuw

September 26, 2006

Abstract

We will discuss the bosonic non-linear sigma-model, which describes bosonicstrings in a curved space-time. We treat this as a two dimensional quantumfield theory, which has interesting UV and IR properties. These were studiedby using the background field method. We will calculate different correlationfunctions with this. We will compute anomalies of the conformal algebra for thebosonic nonlinear sigma model. Using 2-dimensional superspace, we introducethe N = (2,2) non-linear sigma model, which has N = (2,2) supersymmetry. Wewill compute anomalies of the superconformal algebra for the N = (2,2) non-linear sigma-model, which give rise to the Ricci flatness. We will also brieflyconsider the topological twists of the N = (2,2) non-linear sigma model.

Contents

1 Introduction 5

2 The Bosonic Non-Linear Sigma Model 72.1 The Free String in Minkowski Space . . . . . . . . . . . . . . . . 72.2 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 String Propagating in Curved Background . . . . . . . . . . . . . 102.4 Including the B-Field . . . . . . . . . . . . . . . . . . . . . . . . . 112.5 Including the Dilaton . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 The Bosonic Non-Linear Sigma Model and the BackgroundField Method 143.1 The Background Field Method . . . . . . . . . . . . . . . . . . . 143.2 Non-Linear Sigma Model in BFM . . . . . . . . . . . . . . . . . . 17

4 The Computer Program and Calculation Method 224.1 Dictionary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2 Computing the Wick Contractions . . . . . . . . . . . . . . . . . 244.3 Selecting Connected Diagrams . . . . . . . . . . . . . . . . . . . 254.4 Taylor Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . 254.5 Keeping Track of Orders . . . . . . . . . . . . . . . . . . . . . . . 264.6 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5 Weyl Symmetry at Quantum Level 28

6 Calculation of the Conformal Algebra 336.1 Conformal Algebra in Flat Space . . . . . . . . . . . . . . . . . . 336.2 The Conformal Algebra in Curved Background . . . . . . . . . . 35

6.2.1 How to deal with the dilaton . . . . . . . . . . . . . . . . 366.2.2 〈Tr〉 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366.2.3 c, The Anomaly term without X . . . . . . . . . . . . . . 396.2.4 〈T (z)T (w)〉 . . . . . . . . . . . . . . . . . . . . . . . . . . 416.2.5 Putting it all together . . . . . . . . . . . . . . . . . . . . 466.2.6 Obtaining the Einstein Equations . . . . . . . . . . . . . . 46

7 Topological Field Theories 487.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487.2 Chern-Simons Theory . . . . . . . . . . . . . . . . . . . . . . . . 497.3 Cohomological Field Theories . . . . . . . . . . . . . . . . . . . . 50

7.3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2

7.4 Two-dimensional TFTs . . . . . . . . . . . . . . . . . . . . . . . 51

8 The N=(2,2) Non-Linear Sigma Model 548.1 Superspace and its Symmetries . . . . . . . . . . . . . . . . . . . 54

8.1.1 Flat Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 548.1.2 General Case and R-Symmetry . . . . . . . . . . . . . . . 58

8.2 Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608.3 The A-Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628.4 The B-Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

9 The Superconformal Algebra 679.1 The Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679.2 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699.3 Chiral Primary Fields . . . . . . . . . . . . . . . . . . . . . . . . 71

10 Background Field Method for Kahler Manifolds 7410.1 Kahler Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 7410.2 Kahler Normal Coordinates . . . . . . . . . . . . . . . . . . . . . 7510.3 Background Field Method and Application to the NLSM . . . . . 7510.4 Derivation of the Propagators . . . . . . . . . . . . . . . . . . . . 77

11 The Algebra of the N = (2, 2) Non-Linear Sigma Model 7911.1 The Conformal Superalgebra in Flat Space . . . . . . . . . . . . 7911.2 The Superconformal Algebra in Curved Space . . . . . . . . . . . 80

11.2.1 Renormalization . . . . . . . . . . . . . . . . . . . . . . . 8011.2.2 〈TT 〉, 〈TJ〉, etc. . . . . . . . . . . . . . . . . . . . . . . . 8011.2.3 < T >,< G >,< J > . . . . . . . . . . . . . . . . . . . . 8111.2.4 Putting it all together . . . . . . . . . . . . . . . . . . . . 82

A Lorentz Indices 83

B Wick’s Theorem 85

C Alternative Method of Integral Calculations 86C.1 Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86C.2 Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

D References 89

3

Acknowledgements

I would like to thank my supervisor Dr. G. Arutyunov. He has been a greathelp in understanding the material covered. He always had time to discuss ourfindings in detail in sometimes also in great length where needed. He was agreat support and I look forward to working with him. Also a word of thanks toMartijn Kool, with whom I studied most of this material and did the calculationswith. Good luck in Oxford. Finally, a thanks to my family who put up with meduring the writing of this thesis.

4

Chapter 1

Introduction

This is the final result of the work I have done on my master’s thesis at UtrechtUniversity under the supervision of Dr. G. Arutyunov.

One of the main goals of modern theoretical physics is to find a theory thatincorporates both general relativity and quantum field theory. One of the bestcandidates for such a theory is string theory, and it is studied quite intensively.

For a consistent string theory, the theory needs to have so-called conformalsymmetry. This symmetry, in some way, is a symmetry of rescalings of themetric of the world sheet of the string. If the string is propagating in flat space,one can prove that this is indeed the case, but in curved space things change.In this thesis, we study the conformal symmetry of the bosonic string as wellas the N = (2, 2) non-linear sigma model in curved backgrounds. The bosonicstring is described by a model called the bosonic non-linear sigma model.

The conformal symmetry of the bosonic model will be studied by consideringthe conformal algebra as well as making the effective action Weyl symmetric.The N = (2, 2) non-linear sigma model also has supersymmetries and for thisreason we study the superconformal algebra in this case.

For the calculations of the (super)conformal algebra, we have mostly followedand reproduced an article by T.Banks, D. Nemeschansky and A. Sen [BNS] from1986. For the other calculation, we have used [A-GFM] which also discusses thebackground field method.

As is clear from the name of both theories, the string propagating in a curvedbackground is a non-linear system. In order to be able to derive results from thison has to use certain tools. We have used the so-called background field method.

This thesis is organized as follows. We will start with a short discussion onbosonic strings. We will treat it in flat space as well as in curved space. Inthat chapter we will also discuss how to add the dilaton and an anti-symmetricB-field to the action.

In the next chapter the basic tool which we used to derive our results isdiscussed, the background field method (BFM). This method can make non-linear theories suitable for perturbation theory. In this chapter we will rewritethe bosonic string action, including dilaton and B-field by using this method.

We did most of our calculations with a mathematica notebook that we wrotefor this purpose. How this computer program works, will be explained in Chap-

5

ter 4. The notebooks are also included in this thesis, if not they can probablybe found on my website.

In the next chapter the effective action will be discussed to some extend. Wehave reproduced the result of [A-GFM], but we have also calculated the vacuumexpectation value for non-Ricci flat backgrounds. Unfortunately, it turns outthat the result does not allow for a quick correction to the equations obtained.

In Chapter 6 we will study what happens to the conformal algebra in curvedbackgrounds. We obtain, the same equations as in the literature, but we noticedthat the authors of [BNS] have made some mistakes in their article. However, inthis calculation, there are a lot of subtleties involved and they will be discussedthoroughly.

After this we will turn our attention to the supersymmetric case. We willstart by discussing topological field theories. After this we will discuss theN = (2, 2) non-linear sigma model from the superspace point of view. We willalso derive the Noether currents that come with the different symmetries of thetheory. That chapter ends with a discussion on the A-and B-model.

In next chapter will deal with the superconformal algebra. After this, it isdiscussed how the background field method should be adapted in order to applyit to the N = (2, 2) non-linear sigma model. This is discussed and done inChapter 10.

In the last chapter all of the above will be applied to the conformal algebraof the N = (2, 2) non-linear sigma model. It turns out that this model can onlysatisfy the complete superconformal algebra on Ricci flat manifolds. To satisfyto the conformal algebra, however, one gets similar equations as discussed inthe bosonic non-linear sigma model.

6

Chapter 2

The Bosonic Non-LinearSigma Model

In this chapter we discuss the model that we will study in the first part of thisthesis, the bosonic sigma model. This model describes propagation of a bosonicstring in a target space. We will start with a discussion of the free string inMinkowksi space. After this, we will discuss how to quantize this system. Thenext step is the description of a string in a curved background. We finish thischapter by describing how this model can be extended by including additionalfields like the dilaton and the B-field. Most of this material can be found in therelevant chapters in [GSW].

2.1 The Free String in Minkowski Space

A string propagating in Minkowski space is described by the Polyakov action:

S =1

4πα′

∫

Σ

d2σ√

hhαβηij∂αXi∂βXj (2.1)

We are working in the Euclidean setting. This action has a couple of symmetries.First, there are 2 reparametrization invariances, described by:

δXi = λα∂αXi

δhαβ = λγ∂γhαβ − ∂γλαhγβ − ∂γλβhαγ

δ(√

h) = ∂α(ξα√

h).

Secondly, we have symmetry under Weyl rescalings of the metric:

δhαβ = Λhαβ .

Theses are local symmetries. There are, of course, also a few global symmetriescoming from Poincare invariance.

We now define the energy-momentum tensor to be the variation of the actionwith respect to the world sheet metric hαβ :

Tαβ =2√h

δS

δhαβ.

7

For the string propagating in Minkowksi space, described by 2.1, this gives:

Tαβ =1

2πα′(ηij∂αXi∂βXj − 1

2hαβhγβ′ηij∂γXi∂β′X

j) (2.2)

The field equation δSδhαβ = 0 is thus described by the equation Tαβ = 0. Weyl

symmetry gives us that Tαβ is traceless, i.e. Tαα = 0:

0 = δS

=δS

δhαβδhαβ

=12

√hTαβΛhαβ

=12

√hΛTα

α .

One usually uses a gauge called the conformal gauge. This gauge uses thetwo reparametrization symmetries and Weyl symmetry to write, at least locally,hαβ = δαβ . One can always, locally, gauge the metric in the form hαβ =eφηαβ by using reparametrization invariance, this is sometimes also called theconformal gauge, but we will refer to this as the semi-comformal gauge. So,

• conformal gauge: hαβ =(

1 00 1

).

• semi-conformal gauge:hαβ = eφ

(1 00 1

).

These gauges can be done globally for genus zero world sheets, so we willrestrict ourselves to that case from now on.

In this thesis, we will be dealing with closed strings and we will take the worldsheet to be a cylinder. To map the cylinder to the complex plane (punctured)one uses the map, see figure 2.1,

(z, z) := (eτ+iσ√

2 , eτ−iσ√

2 ), (2.3)

where σ and τ are the regular string variables. This map clearly maps σ0 = ± infto the points 0, inf in the complex plane. Furthermore, points with constant σ0

are mapped to circles of constant radius. It is easy to see that under this map,in the conformal gauge, we have:

(dσ)2 + (dτ)2 =dzdz + dzdz

|z|2 ↔ dzdz + dzdz, (2.4)

via Weyl rescalings. From now on we will be working in these coordinates a lot.The gauges become in these coordinates:

• conformal gauge: hαβ =(

0 11 0

).

• semi-conformal gauge:hαβ = eφ

(0 11 0

).

8

Figure 2.1: The Map from the Cylinder to the Plane

We write the derivatives conjugate to these momenta ∂, ∂.Let us now take a closer look at the energy-momentum tensor in the confor-

mal gauge. Clearly, the action reduces to

S =1

4πα′

∫ηij∂αXi∂αXj , (2.5)

which gives the following energy momentum tensor:

Tαβ =1

2πα′ηij∂αXi∂βXj .

In complex coordinates, this gives:

T := Tzz = ηij∂Xi∂Xj

T := Tzz = ηij∂Xi∂Xj .

The other components Tzz, Tzz vanish because of tracelessness. So, the con-straints on our theory now become T = T = 0.

Conservation of the energy-momentum tensor reduces to ∂T + ∂Tzz and thesame equation with the bars interchanged. So, Weyl symmetry will give us inthis case that

∂T = 0∂T = 0.

This is a powerful statement since it gives that T is a holomorphic function andT is an anti-holomorphic function.

Even after making the conformal gauge, there is still symmetry left in thetheory, namely symmetries given by diffeomomrphism that can be countered byWeyl rescalings.

2.2 Quantization

We will a brief recap of quantizing the theory. This can be done in a lot of dif-ferent ways, see for example [GSW]. We will discuss the old covariant approach

9

discusses in Chapter 2 of [GSW] and all details can be found in that chapter.

In the conformal gauge, the string equation of motion can be solved to be:

XiL =

12xi +

12l2pi(τ − σ) +

i

2l∑

n6=0

1n

αine−2in(τ−σ)

XiR =

12xi +

12l2pi(τ − σ) +

i

2l∑

n6=0

1n

αine−2in(τ+σ). (2.6)

The Fourier components of the energy-momentum tensor can then be writtenas:

Lm =12

∞∑−∞

αm−nαn

Lm =12

∞∑−∞

αm−nαn. (2.7)

It can be proved that, in the classical theory, they satisfy the Virasoro algebra

[Lm, Ln] = i(m− n)Lm+n. (2.8)

However, one makes this into a quantum theory by making the αin, αi

n intooperators, one runs into some difficulties. The operators Lm, Lm now needto be defined via normal ordering. One can then calculate the commutator[Lm, Ln] again and one finds quantum mechanical corrections. To be precise,one finds that the Virasoro algebra picks up an anomaly:

[Lm, Ln] = (m− n)Lm+n +112

D(m3 −m), (2.9)

which in a more general setting is written like

[Lm, Ln] = (m− n)Lm+n +c

12(m3 −m), (2.10)

where c is called the central charge. A nice result is that this can be encodedin terms of operator product expansions. Namely, 2.10 follows from the OPE:

T (z)T (w) =c/2

(z − w)4+

2T (w)(z − w)2

+∂wT (w)z − w

+ reg. (2.11)

A proof of this can be found in [Gin], Section 3.3. And in fact, by similararguments as discussed there, one can see that the other implication also holds.

2.3 String Propagating in Curved Background

In this section we will discuss some of the properties of strings propagating ina curved background. First, let’s generalize the action 2.1 to a curved targetspace:

S =1

4πα′

∫

Σ

d2σ√

hhαβgij∂αXi∂βXj .

10

This describes an embedding X : Σ → M of a Riemann surface Σ with metrichαβ in a target space M with metric gij .

On the classical level, all the symmetries discussed above still hold. In theconformal gauge, the action takes the form:

S =1

4πα′

∫

Σ

d2σgij∂αXi∂αXj . (2.12)

The energy-momentum tensor is just given by (in complex coordinates):

T := Tzz = gij∂Xi∂Xj (2.13)T := Tzz = gij∂Xi∂Xj . (2.14)

and again Tzz = Tzz = 0. So, we see that at the classical level nothing muchchanges. The equations of motion are given by:

¤Xi = Γijk∂αXj∂αXk, (2.15)

which are non-linear. So, for one, we cannot quantize the theory in the sameway as we did before. One way to handle this theory and do perturbation theorywith it is called the Background Field Method, which will be discussed in thenext chapter.

There is also another problem involving Weyl symmetry, which only holds intwo dimensions. Note that if this theory needs to be renormalized, Weyl sym-metry is possibly broken. If one, for example, would use dimensional renormali-sation, one needs to work in 2+ ε dimensions, which breaks the Weyl symmetry.How to deal with this will be discussed later.

2.4 Including the B-Field

In this section we will examine how the action 2.12 can be further extended.In 2.12 the world sheet metric hαβ is used, but one could also use the two

dimensional anti-symmetric tensor εαβ . The way to incorporate this is via ananti-symmetric tensor field Bij(X)(= −Bji(X)). The action corresponding tothis term is given by:

SB =1

4πα′

∫

Σ

d2σεαβBij∂αXi∂βXj .

Note that under the gauge transformation

δBij = ∂iΛj(X)− ∂iΛj(X)

the B-field action changes by a total divergence, since∫

d2σ(∂iΛj − ∂iΛj)εαβ∂αXi∂βXj =∫

d2σ∂β(εαβΛ∂αXi), (2.16)

by the chain rule and anti-symmetry of εαβ .

Let us now compute the equations of motion for the non-linear sigma modelincluding the B-field. The Euler-Lagrange equations give:

∂α

(2Gij∂

αXi − 2εαβBij∂βXi)

= εαβ∂jBik∂αXi∂βXk + ∂jGik∂αXi∂αXk,

11

which can be rewritten as

¤Xi = Γijk∂αXj∂αXk − 2Si

jkεαβ∂αXj∂βXk. (2.17)

It is easy to see that the B field does not contribute to the energy momentumtensor. This follows from the fact that it is independent of the world sheetmetric.

2.5 Including the Dilaton

Next we focus on the dilaton. At first sight, one could include a term of theform

χ =14π

∫

Σ

d2σ√

hR(2), (2.18)

where R(2) is the curvature coming from the world-sheet metric hαβ . For thisterm, however, it can be shown that it does not give any dynamics to the metrich. This can be seen as follows. In two dimensions, any anti-symmetric two tensoris proportional to the tensor εαβ . Since the Riemann tensor is anti-symmetricin the last en and the first two indices it follows that:

R(2)αβγδ

∼= εαβεγδ.

In any dimension we have that

εαβεγδ = δαγδβδ − δαδδβγ ,

from which we deduce that in two dimensions:

R(2)αβγδ =

12

(hαγhβδ − hαδhβγ)R(2).

Contracting with hβδ gives

R(2)αγ −

12hαγR(2) = 0. (2.19)

Varying 2.18 with respect to hαβ , however, is precisely proportional to this term,so we see that this term does not give any dynamics. In fact, one can show (see[GSW] 12.5.3), that 2.18 is in fact the Euler characteristic of the world sheet:

χ = 2(1− g),

where g is the genus of Σ.

The way to include the dilaton is by generalizing 2.18 in the following way:

SD =14π

∫d2σ

√hΦ(Xi)R(2). (2.20)

Let us now take a closer look at the symmetries of the dilaton action. It clearlystill satisfies the reparametrization invariance discussed earlier. However, Weylsymmetry is broken. This can be seen in the following way. One can use

12

the reparametrization invariances to bring the world-sheet metric to the formhαβ = eφηαβ . Explicit calculation of the curvature gives R = −2e−φ∂∂φ, sowe see that the action is φ dependent and hence does not have Weyl symmetry.However, note that the dilaton action come without the factor of 1

α′ , which theother actions did have. In fact, we will see later on that we can do perturbationtheory with α′ as parameter. In other words, tree diagrams for the dilaton areof the same order as quantum effects of the B-field and the NLSM. So, eventhough Weyl symmetry is broken at the classical level, it still might be preservedat quantum level. The problem how to deal with this at a quantum levels willbe discussed later on and will be, more or less, the main subject of the first fewchapters to come.

The dilaton has no contribution to the energy-momentum tensor in two di-mensions since varying the dilaton action with respect to the world sheet metricgives a term proportional to equation 2.19.

For future reference, let us conclude by writing down the complete action in2 − ε dimensions of the non-linear sigma model including the B-field and thedilaton in the gauge discussed above.

S =1

4πα′

∫

Σ

d2−εσe−εφgij∂αXi∂αXj + εαβBij∂αXi∂βXj − 2α′e−εφφ∂α∂αΦ.(2.21)

13

Chapter 3

The Bosonic Non-LinearSigma Model and theBackground Field Method

In this chapter we will discuss a method use to bring non-linear theories to a formin which they are suitable to do perturbation theory in. This method is calledthe background field method and it is discussed, for example in [A-GFM]. In thefirst section we will discuss the Background field method (BFM) and derive theneeded formulas. In the next section we will apply this the bosonic non-linearsigma model and derive its propagator.

3.1 The Background Field Method

In the background field method one considers variations around the classicalsolution of a certain action. Consider a manifold M with 2 points Xi

B andXi

B + δXi. We consider them to be close in the sense that there is a geodesicλi(t) connecting the two. λi(t) is described by the geodesic equation:

λi(t) + Γijkλj(t)λk(t) = 0.

We take λi(t) such that λi(0) = XiB and λi(1) = Xi

B + δXi. Finally, we defineξi to be the tangent vector to λi(t) at Xi

B , i.e. ξi = λi(0), see Figure 3.1.

Next, one can solve the geodesic equation as a power series in the parameter t:

λi(t) = λi(0) + λi(0)t +12!

λi(0)t2 + . . . .

By successively differentiating the geodesic equation one obtains the followingexpression:

λi(t) = XiB + ξit− 1

2!Γi

jkξjξkt2 − 13!

(DlΓi

jk

)ξjξkξlt3 + . . . , (3.1)

where the target space covariant derivative Dl only acts on the lower indices of

14

Figure 3.1: Background Field Method

Γijk. For example, the third term is obtained in the following way:

λi,′′′(0) =(Γi

jkλj(t)λk(t))′|t=0

= Γijkλj(0)λk(0) + Γi

jkλj(0)λk(0) + (∂lΓijk)λl(0)λj(0)λk(0)

=(∂lΓi

jk − ΓimkΓm

jl − ΓijmΓm

kl

)ξjξkξl

=(DlΓi

jk

)ξjξkξl.

From 3.1 we clearly see that

XiB + δXi = λi(1)

= XiB + ξi − 1

2!Γi

jkξjξk − 13!

DlΓijkξjξkξl + . . . .

This defines a coordinates transformation between the coordinates XiB+δXi and

the ξis. From the above relation we see that at XB we have ∂(XiB+δXi)∂ξj = δi

j ,which means that this map defines a diffeomorphism between a small neigh-borhood of Xi

B and the tangent space TXiBM . In fact, this is the well-known

exponential map. So, we obtain a new atlas (UXiB, ξi) for M . It is also clear from

this construction that geodesics on M are mapped to straight lines in TXiBM .

From this we actually see that the coordinates ξi are in fact Riemann NormalCoordinates (RNC).

So, we now have a new set of coordinates and we would like to transformtensors and vectors to this new coordinate system. Let’s start with a 2 tensorTij :

Ti′j′ =∂(Xi

B + δXi)∂(ξi′)

∂(XjB + δXj)∂(ξj′)

Tij((XiB + δXi

B)(ξi)).

15

The part ∂(XiB+δXi)

∂(ξi′ )can be calculated via 3.1, which gives:

∂(XiB + δXi)∂(ξi′)

= δii′ − Γi

i′kξk − 13

(DlΓi

i′k)ξkξl − 1

6(Di′Γi

kl

)ξkξl + . . . .

Finally, making a Taylor expansion of Tij gives:

Tij((XiB + δXi

B)(ξi)) = Tij(XB) + ∂kTij(XB)ξk + . . . .

Combining the two gives the following result:

Tij(ξ) = Tij(XB) + DkTij(XB)ξk +12

(DkDlTij + Rm

iklTmj + RmjklTim

)ξkξl + . . . .(3.2)

In the literature, one often derives this formula in a different, more convenient,way. Suppose the coordinates Xi

B + δXi are Riemann normal coordinates al-ready, then one has that Γi

jk = 0 and even

Γi(j1...jn) = 0. (3.3)

Furthermore, the Riemann tensor simplifies and becomes:

Rijkl = ∂kΓi

jk − ∂lΓijk. (3.4)

This can, for instance, be combined with 3.3 to give:

∂kΓijl =

13(Ri

jkl + Rilkj). (3.5)

So, for a tensor Tij these relations imply:

∂kTij = DkTij

∂l∂kTij = DlDkTij − 13Rm

kilTmj + i ↔ j. (3.6)

Remember that these relations are only valid in RNC, but if we use them in theTaylor expansion, we derive

Tij(ξ) = Tij(XB) + DkTij(XB)ξk +12

(DkDlTij + Rm

iklTmj + RmjklTim

)ξkξl + . . .(3.7)

again. But this is a relation between tensors and hence valid in any coordinatesystem. So, we see that the different formulae can also be derived in this way.

We now consider the case in which we that the coordinates depend on pa-rameters σ, e.g. suppose we have an embedding of a world sheet in the targetspace M . It is clear that the quantity ∂αXi is a vector and hence one can applythe same analysis as above. Note, for one thing, that this automatically makesthe fields ξi also dependent on σ. In fact we get new coordinates ξi for every σ.

One of the problems is, however, that one needs to invert 3.1. One can,however, also use the RNC method discussed. Doing this, we obtain:

∂αXi = ∂αXiB + Dαξi +

13Ri

jklξjξk∂X l

B +112

DmRijklξ

jξk∂X lB

+(

160

DmDnRijkl −

145

RikmpR

pjnl

)ξjξkξmξn∂X l

B . . . . (3.8)

16

In the above formula we have used the definition:

Dαξi := ∂αξi + Γijk∂Xjξk. (3.9)

The quantities derived in this chapter are all we need to write the non-linearsigma model in the new fields ξi, in which we only see two tensors (the metricand the B-field) and the derivatives ∂αXi. Let us for completeness give theexpansion of the metric gij , which is easier because of the different propertiesof the metric:

gij(ξ) = gij(XB)− 13Rikjlξ

kξl − 16DmRikjlξ

kξlξm

1120

(−6DnDmRikjl +

163

RnjmpRp

kil

)ξkξlξmξn + . . . .(3.10)

However, it turns out that a small further redefinition of our fields is convenient.It turns out that it is worthwhile to switch to Lorentz indices. For a moredetailed discussion vielbein, spin connection etc. see Appendix A. We use thevielbein ea

i :

ξa = eai ξi. (3.11)

In this new system, the derivative on ξ also changes,

Dαξa := ∂αξa + ωajb∂Xjξb, (3.12)

where ωabj is the spin connection. One of the advantages of this is that one can

prove that the following result:

Dαξi = eiaDαξa, (3.13)

by using formula A.8 (tetrad postulate) from the appendix. Note that since ξi

is a tangent vector at XiB , the vielbein is also evaluated at this point and hence

also σ dependent.The reason for this redefinition is that the following identity holds:

gij(XB)DαξiDαξj = δabDξaDξb, (3.14)

which follows from the identities in the aforementioned appendix. This willmake the quadratic part in the NLSM a lot simpler.

3.2 Non-Linear Sigma Model in BFM

In this section we will study the bosonic non-linear sigma model in the back-ground field method, which was discussed in the previous section. We will alsoinclude the B-field and the dilaton, i.e. we will work with action 2.21. We willstudy its propagator in detail.

We only see two tensors and vectors in this action and hence we can just sub-stitute the formulas 3.2 and 3.8. We choose the background field XB to satisfythe classical equations of motion 2.17 and in the rest of this thesis we will skipthe subscript B to avoid very cumbersome notation.

17

Let us first exclude the dilaton and work in the semi-conformal guage. Doingthese substitution gives us, to the relevant order for our first computations, thefollowing action:

S =1

4πα′

∫dζdζ e−εφ(DαξaDαξa + Rijklξ

jξk∂αXiB∂αX l

B +43Rijkl∂

αXiBξjξkDαξl +

13RijklDαξiDαξlξjξk) + 2εαβ∂βXk

BSijkξiDαξj − εαβDkSijlξjξk∂αXi

B∂βX lB +

23SijkξiεαβDαξjDβξk +

43DkSijlε

αβ∂βXiBξjξkDαξl

−12DkSijlξ

jξkεαβDαξiDβξl (3.15)

where we define:

Sijk = −12(DiBjk + DjBki + DkBij). (3.16)

For closed a B field, Sijk vanishes. It is not trivial that the action can completelybe written in terms of Sijk and its derivatives. More details on this can be foundin an article of Muhki, [Mu] in which he derives an algorithm with which theseterms, as well as the other terms, can be calculated to even higher orders.

We have also omitted the terms which do not depend on the fields ξa. Fur-thermore, we used Formula A.3, to get the quadratic part in this form as men-tioned before.

The terms linear in ξa that vanish because of the equations of motion 2.17.This can be seen by using partial integration. The argument is as follows.

14πα′

∫dζdζ

(2Gij∂αXiDαξj + 2εαβBij∂αXi

BDβξj + εαβDkBij∂αXiB∂βXj

Bξk)

=1

4πα′

∫dζdζ

(−∂α

(2Gij∂

αXiB − 2εαβBij∂βXi

B

)ξj + 2GijΓj

mn∂αXiB∂αXm

B ξn+

2εαβBijΓjmn∂αXi∂βXmξn + εαβDkBij∂αXi∂βXjξk

)

Now use the following version of the equations of motion (this is the versiondirectly from Euler-Lagrange equations)

∂α

(2Gij∂

αXi − 2εαβBij∂βXi)

= εαβ∂jBik∂αXi∂βXk + ∂jGik∂αXi∂αXk.

This gives

14πα′

∫dζdζ

(−εαβ∂jBik∂αXiB∂βXk

Bξj − ∂jGik∂αXiB∂αXk

Bξj + 2GijΓjmn∂αXi

B∂αXmB ξn+

2εαβBijΓjmn∂αXi

B∂βXmB ξn + εαβ∂kBij∂αXi

B∂βXjBξk − εαβΓl

ikBlj∂αXiB∂βXj

Bξk−εαβΓl

kjBil∂αXiB∂βXj

Bξk)

=1

4πα′

∫dζdζ

(−∂jGik∂αXiB∂αXk

Bξj + 2GijΓjmn∂αXi

B∂αXmB ξn

),

where the last expression is easily seen to be zero by using the definition of theChristoffel connection.

18

One can also rewrite the energy-momentum tensor for the non-linear sigmamodel in terms of the background field method. Doing this, gives:

T =1

2πα′(2gij∂XiDξj + DξaDξa + Rijklξ

jξk∂Xi∂X l

+43Rijkl∂XiξjξkDξl +

13RijklDξiξjξkDξl

). (3.17)

Notice that we still have linear terms here since we cannot use partial integrationto obtain the equations of motions for Xi.

We will now study the action 3.15 in more detail. First, note that the freepart of the action is given by:

14πα′

∫d2η∂αξa∂αξa.

So, we will start by calculating the propagator. We will follow Ryder [Ry].First, we include a source J

Z0[J ] =∫Dξ exp i

∫d2η

(1

4πα′∂αξa∂αξa + ξaJa

).

Since we consider a world sheet without boundaries, the boundary term frompartial integration vanishes. This gives us:

Z0[J ] =∫Dξ exp−i

∫d2η

(1

4πα′ξa∂α∂αξa − ξaJa

).

We now write our field ξa like

ξa → ξa0 + ξa.

Doing a double partial integration on the term ξa0 ξa gives:

Z0[J ] →∫Dξ exp−i

∫d2η

(1

4πα′(ξa¤ξa + 2ξa¤ξa

0 + ξa0¤ξa

0 )− ξaJa − ξa0Ja

).

Now choose ξa0 to be a solution of the equation:

¤ξa0 = 2πα′Ja.

This gives∫Dξ exp−i

∫d2η

(1

4πα′ξa¤ξa − 1

2ξa0Ja

).

The solution to the above equation of ξa0 is given by:

ξa0 = −

∫∆F (x− y)J(y),

where ∆F (x− y) is the Feynman propagator satisfying:

¤∆F (x− y) = −2πα′δ(x− y).

19

In complex coordinates, this yields

¤∆F (z − w) = −πiα′δ(z − w).

The inhomogeneous solution is given by:

∆F (z − w) = −α′i2π

∫d2p

eip·(z−w)

|p|2 ,

which can be seen by using the rule∫

d2peip·(z−w)

|p|2 = −(2π)2δ(z − w).

The propagator can also be written as

∆F (z − w) =α′

2ln |z − w|2,

which follows from the fact:

∂1z

=2π

iδ(z).

Note that both formulae for the propagators also give the identity

ln |z − w|2 =1πi

∫d2p

eip·(z−w)

|p|2 ,

which will be used later on.We might run into infrared divergences later on, so it would be wise to see

how things change if a mass is introduced. The whole calculation goes the sameexcept for the fact that our propagator now becomes too:

∆(z − w) =α′

2πi

∫d2p

eip·(z−w)

|p|2 + m2, (3.18)

and cannot be written as an easy ln any more.

Let us finsh by considering the propagator in zero. It is clear that thepropagator at zero seperation suffers from IR and UV divergences, hence wework with . In two dimensions this expression is logarithmically divergent.However, if we work 2−2ε dimensions the expression becomes finite and can becalculated, by standard methodes, to give:

∆(0) = −2π(m2)−ε

(2π)1−εΓ(ε), (3.19)

where Γ is the gamma function. This expression can be expanded in ε to makethe divergence more eminent:

∆(0) = − 12ε− 1

2(γE + ln m2 − ln 4π) +O(ε), (3.20)

where γE ≈ 0.577216 is the Eulergamma constant. From this expansion weclearly see the UV and IR divergence appearing.

20

An important remark is in place now. Note that by taking the quadratic pieceof our action to be ∂αξa∂αξa we explicitly make our interaction part depend onthe spin connection in a non-covariant way. So, if we do calculations in this wayone has to pay close attention to these non-covariant terms. In fact, it turns outthat the results obtained are, in indeed, in some cases, non-covariant, which isquite surprising but not completely unexpected.

21

Chapter 4

The Computer Programand Calculation Method

The calculations have been done with the help of a mathematica notebook. Inthis chapter we will explain how the Mathematica notebook works and howit can be used. We will start by introducing the notation used. After this,we will discuss how all full Wick contractions are used and it is explained howconnected diagrams are selected. We finish with a short discussion on the Taylorexpansions used as well as how we keep track of the different orders of ourcalculation.

4.1 Dictionary

In order for the computer program to function more efficiently, we introduced aslightly different notation. In this subsection we will give the dictionary so thatthe reader can understand the different notebooks.

In the computer program we calculate all full Wick contractions of diagramsthat contain up to two actions and up to two energy momentum tensors. Thefields ξν(θ) are labelled as follows (n ∈ Z>0):

ξν(θ) ↔ ξν [θ, 0]∂nξν(θ) ↔ ξν [θ, n]∂

nξν(θ) ↔ ξν [θ,−n], (4.1)

where ν is a Lorentz index and θ is a complex variable (e.g. z or w). Thebackground fields Xµ and tensors Tµ1...µm depending on it are put in via:

Xµ(θ) ↔ Xµ[θ, 0]∂nXµ(θ) ↔ Xµ[θ, n]

∂nXµ(θ) ↔ Xµ[θ,−n]

Tµ1...µm(θ) ↔ Tµ1,... µm [θ]eνµ(θ) ↔ eν,µ[θ], (4.2)

22

where µ is a space-time index. Furthermore, we use the following conventions.For the first energy momentum tensor T (z):

z ↔ z1

ν1 ↔ a1

...νm ↔ am

µ1 ↔ i1...

µm ↔ im (4.3)

For the second energy momentum tensor T (z):

z ↔ z2

ν1 ↔ b1

...νm ↔ bm

µ1 ↔ j1...

µm ↔ jm (4.4)

For the actions:

ζ ↔ z3

ν1 ↔ c1

...νm ↔ cm

µ1 ↔ k1

...µm ↔ km

(4.5)

and

η ↔ z4

ν1 ↔ d1

...νm ↔ dm

µ1 ↔ l1...

µm ↔ lm. (4.6)

23

So, for example, the quadratic term from the first energy momentum tensorlooks like

δa1a2ξa1 [1, z1]ξa2 [1, z1], (4.7)

and the term

Rijkl∂Xi(ζ)Dξl(ζ)ξj(ζ)ξk(ζ) (4.8)

from the action corresponds in our program with:

Rk1,k2,k3,k4Xk1 [1, z3]ec1,k4 [z3]ξc1 [−1, z3]ec1,k2 [z3]ξc2 [0, z3]ec3,k3 [z3]ξc3 [0, z3],(4.9)

where we have explicitly written out the different vielbeins. Clearly, this can beeasily generalized to higher order by just continuing the labelling in the aboveway. For the calculations we have done however, these conventions suffice.

4.2 Computing the Wick Contractions

We will calculate the correlation functions via Wick contractions. The way thisis implemented is the following. Via a number of for-loops we select all possiblecombinations of two fields. After this we differentiate with respect to both andmultiply with the correct factor. In order for this to work correctly one, ofcourse, has to pay close attention to labels etc.

Let us explicitly write down the replacements:

ξν [n, zi]ξν [m, zj ] → (−1)n+1(n + m− 2)!(zi − zj)n+m

ξν [−n, zi]ξν [−m, zj ] → (−1)n+1(n + m− 2)!(zi − zj)−n−m

ξν [0, zi]ξν [0, zj ] → 12

ln |zi − zj |2

ξν [1, zi]ξν [−1, zj ] → −2π

iδ(zi − zj)

ξν [2, zi]ξν [−1, zj ] → −2π

iδzi(zi − zj), (4.10)

where in the first two steps, m ≥ 0 and n ≥ 0 are not both zero. After this isdone, we repeat the process until all fields are contracted. So, in each step twofields get replaced. We add all steps and set all remaining fields to zero. Thissuccessfully computes the correlation functions.

The run time for the calculations done vary from seconds to hours. Basically,the numbers of steps the program has to do to contract 2N fields is given by(2N2

)(2(N−1)

2

) · . . . · (42

)(22

). For the computers on which we did the calculation,

the complete computation of the full Wick contractions of 10 fields was severalhours. Again, the above procedure can easily be made in such a way that itdeals with more actions or operators, but that will clearly result in long runtimes.

24

4.3 Selecting Connected Diagrams

We only wish to take connected diagrams into account. This is done by thecomputer program in the following way. Suppose we have 3 points, labelledwith z1, . . . , zn. We include in any propagator ∆(zi − zj) a parameter λi,j

(i > j). In other words, if a diagram comes with a factor of λi,j it means thatthere’s a line between zi and zj . It is a simple observation that a diagram isconnected if and only if one can ”reach” any of the points zi starting from z1.This fact will be used.

The next step is to replace all powers of λi,j by λi,j itself, i.e. we replaceλm

i,j → λi,j for all m. This is done to make the ”Replace”-option from Mathe-matica function correctly. We now let Mathematica make the following replace-ments:

λi,jλj,k → λi,jλj,kλi,k, (4.11)

which corresponds to composition of lines. I.e. if there’s a line from zi to zj

and from zj to zk then there is also a line between zi and zk.After this step we have factors of λi,j in front of our diagram for any points

that are connected. So we will select the connected diagrams by differentiat-ing with respect to λ1,2 up to λ1,n for diagrams that depend on the variablesz1, . . . , zn. To avoid wrong numerical factors caused by differentiation, we firstrepeat the step in which we remove higher powers of λi,j , λm

i,j → λi,j for all m.We now give a short example of how this works. Consider diagrams that

arise by contracting

∂ξa(z)∂ξa(z)×Rijkl∂Xi(ζ)∂X l(ζ)ξj(ζ)ξk(ζ)×

∂ξb(w)∂ξb(w). (4.12)

There are two different full contractions possible:

Rijkl∂Xi(ζ)∂X l(ζ) ∂ξa(z)∂ξa(z)ξj(ζ)ξk(ζ)∂ξb(w)∂ξb(w) . (4.13)

and

Rijkl∂Xi(ζ)∂X l(ζ) ∂ξa(z)∂ξa(z)ξj(ζ)ξk(ζ)∂ξb(w)∂ξb(w) . (4.14)

The first one comes with a factor of λ1,2λ3,3λ1,2 and the second has a factor ofλ1,2λ1,3λ2,3. Doing the above procedure, we see that the prefactor of the firstterm does not change and the second term will keep the prefactor λ1,2λ1,3λ2,3.The terms correspond to the diagrams from figure 4.1. And we indeed see thatwe filter out the disconnected diagrams.

4.4 Taylor Expansions

In the conformal algebra, the operators always need to be evaluated in certainpoints. However, the operators are not evaluated in the right point from the

25

Figure 4.1: Connected and Disconnected Diagram

start. In order to cope with this, we use the Taylor expansions of the operators.Note that after applaying the background field method one also has to includethe equations of motion of the background field Xi. Here’s the Taylor expansionof an arbitrary quantity Q up to our order:

Q(Xi(θ)) = Q +2θ − z − w

2(∂iQ)∂Xi +

2θ − z − w

2(∂iQ)∂Xi +

+(2θ − z − w)2

8(∂i∂jQ)∂Xi∂Xi +

(2θ − z − w)2

8(∂i∂jQ)∂Xi∂Xj

+(2θ − z − w)

2(2θ − z − w)

2(∂i∂jQ)∂Xi∂Xj

+(2θ − z − w)

2(2θ − z − w)

2(∂iQ)Γi

jk∂Xj∂Xk

+(2θ − z − w)

2(2θ − z − w)

2(∂iQ)Si

jk∂Xj∂Xk + . . . , (4.15)

where, as before, have omitted the explicit dependence of the righthand side onthe point z+w

2 to avoid too cumbersome notation.The expansions of the derivatives of the background field are given by:

∂Xi(θ) = ∂Xi +(2θ − z − w)

2Γi

jk∂Xj∂Xk +(2θ − z − w)

2Si

jk∂Xj∂Xk + . . .(4.16)

and

∂Xi(θ) = ∂Xi +(2θ − z − w)

2Γi

jk∂Xj∂Xk +(2θ − z − w)

2Si

jk∂Xj∂Xk + . . .(4.17)

In the calculation one needs to uses the latter two expansions as well as thefirst for Q = Gij , Rijkl, e

ai , ωab

i , DiΦ, . . . and these expansions indeed give a con-tribution. These Taylor expansions are literally implemented in the computerprogram, as can be seen in the various notebooks.

4.5 Keeping Track of Orders

All the calculations must be done to a certain order. Of course one can selectby hand the different diagram that contribute to a certain order and put themin the program. However, it is more convenient to let the computer select theappropriate diagrams.

In our calculations there are basically two different expansion parameters.First, there is the string coupling α′. From formula 3.2, we see that every

26

propagator comes with a factor of α′, so if a term contains 2n fields ξa, then,after full Wick contractions, the propagators will give a factor of α′n. This,of course, basically comes down to multiplying every field ξa with a factor of√

α′. This is exactly what we’ve done and in this way one can easily select therelevant diagrams by sending α′m → 0 for too high orders.

The other ”parameter” we use, are derivatives of the background field: ∂Xi

and ∂Xj . The background field does not have a propagator and hence doesn’tget contracted. This is dealt with in a similar fashion as the string couplingconstant. We just multiply every Xi with a constant ρX and we can againselect the appropriate number of ∂Xi and ∂Xj . E.g. we calculate the conformalanomaly c up to order α′ and without Xs. So, in such a notebook, one includesa replace statement that sends ρX → 0 and α′m → 0 for m ≥ 2.

4.6 Final Remarks

After going through the above steps, the computer output will give the answerbut that’s not very tractable yet. We did calculations to go one order higherthan the calculations done, and the output was 250+ pages (A4-size), so theanswer needs to be simplified further. This is done in all the notebooks. Af-ter the above steps have been taken, we pull the output through a number ofreplace loops that, e.g. contract metric and Riemann tensors. After this, theanswer greatly reduces and the only thing left to do is explicit evaluation of theintegrals encountered. We have calculated the integrals separately by hand orvia Mathematica. Substituting the results of the integrals is the final part ofour notebooks.

Finally, let we remark that Mathematica is quite sensitive with replacementand in order to avoid mistakes, the replace loops should be checked thoroughly.We have used this program in all the calculations of the next chapter, but wehave also redone the calculations and checked the output by hand a lot of times,and we have found no discrepancies.

27

Chapter 5

Weyl Symmetry atQuantum Level

In this chapter we will consider the easiest correlation function one can consider〈1〉. It turns out that it is divergent and hence we need to renormalize theaction. We will do this in this chapter. We will, in fact, renormalize the ac-tion in such a way that 〈1〉=1, which turns out to be convenient in later chapters.

We calculate 〈0|eiSint |0〉. We will calculate this by considering all full Wickcontractions of each of the terms of

〈0|eiSint |0〉 = 1 + 〈0|iSint|0〉 − 12〈0|SintSint|0〉+ . . .

The way to go is using the BFM method. So, we use action 3.15 and considerthe fields ξa as our quantum fields. If we only consider terms up to order α′

without Xs, it is not hard to check that the order of the formula 3.15 is indeedcorrect. There will be infrared divergences as well as ultraviolet divergences, sowe will use propagator 3.2.

The calculation will be presented in detail. Let us first start with the NSLMwithout the dilaton and B-field in the conformal gauge. First, write out thecomplete action in terms of spin connections. The pullback Dαξa was definedas:

Dαξa = ∂ξa + ωabi ∂αXiξb.

After explicit substitution of this identity, we see that the only diagram withoutspin connection that will contribute is:and it corresponds to full Wick contractions of

i

4πα′

∫d2ζ[Rijkl∂αXi

B∂αX lBξjξk].

Evaluation of this diagram clearly gives the following contribution:

i

8π2

∫d2ζRij∂αXi

B∂αX lB

∫d2p

1|p|2 + m2

. (5.1)

28

Figure 5.1: Contribution to 〈1〉 Proportional to Rij

Figure 5.2: Spin Connection Contributions

Let us now focus on terms that contain the spin connection ωabi . There are two

diagrams contributing here as depicted in Figure 5.2. The first diagram comesfrom contraction of the following terms:

i

4πα′

∫d2η ωab

i ωacj ∂αXi∂αXjξbξc,

which, by explicitly inserting our propagator, can be seen to be equal too:

18π2

∫d2ηωab

i ωabi ∂αXi∂αXj

∫d2p

1|p|2 + m2

.

The second diagram is described by:

− 2(4πα′)2

∫d2ηd2ζωab

i (η)ωcdj (ζ)∂αXi(η)∂βXj(ζ)∂αξa(η)ξb(η)∂αξc(ζ)ξd(ζ).

The different contractions give, after a Taylor expansion:

24π2(4π)2

∫d2ηd2ζd2pd2q ωab

i (η)ωabj (η)∂αXi(η)∂βXj(η) (pαpβ − pαqβ)

eip·(η−ζ)

|p|2 + m2

eiq·(η−ζ)

|q|2 + m2.

Doing the ζ and the q integration leaves us with:

− 4(2π)2

4π2(4π)2

∫d2ηd2p ωab

i (η)ωabj (η)∂αXi(η)∂βXj(η)

pαpβ

(|p|2 + m2)2.

Using the identity

∂pα

1|p|2 + m2

= −2pα

(|p|2 + m2)2

29

and partial integration gives:

− 2(2π)2

4π2(4π)2

∫d2η ωab

i (η)ωabj (η)∂αXi(η)∂αXj(η)

1(|p|2 + m2)

,

which exactly cancels the contribution from the first diagram. So, we see thatthe spin connection does not contribute.

In other words, for the NLSM, we have:

〈1〉 = 1 +iI

8π2

∫Rij∂αXi∂αXj , (5.2)

where we define:

I :=∫

d2p1

p2 + m2. (5.3)

Note that this is UV and IR divergent. The easiest way to counter these diver-gences is by adding the counter term (before BFM, so X 6= XB here, clash ofnotations)

− I

8π2

∫Rij∂αXi∂αXj (5.4)

to the action. Notice that this term is one order in α′ higher than the NLSMaction. A big advantage of adding this term is that we obtain 〈1〉 = 1. Thisrenormalization of our action will be used in the next chapter.

Now, let us turn on the B-field in two dimensions and in the conformal gauge.The B-field will produce the diagrams that are depicted in figure 5.3 It is easy

Figure 5.3: Contribution from the B-Field

to see that these diagrams are the same as the ones from the spin connectioncalculation, so the first diagram gives:

18π2

∫d2ηDkS k

ij εαβ∂αXi∂βXj

∫d2p

1|p|2 + m2

.

The second diagram gives:

(d− 1)8π2

∫d2η SiklS

klj ∂αXi∂αXj 1

(|p|2 + m2),

30

where the identity:

εαβεγδ = δαγδβδ + δαδδβγ (5.5)

was used. We again encounter UV and IR divergences.

Let us now concentrate on the UV divergent part. For this we should actuallywork in 2− 2ε dimensions and hence use the semi-conformal gauge. In terms ofε, we obtain the following divergences of the vacuum expectation value:

〈1〉 = 1− 1ε

(i

8π

∫d2η

(Rij + SiklS

klj

)∂αXi∂αXj

− i

8π

∫d2ηDkS k

ij εαβ∂αXi∂βXj

).

To counter these divergences, we need to add the term

1ε

18π

(∫d2η

(Rij + SiklS

klj

)∂αXi∂αXj +

∫d2ηDkS k

ij εαβ∂αXi∂βXj

)(5.6)

to our action. However, things become more complicated when working in thesemi-conformal gauge, which we actually should be doing. So, let us considerthe action 2.21. Let us put the dilaton to zero and consider, for simplicity, thecase in which φ is constant. The effect of this is that we get a new propagatorwhich is:

∆ab(z − w) =δabeεφ

2ln |z − w|2. (5.7)

The vertex coming from Rijkl is now multiplied with a factor of e−εφ. If we takethis into account, we see that the diagrams that are contributing get additionalfactors eεφ. As a matter of fact, we now obtain the following divergent terms:

1ε

18π

(∫d2η

(Rij + e2εφSiklS

klj

)∂αXi∂αXj +

∫d2ηeεφDkS k

ij εαβ∂αXi∂βXj

)(5.8)

By carefully renormalizing the theory one can calculate the β functions andthey give ([CFMP],[deAlw]) (they include the dilaton):

βg := Rij + SiklSkl

j − 2DiDjΦ

βB := DkS kij − 2(DkΦ)Sk

ij

βΦ :=14R−D2Φ− (DΦ)2 +

112

SijkSijk (5.9)

In order for the theory to have Weyl symmetry these functions should vanish.In the article [A-GFM], the authors also do the calculation of 〈1〉 to one order

in α′ higher on RIcci flat manifolds. However, with the use of the computer(”VEVHO.nb”) we can also do this calculation for non-Ricci flat manifolds. Wealso put the B-field and the dilaton to zero. The result can be written as follows

31

(after some small tensor manipulations):

1ε2

(− 13

288RkiRkj +

196

D2Rij +124

DkDlRiklj

− 124

RklRiklj

)∂αXi∂αXj

+1ε

(124

RklRiklj − 148

RmkliRmkl

j

− 148

RmkliRmlk

j

)∂αXi∂αXj + finite.

Here, we only look at the UV divergences. Note that it is not clear how torenormalize the action now, since we are dealing with harder divergences. Oneprobably has to consider field redefinitions.

32

Chapter 6

Calculation of theConformal Algebra

In this chapter we discuss the calculation of the conformal algebra of the bosonicnon-linear sigma-model. As said before, we have implemented the biggest partof the computation into a Mathematica notebook. This notebook computesall possible Wick contractions, selects the connected diagrams and processesthe data even further by doing Taylor expansions where needed. The differ-ent Mathematica notebooks are included in this thesis on a CD-ROM. In thischapter, we will discuss the calculation of the conformal algebra. Most of thetime we will just give the computer output, but where there are subtleties, wewill go into more detail. The calculations will be done at lowest order, whichmeans: terms without ∂X to order α′, terms proportional to ∂X to order α′0.Furthermore, we will take the B-field to be closed.

6.1 Conformal Algebra in Flat Space

Before we have a look at the conformal algebra in a curved background, let usstart by studying it in flat space. The basic steps of our calculation, like Wickcontractions and Taylor expansions, are also present in this case. In flat spacehowever, the calculation is a rather trivial procedure, since for gij = ηij we arejust dealing with a free boson.

First off, the action is given by:

S =1

4πα′

∫

Σ

d2σηij∂αXi∂αXj . (6.1)

The corresponding equations of motion are given by:

∂∂Xi = 0. (6.2)

As seen before, the energy momentum tensor is given by:

T (z) =ηij

2πα′∂Xi∂Xj . (6.3)

33

We will, however, use the normal ordered version:

T (z) =ηij

2πα′: ∂Xi∂Xj : . (6.4)

As mentioned before, the propagator is given by:

∆ij(z − w) =α′δij

2ln|z − w|2. (6.5)

One can now calculate the OPE of T with itself by using Wick’s theorem, seeformula B.1.

: T (z)T (w) : =1

(2πα′)2: ηij∂Xi(z)∂Xj(z) :: ηkl∂Xk(w)∂X l(w) :

=ηijηkl

(2πα′)2

(: ∂Xi(z)∂Xj(z)∂Xk(w)∂X l(w) : + : ∂Xi(z)∂Xj(z)∂Xk(w)∂X l(w) :

+ : ∂Xi(z)∂Xj(z)∂Xk(w)∂X l(w) : + : ∂Xi(z)∂Xj(z)∂Xk(w)∂X l(w) : +

+ : ∂Xi(z)∂Xj(z)∂Xk(w)∂X l(w) : + : ∂Xi(z)∂Xj(z)∂Xk(w)∂X l(w) :

+ : ∂Xi(z)∂Xj(z)∂Xk(w)∂X l(w) : + : ∂Xi(z)∂Xj(z)∂Xk(w)∂X l(w) :

+ reg. (6.6)

We now explicitly substitute the expression for the propagator and obtain thefollowing:

T (z)T (w) =D

4π2(z − w)4+

24π2α′(z − w)2

: ηij∂Xi(z)∂Xj(w) :

+ reg. (6.7)

In order to recognize T (w) we have to make a Taylor expansion of T (z) aroundthe point w. This yields

T (z)T (w) =D

4π2(z − w)4+

24π2α′(z − w)2

: ηij∂Xi(w)∂Xj(w) : +

24π2α′(z − w)

: ηij∂2Xi(w)∂Xj(w) : + reg. (6.8)

From the above we deduce:

T (z)T (w) =D

4π2(z − w)4+

T (w)π(z − w)2

+∂T (w)

2π(z − w)+ reg. (6.9)

In this we recognize the conformal algebra (up to redefinition of T → T2π by

scalar multiplication). The central charge is given by 2D.One of the main goals of this thesis is to investigate if and under which

circumstances these properties carry over to the case of a curved background.To study this, one needs more tools but the main construction will be verysimilar to the one discussed above.

34

6.2 The Conformal Algebra in Curved Background

We now continue with the calculation of the conformal algebra for the non-linearsigma model in a curved background. To this end we use the action and energy-momentum tensor derived in the chapter on the background field method, i.e.action 3.15 and corresponding energy momentum tensor 3.17. However, in [BNS]the authors give a different way of dealing with the dilaton. It is this way thatwe will use and it is discussed briefly in the next subsection.

Recall that the conformal algebra is given by:

T (z)T (w) =c

(z − w)4+

2T ( z+w2 )

(z − w)2+ reg. (6.10)

We have set up our computer so that we can calculate correlation functions. Wecan compute different terms from the algebra by considering:

〈T (z)T (w)〉 =c〈1〉

(z − w)4+

2〈T ( z+w2 )〉

(z − w)2+ reg. (6.11)

We see that we need to calculate two different correlation functions involvingthe energy-momentum tensor, namely 〈T (z)T (w)〉 and 〈T ( z+w

2 )〉. We take theB-field to be closed, so it vanishes from our discussion. We also take the renor-malized action, i.e. we want 〈1〉 = 1.

Note that in their article, [BNS] do not renormalize the action and hencemiss Feynman diagrams. In the previous chapter we saw the we need to add acounter term:

I

8π2Rij

√hhαβ∂αXi∂βXj , (6.12)

where this expression is taken before the background field method is used. Notethat this also gives a contribution to T , namely

Tr = T +I

4π2Rij∂Xi∂Xj . (6.13)

This is the energy momentum tensor we will use. In the calculations we willmake a distinction between T and Tr, so that it becomes clear which terms comefrom renormalization.

The calculation is done with our computer program. It turns out that thereare two different non-zero contributions, namely

c =14π

6(z − w)4

(D

12π+

α′

2π

−D2Φ− (DΦ)2 +

14R

). (6.14)

and we have a contribution

14π2

∂Xi∂Xj z − w

(z − w)3(Rij − 2DiDjΦ) . (6.15)

We must note, however, that the S2 term in c was not calculated by us dueto lack of time, but rather quoted from [BNS], formula 3.24. The dimensionalterm in c can be countered by ghosts as usual.

35

So, in order for the bosonic sigma model to still satisfy the ”normal” confor-mal algebra with central charge 2D, we need the above terms to cancel. Thisgives us equations for the background field Xi. This will be discussed at theend of this chapter. We will continue with the details of the calculation. Thecalculation is not straightforward, even though not many authors seem to gointo details.

6.2.1 How to deal with the dilaton

In the article [BNS] the authors prove that, by adding ghosts, one can simplifythe calculation at hand. In fact, the authors prove that one can completelyremove the dilaton from the action at the cost of adding the following term tothe energy momentum tensor:

− 12π

∂∂Φ. (6.16)

This has the following effect after applying the background field method.

T =1

2πα′

(2gij∂XiDξj + DξaDξa + Rijklξ

jξk∂Xi∂X l +43Rijkl∂XiξjξkDξl

+13RijklDξiDξlξjξk − α′2DjDiΦ∂Xjξi + DkΦDDξk

+12(DiDjΦ)DD(ξiξj)

). (6.17)

In what follows, we will use this convention.

6.2.2 〈Tr〉In order to evaluate the algebra, we also need to compute 〈Tr〉. There are anumber of subtleties involved and they will be discussed in this section. Theonly thing that is mildly disturbing is that it turns out that 〈Tr〉 is non-zero.

For this calculation, there are four different cases, obviously. The completecalculation can be found in the notebook ”T.nb”.

Case 1: No X

This case is fairly easy. There are only covariant terms involved, since all non-covariant terms like the spin connection always come with a derivative of thebackground field. It is easy to see that there is only one contribution, depictedin the Figure 6.1 below. The cross stands for an insertion of T .It is proportional to (in coordinate space)

α′R∆(0)∫

d2ζδ(ζ)ζ2

, (6.18)

which is proportional (transformed to momentum space)

α′R∆(0)∫

d2pp2

|p|2 . (6.19)

By going to polar coordinates we see that it is quadratically divergent and thatits corresponding angle integral vanished. Hence we discard it.

36

Figure 6.1: Contribution to 〈T 〉 Proportional to R.

Case 2: ∂Xi∂Xj

Let us first consider the covariant terms. Also in this case there is a very limitednumber of possible diagrams. The diagrams coming from T that contribute aredepicted in Figure 6.2:

Figure 6.2: Covariant Contribution to 〈T 〉 Proportional to ∂Xi∂Xj .

Let us compute the diagrams. Note that evaluation of the first diagram is trivialand gives

−∆(0)2π

Rij∂Xi∂Xj . (6.20)

The last diagram also gives a non-zero contribution. This diagram gives:

i

3π2gijRklmn ∂Xi∂ξj∂Xkξlξm∂ξn, (6.21)

which amounts to

∆(0)3π

Rij∂Xi∂Xj . (6.22)

Adding gives the following, contribution:

−∆(0)6π

Rij∂Xi∂Xj . (6.23)

For the last step we look at all non-covariant terms. These terms come from anumber of diagrams. The computer gives the following output:

ωabi ωab

j ∂Xi∂Xj

(∆(0)2π

+i

8π2|ζ|2 −1

64π3(η − ζ)2ζη− ln |ζ|2δ(ζ)

4π

− i

32π2

δ(ζ)(η − ζ)η

− i

32π2

δ(η)(η − ζ)ζ

+ln |ζ − η|2δ(ζ)δ(η)

16π

). (6.24)

37

Using partial integration as well as the fact that∫

d2ζ1|ζ|2 =

2π

i∆(0), (6.25)

we see that these terms cancel among themselves, so we only have a non-zerocovariant contribution in this case.

Obviously, there are two contributions coming from the renormalization,namely the constant term:

− I

2π2Rij∂Xi∂Xj (6.26)

and a term coming from the action

I

4π2Rij∂Xi∂Xj

∫δ(ζ). (6.27)

We see that they cancel and hence that our renormalization has no influencehere.

Case 3: ∂Xi∂Xj

Let us split up this case also in covariant and non-covariant terms. The covariantcase is rather easy. There are two diagrams from T that contribute. Thecontributions can easily calculated to be:

i

8π2Rij∂Xi∂Xj

∫d2ζ

1ζ2

+i

6π2Rij∂Xi∂Xj

∫d2ζ∆(0)

1ζ2

, (6.28)

which vanishes for the same reasons as before. The counter terms come with thesame integrals and hence vanish also. The non-covariant terms go completelyanalogously as in the previous case. After evaluation of the integrals and somepartial integration where needed, we see that this term is proportional to theintegral:

∫d2ζ

1ζ2

, (6.29)

which was discarded. The term coming from the counterterm is also propor-tional to this integral and hence vanishes for the same reason.

Case 4: ∂Xi∂Xj

It is easy to see that the only terms contributing to this are diagrams thatcontain two insertions of the action, since the energy-momentum tensor onlycontains ∂X. As a consequence, we see that only one diagram will contributeExplicitly, we must compute the full Wick contractions of:

∂ξa(z + w

2)∂ξa(

z + w

2)ωbc

i (ζ)∂Xi(ζ)ξc(ζ)∂ξb(ζ)ωdei (η)∂Xj(η)ξe(ζ)∂ξd(ζ).(6.30)

Evaluation is fairly straightforward and it only gives rise to vanishing integrals.As a consequence, this term does not contribute.

38

Figure 6.3: Spin Connection Contribution Proportional to ∂Xi∂Xj in 〈T 〉

In brief, in this section we have calculated 〈T 〉 which turns out to be:

〈T 〉 = − I

12π2Rij∂Xi∂Xj − I

4π2Rij∂Xi∂Xj , (6.31)

which is non-zero and even divergent.

6.2.3 c, The Anomaly term without X

In this section we discuss the calculation of the anomaly term that does not con-tain partial derivatives of X. This term is proportional to 1

(z−w)4 . Here, thereare some subtleties concerning dimensional regularization. This calculation isin ”TTnoX.nb”.

The order to which this term is calculated is α′. First, let us do the cal-culation the same way [BNS] did, so we only do diagrams that do not containcounter terms. The computer program gives us the following output:

D

8π2(z − w)4− α′R∆(0)

12π2(z − w)4− 3α′DiDiΦ

4π2(z − w)4− 3α′DiΦDiΦ

4π2(z − w)4+

α′Rδ(z − ζ)48π2(z − ζ)(ζ − w)3

+α′Rδ(z − ζ)∆(0)

24π2(z − w)(ζ − w)2+

α′Rδ(z − ζ)48π2(z − ζ)2(ζ − w)2

+

α′Rδ(ζ − w)48π2(z − w)2(z − ζ)2

+α′Rδ(ζ − w)∆(0)

24π2(z − w)2(z − ζ)2+

α′Rδ(ζ − w)48π2(z − ζ)2(ζ − w)2

.(6.32)

Now, terms 5,7,8 and 10 are evaluated using the tools from Appendix C. Theygive a total contribution of

5α′R48π2(z − w)4

. (6.33)

Clearly, we are dealing with UV-divergent diagrams, since we are dealingwith internal contractions ∆(0). We deal with these singularities by using di-mensional regularization. The diagrams involved are depicted in Figure 6.4.Evaluation of diagrams c,d are straightforwardly given by:

−∆(0)1

12π2

R

(z − w)4. (6.34)

39

Figure 6.4: The Contributions to R in c.

Diagram a, however, also involves an integral that needs to be evaluated. Usingform 3.2 of the propagator, we obtain the following expression:

Ic =112

∆(0)(z − w)2

∂z∂w

∫d2ζd2pd2q

δαβpαqβeip(z−ζ)eiq(ζ−w)

|p|2|q|2 . (6.35)

Collins ([Co]), gives

Ic =112

δαβ

(z − w)2∆(0)∂z∂w

∫d2ζd2pd2q

pαqβαeip(z−ζ)eiq(ζ−w)

D|p|2|q|2 . (6.36)

We can now integrate out our the ζ-integral to give a delta function.

Ic =112

δαβ ∆(0)(z − w)2

∂z∂w

∫d2p

pαpβeip(z−w)

D(|p|2)2 . (6.37)

By using partial integration, together with D = 2 − 2ε, this can easily seen tobe

Ic =112

(1 + ε)∆(0)

(z − w)4. (6.38)

By also expanding ∆(0) in 2− 2ε dimensions,

∆(0) = − 12ε

+O(ε0), (6.39)

we see that the divergent terms (also the IR divergences) cancel and we are thusleft with only a finite contribution

Ic = − 124

R

(z − w)4. (6.40)

All the other terms/diagrams are all finite and can be calculated by using themethods from the Appendix and are also listed there. Summing up all thesecontributions finally gives us the result quoted before:

14π

6(z − w)4

(D

12π+

α′

2π

−D2Φ− (Dφ)2 +

112

R

). (6.41)

Which does not coincide with [BNS]. Note that the IR divergences cancel amongstthemselves.

40

However, we still need to include the contributions from the counter terms. Thecontribution from the counter terms is given by:

I

8π3

R

(z − w)4− IR

8π3

δ(z − ζ)(z − w)2(ζ − w)2

. (6.42)

These integrals are the same as encountered above and are easily seen to give acontribution of

R

8π2

1(z − w)2

. (6.43)

Note that the UV-divergences and IR again cancel amongst themselves andleave a finite contribution just as before. Adding this to the former equation wefinally obtain the correct result

14π

6(z − w)4

(D

12π+

α′

2π

−D2Φ− (Dφ)2 +

14R

). (6.44)

So, we see that [BNS] did not do this calculation right. The counter termsthat arise when one renormalizes the action leave a finite contribution and thiscontribution is needed in order to obtain the right answer.

6.2.4 〈T (z)T (w)〉In this section we will discuss the calculation of terms from 〈T (z)T (w)〉 that docontain derivatives of Xi. These calculations can be found in ”TTwithXX1.nb”and ”TTwithXX2.nb”.

Case 1:∂Xi∂Xj

In this section we discuss in detail the calculation of terms proportional to∂Xi∂Xj from 〈TT 〉. We have split up this section in two different parts. In thefirst part we treat the covariant terms and in the second part we treat termsproportional to the spin-connection, Christoffel symbols etc.

First, we take a look at the covariant terms. The computer gives:(

1α′2π2

− Rij

12π2− ∆(0)Rij

6π2− Rij

6π2− DiDjΦ

π2

)∂Xi∂Xj

(z − w)2

+(− 1

12π2

δ(z − ζ)(z − ζ)(ζ − w)

− 112π2

δ(ζ − w)(z − ζ)(ζ − w)

)Rij∂Xi∂Xj .(6.45)

The term Rij

6π2 is a left over finite contribution from the divergent terms, just asin the previous section. The term 1

α′2π2 is somewhat disturbing, but it vanishesif we take our background field X to satisfy the usual constraint T (X) = 0 nextto the equations of motion. The integrals give a net contribution of 1

12π2 , soafter evaluation of the integral, we are left with:

(−∆(0)Rij

6π2− Rij

6π2− DiDjΦ

π2

)∂Xi∂Xj

(z − w)2. (6.46)

41

Secondly, there non-covariant terms that are proportional to the dilaton. Theyare given by:

(DiDjΦ

2π2− 1

2π2Γk

ijDkΦ)

∂Xi∂Xj

(z − w)2(6.47)

where we’ve used the relations

eai ea

j = gij

ωaµb = ea

νeλb Γν

µλ − eλb ∂µea

λ

DiDjΦ = ∂iDjφ− ΓkijDkΦ, (6.48)

from Appendix A.

We will now take a closer look at the term proportional to ωabi ωab

j ∂Xi∂Xj . Thisterm will be calculated explicitly, since it is non-zero, which is unsuspected. Wewill do the calculation in steps, in each step we will include an extra Lagrangian.

The first step consists of the evaluation of diagrams a and b from figure 6.5.

Figure 6.5: Spin Connection Contribution

Diagram a contains the full Wick contractions of the following term:

1(2πα′)2

ωabi (z)ωac

j (z)∂Xi(z)∂Xj(z)ξb(z)ξc(z)∂ξd(w)∂ξc(w).

After making a Taylor expansion of ωabi (z)ωac

j (z)∂Xi(z)∂Xj(z) in the pointz+w

2 and doing full Wick contractions, we are left with

14π2

1(z − w)2

ωabi ωab

j ∂Xi∂Xj ,

where we have also added the diagram obtained by interchanging z ↔ w and wehave omitted the evaluation in the point z+w

2 to avoid cumbersome notation.Diagram b comes from:

1(2πα′)2

2ωabi (z)∂Xi(z)ξb(z)∂ξa(z)2ωcd

j (w)∂Xj(w)ξd(w)∂ξc(w).

Doing the same as above, we arrive at:

14π2

ωabi ωab

j ∂Xi∂Xj

(1

(z − w)2+

ln |z − w|2(z − w)2

).

42

Summing the two contribution, finally gives us the result:

12π2

ωabi ωab

j ∂Xi∂Xj 1(z − w)2

+1

4π2ωab

i ωabj ∂Xi∂Xj ln |z − w|2

(z − w)2. (6.49)

Figure 6.6: Spin Connection Contribution

We now move on to the more difficult case of figure 6.6. This diagram involvesone Lagrangian and consists of full Wick contractions of

i

4π3α′3∂ξa(z)∂ξa(z)ωde

j (w)∂Xj(w)ξe(w)∂ξd(w)∫

Σ

d2ζωbci (ζ)∂Xi(ζ)ξc(ζ)∂ξb(ζ)

+ z ↔ w (6.50)

Evaluation is fairly straightforward and gives:

−i

8π3ωab

i ωabj ∂Xi∂Xi

∫

Σ

d2ζ

(2π

i

δ(ζ − w)z − ζ

1z − w

+2π

i

δ(ζ − z)ζ − w

1z − w

1z − ζ

1(z − w)2

1ζ − w

+1

ζ − w

1(z − w)2

1z − ζ

).

These graphs do not suffer from infrared divergences and the integrals are eval-uated in Appendix A. The result is:

− 12π2

ωabi ωab

j ∂Xi∂Xj 1(z − w)2

− 12π2

ωabi ωab

j ∂Xi∂Xj ln |z − w|2(z − w)2

. (6.51)

The final diagram that needs to be evaluated is the diagram from figure 6.7.In this case we bring down the action twice, which gives:

−132π4α′4

∂ξa(z)∂ξa(z)∂ξb(w)∂ξb(w)∫

Σ

d2ζωcdi (ζ)∂Xi(ζ)ξd(ζ)∂ξc(ζ)

∫

Σ

d2ηωefi (η)∂Xi(η)ξf (η)∂ξe(η).

43

Figure 6.7: Spin Connection Contribution

Doing all the Wick contractions gives a total of 12 terms, which are given by:

132

δ(z − ζ)δ(z − η)1

ζ − w

1η − w

+132

δ(ζ − w)δ(z − η)1

z − ζ

1η − w

+132

δ(z − ζ)δ(η − w)1

ζ − w

1z − η

− 132

δ(ζ − w)δ(η − w)1

z − ζ

1z − η

− 164π3i

1(z − w)2

δ(z − ζ)1

ζ − η

1η − w

+1

32π2

1(z − w)2

δ(z − ζ) ln |ζ − η|2δ(η − w)

+1

128π4

1(z − w)2

1z − ζ

1(ζ − η)2

1η − w

− 164π3i

1(z − w)2

1z − ζ

1ζ − η

δ(η − w)

+1

64π3i

1(z − w)2

δ(z − η)1

ζ − η

1ζ − w

− 132π2

1(z − w)2

δ(z − η) ln |ζ − η|2δ(ζ − w)

+1

128π4

1(z − w)2

1z − η

1(ζ − η)2

1ζ − w

+1

64π3i

1(z − w)2

1z − η

1ζ − η

δ(ζ − w).

Using the integrals from the Appendix leaves us with:

18π2

ωabi ωab

j ∂Xi∂Xj 1(z − w)2

+1

4π2ωab

i ωabj ∂Xi∂Xj ln |z − w|2

(z − w)2. (6.52)

So, finally, summing all diagrams will give:

18π2

ωabi ωab

j ∂Xi∂Xj 1(z − w)2

. (6.53)

Now, this is a problem since we would like our algebra to be covariant. Finallylet us consider the contributions from the counter terms. They are given by:

− I

4π3

Rij∂Xi∂Xj

(z − w)2(6.54)

Summing all different contributions gives us:(− IRij

12π3− I

4π3

Rij∂Xi∂Xj

(z − w)2− Rij

6π2− ∂iDjΦ

2π2+

ωabi ωab

j

8π2

)∂Xi∂Xj

(z − w)2. (6.55)

44

Case 2:∂Xi∂Xj

This case is very similar to the above case. A big advantage of this term isthat all the integrals encountered are finite. Therefore, there are very littlesubtleties with this term and we can be quite brief. Taylor expansions and theequations of motion, however, do play a prominent role in this computation.The integral that are used can all be found in the appendix. It is easy to seethat the renormalization terms have no effect on this calculation since they canonly come with vanishing integrals. The covariant part of this anomaly wascalculated to be:

14π2

∂Xi∂Xj z − w

(z − w)3(Rij − 2DiDjΦ) (6.56)

The B-field term was taken from [BNS]. Let us now take a closer look at thenon-covariant parts. These terms also have trivial integrals and the sum of themis given by:

(2Γk

ijDkΦ + 2∂jDiΦ + 4eai ek

bωbjaDkΦ− 2ek

a∂jeai + 2(∂je

ka)ea

i

) ∂Xi∂Xj

4π2. (6.57)

At first sight does not seem to be zero, but if we take a closer look it actuallyvanishes. This can be seen by using the following relations, that are proved inAppendix A:

eai ea

j = gij

ωaµb = ea

νeλb Γν

µλ − eλb ∂µea

λ

DiDjΦ = ∂iDjφ− ΓkijDkΦ, (6.58)

which is a highly non-trivial cancellation.

Case 3:∂Xi∂Xj

Here we discuss the terms from 〈TT 〉 proportional to ∂Xi∂Xj . Again, renormal-ization does not play a role here. As stated before, without Taylor expansions,this can only come from diagrams containing two actions. Furthermore, as seenbefore, these terms will become proportional to ωab

i ωabj ∂Xi∂Xj . There are, in

principle, also terms that arise from doing Taylor expansions from terms thatcontain only one ∂Xi or no ∂Xis.

However, it is not hard to see that, to this order, there are no diagrams with-out ∂Xi that can contribute. Also, diagrams that are proportional to ∂Xi areeasily seen cancel also, since they are proportional to the full Wick contractionsof:

∂ξa(z)∂ξa(z)ωcdi (ζ)∂Xi(ζ)ξd(ζ)∂ξc(ζ)∂ξb(w)∂ξb(w), (6.59)

which is proportional to ωaai , and this vanishes because of anti-symmetry of the

spin connection. So, this leaves us with just one diagram, Figure 6.8

∂ξa(z)∂ξa(z)ωcdi (ζ)∂Xi(ζ)ξd(ζ)∂ξc(ζ)ωef

i (η)∂Xi(η)ξf (η)∂ξe(η)∂ξb(w)∂ξb(w).(6.60)

Letting the computer do the Wick contractions, we see that all the integrals thatare seen in this expression can be seen to be products of the integrals from theappendix or they can be computed by repeatedly doing the integrals from thesame appendix. Doing this, one sees that these terms cancel among themselves.

45

Figure 6.8: Spin Connection Contribution to ∂Xi∂Xj

6.2.5 Putting it all together

From all the calculations above, we see that there are only three types of termsthat remain. First, we have the central charge:

c =14π

6(z − w)4

(D

12π+

α′

2π

−D2Φ− (Dφ)2 +

14R

). (6.61)

Next, we have a term:

14π2

∂Xi∂Xj z − w

(z − w)3(Rij − 2DiDjΦ) . (6.62)

So, we see that in order for T to satisfy the ”normal” conformal algebra of flatspace, we need to make a redefinition T → T

2π . Furthermore, the backgroundfield should satisfy the following equations:

−D2Φ− (Dφ)2 +14R = 0

Rij − 2DiDjΦ = 0 (6.63)

We also had divergent contributions (IR and UV) proportional to ∂Xi∂Xj

coming from 〈T 〉 and 〈TT 〉, but we see that they appear on both sides of thealgebra with the same prefactors. In other words, these terms have no effect onthe algebra. There were, however, also non divergent terms, some of which wereeven non-covariant. However, these terms can easily be countered by shifting Twith a constant (with which we mean only depending on the background field)term. Such a term is easily seen not to affect the algebra. In other words, wesee that from the algebra all UV and IR divergences cancel and we are only leftwith equations 6.63 as restrictions on our background field.

6.2.6 Obtaining the Einstein Equations

Now that we have obtained equations that the background field has to satisfyin order for our bosoinc string to satisfy the conformal algebra, let’s see howthis affects some of the properties of our background field. This is discussed indetail in [CFMP].

For example, one can combine equations 6.14 and 6.15 to obtain the Einstein

46

equations:

0 = βGij +

8π2

α′gijβ

Φ

= Rij − 12gijR− Tij , (6.64)

where we have defined:

Tij = −2DiDjΦ+2gijDkDkΦ− 2gijDkΦDkΦ. (6.65)

From this we see that if we compute the anomalies to higher orders in α′, weobtain string corrections to the Einstein equations. This could be interestingfor further study. THis can be easily done by using the methods set up here.Also the computer program can do this calculation, however it takes some timeto process all the data.

47

Chapter 7

Topological Field Theories

In this section we will discuss aspects topological field theories. We will start bygiving some basic definitions and facts about topological field theories. After thiswe will discuss two examples: Chern-Simons theory followed by cohomologicalfield theories. We will finish by giving some basic properties of two dimensionaltopological field theories. Pretty much all that will be discussed here can befound in [Vonk]. Also a good reference is [DVV].

7.1 Definitions

The observables of any quantum field theory is given by correlation functionsof different physical operators Oi

〈O1 . . .On〉. (7.1)

What the physical operators are, of course, depends on the theory one considers.As we have seen in the previous chapters, a quantum field theory might dependon the choice of background manifold M . As we saw in the bosonic non-linearsigma-model, certain quantities like 〈TT 〉, depended explicitly on the curvatureof the background field.

Definition 7.1. A topological field theory is a quantum field theory in whichthe observables do not depend on the metric of the background manifold M .

From this definition we, for example, see that the bosonic non-linear sigmamodel is not a topological field theory on any manifold M . Note that the termtopological is not completely correct, since topology entails more than only themetric.

Finally, topological field theories have an interesting property. If the theoryis also coordinate invariant, which quantum field theories normally are, thenthe observables do not depend on the insertion points of the operators. Thiscan be seen by first doing a coordinate transformation, which transforms thecoordinates as well as the metric. But since the theory is metric independent,we see that the net effect is only a coordinate change.

48

7.2 Chern-Simons Theory

The easiest way to construct a topological field theory is by taking a theory inwhich the action measure (eiS) and the fields/operators do not contain the met-ric at all. These theories are called Schwartz-type topological field theories.

An example of such a theory is Chern-Simons theory on a threefold M . As iscommon in physics, we suppose that our manifold does not have a boundary, i.e.total derivatives vanish after integration. In particular, Chern-Simons theory isa gauge theory, so it is constructed from a vector bundle E over the base space Mwith structure group G and connection A. The Lagrangian for a Chern-Simonstheory is given by:

L = Tr(A ∧ dA +23A ∧A ∧A). (7.2)

We define the action to be:

S =k

4π

∫

M

L, (7.3)

with k ∈ Z. Under the gauge transformation A → gAg−1 − gdg−1, this La-grangian transforms like

L = Tr(A ∧ dA +23A ∧ A ∧ A)

= L+ dTr(gA ∧ dg−1) +13Tr(gdg−1 ∧ dg ∧ dg−1).

Plugging this in the action and using that the second term is a total derivativegives

δS =k

4π

∫

M

Tr(gdg−1 ∧ dg ∧ dg−1). (7.4)

It can be shown that this is a topological invariant of the map g(x) called thehomotopy class of the gauge transformation and that this gives

δS = 2πkm,

for some m ∈ Z. In other words, we see that, under these transformations,the action measure eiS is indeed invariant. From this we see that the partitionfunction

Z =∫DAeiS[A], (7.5)

is in fact a topological invariant of our manifold M .We will now consider the classical solution of the Chern-Simons action. Con-

sider a one-parameter family of connections A → A + εB. The fact that theaction S is extremal at a connection A corresponds to saying that d

dεS(A +εB)|ε=0 = 0 for any B. It is not hard to show that:

d

dεS(A + εB) =

k

2π

∫

M

Tr(B ∧ FA),

where FA = dA− A ∧ A is the field strength. Since B is arbitrary, we see thatflat connections are the classical solutions. Chern-Simons theory has importantrelations to knot theory. The interested reader could consider reading [Hu].

49

7.3 Cohomological Field Theories

7.3.1 Definitions

In this section we will discuss a type of topological field theories in which themetric can play a role. The theories discussed here are called cohomologicalfield theories or topological theories of Witten type. These theories look likeBRST theories.

Definition 7.2. A cohomological field theory (CHFT) consists of the fol-lowing data:

1. a fermionic symmetry operator Q, such that Q2 = 0 and δQδhαβ = 0

2. physical operators, O, in the theory satisfy Q,O = 0

3. the vacuum |0〉 is symmetric under Q, i.e. Q|0〉 = 0

4. the energy momentum tensor Tαβ := δSδhαβ can be written as Q, Gαβ for

some operator Gαβ.

Note that the brackets depend on the nature of the operator, they can be acommutator or anti-commutator.

The above definition allows us to define an operation d := Q, · on operators.By property 2, the physical operators are given by ”d-closed” operators, i.e.operators P for which dP = 0. Furthermore, since our operator Q annihilatesthe vacuum, we see that changing an operator by a d-exact operator will notchange correlation functions:

〈O1 . . .Oi + Q,O . . .On〉 = 〈O1 . . .On〉+ 〈O1 . . . QO . . .On〉 ± 〈O1 . . .OQ . . .On〉= 〈O1 . . .On〉+ 〈QO1 . . .O . . .On〉 ± 〈O1 . . .O . . .OnQ〉= 〈O1 . . .O1〉. (7.6)

Where, in step 2 we used property 2 and in step 3 we used property 3. In otherwords, we are only interested in closed operators modulo exact operators.

So, this means we only consider cohomology classes, hence the name co-homological field theories. From the above discussion, we see that we can re-formulate property 4 by requiring that the energy-momentum tensor is Q-exact.

We now have to check that this theory is a topological field theory, i.e. wehave to check that the observables are metric independent. Suppose that ourobservables and measure Dφ do not depend on the metric. So, we consider thefunctional derivative of an observable:

δ

δhαβ〈O1 . . .On〉 =

δ

δhαβ

∫DφO1 . . .OneiS[φ]

=∫DφO1 . . .On

δ

δhαβeiS[φ]

=∫DφO1 . . .OnQ, GαβeiS[φ]

= 〈O1 . . .OnQ, Gαβ〉= 0.

50

Note we have assumed that the operators are well-ordered in the sense that theirordering does not change if we go from operator formalism to the path-integralformalism. However, we see that the ordering of the operators does not playa role in this derivation, so this result also holds if the operators are ordereddifferently. So, we indeed see that this theory is a topological field theory.

An easy way to make sure that property 4 is satisfied, is to use a Lagrangianthat is Q-exact itself:

L = Q, V ,

for some operator V . From this choice it follows that property 4 holds if ouroperator commutes with integration, i.e. Q

∫M

=∫

MQ. In this case, the

quantum measure from the path integral can be written as

ei~Q,

RM

V .

If our operators do not depend on ~, we see by the same reasoning as abovethat

d

d~〈O1 . . .On〉 = 0.

In other words, we see that we can calculate correlation functions exactly in theclassical setting.

7.4 Two-dimensional TFTs

In this section we will restrict ourselves to the two-dimensional case. This is thesetting we are working in constantly.

Normally, if one calculates correlation functions, one must compute a path in-tegral like:

∫

BC,Σ

Dφ . . . eiS ,

where Σ is the world sheet (with boundary) and BC stands for the boundaryconditions. In this case, this can be written as

∫eΣDφ . . .OaeiS ,

where the operator Oa can, more or less, be seen as a characteristic functionthat goes with the boundary condition BC. Furthermore, Σ is obtained from Σby gluing a hemisphere onto each boundary circle. In other words, it suffices tostudy Riemann surfaces without boundaries. Some more details on this can befound in [Vonk], Chapter 3. So, from now on we will consider Riemann surfaceswithout boundaries.

Let’s now consider correlation functions in a two dimensional topologicalfield theory. Recall that in we are considering theories in which the correlation

51

functions do not depend on the insertion points. So, in the case of a two di-mensional theory on a Riemann surface Σ, the correlation function 〈φ1 . . . φn〉Σonly depends on the labels 1, . . . , n and on the topology of Σ.

The next property of a two dimensional TFT is that all correlation functionscan be factorized. First, consider the two point correlation function on a sphere:

ηij = 〈φiφj〉Σ0 .

It turns out that this correlation function has some metric like properties (whencethe notation). For example, one can write unity as

1phys = |φi〉ηij〈φj |,

where ηij is the inverse of ηij (sum convention is used). This equation allows usto factorize other correlation functions. There are basically two different rulesfor this.

Consider n point correlation function on a Riemann surface Σ of genus g.The first one disconnects the Riemann surface Σ

Figure 7.1: Rule of TFT.

52

〈φ1 . . . φn〉Σ = 〈φ1 . . . φkφi〉Σ1ηij〈φjφk+1 . . . φn〉Σ2 , (7.7)

where the genera of Σ1,2 is g1,2, with g1 + g2 = g. The other rule reduces thegenus of Σ:

〈φ1 . . . φn〉Σ = (−1)Fiηij〈φiφjφ1 . . . φn〉Σ′ , (7.8)

where Σ′ is a Riemann surface of genus g − 1 and Fi is the fermion number ofthe field φi.

Figure 7.2: Rule 2 of TFT.

From these rules we easily deduce the remarkable property that every cor-relation function on a Riemann surface of a topological field theory can becompletely written in terms of two and three point correlation functions on thesphere.

53

Chapter 8

The N=(2,2) Non-LinearSigma Model

In this chapter we will discus the N = (2, 2) non-linear sigma-model. We willmainly do this from the superspace perspective. We will start by constructingthis sigma model. We will also discuss the symmetries and different currents.

8.1 Superspace and its Symmetries

We will start by considering N = (2, 2) supersymmetry in two dimensions.

8.1.1 Flat Case

Consider C with (real) coordinates z, z. We can construct a so called superspacefrom this by adding four real Grassmann variables θ±, θ

±. Under complex

conjugation θ+ maps to θ−

and θ− maps to θ+. They transform as spinors,

namely if z 7→ eiαz, then

θ± 7→ e±iα/2θ±

θ± 7→ e±iα/2θ

±. (8.1)

A superfield is a function Φ(z, z, θ±, θ±

) of these variables. If we consideran analogue of Taylor expansions in the superspace, i.e. we write:

Φ(z, z, θ±, θ±

) = φ(z, z) + ψ+(z, z)θ+ + ψ−(z, z)θ− + F (z, z) θ+θ− + . . . .

Since the θ-variables are anti-commuting, this is a finite expression, consistingof 16 terms. The fields acting as coefficients in the above expansion can benormal or Grassmann valued. In the rest of this paper we will only considerthe case in which the fields have the same statistics as the variables they are infront of, e.g. φ(z, z) is an ordinary function and ψ±(z, z) are Grassmann valuedfunctions.

Next, we consider actions consisting of superfields. Consider the action:

SD =∫

d2zd4θK(Φi, Φi), (8.2)

54

for some scalar function K. Terms in the action that are of this form are calledD-terms.

Let us now consider symmetries that leave the measure

dzdzdθ+dθ−dθ+dθ−

invariant. Since K is a scalar function, these symmetries also leave the actionSD invariant. It is easy to see that the ordinary Poincare group acting on z, zpreserves this measure. Write z = ix0+x1, then we can write the correspondinggenerators as follows:

H = −id

d(ix0)= −i(∂ − ∂)

P = −id

dx1= −i(∂ + ∂).

These symmetries just correspond to ordinary translations of the coordinatesz, z. Furthermore, there is a Lorentz rotation generator:

M = 2z∂ − 2z∂ + θ+ d

dθ+− θ−

d

dθ−+ θ

+ d

dθ+ − θ

+ d

dθ+ . (8.3)

These operators satisfy the following (anti-)commutation relations:

[M,H] = −2P

[M,P ] = −2H. (8.4)

Since we are also dealing with fermionic coordinates, we have other interestingsymmetries. We could, for example, shift the bosonic variable by a fermion, e.g.z 7→ z + cθ. Since θ is anti-commuting, we see that under this transformationdzdθ 7→ (dz + cdθ)dθ = dzdθ since θ is anti-commuting. Hence this defines asymmetry. Similarly, we also can shift the fermionic coordinates by a bosonicconstant θ 7→ θ + c. These symmetries are generated by the vectorfields:

Q± =∂

∂θ±+ iθ

±∂±

Q± = − ∂

∂θ± − iθ±∂±

D± =∂

∂θ±− iθ

±∂±

D± = − ∂

∂θ± + iθ±∂±. (8.5)

It is easily checked that these operators satisfy the following (anti-)commutationrelations:

Q±,Q± = P ±H

D±, D± = −(P ±H), (8.6)

55

the other anti-commutators vanish. These operators also have non-vanishingcommutators with M :

[M,Q±] = ∓Q±[M,Q±] = ∓Q±[M,D±] = ∓D±[M, D±] = ∓D±. (8.7)

So far, we have not made a real distinction between the D and the Q opera-tors. The action is still invariant under both transformations. Both types ofgenerators give us a supersymmetry, so now we would have something like aN = (4, 4) supersymmetry. As the title of this section indicates, we would liketo bring down the number of supersymmetries, so we place a restriction on thesuperfields. We define a chiral superfield to be a superfield Φ satisfying thefollowing condition:

D±Φ = 0.

Similarly, we can define anti-chiral superfields to be superfields Φ that satisfythe condition:

D±Φ = 0.

From now on, we shall impose the condition that we will be dealing with chiralfields. It is easy to see that this reduces the number of degrees of freedom ofΦ from 16 to 4. The chirality condition can be worked out explicitly in termsof the coefficients of the superfield. Doing this gives that the only degrees offreedom that are left are the fields φ, ψ+, ψ− and F , where the latter is the termstanding in front of θ+θ−. All the other fields depend on these fields in thefollowing way:

Φ++ = −i∂+φ

Φ−− = −i∂−φ

Φ+−+ = i∂+ψ−Φ+−− = −i∂−φ+

Φ+−+− = ∂+∂−φ,

where Φ++ stands for the field in front of the term θ+θ+

etc. The fields thatdo not occur in the equations above vanish identically.

From the anti-commutator between the Q- and D-operators, we see that aQ transformed chiral superfield is still chiral. We also see that the operatorsP, H and M preserve chirality.

By the above, we can check how chiral fields transform under theQ-transformationsby only considering its action on the fields φ, ψ±, F . The transformations of theother fields are given by the relations chirality imposes. If we write out thetransformations explicitly we see that the above operators act on chiral fields

56

as follows:

Q+ : δφi = α−ψi+

δψi− = α−F i

δψi

+ = 2iα−∂+φi

δFi

= 2iα−∂+ψi

−

Q− : δφi = α+ψi−

δψi+ = −α+F i

δψi

− = 2iα∂−φi

δFi

= −2iα+∂−ψi

+

Q+ : δψi+ = −2iα−∂+φi

δF i = 2α−∂+ψi−

δφi

= −α−ψi

+

δψi

− = −α−Fi

Q− : δψi+ = 2iα−∂−φi

δF i = 2α−∂−ψi+

δφi

= −α−ψi

−

δψi

+ = α−Fi. (8.8)

The parameters α±, α± are chosen in such a way that the symmetries corre-spond to those given by Aspinwall [As]. Unfortunately, the ± part is a bitcounterintuitive here.

We will now evaluate the action 8.2 further. First, notice that by the rules ofintegration of Grassmann-variables, we see that the only term in the Taylor-expansion that remains in the action is the one standing in front of θ+θ−θ

+θ−

.De field F only appears quadratically and hence, we replace it with its equationsof motion, which are given by:

F i = −Γijkψjψk. (8.9)

Doing the explicit Taylor-expansion of K(Φ,Φ), baring in mind the fact that thefields involved are chiral and adding some total derivatives, gives the followingLagrangian:

L = −gi∂αφi∂αφ

i − 2igiψi

−∆+ψj− − 2igiψ

i

+∆−ψj+ −Riklψ

i+ψk

−ψj

+ψl

−.(8.10)

57

In the above expression, we used the notation:

gi =∂2K

∂φi∂φj

Γijk = gilglj,k

Rikl = gmngm,lgni,k − gi,kl

∆±ψi = ∂±ψi + Γijk∂φiψk. (8.11)

We see that gi has the form of a Kahler metric with Kahler potential K andthe other expressions are the corresponding Christoffel symbols and Riemanntensor. For more details, see Appendix ??. So, we see thatif one considersD-terms of chiral superfields, one naturally gets a theory defined on a Kahlermanifold.

Since we have replaced F with its equations of motion, let us write downthe supersymmetry transformations with this substitution:

δφi = iα−ψi+ + iα+ψi

−

δφi

= iα−ψi

+ + iα+ψi

−δψi

+ = −α−∂φi − iα+Γijkψj

−ψk+

δψi

+ = −α−∂φi − iα+Γı

kψ

j

−ψk

+

δψi− = −α+∂φi − iα−Γi

jkψj−ψk

+

δψi

− = −α+∂φi − iα−Γı

kψ

j

−ψk

+. (8.12)

Action 8.10 clearly inherits all the supersymmetries from superspace. However,one can of course also check explicitly that it is invariant under 8.12.

8.1.2 General Case and R-Symmetry

The map φ(z, z) := (φ1, . . . , φj) above can be thought of as a map going from Cto Cn with Kahler metric gi. To generalize this, we consider a Riemann surfaceΣ and a Kahler manifold M of dimension n. Consider the following map:

φ : Σ −→ M.

Consider a coordinate patch (U, xi, xj) on M . We can now define the fields φon Σ as pullbacks of the coordinates:

φi := φ∗(xi)

φi:= φ∗(xi). (8.13)

Similarly, the fields ψ can be viewed as sections of the following bundles on Σ:

ψi+ ∈ H0(K

12 ⊗ φ∗T 1,0M)

ψi− ∈ H0(K

12 ⊗ φ∗T 1,0M)

ψi

+ ∈ H0(K12 ⊗ φ∗T 0,1M)

ψi

− ∈ H0(K12 ⊗ φ∗T 0,1M), (8.14)

58

where K is the canonical bundle of Σ, i.e. the holomorphic part of the cotan-gent bundle and K is the anti-holomorphic part of the cotangent bundle. Themeaning of these notation is as follows. The H0 stands for global sections.The φ∗T 1,0M resp. φ∗T 0,1M stand for the pull-back of the (anti-)holomorphictangent bundle of M and this just makes sure that our fields here have upperindices, e.g. φ∗T 1,0M means we have an upper index i and φ∗T 0,1M means wehave an upper index ı. The bundle K

12 just means that we are dealing with

spinors.Note that since Σ is a Riemann surface, it is Kahler and hence K−1 = K.

Other choices of bundles can be made also and they lead to the so-called A andB-model.

Clearly, locally one can write down the Lagrangian 8.10. In order to be welldefined, we need to check that it is invariant under change of charts. One must,however, note that the charts involved are not only the coordinate charts ofΣ. Since we are dealing with pullback bundles, we also get transition functionscoming from their trivialisations. It is a tedious, yet straightforward calculationto show that Lagrangian 8.10 is globally defined.

Note that the supersymmetry transformations 8.8 still make sense locally.We shall assume that they still define global symmetries. Note, that from thisassumption we see that our spinors α±, α± are now sections of the followingbundles:

α−, α− ∈ H0(M, K−1/2)α+, α+ ∈ H0(M, K1/2)

In general, however, it is not always possible to choose our parameters covari-antly constant, globally defined spinors, since there simply might not be globalsections.

In the sections below, the A- and B-model will be discussed and a similarchecks considering global aspects have to be made there also. In order to avoiddoing the, more or less, same calculation three times, we will only do the explicitcalculation in the case of the B-model.

Finally, there is a third symmetry that one can consider, called R-symmetry.We will first discuss this symmetry locally, but we will see that imposing thissymmetry at the quantum level this symmetry will have consequences globally.Consider the following U(1)-rotations:

RV (α) : (θ+, θ+) 7→ (e−iαθ+, eiαθ

+),

(θ−, θ−

) 7→ (e−iαθ−, eiαθ−

)

RA(α) : (θ+, θ+) 7→ (e−iαθ+, eiαθ

+),

(θ−, θ−

) 7→ (eiαθ−, e−iαθ−

).

These rotations clearly leave the D-term 8.2 invariant. These symmetries aregenerated by the following vector fields:

FV = −θ+ d

dθ+− θ−

d

dθ−+ θ

+ d

dθ+ + θ

− d

dθ−

FA = −θ+ d

dθ++ θ−

d

dθ−+ θ

+ d

dθ+ − θ

− d

dθ− .

59

Hence, we can write:

RV : δψ+ = −αψ+

δψ− = −αψ−δψ+ = αψ+

δψ− = αψ−

RA : δψ+ = −αψ+

δψ− = αψ−δψ+ = αψ+

δψ− = −αψ−, (8.15)

where α is a bosonic parameter. Again, for these symmetries to be global, weneed to choose α to be a scalar function.

Let us also write down the commutation relations:

[FV ,Q±] = Q±[FV ,Q±] = −Q±[FA,Q±] = ±Q±[FA,Q±] = ∓Q±, (8.16)

and similar expressions for the commutators with D. Now, the N = (2, 2)conformal algebra is defined by the (anti-)commutators between P, H,M,Q±,Q±, FA, FV .

By using the explicit form of the Lagrangian 8.10, one can prove that RV -symmetry is always present at the quantum level and that RA-symmetry is onlyvalid if our target space M , is a Calabi-Yau manifold. A discussion on this canbe found in [Vonk], Chapter 5.

Let us finish this discussion by adding a B-field term. Let B be a closed (1, 1)-form (i.e. dB = 0) on M and consider its pull-back. This pull back will beadded to the action:

SB =∫

Σ

d2σBij(∂φi∂φj − ∂φi∂φ

j)

=∫

Σ

d2σBijεαβ∂αφi∂βφ

j. (8.17)

One can prove that even when this term is added, the action is still supersym-metric under 8.12. Closedness of the B-fields plays a role in this.

8.2 Currents

For each of the symmetries discussed above, we can calculate its classical Noethercurrent. It is, however, a tedious calculation to vary the action corresponding to8.10. One has to bare in mind that δgi = gi,kδφk + gi,kδφ

kand write the Rie-

mann tensor and the covariant derivatives ∆± explicitly in terms of the metric.

60

To avoid tedious calculations we will explicitly derive the currents correspondingto the R-symmetry and leave the rest for the reader.

First the RV -transformations. Suppose the parameter α is local, i.e. itdepends on the local coordinates. Varying the action gives:

δS = −∫

Σ

(giψj

−ψi−)∂+α + (giψ

j

+ψi+)∂−α.

If we define J(z, z) = giψj

−ψi− and J(z, z) = giψ

j

+ψi+, then partial integration

gives:

δS =∫

Σ

α(∂+J + ∂−J).

Similarly, the RA-symmetry gives

δS =∫

Σ

α(−∂+J + ∂−J).

So, from this we conclude that J is holomorphic and J is anti-holomorphic. Toconclude we have a conserved current given by

J(z) = giψi−ψ

j

−

J(z) = giψi+ψ

j

+.

As said above, this discussion can also be done for the other symmetries, butthey involve quite some work. We will list the results here.

The current corresponding to Q+:

G+(z) =12giψ

i+∂φ

j.

The current corresponding to Q−:

G−(z) =12giψ

j

+∂φi.

The current corresponding to diffeomorphism invariance:

T (z) = gi∂φi∂φj+

i

2giψ

i+∂ψ

j+

i

2giψ

j

+∂ψi.

The fermionic part of T best be derived by writing noting that the fermionicpart is proportional to

hαβψjγα∂βψi, (8.18)

where

ψ =(

ψ−ψ+

)(8.19)

is a Majorana fermion. Furthermore, in two dimensions, the gamma matricesfor the Euclidean metric are given by:

ρ0 =(

0 11 0

), ρ1 =

(0 i−i 0

). (8.20)

61

Each of the above currents has a similar right-moving counterpart, just like J(z)has J(z).

One of the most important goals for the remainder of this thesis is to studywhen these currents satisfy the N = (2, 2)-conformal algebra, or to be moreprecise study when when the currents satisfy the OPEs corresponding to thissuperconformal algebra. In order for them to satisfy the conformal algebraprecisely, we actually need to make a small redefinition as follows:

T =1α′

(gij∂Xi∂Xj+

i

2gijψ

i+∂ψ

j+

i

2gijψ

j

+∂ψi)

G+ =

√2i

1α′

gijψi+∂φ

j

G− =

√2i

1α′

gijψj

+∂φi

J = − i

α′gijψ

i+ψ

j. (8.21)

8.3 The A-Model

As said before, we could choose our ψs to be sections different bundles than theone in 8.14. We can, more or less ’twist’ the bundles by K1/2 and consider thefollowing case:

ψi+ ∈ H0(K ⊗ φ∗T 1,0M)

ψi− ∈ H0(φ∗T 1,0M)

ψi

+ ∈ H0(φ∗T 0,1M)

ψi

− ∈ H0(K ⊗ φ∗T 0,1M). (8.22)

Again, locally, one can write down Lagrangian 8.10 and it still makes senseand one can check that it is indeed globally defined. It also follows naturallyfrom the supersymmetry transformations that the parameters α±, α± shouldlive in different bundles now in order for it to define a global symmetry. To beprecise, α−, α+ should be functions, α+ should be a section of K and α− shouldbe section a of K−1. In general, however, K and K−1 need not have globalsections. It is custom to rename the fields involved as:

ψiz := ψi

+

χi := ψi−

χı := ψi

+

ψız := ψ

i

−,

which better underlines their properties. Note that from this notation we cansee χ := (χi, χ) as a section of φ∗TM . After partial integration, we can rewriteour Lagrangian as:

L = −2t(gi∂zφi∂zφ

j+ gi∂zφ

i∂zφj+ igiψ

iz∆zχ

+ igiψz∆zχ

i +12Riklψ

izψ

zχ

kχl)

62

Next, we want to make the A-model into a topological field theory by makingit into a cohomological field theory. So, we need an fermionic operator thatsquares to zero. We can consider the following operator:

QA = Q+ −Q−.

This choice of operator comes down to setting α+ = α− = 0 and α− = α+ = α.This makes sense because as discussed above, α+ and α− need not exist. Fromthe definitions from the previous paragraph, we see that QA squares to zero. Infact, we reduce the number of super symmetries to one, namely QA, which isalways globally defined.

It turns out that the Lagrangian is almost QA exact. Consider the operator:

VA := gi(ψiz∂zφ

j+ ∂zφ

iψz).

Up to multiples of the ψ equations of motion, we can write:

L = −itQA, V + 2tφ∗ω.

It is clear that the second term does not depend on the metric and by thestandard arguments, given in the previous chapters, we see that we are dealingwith a topological field theory. Also notice that the second term only dependson the Kahler structure of our target space. Furthermore, notice that since, asalways, we consider M without boundaries, the integral of the Kahler form onlydepends on its cohomology class and the homotopy class of φ. The dependenceof the complex structure is in V , but since this part of the Lagrangian is QA-exact it does not contribute to correlation functions. So if we consider the pathintegral, we can calculate this part exactly by taking the classical t →∞ limit.In other words, we only integrate over fields that satisfy:

∂−φ(z, z) = 0∂+φ(z, z) = 0,

i.e. we only integrate over holomorphic maps φ.Let us now take a closer look at the physical operators of this model. We

will only consider so-called local operators, i.e. operators that only dependon the fields φ and χ and their anti-holomorphic counterparts. Let W =WI1...IndφI1 . . . φIn be an n-form on X. We can assign a local operator to thisform in the following way:

OW = WI1...InχI1 . . . χIn .

By the explicit form of the supersymmetry transformation, it is easy to see that

Q,OW = −OdW ,

where d is the ordinary exterior derivative. So by the above identification, wesee that we can identify physical operators with the de Rahm cohomology:

Hphys∼=

6⊕

i=1

HidR(X).

63

8.4 The B-Model

For the B-model we consider a different twist, i.e. we take

ψi+ ∈ H0(K ⊗ φ∗T 1,0M)

ψi− ∈ H0(K ⊗ φ∗T 1,0M)

ψi

+ ∈ H0(φ∗T 0,1M)

ψi

− ∈ H0(φ∗T 0,1M). (8.23)

Just as in the A-model is custom to rename our fields:

ηı := ψi

+ + ψi

−

θi := gi(ψj

+ − ψj

−)

ρiz := ψi

+

ρiz := ψi

−.

If we now rewrite our Lagrangian in terms of these fields, we obtain (afterincluding a coupling constant t):

L = −t(gi∂αφi∂αφ

i+ igiη

(∆zρiz + ∆zρ

iz) + iθi(∆zρ

iz −∆zρ

iz) +

12R l

ik ηρizρzkηθl).

Again, in this case one can explicitly check that 8.24 is globally defined.

We would like to fit this model in a topological field theory, just like theA-model. This time, we consider the operator:

QB := Q+ +Q−.

It clearly squares to zero.Just as in the case of the A-model we would like to see that our Lagrangian

is QB-exact, but just as in the A-model we can only partly succeed in doingthis. Consider the operator:

VB = gi(ρiz∂zφ

j+ ρi

z∂zφj).

By computing the commutator

L′ = −itQB , V = −itδV,

we see that up to partial integration, we can write:

L = L′ − t(iθi(∆zρiz −∆zρ

iz) +

12R l

ik ηρizρ

kzηθl). (8.24)

In fact, this shows that we can write our action as:

S = −it

∫

Σ

d2zQB , V − t

∫

Σ

d2zU.

One can show that the second term in 8.24 is metric independent (allthoughit isn’t known to us how to do this), so we indeed see that our theory is topo-logical.

64

Just as in the A-model, we can be more explicit about the physical operatorsand the correlation functions. In this case, consider an operator of the form:

OW := Wj1...jq

ı1...ıpηı1 . . . ηıpθj1 . . . θjq

.

Let us now make the following identification:

ηı 7→ dφı

θi 7→ d

dφi,

so our operator OW corresponds to a∧q

T (1,0)M -valued (0, p)-form. From theexplicit transformations induced byQB , we now see that our symmetry operatorcorrespond to the Dolbeault operator:

QB ,OW = −O∂W .

So, from this we conclude that the physical operators correspond to the Dol-beault cohomology on form valued in the exterior powers of the holomorphictangent bundle.

As is clear from the above discussion, we can again calculate correlationfunctions by considering the classical t → ∞ limit. This time, however, theequations for φi and φı become even better:

∂+φi = ∂−φi = ∂+φı = ∂−φı = 0.

This means that we only integrate over globally constant functions, i.e. over themanifold M itself. So, we can now calculate correlation functions by integratingthe corresponding operators (

∧qT (1,0)M -valued (0, p)-forms) over the manifold

M . Well, clearly the only non-zero correlators are the ones in which (0,m)-forms are integrated. Now, however, we encounter a difficulty. Since we onlycan integrate (m,m) forms, we need to adjust the

∧mT (1,0)M -valued (0,m)-

form so that it becomes a (m,m) form. Since our target space is Calabi-Yau,we have a natural (m, 0) form Ω on it. We can now make a (m,m) form by firstcontracting with Ω and then wedging with Ω:

W j1...jm

ı1...ım→ W j1...jm

ı1...ımΩj1...jmΩk1...km .

It turns out that this is indeed the correct way to solve this problem. So,we arrive at the conclusion that correlation functions in the B-model are justgiven by integrals of wedge products of forms over the target manifold. So, itis much simpler to calculate correlation functions in the B-model than in theA-model. The A-model is, however, mathematical much more interesting thanthe B-model. So, one would like to calculate things in the B-model and thenrelate them to the A-model. In a lot of case such things can be done. This isthe result of mirror symmetry.

Remark. As is clear from the above discussion, locally the A and B modelare exactly the same. They have the same Lagrangian and hence the sameequations of motion. The only point in which they differ is the BRST-symmetryone considers. As one sees from the definition of a topological field theory, this

65

only plays a role in the definition of what our physical operators are. That thisstill have major consequences is clear from the fact that correlations functionsin the B model are much more easy to calculate than correlation functions ionthe A model.

66

Chapter 9

The SuperconformalAlgebra

In this chapter we will discuss the superconformal algebra. A good reference forthis is [Greene] and the reader can find more details there..

9.1 The Algebra

A convenient method of writing the algebra is by giving operator products ex-pansions.

Let’s start with the N = 0 conformal algebra. This has already been dis-cussed in the first chapter. This conformal algebra is generated by the energy-momentum tensor T (z). It has the following operator product expansion (OPE):

T (z)T (w) =c/2

(z − w)4+

2T (w)(z − w)2

+∂wT (w)z − w

+ . . . ,

where c is called the central charge of the theory. We can now expand T (z) inthe modes Ln as follows:

T (z) =∑

n∈ZLnz−n−2.

From the above operator product expansion, we then see that the we get thefollowing commutator relations:

[Ln, Lm] = (n−m)Ln+m +c

12n(n2 − 1)δn+m,0, (9.1)

which is the well-known Virasoro algebra from string theory.If we now continue to the N = 1 superconformal, then we have to include

the worldsheet superpartner of T (z). If denote this generator by G(z), then theN = 1 superconformal algebra, by definition, is given by the following OPEs:

T (z)G(w) =3/2

(z − w)2G(w) +

∂wG(w)z − w

+ . . .

G(z)G(w) =2c/3

(z − w)3+

2T (w)z − w

+ . . . . (9.2)

67

The OPE of T with itself remains unchanged.We are however mainly interested in the N = 2 superconformal algebra.

This algebra contains, next to the energy-momentum tensor T (z), two weight3/2 supercurrents G1(z) and G2(z) and an additional field J(z). The field Jhas conformal weight one and is a U(1) current. These currents each satisfy therelations given above, i.e. both G1 and G2 satisfy the relations given above forG. The product between the G1 and G2 is given by:

G1(z)G2(w) =2c/3

(z − w)3+

2T (w)z − w

+ i

(2J(w)

(z − w)2+

∂wJ(w)z − w

).

The OPEs including J are given by:

T (z)J(w) =J(w)

(z − w)2+

∂wJ(w)z − w

+ . . .

J(z)G1(w) =iG2(w)z − w

+ . . .

J(z)G2(w) =−iG1(w)

z − w+ . . .

J(z)J(w) =c/3

(z − w)2+ . . . .

The OPEs between the J and the G currents can be written down more elegantlyby defining:

G± =1√2(G1(z)± iG2(z)).

Under this relabelling we get:

J(z)G±(w) = ±G±(w)z − w

+ . . . .

It is also not hard to check that the OPE of G± with itself is still given by9.2. So to summarize, the N = 2 superconformal algebra is generated by thecurrents, T (z), G±(z), J(z) and the OPEs are given by:

T (z)T (w) =c/2

(z − w)4+

2T (w)(z − w)2

+∂wT (w)z − w

+ . . .

T (z)G±(w) =3/2

(z − w)2G±(w) +

∂wG±(w)z − w

+ . . .

T (z)J(w) =J(w)

(z − w)2+

∂wJ(w)z − w

+ . . .

G±(z)G±(w) =2c/3

(z − w)3+

2T (w)z − w

+ . . .

G+(z)G−(w) =2c/3

(z − w)3+

2J(w)(z − w)2

+2T (w) + ∂wJ(w)

z − w+ . . .

J(z)G±(w) = ±G±(w)z − w

+ . . .

J(z)J(w) =c/3

(z − w)2+ . . . .

68

We will now expand the currents into modes and extract the commutationrelations from the above OPEs. First of all, we have the expansions:

J(z) =∑

n

Jnz−n−1

G±(z) =∑

n

G±n±az−(n±a)−3/2.

Notice that in the expansion of the G currents, there is a parameter 0 ≤ a < 1.This parameter controls the boundary conditions on these currents. Considerthe map z 7→ e2πiz. Under this map, we have:

G±(z) 7→ G±(e2πiz) = −e∓2πiaG±(z).

In the rest of this section we shall restrict our attention to the cases in which a isintegral or half-integral. These cases correspond to periodic (Neveu-Schwartz)and anti-periodic (Ramond) boundary conditions.

In terms of these modes, the N = 2 superconformal algebra is given by:

[Lm, Ln] = (m− n)Lm+n +c

12m(m2 − 1)δm+n,0

[Jm, Jn] =c

3mδm+n,0

[Ln, Jm] = −mJm+n

[Ln, G±m±a] =(n

2− (m± a)

)G±m+n±a

[Jn, G±m±a] = ±G±m+n±a

G+n+a, G−m−a = 2Lm+n + (n−m + 2a)Jn+m +

c

3

((n + a)2 − 1

4

)δm+n,0.

(9.3)

9.2 Representations

Physical properties of theories that contain the N = 2 superconformal symme-try can be studied by considering the representation theory of the correspondingalgebra. The representation will be studied by using creation and annihilationoperators.

As in the previous section, let’s start with the N = 0 case. In this case weneed to consider the algebra 9.1. Consider an eigenstate |s〉 of L0, such thatL0|s〉 = s|s〉. Let us also assume the following conditions:

L†m = L−m, L†m = L−m, (9.4)

from which we see that L0 is Hermitean.From the algebra we see that the state Lm|s〉 is again an eigenstate of L0,

but this state has eigenvalue s−m. From this we see that the operators Lm forpositive m can be viewed as annihilation operators and the operators Lm fornegative m can be viewed as creation operators.

If we now assume that the eigenvalues of L0 have a minimum hφ, withcorresponding eigenstate |φ〉, we have that Lm|φ〉 = 0 if m > 0. The state |φ〉

69

is called the highest weight state. So, if we have a highest weight state, wecan really view the operators Lm for m > 0 as annihilation operators and theoperators Lm for m < 0 as creation operators.

The same reasoning can be done for L0. Since L0 and L0 commute, we canpick a basis of simultaneous eigenstates. So, from now on we will assume thatwe have a state |φ〉, called the highest weight state such that:

L0|φ〉 = hφ|φ〉L0|φ〉 = hφ|φ〉,

which is annihilated by Lm and its anti-holomorphic counterpart for m > 0.The representation we now consider consists of states of the form

∏L−ni |φ〉

with ni < 0.

In conformal field theory there is a correspondence between operators and fields.Let φ be the operator corresponding to |φ〉, the condition that the state |φ〉 isa highest weight state is equivalent to the following OPE:

T (z)φ(w, w) =hφ

(z − w)2+

∂wφ(w, w)z − w

+ . . .

T (z)φ(w, w) =hφ

(z − w)2+

∂wφ(w, w)z − w

+ . . . .

Let us now generalize this to the N = 2 picture. Just as above, we would like todivide the operators into annihilation and creation operators. However, this timewe also have the operators Jn, G±n±a and their anti-holomorphic counterparts toconsider. We again make some sort of Hermiticity assumptions:

L†m = L−m, J†m = J−m, (G±m±a)† = G∓−m∓a. (9.5)

It turns out that we again take the ones with positive index n as annihilationoperators and the ones with negative index n as creation annihilators.

From the algebra 9.3 we see that L0, J0, L0 and J0 commute, i.e. we haveeigenstates |φ〉 such that:

L0|φ〉 = hφ|φ〉J0|φ〉 = Qφ|φ〉L0|φ〉 = hφ|φ〉

J0|φ〉 = Qφ|φ〉. (9.6)

It is clear that the discussion from above still holds for the Ln and we see that Ln

with n > 0 can still be seen as annihilation operators and the operators Ln withn < 0 can still be seen as creation operators. By considering the commutator[L0, Jn], we see that

L0(Jm|φ〉) = (hφ −m)Jm|φ〉.So, by the same discussion as in the case of the Ln, we conclude that the Jn forn > 0 are annihilation operators and Jn for n < 0 are creation operators.

Finally, we consider the commutator [L0, G±n±a], which gives:

L0G±n±a|φ〉 = (hφ − (n± a))|φ〉.

70

So, again we conclude that the G±n±a for n ± a > 0 are annihilation opera-tors and G±n±a for n ± a < 0 are creation operators. It is clear that the samearguments also apply to the anti-holomorphic case. So, we see that the oper-ators Lm, Jm, G

±n±a can be see as creation operators for m,n ± a < 0 and as

annihilation operators for m, n± a > 0.Now notice that if a = 0, we also have to pay attention to the G±0 operators.

From the above commutation relations we see that for any state |φ〉 satisfying9.6, the state G±0 |φ〉 also satisfies 9.6. Furthermore, if |φ〉 is also a highestweight state, then so is G±0 |φ〉. In other words the highest weight state becomesdegenerate. We define a state to be a Ramond ground state if it is annihilatedby G±0 .

Again, we can assign field to the highest weight states and these fields satisfythe following OPEs:

T (z)φ(w, w) =hφ

(z − w)2+

∂wφ(w, w)z − w

+ . . .

J(z)φ(w, w) =Qφ

z − w+ . . .

G±(z)φ(w, w) =G±−1/2φ(w, w)

z − w+ . . .

T (z)φ(w, w) =hφ

(z − w)2+

∂wφ(w, w)z − w

+ . . .

J(z)φ(w, w) =Qφ

z − w+ . . .

G±

(z)φ(w, w) =G±−1/2φ(w, w)

z − w+ . . . .

9.3 Chiral Primary Fields

In this section we will discuss a finite algebra of fields by imposing some addi-tional constraints on our primary fields.

Suppose that we are in the Neveu-Schwarz sector. A field φ is called a chiralprimary field (in the holomorphic sense) if φ is a superconform primary fieldthat creates a state |φ〉 that satisfies:

G+−1/2|φ〉 = 0.

From the previous section we see that this corresponds to the OPE:

G+(z)φ(w, w) = reg.

We can also impose the condition

G−−1/2|φ〉 = 0.

The superconformal primary fields that satisfy this equation are called anti-chiral fields (in the holomorphic sense). Finally, we also have to take explicitlyinto account the anti-holomorphic side of things. It is clear that with these

71

operators one can also find (anti-)chiral operators. So, we have (anti-)chiralityin the holomorphic sense and in the anti-holomorphic sense. We will denotefields that are chiral in both senses by (c, c), i.e. a state |φ〉 which is (c, c)satisfies:

G+−1/2|φ〉 = 0

G+

−1/2|φ〉 = 0.

Equivalently, fields that are chiral in the holomorphic sense and anti-chiral inthe anti-holomorphic sense by (c, a) etc.

It turns out that the fields from (c, c), (c, a), (a, c) and (a, a) are finite innumber if the N = 2 theory is non-degenerate and they form a ring. We willdiscuss the (c, c) case in full detail.

First of all, notice that from 9.3 we have the following anti-commutator:

G+−1/2, G

−−1/2 = 2L0 − J0.

Since G−1/2 = (G+−1/2)

†, we see that for any field |φ〉 we have that

0 ≤ 〈φ|G+−1/2, G

−−1/2|φ〉

= 2〈φ|L0 − J0|φ〉= 2hφ −Qφ,

with equality occurring for chiral primary fields. In fact, for primary chiralprimary fields we have:

hφ =Qφ

2,

and for other states |ψ〉

hψ >Qψ

2.

From the anti-holomorphic part, we get a similar relation:

hψ ≥Qψ

2,

with equality for chiral superconformal primary fields.Next we define a multiplication between chiral primary fields. Consider the

OPE of a product of two chiral primary fields:

φ(z, z)χ(w, w) =∑

i

(z − w)hψi−hφ−hχ(z − w)hψi

−hφ−hχψi(w, w), (9.7)

where the fields ψi are of weight hψi .Since U(1) charges add, i.e. Qψi = Qφ +Qχ, we conclude that hψ ≥ hφ +hχ

as well as hψ ≥ hφ + hχ. By taking the limit (w, w) → (z, z) we see that thesum in 9.7 reduces to a sum over the chiral primary fields. In other words, wesee that the set of chiral primary fields is closed under the multiplication

φ(z, z) · χ(z, z) = lim(w,w)→(z,z)

φ(z, z) · χ(w, w).

72

So, we see that we have a multiplicative structure.

However, it turns out that there are only a finite number of chiral primary fields.From 9.3 we see that for every chiral primary field:

0 ≤ 〈φ|G−3/2, G+−3/2|φ〉

= 〈φ|(2L0 − 3J0 + 2c/3)|φ〉= −4hφ +

2c

3.

So, for chiral primary fields we have an upper bound for the conformal weighthφ ≤ c/6. Since the lower bound is given by the highest weight we concludethat for every non-degenerate N = 2 theory the chiral primary fields form afinite ring.

To conclude we remark that the above discussion can be easily generalizedto anti-chiral fields.

73

Chapter 10

Background Field Methodfor Kahler Manifolds

We would like to apply a procedure close to the one discussed in Chapter 3 tothe N = (2, 2)-nonlinear sigma-model. Since we are now dealing with Kahlermanifolds, it turns out that instead of Riemann normal coordinates, we can useKahler normal coordinates. But apart from that, we can again go through thesame steps as before. We will start by giving a brief recap of the geometry ofKahler manifolds. After this we will discuss what Kahler normal coordinatesare. In the next section we will apply those to the supersymmetric non-linearsigma-model.

10.1 Kahler Manifolds

We will start by discussing the basic definitions and facts on Kahler manifolds.A good reference to this is [GH], but there are a lot more books on the subject.

Definition 10.1. A Kahler manifold is a complex manifold, with a Hermitianmetric gij, such that the Kahler from Ω = gidzidzj is closed, i.e. dΩ = 0. Fromthis condition it follows that the metric satisfies (by the Poincare lemma):

gi =∂2K

∂zi∂zj. (10.1)

and the other components vanish. The real function K is called the Kahlerpotential.

Since the metric has this nice form, we can derive a number of interestingproperties for the Riemann tensor and the Ricci tensor. For instance, we haveRij = Rı = 0. Furthermore, the only non-zero Christoffel symbols are Γi

jk andΓı

k. The explicit forms of these quantities are already given in 8.11. Apart

from the normal symmetries of the Riemann tensor, there are more symmetriesof the Riemann tensor on Kahler manifolds, namely:

Rikl = Rilk. (10.2)

74

10.2 Kahler Normal Coordinates

To study the super symmetric non-linear sigma model, we have to use a methodsimilar to the previous background field method. But, where in the bosonicnon-linear sigma model we use Riemann normal coordinates (remember thatin our new coordinates the geodesics where mapped to straight lines), it turnsout to be more convenient to work in so-called Kahler normal coordinates. Fordetails, the interested reader can read [HIN].

The authors of [HIN] argue that Riemann normal coordinates are not holomor-phic and hence not suited to be used for the supersymmetric non-linear sigmamodel, which has to be defined on Kahler manifolds. Since Kahler manifoldsalso complex manifolds it is far more natural to work with holomorphic coor-dinate transformation instead of just normal diffeomorphisms. Hence we willwork with Kahler normal coordinates. Kahler normal coordinates are definedas follows:

Definition 10.2. Let M be the Kahler manifold and let K be its Kahler poten-tial. We fix a point φ and we, locally, write the coordinates as zi = Xi

B + δXi.We now define Kahler normal coordinates ξi at the point Xi

B, to be the coordi-nates such that gi,i1...(ξ)|0 = 0. Here |0 stands for evaluation in Xi

B.

The coordinate transformation between to these new coordinates is given by atransformation similar to the one used in the chapter on the background fieldmethod.

ξi =∞∑

m=1

1m!

gi(XB)gi1,i2...im(XB)ξ1 . . . ξm. (10.3)

In Kahler normal coordinates, just as in Riemann normal coordinates, the ex-pression for the Riemann tensor simplifies and becomes:

Rikl = gi,kl, (10.4)

as follows from formula 8.11. We finish this section by giving the inverse coor-dinate transformation

δXi = ξi −∞∑

m=2

1m!

Γij1...jm

(XB)ξj1 . . . ξjm , (10.5)

where

Γij1...jm

= Dj1 . . . Djm−2Γijm−1jm

, (10.6)

and where the covariant derivative only works on the lower indices.

10.3 Background Field Method and Applicationto the NLSM

In this section we will discuss how the Kahler normal coordinates will be usedin a Background field method related to the supersymmetric NLSM. For a moredetailed account, see [HIN].

75

As before, we will take the point X to be the background field and we writefor our coordinates zi = Xi

B + δXi. Then, following the same steps as inChapter 3, we arrive at the following formulae for tensors and vectors in our newcoordinate system. Again, all quantities on the right hand side are evaluated inthe background field X as before.

gi(zi) = gi + Riklξkξ

l+O(ξ3)

∂αXi = ∂αXB + Dαξi − 12∂αXB

jRi

klξkξ

l+O(ξ3). (10.7)

We already see different coefficients compared to the RNC case. The next stepis to take a look at the action for the NSLM and transform it to these newcoordinates. Recall that the bosonice part of the action is given by:

L =1

4πα′

(gi∂αXi∂αX

j)

(10.8)

So, the boson part of the action in the BFM is given by (we again drop thesubscript B for the background field from now on):

SBoson =1

4πα′

∫∂αξa∂αξ

a+ Rikl

(∂αXi∂αX

jξkξ

l − 12∂αXi∂αXkξ

jξ

l

−12∂αX

j∂αX

lξiξk + ξkξ

lDαξi∂αX

j+ ξkξ

lDαξ

j∂αXi

−12∂αX

jξiξkDαξ

l − 12∂αXiξ

jξ

lDαξk + DαξiDαξ

jξkξl

)

+O(ξ5). (10.9)

We again see quite different coefficients compared to the previous backgroundfield method. For the fermion part we also need to transform the Christoffelsymbols. This can be done in the following way:

Γijk = Ri

klξ

l+O(ξ2). (10.10)

Using this, we see that the fermion action is given by:

Sfermion = − i

2πα′

∫d2σgiψ

j

−∂ψi− + Riklξ

kξlψ

j

−∂ψi− + Riklξ

l∂Xiψ

j

−ψk−

+RiklξlDξkψ

j

−ψi− + giψ

j

+∂ψi+ + Riklξ

kξlψ

j

+∂ψi+

+Riklξl∂Xiψ

j

+ψk+ + Riklξ

lDξkψ

j

+ψi+

− i

2Riklψ

i+ψk

−ψj

+ψl

− +O(ξ3). (10.11)

We would also like to transform the currents to this new system in order to cal-culate the superconformal algebra similar as is done in Chapter 6. The currentsconsist of the same quantities as the action and hence we can easily transformthem. The result is, to the relevant order:

J = giψi−ψ

j

− + Riklξkξ

lψi−ψ

j

− (10.12)

76

for the supercurrents:

G+ =12gijψ

i+∂X

j+

12Riklξ

kξlψi

+∂Xj+

12gijψ

i+Dξ

j+

12Riklξ

kξlψi

+Dξj

−14Riklξ

jξ

lψi

+Dξk. (10.13)

and

G− =12gijψ

j

+∂Xi +12Riklξ

kξlψ

j

+∂Xj +12gijψ

j

+Dξj +12Riklξ

kξlψ

j

+Dξj

−14Riklξ

jξlψi

+Dξk. (10.14)

Let us finally give the energy-momentum tensor:

T =1

4πα′

∫∂αξa∂αξ

a+ Rikl

(∂αXi∂αX

jξkξ

l − 12∂αXi∂αXkξ

jξ

l

−12∂αX

j∂αX

lξiξk + ξkξ

lDαξi∂αX

j+ ξkξ

lDαξ

j∂αXi

−12∂αX

jξiξkDαξ

l − 12∂αXiξ

jξ

lDαξk + DαξiDαξ

jξkξl

)

+i

2gijψ

j

+∂ψi+ +

i

2gijψ

j+∂ψ

i

+ +i

2Riklψ

j

+∂ψi+ξkξ

l

+i

2Riklψ

i+∂ψ

j

+ξkξl. (10.15)

10.4 Derivation of the Propagators

We have now set up a system which is, more or less, equivalent to the one dis-cussed for the bosonic non-linear sigma model. Therefore, we can apply thesame machinery as before. The only thing that needs to be done is the deriva-tions of the different propagators, which will be done in this section.

We will do the same derivation as in Chapter 3. For the bosonic part we canimmediately read of the propagator:

〈ξa(z)ξb(w)〉 = α′δab ln |z − w|2, (10.16)

since the prefactors coincide with the ones from the previous calculation butsince we are dealing with both ξa and ξ

again a factor 2. This propagator will

also be evaluated at zero seperation, just like before. The same reasoning as inthe relevant chapter gives:

∆(0) = −1ε

+O(ε0). (10.17)

Note that, just as before, the O(ε0) still contains a IR divergent term ln m2.

For the fermions, we introduce fermionic currents:

Ja

±ψa± + ψ

a

±Ja±. (10.18)

77

By just going through the entire derivation again, we come to the followingequation:

∂∆Fermion(z − w) = 2πα′δ(z − w), (10.19)

which is solved by:

〈ψa±(z)ψ

b

±(w)〉 = iα′δab

z − w. (10.20)

These are the only propagators that we will need. By anti-commutivaty of thefermions we do have that the contraction

〈ψi±(z)ψ

j

±(z)〉 = 0. (10.21)

78

Chapter 11

The Algebra of the N = (2, 2)Non-Linear Sigma Model

In this chapter we will discuss the superconformal algebra and we will calculate itin a curved background by using similar techniques as displayed in the previouschapter. We will start by considering the algebra in flat space. After this wewill shortly discuss renoralization of the action. We end with the calculation,which will show that the superconformal algebra is only satisfied on Ricci flatKahler manifolds. Throughout this chapter we, again, will assume that we areworking in the conformal gauge.

11.1 The Conformal Superalgebra in Flat Space

In Chapter 6 we discussed conformal field theory and the conformal algebra.For the bosonic non-linear sigma model we were able to calculate the conformalalgebra in flat space exactly. The same thing can be done for the N = (2, 2)non-linear sigma model, by using the same techniques. The only difference isthat we are dealing with fermions and hence one needs to be careful with signswhen contracting. In this section we will discuss an example of this calculation.

Recall that the fermion propagator is given by

∆F (z − w) = iα′δab 1z − w

. (11.1)

Let us first start with the OPE of JJ . Wick’s theorem gives:

JJ = ηiψiψηklψ

kψl + ηiψiψηklψ

kψl + ηiψiψηklψ

kψl + reg. (11.2)

Using the explicit form of the propagator, we obtain:

JJ = D1

(z − w)2+ reg (11.3)

So, we see that we get the correct OPE for the superconformal algebra withc = 3D. In a similar way, one obtains all other OPEs of the superconformal

79

algebra, one just has to be careful about contraction fermions. So, just as inthe case of the bosonic sigma model we see that the algebra is satisfied in flatspace and we will study how this works for curved backgrounds.

11.2 The Superconformal Algebra in Curved Space

We will now calculate the superconformal algebra of the N = (2, 2) NLSM in acurved background. We will basically do the same as in the case of the bosonicNLSM, as was done by [BNS]. We will also go to the same orders. Since bothcalculations are very similar, we will not explicitly go into all the details, sincethe integrals are the same and the procedure does not have new difficulties. Wehave only encountered one new integral and that one is listed in the Appendix.

11.2.1 Renormalization

Just as in the case of the bosonic NLSM, we would like to have that 〈1〉 = 1.Doing a similar calculation as in Chapter 5. So, let us calculate 〈1〉 = 1. It iseasy to see that there is only one relevant contribution, which is given by:

∆(0)4π

Ri∂αXi∂αXj

(11.4)

So we, just as before, add the counter term

−∆(0)4π

Ri∂αXi∂αXj

(11.5)

to the action (again this is before the BFM method, again clash of notations).This indeed gives that 〈1〉 = 1.

However, by doing this, our currents also pick up a contribution:

Tr = T − ∆(0)2π

Ri∂Xi∂Xj

G+r = G+ − ∆(0)

2πRiψ

i+∂X

j

These will give non-trivial contributions.

11.2.2 〈TT 〉, 〈TJ〉, etc.

Terms without X

The computer gives the following results (”SUSYNLSM1.nb”):

〈T (z)T (w)〉 =3D/2 + 2α′R

(z − w)4

〈J(z)J(w)〉 =D + 2α′R(z − w)2

〈G+(z)G−(w)〉 =2D

(z − w)3. (11.6)

All the other ones cancel among themselves or are just zero. Note that, just inthe bosonic case all divergences cancel after dimensional regularization.

80

Terms with X

There are no terms proportional to ∂Xi∂Xj

and no terms proportional to∂Xi∂X

j. In total we find (”SUSYNLSM2.nb”):

〈T (z)T (w)〉 =z − w

(z − w)3Rij∂αXi∂αX

j

〈T (z)J(w)〉 = − z − w

(z − w)2Rij∂Xi∂X

j

〈G+(z)G−(w)〉 = − z − w

(z − w)2Rij(3∂Xi∂X

j+ ∂Xi∂X

j). (11.7)

The asymmetry may seem a bit odd, but it follows from the BFM ex-pansion of the fermions. There is an asymmetry there with respect to the(anti)holomorphic derivative of the background field X. And we indeed onlysee the asymmetry with operators involving fermions. Again, all divergencescancel among themselves.

11.2.3 < T >, < G >,< J >

In order to completely determine the algebra, we need to compute to quantitiesmentioned in the title of this section.

Let us first note that for the supercurrents 〈G±〉 = 0. This can be seen im-mediately because G± only contains one fermion. In the action fermions alwaysappear in pairs and hence 〈G±〉 always contains an odd number of fermions,which will vanish when doing Wick contractions.

Next, let us consider 〈J〉. Since this term contains two fermions, the onlyway this can contribute is by coupling it to a term from SInt that also containstwo or more fermions. It is easy to see that, to our order, the only diagram thatcontributes is figure 11.1 (the dotted lines stand for fermions).

Figure 11.1: Contribution to 〈J〉.

It is proportional to:

giψi(

z + w

2)ψ

j(z + w

2)∫

d2ζ Rikl(ζ)ψψξk(ζ)ξl(ζ). (11.8)

Evaluation is straightforward and gives zero by the same reasons as discussedbefore.

81

The last term that needs to be evaluated is 〈T 〉. First the case without Xs. Thecomputer program gives the following result

iα′R32π2ε

(δ( z+w

2 − ζ)( z+w

2 − ζ)2−

∂ z+w2

δ( z+w2 − ζ)

( z+w2 − ζ)

), (11.9)

which can easily be seen to vanish after partial integration. The more interestingterm is the one containing two derivatives on X. Remember that this term was,unexpectedly, non-vanishing in the bosonic non-linear sigma model. However,it turns out that in this case, matters are a lot nicer. To be precise, we see fromthe computer program that all the divergent terms cancel among themselvesand that the remaining, finite, terms are proportional to

Rij

∫d2ζ

1(ζ)2

, (11.10)

which is discarded. Maybe this might not seem to be very surprising to thereader since we have added fermions, but a closer inspection learns that thefermions do not contribute to this term. In other words, this is a purely bosoniceffect and the discrepancy between this result and the result previously obtainedis the fact that we are now working in Kahler normal coordinates and before weworked in Riemann normal coordinates. In other words, the fact that we are ona Kahler manifold and that we can use Kahler normal coordinate, more or less,automatically takes care of the renormalization of T .

11.2.4 Putting it all together

If we now use the results obtained above, we can put together the total conformalalgebra and we can see the different anomalies on it. Doing this, basically, leadsto one, not unexpected, conclusion, namely that the N = (2, 2) non-linearsigma-model and its twists can only satisfies the superconformal algebra on aRicci flat Kahler manifold, i.e. Ri = 0. We also see that a number of problemsencountered in the bosonic sigma model are, somewhat miraculously, solvedsince we are working on Khaler manifolds. One final remark is in place. Wehave not taken into account the non-covariant terms like spin connections. Theyshould vanish from our algebra, but we have not checked this explicitly like inthe bosonic case.

The dilaton is harder to include int the supersymmetric case, since it willalso couple to the fermions. However, this has been done by Maldacena et al.[Mal].

82

Appendix A

Lorentz Indices

In this appendix we give a short description of how one can change from space-time indices to Lorentz indices by using the vielbein. We will also discuss somebasic properties and results. A good reference is for example [Carroll], Chapter3.

Consider a space time M , with metric gµν . Normally, one uses the canonicalbase eu = ∂µ resp. dxµ to describe the tangent space TpM resp. cotangentspace T ∗p M . However, one can choose a different base ea, which satisfies:

g(ea, eb) = ηab, (A.1)

where ηab is the signature of the metric. So, e.g. in the Euclidean setting wehave ηab = δab. We now have two different bases and we transform one into theother via:

eµ = eaµeµ. (A.2)

The components eaµ of the coordinate transformation for an invertible n × n

matrix and are called the vielbein or vielbeins (plural). Since the matrix isinvertible we can also define an inverse vielbein eµ

a . In terms of the vielbeinsequation A.1, gives:

gµνeµaeν

b = ηab

gµν = eaµeb

νηab. (A.3)

Similarly, one can also set this formalism up in the cotangent space.

In normal space-time indices, one uses the Christoffel connection to definecovariant derivatives. Since we now have different basis vectors, one obtains adifferent connection, called the spin connection ωa

µb. So, for example we have

DµXa = ∂Xa + ωaµbX

b. (A.4)

One of course, wants that a tensor is independent of the coordinate system itis written in. This allows for a comparison between the Christoffel connectionand the spin connection. To see this we calculate:

DX = DµXνdxµ ⊗ ∂ν

= (∂µXν + ΓνµλXλ)dxµ ⊗ ∂ν . (A.5)

83

But we have also

DX = DµXadxµ ⊗ ea

= (∂µXa + ωaµbX

b)dxµ ⊗ ea

= (∂µ(eaνXν) + ωa

µbebνXν)dxµ ⊗ eσ

a∂σ. (A.6)

Comparing the two gives us the following relationship:

Γνµλ = eν

a∂µeaλ + eν

aebλωa

µb (A.7)

or

ωaµb = ea

νeλb Γν

µλ − eλb ∂µea

λ. (A.8)

This relation proves to be useful when comparing covariant derivatives etc. Infact one can prove the relation:

Dµeaν = 0, (A.9)

which is known as the tetrad postulate.

84

Appendix B

Wick’s Theorem

In this chapter we will discuss the method used in calculating the different cor-relation functions.

The method used is the method of Wick contractions.

Theorem B.1 (Wick’s Theorem). Consider a quantum theory with operatorsφ1(z1), . . . , φn(zn). Let : · : denote normal ordering. We then have the followingresult:

φ1 . . . φn = : φ1 . . . φn : +∑

i<j,zi 6=zj

〈φi(zi)φj(zj)〉 : φ1(z1) . . . φi(zi) . . . φj(zj) . . . φn(zn) : +

∑i<j,zi 6=zj

k<l,zk 6=zl

〈φi(zi)φj(zj)〉〈φk(zk)φl(zl)〉 : φ1(z1) . . . φi(zi) . . . φk(zk) . . . φj(zj) . . . φl(zl) . . . φn(zn) :

+ . . . +∑

i1,...,in

〈φi1(zi1)φi2(zi2)〉 . . . 〈φin−1(zin−1)φin(zin)〉,

where φi stands for omitting φi.

85

Appendix C

Alternative Method ofIntegral Calculations

In the appendix we discuss how the encountered integrals can be computed. Inthe first part we discuss an alternative method of integral computing. In thesecond part we use the integral representation of our propagators.

C.1 Part I

Consider for example the integral∫

d2η1

(u− η)(η).

This integral clearly has poles around η = 0, u and hence the integral is notwell-defined. writing this out in real coordinates η = x + iy, u = a + ib, weobtain

2i

∫dxdy

(x− iy)((x− a)− i(y − b))(x2 + y2)((x− a)2 + (y − b)2)

.

The above form shows some similarities with the regular propagator in Euclidiansignature. Inspired by this, we now introduce our cut-off parameter m in thefollowing way:

2i

∫dxdy

(x− iy)((x− a)− i(y − b))(x2 + y2 + m)((x− a)2 + (y − b)2 + m)

.

I turns out that the integrals encountered do not contain poles in m and hencewe let m → 0. The integrals are further evaluated by using Feynman parametersand splitting of squares. Let us do this explicitly for the integral quoted above.First, we introduce the Feynman parameter t:

2Γ[2]i∫

dxdy(x− iy)((x− a)− i(y − b))

(t(x2 + y2 + m) + (1− t)((x− a)2 + (y − b)2 + m))2.

By shifting our integration variables (x, y) → (x− ta, y − tb),we get

2i

∫dxdy

(x− iy)((x− a)− i(y − b))(x2 + y2+)2

.

86

Now introduce polar coordinates and evaluate the integral to obtain:

1u2

+O(m).

One might worry why these integrals suddenly do not contain poles in m, sincethe integrand clearly has poles. The reason for this is that we interchange takinglimits and integrations:∫

d2η1

(u− η)(η)=

∫dxdy lim

m→0

(x− iy)((x− a)− i(y − b))(t(x2 + y2 + m) + (1− t)((x− a)2 + (y − b)2 + m))2

,

while with our method we compute

limm→0

∫dxdy

(x− iy)((x− a)− i(y − b))(t(x2 + y2 + m) + (1− t)((x− a)2 + (y − b)2 + m))2

,

which turns out to be finite and actually gives the right answer. Hence, thismethod gives a prescription how to deal with singularities encountered.

This method of calculating integrals turn out to be convenient in some cases.One specific case in which this method is very useful is when one needs, incoordinate space, to evaluate the integral:

∫d2η

δ(η)η(u− η)

.

One gets rid of the delta function by using the rule δ = 2πi ∂ 1

η and partialintegration.

C.2 Part II

We have encountered integrals of the form:

I :=∫

d2ζ

(z − ζ)m(ζ − w)n.

We will now show how to evaluate them. We will do this by using Fourierrepresentations. First, write:∫

d2ζ

(z − ζ)m(ζ − w)n=

(−1)m−1

(m− 1)!(n− 1)!

∫d2ζ∂n

z ∂nw

(ln |z − η|2 ln |η − w|2) .

Using the fact:

ln |z − η|2 =1πi

∫d2p

eip·(z−η)

|p|2 ,

we get

I =−1π2

(−1)m−1

(m− 1)!(n− 1)!

∫d2p

∫d2q

∫d2η

(ip)m(−iq)neip·(z−η)eiq·(η−w)

|p|2|q|2 .

We also have:

−(2π)2δ(x− y) =∫

d2keik·(x−y).

87

So, doing the ζ and q integral leads us to

I = 4(−1)m−1

(m− 1)!(n− 1)!

∫d2p

(ip)m(−ip)neip·(z−w)

|p|2|p|2

=4(−1)n

i

(−1)m−1

(m− 1)!(n− 1)!∂n+m−1(z−w)

∫d2ζ

peip·(z−w)

|p|2|p|2 .

From |p|2 = 2pp, we see that

p

|p|2|p|2 = −12∂p

1|p|2 .

So, if we use this, combined with partial integration, we find:

I =2(−1)n

i

(−1)m−1

(m− 1)!(n− 1)!∂n+m−1(z−w) i(z − w)

∫d2ζ

eip·(z−w)

|p|2

=2(−1)n

i

(−1)m−1

(m− 1)!(n− 1)!∂n+m−1(z−w) i(z − w)πi ln |z − w|2

=2π

i

(n + m− 2

m− 1

)z − w

(z − w)m+n−1.

So in brief, we get:∫

d2ζ

(z − ζ)m(ζ − w)n=

2π

i

(n + m− 2

m− 1

)z − w

(z − w)m+n−1.

In a similar fashion we obtain:∫

d2ζ

(z − ζ)(ζ − w)= −2πi ln |z − w|2

and∫

d2ζd2η

(z − η)(ζ − η)2(ζ − w)= 4π2 ln |z − w|2.

88

Appendix D

References

[GSW], M.B.Green, J.H. Schwartz, E.Witten, String Theory, Volume I & II,Cambridge University Press

[BNS], T. Banks, D. Nemeschansky, A. Sen, Dilaton Coupling and BRST Quan-tization of Bosonic Strings, Nucl.Phys.B277:67,1986.

[A-GFM], L. Alvarez-Gaume, D.Z. Freedman and S. Mukhi,The BackgroundField Method and the Ultraviolet Structure of the Supersymmetric Nonlinear σmodel, Ann. of Phys. 134 (1981) 85.

[Gin], P. Ginsparg, Applied Conformal Field Theory, hep-th/9108028.

[Greene], B. Greene, String Theory on Calabi-Yau manifolds, hep-th/9702155.

[Aspinwall], P. Aspinwall, D-Branes on Calabi-Yau Manifolds, hep-th/0403166.

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