Symmetries and Dualities: A Reformulation of the String Worldsheet

Felipe Dıaz-Jaramillo

Student ID: 602533

Submitted in fulfilment of the requirements for the degree of:Master of Science (M. Sc.) in Physics

Supervisor

Dr. Olaf Hohm

Second examiner

Prof. Dr. Jan Plefka

Humboldt Universitat zu Berlin

Faculty of Natural Sciences

Physics Institute

Berlin, Germany

September 10 2020

A mi familia y amigos, por sus

aportes, consejos y apoyo.

Acknowledgements

First, I would like to thank Dr. Olaf Hohm for the opportunity to do this project under his

supervision. I have learned a lot during my time in his group. He got very involved in the project,

making the work there an excellent experience personally and professionally. I am especially

grateful to Dr. Roberto Bonezzi for his patience, teachings and his enormous contributions to

this project. I would also like to thank Julien Barrat for the coffee breaks and very helpful

observations. Finally, I would like thank Allison for her advice and help.

This thesis is based on a collaboration with R.B and O.H, leading to the results in Ref. [24].:

Roberto Bonezzi, Felipe Diaz-Jaramillo, and Olaf Hohm. Old Dualities and New Anomalies 8

2020, in the arXiv preprint.

iii

Agradecimientos

Quisiera agradecerle principalmente a mi familia por su apoyo incondicional en todas las deci-

siones que he tomado a lo largo de mi carrera y de mi vida. Sus consejos, amor y apoyo me

motivan cada vez mas a seguir persiguiendo mis suenos. Tambien quisiera agradecerle a Mariano

por su companıa todo este tiempo que llevamos en Berlın. Finalmente, quisiera agradecerle a

Simon, Juan Jose, Mateo, Pablo y Santiago, por su amistad y apoyo durante tantos anos.

iv

Contents

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

1 Introduction 1

2 Theoretical background 6

2.1 Einstein’s gravity and the point particle . . . . . . . . . . . . . . . . . . . . . . 6

2.2 The non-linear sigma model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 T-duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 The Target Space Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Towards an O(d, d) Invariant Worldsheet Action 23

3.1 A non-local field redefinition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 Mode expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3 Coupling the string . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 Applications 42

4.1 The anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2 A cosmological spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3 The cosmological worldsheet action . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 Discussion and conclusions 59

References 61

v

Chapter 1

Introduction

It is well known that attempting to quantize Einstein’s theory of General Relativity (GR) em-

ploying standard quantum-field-theoretical methods leads to a non-renormalizable theory. As a

consequence, the resulting theory can only be trusted as an effective field theory describing low

energy phenomena. Thus, GR fails to describe physics at high energies, or equivalently small

distance scales, for instance at the Big-Bang or in the center of a Black Hole. This suggests

that in order to have a UV-complete quantum theory with which we can describe gravity at

the smallest scales, a new framework beyond standard quantum field theory is needed. String

theory is a particularly promising candidate for such a framework, since gravity arises naturally

in the process of quantizing the string. For instance, in the low energy limit of string theory,

on grounds of consistency of the string’s quantization a (super) gravity theory emerges. This

field theory describes the low energy dynamics of the background where the string is moving, in

the same spirit of the Einstein-Hilbert action in GR which describes the dynamics of spacetime.

In this thesis we will examine bosonic string theory, and for this particular type of string the

massless sector of the background consists of the target space metric, an antisymmetric B-field

(or Kalb-Ramond field) and a scalar field called the dilaton. This low energy effective field the-

ory is also referred to as spacetime, or target space theory, and in contrast to Einstein’s gravity,

it acquires higher derivative corrections governed by the inverse string tension α′ that originate

from the extended nature of the string.

Strings, being extended objects, probe spacetime geometries differently than point particles

leading to dualities that display a physical equivalence between string theories defined on seem-

ingly different backgrounds. Arguably, the most prominent example of duality in string theory

is T-duality. In its simplest form, it states that the theory of a closed quantum string moving in

a spacetime with one coordinate compactified on a circle of radius R, around which the string

can wind, is physically equivalent to the theory of a string moving in a spacetime where the

compact coordinate is a circle of radius α′/R, upon exchanging the roles of momentum and

winding. This equivalence can be observed explicitly in the mass spectrum of the string, which

1

is invariant under these exchanges. Moreover, compactifying d coordinates on a d dimensional

torus and including constant background fields yields an invariance of the mass spectrum under

the discrete group O(d, d,Z). Similarly, the spacetime theory is invariant under the continuous

group O(d, d,R) when the target space fields do not depend on d coodinates of a higher dimen-

sional space. Nevertheless, it is not clear a priori whether the symmetry of the spacetime theory

is related to T-duality, given that the latter is governed by the discrete group O(d, d,Z).

O(d, d,R) invariance of the low energy effective field theory to lowest order in α′ has its

origin in studying this symmetry in a cosmological setting [1], where the spacetime fields are

independent of d spacelike coordinates, but do depend on time. In the context of cosmology,

this symmetry implies that the theory is invariant under the exchange a(t)↔ 1/a(t), where a(t)

is the scale factor that characterizes the expansion of the universe. This scale factor duality is

intriguing because: (i) Similarly to T-duality, it states a special connection between large and

small spaces, and (ii) String theory sets the string length as the minimal length scale of the

universe. The latter implication is related to the infinite α′ corrections, which possibly could

smooth out singularities present in conventional GR, avoiding the breakdown of the theory at

high energies. In Ref. [2] it is proved that, for a cosmological spacetime, all α′ corrections are

O(d, d) invariant, and more recently in Ref. [3] a classification of all possible O(d, d) invariant

α′ corrections relevant for the cosmological case was successfully developed. For this reason

string theory offers ways to investigate scenarios in cosmology inaccessible to point particles in

conventional gravity, for instance the Big-Bang singularity, by means of the connection between

small and large spaces.

Even though O(d, d) symmetry of the spacetime theory in the absence of matter has been

thoroughly studied, in the transition from standard GR to cosmology the introduction of matter

to the model is required, in order to find cosmological solutions for the scale factor a(t) and

describe the dynamics of the universe. We need to achieve something similar in the string

theory context. Due to the interesting possible applications of O(d, d) invariance in cosmology,

it is natural to insist that the action of the matter fields has this symmetry, as in Ref. [4], where

O(d, d) invariant cosmological solutions including matter sources were found to all orders in α′

using the classification in Ref. [3]. This was achieved without giving an explicit expression for

the matter action.

The first natural candidates for the matter fields are strings, now viewed as classical macro-

scopic objects moving in spacetime. Their ability to extract additional information from space-

time geometries due to their extended nature, as apparent in T-duality, makes them appealing

to incorporate to the model as matter compatible with this symmetry of the target space the-

ory. However, O(d, d,Z) in T-duality, in contrast to O(d, d,R) invariance of the target space,

displays a physical equivalence between strings moving in different backgrounds related by such

transformations, rather than being a locally realized symmetry of the worldsheet action. More-

2

over, O(d, d,Z) invariance of the spectrum has a quantum origin: the exchange or mixing of

momentum and winding numbers can only happen if momentum is quantized around the com-

pact coordinates due to the topology of the torus, which for closed strings only admits integer

winding numbers. O(d, d,R) for the target space, on the other hand, does not consider any

quantum effects. Motivated by these discrepancies between worldsheet and target space, in this

thesis we are interested in exploring the possibility of making O(d, d), either in its discrete or

continuous version, a manifest symmetry of the string action in the interest of constructing a

completely O(d, d) invariant coupled theory. We will not restrict ourselves to the cosmological

case in order to have a general discussion.

In a more general context than cosmology the target space theory has interesting features. In

Ref. [5], it was found that to zeroth order in α′, dimensional reduction from n+d to n coordinates

yields an O(d, d) invariant n dimensional action, generalizing the cosmological case in Ref. [1] to

arbitrary spacetime dimensions. This is achieved after using the usual Kaluza-Klein Ansatz for

the spacetime fields, in combination with a truncation of the theory where only massless Kaluza-

Klein modes are considered, upon compactifying d coordinates on a torus. Following a similar

program, in Ref. [6] dimensional reduction of the spacetime theory was performed to first order in

α′. However, the resulting n dimensional target space theory is not O(d, d) invariant in the same

sense as the theory to lowest order in α′, but it is possible to bring the resulting lower dimensional

action to a manifestly O(d, d) invariant form by implementing a Green-Schwarz (GS) mechanism,

structurally identical to the one used in heterotic string theory for anomaly cancellation [7],

which requires a non-standard transformation of the (originally singlet) B-field of order α′ under

O(d, d)1. The GS mechanism in heterotic string theory has a worldsheet interpretation: the beta

functions arising from the string’s quantization, from which the heterotic target space theory

is constructed, are not invariant under gauge transformations, thus requiring the B-field to

transform under these symmetries in order to restore gauge invariance of the effective field

theory. This observation, together with the construction of a coupled O(d, d) invariant model,

motivate us to find an O(d, d) invariant worldsheet action, which could potentially justify, upon

quantization, the appearance of the GS mechanism for O(d, d) in the target space theory to first

order in α′.

Early indicators of O(d, d) being a symmetry of the worldsheet theory can be found in Refs.

[5, 8], where O(d, d) covariant equations of motion were found for the string. However, these

equations come from first order self-duality relations rather than the variational principle of an

O(d, d) invariant action. A valid action principle for these equations appeared in Ref. [9], where

a non local field redefinition due to Tseytlin in Refs. [10, 11] was used, in order to construct an

O(6, 22,Z) invariant action for the bosonic part of the heterotic string after compactification on

a 28 dimensional torus. Interestingly, the action of the bosonic fields in heterotic string theory

1In order to match the structure of the original GS mechanism, the B-field acquires a transformation underSO(d)× SO(d) in a frame field formulation.

3

after compactification has the same form as the action of a purely bosonic string after utilizing

the Kaluza-Klein split of Ref. [5] directly on the worldsheet action. This formally means that

the generalization from heterotic string theory (O(6, 22,Z)) to the purely bosonic one (O(d, d))

is straightforward. In Ref. [12], this O(d, d) invariant action for the bosonic string was obtained

in the framework of double field theory.

In the literature there seems to be a lack of clarity with regard to the symmetry group of the

worldsheet action, namely whether it is O(d, d,R) or O(d, d,Z), which is closely related to the

issue of the discrete momentum and winding modes. For instance, in Ref. [13] based on Tseytlin’s

approach, it is argued that, in the action, one needs to consider winding to be dynamical, in

order to pair it up with the zero mode of the momentum in an O(d, d) multiplet. This indeed

yields an O(d, d) invariant action, but explicitly breaks the torus boundary conditions for closed

strings.

In this thesis we prove that a truncation of the worldsheet theory is necessary in order to have

a truly O(d, d,R) invariant action for the string, which can be associated to the truncation of

the target space theory, responsible for O(d, d,R) invariance of the low energy effective action.

The origin of this truncation lies in the violation of the torus boundary conditions upon rotating

winding modes and center-of-mass-momenta. Additionally, we explicitly construct an O(d, d,R)

invariant worldsheet action with the same structure as in Refs. [9,12], and discuss its consistency

with gauge symmetries and worldsheet diffeomorphism invariance. Furthermore, and as our main

result, we quantize the O(d, d,R) invariant worldsheet action. We find that, after formulating

the theory in terms of vielbeins based on the coset space O(d, d,R)/SO(d)L × SO(d)R, the

SO(d)L × SO(d)R gauge symmetry is anomalous, and requires the Green-Schwarz mechanism

found for the target space theory to first order in α′ in Ref. [6]. This anomaly comes from the

fact that upon employing Tseytlin’s approach, one deals with chiral or self-dual bosons. This

situation is similar to heterotic string theory, where the original GS mechanism was introduced to

cancel anomalies due to the presence of chiral fermions. Finally, we specialize to a cosmological

spacetime, and find the string’s equations of motion in such a background.

The outline of the thesis is the following:

• In chapter 2, we introduce important aspects of general relativity and string theory. This

will be done by presenting and discussing relevant features of the different actions that we

will encounter throughout the thesis. We will pay special attention to various symmetries,

particularly O(d, d) invariance.

• After having reviewed all these concepts, in chapter 3 we present the construction of an

O(d, d) invariant worldsheet action. We will discuss the different conditions and truncation

necessary for O(d, d) invariance of the action, and towards the end of the chapter, we

will focus on various symmetries of this theory, proving that it is consistent with all the

4

symmetries of the spacetime theory, and the original string action.

• In chapter 4 we show some potential applications of the O(d, d) invariant theory devel-

oped in the previous chapter. We start with a worldsheet-theoretical justification for the

GS mechanism needed in the target space theory for O(d, d) invariance, after quantizing

the O(d, d) invariant worldsheet action. Afterwards we focus on cosmology. We present the

cosmological spacetime theory, and finalize with the worldsheet theory in a cosmological

setting.

• To conclude this thesis, we carry out a discussion about the findings of this thesis in

chapter 5. This discussion will be followed by some conclusions and outlook.

5

Chapter 2

Theoretical background

We begin this thesis by discussing important aspects of general relativity, the worldsheet theory,

and its corresponding low energy effective description: the target space theory. In this chapter

we will review the dynamics of point particles in the context of General Relativity, and how

we probe a given spacetime with these point particles. This will be useful to understand the

main objective of this thesis: to construct an O(d, d) invariant worldsheet theory, which can

be consistently coupled to the spacetime theory. Afterwards, we introduce the standard non-

linear sigma model and study its symmetries and key features. This will allow us to proceed

and examine O(d, d) invariance in the context of T-duality. Once we understand how O(d, d)

invariance arises for the string, we will analyze the low energy effective action of bosonic string

theory in n + d dimensions, and we will see how dimensional reduction of d coordinates yields

an O(d, d) invariant n dimensional spacetime action. We will not cover in detail all these topics,

as this chapter is only intended to give the general picture and necessary background to follow

this thesis.

2.1 Einstein’s gravity and the point particle

In this section we will review two actions of General Relativity: the massless point particle freely

moving in a gravitational field and the Einstein-Hilbert action. This brief review is intended to

be a guide to what we intend to do later in the context of string theory.

The action of a massless point particle moving in a curved spacetime with metric Gµν is given

by

SPP [Xµ, n] =

∫dτ

1

2nGµν(X)XµXν , (2.1)

where τ is an arbitrary parameter that parametrizes the worldline of the particle, ˙≡ ddτ

, Xµ(τ)

6

the coordinates on the worldline and n = n(τ) is a frame field that transforms as a density under

τ -reparametrizations. In this action, we see that the dynamical variables are the coordinates Xµ

and n; we are considering a fixed background where the particle is moving. Under reparametriza-

tions of the form τ → τ = τ − λ(τ) with λ infinitesimal, we have the transformations

δλXµ = λXµ,

δλn =d

dτ(λn).

(2.2)

One can check that the action is invariant under these transformations, which are known as

worldline diffeomorphisms, by noting that due to the dependence of Gµν on Xµ, one has δλGµν =

λXρ∂ρGµν .

Another important action of general relativity is the Einstein-Hilbert action

SEH [Gµν ] =1

16πGN

∫d4x√−GR, (2.3)

where GN is Newton’s constant, G ≡ det(Gµν) and R is the Ricci scalar of Gµν . This action

describes the dynamics of spacetime. For this reason, the dynamical variable of the action is the

metric Gµν , and from a geometrical point of view it transforms as a tensor field under spacetime

diffeomorphisms xµ → xµ − ξµ(x), with ξ infinitesimal, namely

δξGµν = ξρ∂ρGµν + ∂µξρGρν + ∂νξ

ρGµρ ≡ LξGµν , (2.4)

with L the Lie derivative. We can check that this is a symmetry of the theory by noting that

δξSEH [Gµν ] =1

16πGN

∫d4x∂ρ

(ξρ√−GR

).

This is a gauge symmetry because any two metrics related by the coordinate transformation

xµ → xµ − ξµ give the same physics, indicating that there is a redundancy in the theory, some

times denoted by general covariance or general coordinate invariance.

In the interest of probing a given spacetime determined by the metric Gµν , we can include

a point particle to the spacetime theory and see how this particle behaves. This will tell us

interesting things about the spacetime geometry and how the dynamics of the particle and also

the dynamics of this particular spacetime are affected. We can couple a massless point particle

to the Einstein Hilbert action as

S[Gµν , Xµ] = SEH +

1

2n

∫dτ Gµν(X)XµXν . (2.5)

Given that the spacetime action has the gauge symmetry xµ → xµ− ξµ(x), we would expect the

7

coupled theory to have it as well; we cannot afford to loose this gauge symmetry once we couple

the point particle. These coordinate transformations imply Xµ → Xµ− ξµ(X) for the worldline

fields, yielding the transformation of the action

δξS[Gµν , Xµ] = δξSEH+

1

2n

∫dτ(−ξρ∂ρGµν(X)XµXν + LξGµν(X)XµXν + 2Gµν(X)XµδξX

ν),

(2.6)

where the first term in the integrand comes from the fact that Gµν depends on Xµ and as Xµ

transforms, the metric transforms accordingly as a function of Xµ. The second term comes

from noting that now the metric is a dynamical variable in this coupled theory. Expanding all

terms, this variation vanishes identically, which means that the coupled theory is invariant under

spacetime diffeomorphisms.

Now that we have checked that the coupled theory has the desired symmetries, we can go

further and look at the equations of motion. The equations of motion for the particle arise

when we vary the action 2.5 with respect to Xµ. In a straightforward computation and with the

appropriate worldline parametrization, we find the geodesic equation

Xµ + Γµνρ XνXρ = 0, (2.7)

where Γµνρ are the Christoffel symbols 1. These equations have to be supplemented by the frame

field’s equations of motion

Gµν Xµ Xν = 0. (2.8)

We can rewrite 2.5 in a more convenient way in order to find the equations of motion for the

metric Gµν by inserting a δ-function to the point particle term, i.e

S[Gµν , Xµ] = SEH +

1

2

∫d4xGµν(x)T µν(x),

with T µν =

∫dτ

1

nXµXνδ(x−X(τ)).

(2.9)

The variation with respect to the metric Gµν of this action, yields the famous Einstein equations

Rµν − 1

2GµνR = 8πGNT

µν . (2.10)

The equations of motion 2.7 and 2.10 respectively tell us how the geometry of this spacetime

affects the path followed by the point particle, and how the presence of a point particle affects

1In this section we consider everything to be torsion free.

8

the curvature of spacetime.

We do not have to stop here. We can, for instance, consider a point particle moving in a

curved spacetime with metric Gµν in the presence of an electromagnetic gauge potential Aµ.

The action for a point particle in curved spacetime in the presence of a gauge potential (with

electric charge e = 1) is given by

SPPEM [Xµ] =

∫dτ

(1

2nGµν(X)XµXν + Aµ(X)Xµ

). (2.11)

This action has the new gauge symmetry

δΛAµ = ∂µΛ(X), (2.12)

and all other fields are inert under these transformations. Then, the spacetime action encod-

ing the dynamics of Gµν and Aµ that respects this new gauge symmetry and diffeomorphism

invariance2 is given by

SSpacetime[Gµν , Aµ] =

∫d4x√−G

(1

16πGN

R− 1

4GµρGνσFµνFρσ

),

with Fµν = ∂µAν − ∂νAµ,(2.13)

where Fµν is the field strength of the 1-form Aµ. Now, the coupled theory is

SCoupled[Gµν , Aµ, Xµ] = SSpacetime + SPPEM . (2.14)

In this thesis we wish to accomplish something similar: In order to better understand the

nature O(d, d) invariance we will probe, with a string, a target space with a spacetime metric

Gµν , in the presence of a Kalb-Ramond field Bµν and a dilaton scalar field φ. These fields are

the massless fields of bosonic string theory and the action that describes their dynamics is called

the low energy effective action of bosonic string theory, which can be seen as the analog or string

theory generalization of the action 2.13. Analogously to the point particle discussed above, we

need to guarantee that the symmetries of the string action are compatible with the symmetries

of the spacetime theory. To this end, in the remainder of this chapter we will present both

actions and discuss their symmetries and dualities.

2Note that Aµ transforms as a 1-form, δξAµ = LξAµ.

9

2.2 The non-linear sigma model

The action that describes a bosonic string moving in a spacetime with with metric Gµν , anti-

symmetric tensor field (2-form, Kalb-Ramond field) Bµν and dilaton φ is called the non-linear

sigma model, or worldsheet action, and is given by

Sstring[Xµ, hαβ] = − 1

4πα′

∫d2σ√−h(hαβGµν(X)∂αX

µ∂βXν + εαβBµν(X)∂αX

µ∂βXν

+α′φ(X)R(2)),

(2.15)

where hαβ is the worldsheet metric with Minkowski signature (−,+), εαβ is the totally antisym-

metric tensor3, α′ is a parameter related to the string length, α and β are worldsheet indices

corresponding to the worldsheet parameters4 σα = (τ, σ), R(2) is the Ricci scalar of the world-

sheet metric hαβ and Xµ(σ, τ) are the spacetime coordinates on the worldsheet. Note that the

background fields are fixed in the same sense as discussed in section 2.1. We will ignore the

coupling to the dilaton for the moment because it is of order α′ and we will work with the

non-liner sigma model action

Sstring[Xµ, hαβ] = − 1

4πα′

∫d2σ√−h(hαβGµν(X)∂αX

µ∂βXν + εαβBµν(X)∂αX

µ∂βXν), (2.16)

which can be regarded as the 2-dimensional generalization of the point particle action 2.11. Note

that the coupling of the B-field and the worldsheet can be viewed as the two-dimensional gener-

alization of coupling a gauge potential Aµ to the worldline. Under worldsheet diffeomorphisms

σα → σα − ωα(σ, τ), the fields transform as

δωXµ = ωα∂αX

µ,

δωhαβ = ∇α ωβ +∇β ωα,

∇αωβ = ∂α ωβ − Γγ (2)αβ ωγ,

(2.17)

where Γγ (2)αβ are the Christoffel symbols corresponding to the worldsheet metric hαβ. The non-

linear sigma model 2.16 is invariant under these gauge transformations, by noting that analo-

gously to the point particle, the metric and B-field transform as δωGµν = ωα∂αGµν , δωBµν =

ωα∂αBµν . These transformations correspond to the τ -reparametrization invariance of the point

particle action. Analogous to the gauge symmetry Aµ → Aµ + ∂µΛ, the sigma model has the

gauge symmetry

3Note that√−hε01 = −1.

4τ is timelike and σ is spacelike and σ ∈ [0, 2π].

10

δζBµν = ∂µζν(X)− ∂νζµ(X). (2.18)

This action has an extra gauge symmetry that the point particle action does not have: Weyl

invariance. The fields of the sigma model action transform under Weyl transformations as

Xµ → Xµ,

Gµν → Gµν ,

Bµν → Bµν ,

hαβ → Ω2(σ, τ)hαβ,

(2.19)

with Ω a local scale parameter. This gauge symmetry and 2 dimensional diffeomorphism invari-

ance give us enough gauge freedom to set hαβ to a flat Minkowski metric ηαβ. This is known

as conformal gauge and occasionally we will work with it in order to simplify the computations.

Other times we will keep the metric hαβ because it will let us keep track of the 2 dimensional

diffeomorphism invariance of the theory. We have to keep in mind that since the worldsheet

metric hαβ is a dynamical variable in the sigma model, it has equations of motion, in a complete

analogy to the frame field n in the point particle example. The equations of motion that arise

after varying the action 2.16 with respect to hαβ are constraints called Virasoro constraints, and

they supplement the equations of motion of the string. These constraints can be seen as the

generators of 2 dimensional diffeomorphisms and we should always keep in mind that they exist,

even after fixing a gauge for hαβ. On the other hand, if we vary the action 2.16 with respect to

Xµ, we find the equations of motion (in conformal gauge) for the string that read

Gµν ∂α∂αXν +Gµν Γνλρ ∂αX

λ∂αXρ − 1

2εαβHµνλ ∂αX

ν∂βXλ = 0 (2.20)

where Hµνρ is the field strength of the B-field. To find these equations of motion, we have used

the fact that we are dealing with closed strings. The boundary conditions for closed strings are

Xµ(σ + 2π, τ)−Xµ(σ, τ) = 0,∫ 2π

0

dσ ∂σXµ = 0,

which in words means that the coordinates Xµ are periodic in σ. This allows for a mode

expansion in σ of the form

Xµ(σ, τ) =∑n∈Z

Xµn (τ) einσ.

11

2.3 T-duality

All the symmetries that we have seen so far for the non-linear sigma model are symmetries

common to many field theories, we have barely used the fact that we are dealing with strings. We

can exploit the stringy nature of string theory to find new physics. One very famous phenomenon,

which makes use of this stringy nature, is T-duality.

Let us examine a bosonic string moving in a flat 26 dimensional spacetime in the absence of

spacetime fields. We want to compactify one of these coordinates, for instance X25, on a circle

of radius R, so that the resulting spacetime is R1,24×S1. So far we assumed the coordinates Xµ

to be periodic. Here, we will relax the boundary conditions for this compact coordinate, and we

will allow the string to wind around it. Compactification on a circle amounts to assigning the

identification

X25(σ + 2π, τ)−X25(σ, τ) = 2πwR, w ∈ Z, (2.21)

where w is called winding number, an integer that tells us how many times the string winds

around the circle. In this section, we will consider a quantum string so that its center of mass

momentum along the compact coordinate is quantized because of the single-valuedness of its

wave function. The center of mass momentum along X25 is then given by

p25 =n

R, n ∈ Z. (2.22)

From the boundary condition 2.21 and the quantized momentum 2.22, we find that the mass

spectrum of the string is

M2 =n2

R2+w2R2

α′2+ Oscillators. (2.23)

which is invariant under the exchanges

n↔ w, R↔ R, R =α′

R.

This is the simplest case of T-duality. It tells us that a string moving around a circle with

radius R has the same mass spectrum as a string moving around a dual circle of radius R if we

exchange momentum and winding. A more detailed and rigorous derivation of this can be found

in Ref. [14].

It is worth briefly introducing the concept of dual coordinates. If we wanted to reproduce

the spectrum of a string moving around the dual circle of radius R, the coordinate that would

yield the expected result is the dual coordinate X25, related to the original coordinate X25 by

the self-duality relation

12

∂αX25 = εαβ∂

βX25. (2.24)

We can analyze a more general situation by compactifying d coordinates, and introducing

spacetime fields. For simplicity, we will work with constant background fields, in conformal gauge

and in n+d dimensions rather than 26. The spacetime coordinates are now X µ, µ = 0, ..., n+d.

We will use the index split

X µ = (Xµ, Y i), µ = 0, .., n− 1, i = 1, ...d.

The coordinates Xµ are called external coordinates and the coordinates Y i are called internal

coordinates, which we take to be compact. Compactifying d coordinates on independent circles

(S1× S1× ...× S1, d times), amounts to compactifyng on a d dimensional torus Td, which yields

the spacetime R1,n−1×Td. We will consider a constant metric gij on the torus Td, and a constant

B-field bij. The action for the internal sector is then

SCompact = − 1

4πα′

∫d2σ

(gij ∂αY

i∂αY j + εαβbij ∂αYi∂βY

j), (2.25)

and the boundary conditions generalize to

Y i(σ + 2π, τ)− Y i(σ, τ) = 2π√α′wi, wi ∈ Z,

where wi is the winding around each one of the compact coordinates. We will still examine a

quantum string, which means that the center of mass momentum along each compact coordinate

in quantized, i.e

√α′pi = ni, ni ∈ Z.

We can go to the Hamiltonian formulation to obtain the spectrum contribution of the compact

coordinates (up to multiplicative constants and omitting the oscillator’s contributions),

M2 ∼(gijninj +

(gij − bikgklblj

)wiwj − 2gikbkjniw

j), (2.26)

which we formally derive in the next chapter. This mass spectrum can be rewritten more

concisely by introducing the element of the non-compact indefinite orthogonal group O(d, d), H,

defined as

HMN =

((g − bg−1b)ij bikg

kj

−gikbkj gij

).

13

H is called generalized metric and the capital Latin indices M are O(d, d) indices. Being an

element of O(d, d), H obeys

HMP ηPQHQN =ηMN , HMN = ηMP HPQ η

QN

ηMN =

(0 δi

j

δij 0

),

(2.27)

where η is the invariant metric of O(d, d) in its off-diagonal form and it lowers and raises indices

and HMN is the inverse generalized metric. In order to ensure group closure, if we perform any

transformation on the generalized metric by any other element of O(d, d), this transformed H′

has to obey equation 2.27. This implies the transformation in matrix form of H

H′ = ΩTHΩ, Ω ∈ O(d, d). (2.28)

We can also introduce the O(d, d) vector

LM =

(wi

ni

), (2.29)

which transforms under O(d, d) (to ensure invariance of the scalar product) as

L′ = Ω−1L, Ω ∈ O(d, d). (2.30)

This yields the mass spectrum

MCompact ∼ LM HMN LN , (2.31)

which is invariant under O(d, d) transformations. There is a subtlety here. We have considered

quantized center of mass momenta around the compact coordinates, and we are examining closed

strings which means that the winding number has to be an integer. In order to keep integer

momenta and winding, the transformation matrix Ω has to have integer elements. Thus, the mass

spectrum of the theory is invariant under the subgroup O(d, d,Z) of O(d, d,R). Furthermore,

it is possible to prove that the correlation functions, and amplitudes of the theory are O(d, d)

invariant, however, these are out of the scope of this thesis. For a more rigorous discussion on

the topic, the reader may find chapter 2 of Ref. [15] useful. For a more simple (but more detailed

than here) discussion, chapter 7 of Ref. [16] may be useful.

The name T-duality (T stands for target space) comes from the fact that one can relate

different string theories with different backgrounds and get the same physical results, hence

the word duality. We will not see O(d, d) invariance from this perspective. Given that we are

interested in writing a theory that describes the dynamics of the spacetime and the string, the

14

spacetime fields are dynamical and O(d, d) invariance should be viewed as a symmetry of the

theory, as we will show next.

2.4 The Target Space Theory

So far we have only discussed the non-linear sigma model and some of its symmetries. In the

previous section we found an interesting invariance of the mass spectrum under the discrete

group O(d, d,Z) that arises from the stringy nature of the model and from considering quantum

mechanical effects. As we will prove next, the action that describes the dynamics of the massless

fields has the more general symmetry O(d, d,R). This symmetry arises upon dimensional reduc-

tion of d coordinates of a higher dimensional theory. In this section we will introduce the low

energy effective action of bosonic string theory, and we will see how this O(d, d,R) invariance

arises from diffeomorphism and gauge invariance. Furthermore, we will explore the first order

α′ correction of this action, and see how the need for a Green-Schwarz mechanism arises to

maintain O(d, d) invariance to this order. The discussion here will not be very technical and we

will not carry out most of the relevant computations. If the reader is interested in the technical

details of some of the computations, he or she should also look at chapter 2 of Ref. [5], chapter

2 of Ref. [17] and chapter 3 of Ref. [18].

The n+ d dimensional low energy effective action of bosonic string theory to lowest (zeroth)

order in α′ is given by

S0[Gµν , Bµν , φ] =1

2κ20

∫dn+dx

√−Ge−2φ

(R − 1

12H2 + 4(∇φ)2

), (2.32)

with G ≡ det(Gµν), Hµνρ = 3∂[µBνρ], R the Ricci scalar of the metric Gµν , coordinates xµ,

µ = 0, ..., n + d and κ20 a constant5 for dimensional reasons. This is the action that describes

the dynamics of Gµν , Bµν and φ and can be seen as the string theory generalization6 of the

action 2.13. We will consider an external space, with coordinates xµ, and an internal space with

coordinates yi, compactified on a torus. We use the index split

xµ = (xµ, yi), µ = 0, .., n− 1, i = 1, ...d,

and we will use the standard Kaluza-Klein reduction ansatz for the metric Gµν and its inverse

Gµν and the ansatz for the B-field Bµν

5In this work, Gµν , Bµν and φ are dimensionless.6One can also include in this action vector fields Aµ, but we will not consider this case.

15

Gµν =

(Gµν + Aiµ gij A

jν Akµ gkj

gik Akµ gij

), Gµν =

(Gµν −AiρGρµ

−AjρGρν gij + AiµGµν Ajν

)

Bµν =

(Bµν − Ak[µAν]k + Aiµ bij A

jν Aµi − Akµ bik

−Aνi + Akν bjk bij

),

(2.33)

where Aiµ and Aµi are n dimensional one forms, Gµν and Bµν are the n dimensional external

metric and B-field and gij and bij are the d dimensional metric and B-field.

The spacetime metric Gµν transforms under n + d dimensional spacetime diffeomorphisms

xµ → xµ − ξµ, for a diffeomorphism parameter ξµ = (ξµ, ξi) as

δGµν = LξGµν , (2.34)

where L is the Lie derivative in n+d dimensions. Equation 2.33, together with the truncation of

the Kaluza-Klein modes to the massless sector is what we mean by dimensional reduction. We

will not present nor discuss here what the massive Kaluza-Klein modes are, but for the fields of

the low energy effective action, this truncation means that they are independent of the internal

coordinates yi. Then, if we perform a transformation of the massless fields, the transformed

fields should not depend on these internal coordinates either. This will impose some constraints

on the diffeomorphism parameters. Using equation 2.34 and taking the spacetime fields to be

independent of yi, we find that ξµ decomposes into

ξµ = ξµ(x),

ξi = Λik y

k + λi(x),(2.35)

where Λik is an invertible constant matrix that belongs to the general linear group GL(d,R) and

λi is an arbitrary xµ dependent parameter. Bµν transforms under diffeomorphisms and gauge

transformations as

δBµν = LξBµν + ∂µζν − ∂ν ζµ, (2.36)

where ζµ = (ζµ, ζi) is a gauge parameter. Similarly to the treatment followed to find the diffeo-

morphism parameter ξµ, we find

ζµ = ζµ(x),

ζi = Λik yk + λi(x) , Λij = Λ[ij] ,

(2.37)

where a symmetric contribution to Λij has been dropped, being a trivial gauge parameter.

Then, the new fields introduced in the ansatze for the n+ d dimensional fields transform under

diffeomorphisms and gauge transformations as

16

δGµν = LξGµν ,

δBµν = LξBµν + 2 ∂[µζν] + ∂[νλkAµ]k + ∂[νλk A

kµ],

δgij = ξρ∂ρgij + Λki gjk + Λk

j gik,

δbij = ξρ∂ρbij + Λki bkj + Λk

j bik − 2 Λij,

δAiµ = LξAiµ + ∂µλi − Λi

k Akµ,

δAµi = LξAµi + ∂µλi + Λki Aµk − 2 Λik A

kµ,

(2.38)

where L is the Lie derivative in n dimensions corresponding to the indices µ. We can classify these

transformations by noting that terms involving the n dimensional diffeomorphism parameter ξµ,

are n dimensional diffeomorphisms. The external fields Gµν , Bµν and the one-forms Aiµ and

Aµi transform as tensor fields, whereas the internal fields gij and bij transform as scalars. From

dimensional reduction, we see that we also get new gauge transformations involving λi and λi,

under which the vector fields Aiµ and Aµi transform as usual U(1) gauge fields. The GL(d,R)

rotations Λik and the shifts by the antisymmetric matrix Λik on the other hand, combine to

form a subset of O(d, d,R), which suggests that the fields with transformations involving these

objects transform under O(d, d). It can be seen more easily if we rewrite the action in an O(d, d)

covariant way. The O(d, d) covariant lower dimensional action is

I0 =1

2κ2

∫dnx√−Ge−Φ

(R+ ∂µΦ∂µΦ− 1

12H2 +

1

8∂µHMN∂µHMN −

1

4HMNFMµνFµν N

),

(2.39)

where the integration with respect to the internal coordinates and κ0 are included in κ, Φ is

the shifted dilaton and is defined as e−Φ =√−ge−2φ, and the field strengths Hµνρ and FMµν are

defined as

FMµν = 2∂[µAMν] ,

AMµ =

(AiµAµi

),

Hµνρ = 3(∂[µBνρ] −1

2AM[µFνρ]M).

(2.40)

The fields Φ, Gµν and Bµν are inert under O(d, d) transformations, and, by the way that

the indices of all the O(d, d) objects are contracted, the action 2.39 is O(d, d) invariant. In this

situation, we do not have the restrictions we had before for the mass spectrum of the string,

namely integer momenta and winding, which means that this theory is invariant under the full

group O(d, d,R).

The fields in the action 2.39 transform under n dimensional diffeomorphisms and gauge

17

transformations as

δGµν = LξGµν ,

δBµν = LξBµν + 2 ∂[µζν] + ∂[νλMAµ]M

δHMN = ξρ∂ρHMN ,

δAMµ = LξAMµ + ∂µλM ,

(2.41)

where the variation δ corresponds to any local transformation in xµ, and we have introduced the

O(d, d) gauge parameter

λM =

(λi

λi

). (2.42)

Using these transformations, it can be shown that the reduced low energy effective action 2.39

is gauge and n dimensional diffeomorphism invariant. We can extend this analysis to first order

in α′. In order to do so, we will gloss over the most relevant aspects of the findings in Ref. [6],

and we will omit some details and explicit expressions of some terms. If the reader is interested

in the details of the computations, and wishes to have a detailed description of all the steps that

will be carried out next, he or she should look into Ref. [19].

The first order correction to the n+ d dimensional action 2.32 is given by [20]

S1 =α′

4

∫dn+dx

√−G e−2φ

(Rµνρσ Rµνρσ − 1

8H2µν H

2 µν − 1

2H µνλ H ρσ

λ Rµνρσ

+1

24Hµνρ H

µσλ H ν

λτ H ρ

τσ

),

(2.43)

where H2 µν ≡ H µρλ H νρλ. Dimensional reduction of d coordinates, in combination with appro-

priate field redefinitions yield the n-dimensional action [6]

I1 = I1O(d,d) +O1, (2.44)

with an O(d, d) invariant term I1O(d,d), and a non-invariant term O1, given by

O1 =α′

6

∫dnx√−Ge−Φ Hµνρ Ωµνρ, (2.45)

where

Ωµνρ = −3

4Tr(∂[µg

−1g∂νg−1∂ρ]b

)+

1

4Tr(∂[µbg

−1∂νbg−1∂ρ]bg

−1), (2.46)

18

with the trace over the internal indices i. While Ωµνρ is not O(d, d) invariant, its external

derivative is, i.e

4∂[µΩνρσ] =3

8Tr(S∂[µS∂νS∂ρS∂σ]S

), (2.47)

with SMN = ηNP HMP , and the trace is over the O(d, d) indices M . From equation 2.47, we

may conclude that the infinitesimal O(d, d) transformation of Ωµνρ is given by

δΩµνρ = 3∂[µXνρ], (2.48)

where Xνρ is a 2-form. The form of this transformation, in combination with the structure of the

non-invariant term 2.45 suggest that the theory can be made O(d, d) invariant by deforming the

curvature Hµνρ, and including a non-trivial transformation of the external Bµν under O(d, d),

resembling the Green-Schwarz mechanism in heterotic string theory. We can deform Hµνρ as

Hµνρ = Hµνρ − α′Ωµνρ. (2.49)

Note that the additional term is of order α′. This deformation has the following effect on the

action:

I0 + I1O(d,d) +O1 → I0 + I1O(d,d) +O(α′2), (2.50)

where in I0 and I1O(d,d), H is replaced by H. Notice that we have eliminated the non-invariant

term with this deformation. O(d, d) invariance of the action is manifest if H is O(d, d) invariant,

which can be guaranteed if the B-field acquires the following transformation under O(d, d):

δBµν = α′Xµν . (2.51)

With this non-standard transformation of the B-field the n dimensional theory is manifestly

O(d, d) invariant. The deformation of H and the non-trivial transformation of the B-field, how-

ever, do not match the structure of the objects present in the usual GS mechanism. Nevertheless,

it is possible to formulate the theory in a way that the same structures arise for O(d, d) (or rather

GL(d)×GL(d), as we shall see next) by writing the theory in a frame formalism [21,22]. In this

formalism, the O(d, d) metric η can be expressed in terms of vielbeins as [23]

ηMN = EMA(x) ηAB EN

B(x). (2.52)

It is obvious that the vielbeins EMA are O(d, d) elements, given that they preserve the O(d, d)

metric η. In addition, introducing the inverse vielbeins EAM such that

19

EMAEB

M = δAB, (2.53)

it is possible to see that the flat indices are raised (and lowered) with ηAB (and ηAB), by noting

that the following equation holds:

EAM = ηAB η

MN ENB. (2.54)

Similarly, we can write the generalized metric as

HMN = EMA(x)κAB EN

B(x), (2.55)

where we assume κAB to be block diagonal which, without further assumptions, implies a local

gauge symmetry under the group GL(d)×GL(d). With this choice, we can split the flat indices

as A = (a, a), where a and a correspond to independent copies of GL(d).

Let us turn to the local GL(d)×GL(d) symmetry. Under these transformations, the vielbeins

transform as

δΛEAM = ΛA

B EBM , (2.56)

for an infinitesimal gauge parameter Λ, which in matrix form is given by

ΛAB =

(Λa

b 0

0 Λab

). (2.57)

We define the Maurer-Cartan form WµAB as

WµAB ≡ EA

M ∂µEMB. (2.58)

The Maurer-Cartan form obeys the Maurer-Cartan equation, i.e has a vanishing non-abelian

field-strength, is antisymmetric in the flat indices and transforms as a connection, namely

δΛWµAB = −DµΛA

B, (2.59)

where we have introduced the covariant derivative

DµΛAB ≡ ∂µΛA

B + [Wµ,Λ]AB. (2.60)

The Maurer-Cartan form can be decomposed, in matrix form, as

20

WµAB ≡

(Qµa

b Pµab

Pµ ab Qµ a

b

), (2.61)

where Pµab and Pµ a

b are GL(d) × GL(d) tensors, and Qµab and Qµ a

b are GL(d) × GL(d)

connections, i.e

δΛQµab = −DµΛa

b, δΛQµ ab = −DµΛa

b, (2.62)

with the covariant derivative Dµ defined as

DµΛab ≡ ∂µΛa

b + [Qµ,Λ]ab, DµΛa

b ≡ ∂µΛab + [Qµ, Λ]a

b. (2.63)

We can now consider the Chern-Simons-form of the connection Qµ

CSµνρ(Q) ≡ Tr

(Q[µ∂νQρ] +

2

3Q[µQνQρ]

), (2.64)

with the trace over the gauge group indices, and similar for the barred connection. If we plug

into equation 2.64 the explicit expression of the connections Qµ, which can be easily worked

out by choosing the appropriate gauge for the tangent space metric, the deformed field-strength

Hµνρ can be written as

Hµνρ = Hµνρ −3

2α′(CSµνρ(Q)− CSµνρ(Q)

), (2.65)

coinciding precisely with the usual GS mechanism of heterotic string theory. Furthermore, the

Chern-Simons-form for the Q connection transforms under gauge transformations as

δΛCSµνρ(Q) = ∂[µ Tr(∂νΛQρ]

). (2.66)

The new deformed Hµνρ, together with the transformation 2.66 imply that, in order to guar-

antee GL(d)×GL(d) symmetry, the B-field has to acquire a non-trivial gauge transformation. In

this case O(d, d) invariance is manifestly realized, whereas the GL(d)×GL(d) gauge symmetry

gets deformed. The transformation 2.66 implies that the B-field transforms as

δBµν =1

2α′Tr

(∂[µΛQν]

)− 1

2α′Tr

(∂[µΛQν]

), (2.67)

under GL(d) × GL(d) transformations. This resemblances, once again, the non-trivial trans-

formation that the B-field acquires in the heterotic version of the GS mechanism. We do not

necessarily have to choose a GL(d)×GL(d) gauge symmetry; we can make different choices for

the flat metric. We will see later, in the formulation of the worldsheet theory that we propose,

21

that choosing SO(d)× SO(d) gauge symmetry is more convenient.

Up to this point in our discussion of the spacetime theory we have not mentioned any quantum

effect. Recall that α′ corrections are related to the extended nature of strings rather than to

quantum effects. O(d, d) invariance arises from purely classical considerations such as gauge and

diffeomorphism invariance of the higher dimensional theory. This differs from the worldsheet

theory, where O(d, d) invariance appears due to the quantized momentum. Consider for example

an O(d, d,Z) transformation of LM defined in the previous section. Such a transformation for a

matrix Ω yields

w′i = Aij wj +Bij nj,

n′i = Cij wj +Di

j nj,

for ΩMN =

(Aij Bij

Cij Dij

).

(2.68)

If the momenta nj were not integers, it would imply that upon an O(d, d,Z) transformation

the transformed winding numbers w′i break the boundary conditions that we imposed in the

first place, because by dealing with closed strings the winding has to be an integer number.

This motivates us to investigate the relevance of quantum effects in O(d, d) invariance, when

constructing a manifestly O(d, d) invariant action for the string.

22

Chapter 3

Towards an O(d, d) Invariant Worldsheet

Action

In the previous chapter we reviewed how O(d, d,Z) invariance arises at the worldsheet level

from the quantum nature of the momenta along compact coordinates and also how the more

general O(d, d,R) invariance arises when one compactifies d dimensions of the n+d dimensional

spacetime action. We saw as well, that the condition of quantized momenta is crucial for the

symmetry in the worldsheet case, and that non-integer momenta break the torus boundary

conditions. In this chapter we will carry out a fully classical analysis of the non-linear sigma

model in which we will find whether O(d, d) is a symmetry of the classical theory. We will present

an analysis which will allow us to explain under which considerations the non-linear sigma model

is O(d, d) invariant. Finally, we will discuss the procedure that we have to follow in order to

couple a classical string to the reduced n dimensional spacetime action in an O(d, d) covariant

fashion. Coupling, in this case, refers to the construction of an action compatible with all the

symmetries realized in the target space theory, so that it is possible to obtain field equations for

the spacetime fields as well as for the string.

3.1 A non-local field redefinition

In this section we will explore a method to make O(d, d) invariance of the non-linear sigma

model action manifest by means of a non-local field redefinition. This method is due to Tseytlin

in reference [10]. We begin by considering a d dimensional non-linear sigma model action in

conformal gauge with constant background fields gij and bij, with i = 1, ...d, where all the

coordinates Y i are periodic in the sense mentioned at the end of section 2.2. The action is given

by

23

SString = − 1

4πα′

∫d2σ

(gij ∂αY

i∂αY i + εαβbij ∂αYi∂βY

j). (3.1)

When we discussed T-duality, we learned that O(d, d) invariance appears in the spectrum of

the string. Thus, in order to realize O(d, d) it is convenient to write the action in Hamiltonian

form. By doing a Legendre transformation, we get the action

SString =1

2πα′

∫d2σ

(2πα′Pi∂τY

i −H), (3.2)

where Pi are the canonical momenta of Y i

2πα′Pi = gij ∂τYj + bij ∂σYj, (3.3)

and H is the O(d, d) invariant Hamiltonian density given by

H =1

2ZMHMNZN ,

ZM =

(2πα′Pi

∂σYi

),

(3.4)

with the O(d, d) vector ZM . We see that even though the Hamiltonian density is O(d, d) in-

variant, we do not know how the term Pi∂τYi transforms. We do know however, how ∂σY

i

transforms, which suggests that we could integrate with respect to σ and find how ∂τYi trans-

forms. This would lead to a non-local relation between Pi and Y i. Rather than doing this, we

will introduce a non-local field redefinition of Pi:

2πα′Pi = ∂σYi, (3.5)

where Yi are a priori unknown fields. We will take these new fields to be periodic as well. By

using this redefinition, we can write the O(d, d) invariant action

SString =1

4πα′

∫d2σ

(∂σY

M∂τY M −HMN∂σYM∂σY

N), (3.6)

where we have performed an integration by parts without any boundary contributions due to

the periodic nature of both Y i and Yi and we have introduced the O(d, d) vector Y M defined as

Y M =

(Y i

Yi

). (3.7)

We see that in order to get O(d, d) invariance, we sacrificed 2 dimensional Lorentz invariance.

24

Varying the action 3.6 with respect to Y M , we find the equations of motion

∂σ (∂τY M) = ∂σ(HMN∂σY

N). (3.8)

In this equation we have equations for both Y i and Yi but in our original theory the only

dynamical degrees of freedom were Y i. This equations can provide a way to relate Y i and

Yi, but to achieve that we need to make use of a symmetry that the O(d, d) invariant action

has, namely shifts by τ dependent functions. Note that the action 3.6 is invariant under the

transformation

Y M → Y M + fM(τ). (3.9)

This is a gauge symmetry, and we can use it to gauge fix any τ dependent function that arises

from integrating equation 3.8 with respect to σ. That integration yields the equations

∂τY M = HMN∂σYN ,

or ∂αY M = εαβHMN∂βY

N ,(3.10)

which are the O(d, d) covariant version of the self-duality relations 2.24. They relate the fields

by

∂σYi = gik ∂τYk + bik ∂σY

k, (3.11)

which coincides with the equations of the canonical momenta 3.3. We can use the relation 3.10

in the equations of motion 3.8 and see that these are the same as the Hamiltonian equations of

motion that we would get from the action 3.2 after noting that ∂τ∂σYi = 2πα′∂τPi. This shows

that both actions 3.2 and 3.6 are physically equivalent. Except, that is not entirely true. We

have redefined Pi in terms of new fields Yi that we assumed to be periodic and that periodicity

allowed us to integrate by parts without boundary contributions and express the term Pi∂τYi

in terms of O(d, d) covariant objects. If we assume Yi to be periodic, we are implicitly assuming

that the center of mass momentum of the string vanishes. This is not the case in the original

non-linear sigma model. Periodicity of Yi implies

∫ 2π

0

dσ ∂σYi = 0 =

∫ 2π

0

dσ Pi, (3.12)

indicating a vanishing zero mode of Pi. With this in mind, we can see that precisely the vanishing

of this zero mode is what gives the action 3.6 the gauge invariance that let us integrate the

equations of motion to get the self-duality relations. For this reason it is worth looking at the

mode expansion of the sigma model in order to elucidate this point and O(d, d) invariance.

25

3.2 Mode expansion

With the following analysis we intend to explain under what circumstances and assumptions the

classical non-linear sigma model is O(d, d) invariant. To start, we will consider a d dimensional

sigma model on a torus with constant background fields gij and bij, but for the following analysis

we will keep the worldsheet metric hαβ. The boundary conditions for a torus are

Y i(σ + 2π, τ)− Y i(σ, τ) = 2πLi, Li =√α′wi, with wi ∈ Z,

where wi is the winding number around each dimension Y i. With these boundary conditions

the mode expansion of Y i is

Y i(σ, τ) = Liσ +∑n∈Z

yin(τ) einσ. (3.13)

The action for such a string is

SString = − 1

4πα′

∫d2σ√−h(hαβgij ∂αY

i∂βYi + εαβbij ∂αY

i∂βYj), (3.14)

which in the Hamiltonian formalism takes the form

SString =1

2πα′

∫d2σ

(2πα′Pi∂τY

i − uN − eH), (3.15)

with the canonical momenta given by

2πα′Pi =1

egij(∂τY

j − u∂σY j)

+ bij ∂σYj. (3.16)

In equation 3.15 we have introduced the gauge fields e and u and the Virasoro constraints N

and H, the canonical generators of diffeomorphisms on the worldsheet, given by

e = − 1√−hh00

, u = −h01

h00

N =1

2ZMZM , H =

1

2HMNZMZN .

(3.17)

Note that conformal gauge in this formalism corresponds to setting u = 0 and e = 1.

We see that the constraints 3.17 are, at least at first sight, O(d, d) invariant. But since they

are written in terms of the O(d, d) vector ZM , we need to be cautious when performing O(d, d)

transformations. On the one hand, ZM includes the winding numbers and we need to ensure

that upon transforming these winding numbers we do not break the boundary conditions. On

the other hand, ZM includes the zero mode of the canonical momenta Pi, which, as we previously

26

saw, is problematic once we do the non-local field redefinition. Besides these considerations it is

also worth mentioning that the term Pi∂τYi has to be treated carefully as well since it includes

the zero mode of Pi. For these reasons, the best approach to analyzing O(d, d) invariance of the

action is by expanding in σ all fields and constraints in the action [24]. The Fourier modes read

Y i(σ, τ) = Liσ +∑n∈Z

yin(τ) einσ,

Pi(σ, τ) =∑n∈Z

pi n(τ) einσ,

e(σ, τ) =∑n∈Z

en(τ) einσ,

u(σ, τ) =∑n∈Z

un(τ) einσ,

(3.18)

with reality condition ϕ∗n = ϕ−n for ϕn := (yin, pi n, en, un). In terms of the Fourier expansions

for all fields, the action 3.15 becomes

SString =1

α′

∫dτ∑n∈Z

(2πα′pi n∂τy

i−n − u−nNn − e−nHn

). (3.19)

In order to write the mode expansion of the Virasoro constraints, let us first expand ZM as

ZM(σ, τ) = LM(τ)+∑n6=0

ZM n(τ) einσ,

with ZM n(τ) =

(2πα′pi n(τ)

in yin

), n 6= 0,

LM(τ) =

(2πα′pi 0(τ)

Li

).

(3.20)

We can do the invertible field redefinition of the non-zero modes of Pi

2πα′ pi n(τ) = in yi n(τ) for n 6= 0, (3.21)

which corresponds to the non-local field redefinition introduced in the previous section

2πα′ Pi = ∂σYi,

and upon integration, we obtain the mode expansion for Yi

Yi = 2πα′ pi 0(τ)σ + yi 0(τ) +∑n6=0

yi n einσ. (3.22)

27

At this point, it is worth mentioning that the zero modes yi 0(τ) do not appear in the action

because only terms containing derivatives of Yi with respect to σ are present, and the dual

coordinates Yi do not describe a string winding around a dual compact space. We can observe

this by noting that pi 0(τ) are not integers; they depend on the worldsheet parameter τ . On the

other hand, from 3.20 and 3.21, we can define the non-zero modes of the O(d, d) vector Y M as

yMn (τ) =

(yin(τ)

yi n(τ)

)for n 6= 0, (3.23)

and subsequently, we can write the mode expansion of the Virasoro constraints 1

N0 =1

2

(ηMNLMLN + ηMN

∑k 6=0

k2yMk yN−k

),

H0 =1

2

(HMNLMLN +HMN

∑k 6=0

k2yMk yN−k

),

Nn =1

2ηMN

(2inyMn L

N −∑k 6=0

k(n− k)yMk yNn−k

), n 6= 0,

Hn =1

2HMN

(2inyNn L

N +∑k 6=0

k(n− k)yMk yNn−k

), n 6= 0.

(3.24)

Let us discuss the term Pi∂τYi. In the action, upon Fourier expanding and using 3.21 for the

non-zero modes, it takes the form

∫d2σPi∂τY

i = 2π

∫dτ

(pi 0∂τy

i0 +

1

4πα′ηMN

∑n6=0

inyMn ∂τyN−n

), (3.25)

where the first term is not O(d, d) covariant, but the second term can transform under arbitrary

O(d, d,R) rotations. This mode expansion seems to tell us that O(d, d) is not a symmetry of the

full action. Moreover, with this mode expansion we have managed to isolate the terms potentially

inconsistent with O(d, d) transformations involving the vector LM , which contains the center-of-

mass momentum and winding numbers. Let us now examine the behavior of this vector under

O(d, d) transformations. Consider an O(d, d) transformation matrix Ω, with components

ΩMN =

(Aij Bij

Cij Dij

), (3.26)

such that the transformed L′i is given by

1Here we have taken the liberty to raise and lower indices in a way that facilitates notation and matrixmultiplication. The constant nature of ηMN allows us to do this.

28

L′i = Aij Lj +Bijpj 0(τ). (3.27)

The Bij transformation yields a non-acceptable boundary condition for closed strings on a torus.

We start from a constant integer winding number and upon rotating it we obtain a winding

number that depends on τ . Furthermore, if we were interested in transforming only classical

solutions where pi 0 = ki, ki ∈ Rd, the boundary conditions would be violated even if Ω belonged

to the discrete subgroup O(d, d,Z).

This analysis suggests that neither the classical action for the string nor its classical solutions

are O(d, d) invariant. Even if we were interested in non-compact spaces (Li = 0), O(d, d) would

be broken for a constant non-vanishing center of mass momentum pi 0. However, we can focus

on the particular sector of the classical theory in which there is no winding Li and no center of

mass momentum pi 0. With these assumptions, LM = 0 and the term pi 0∂τyi0 vanishes. As a

consequence the rest of the terms in the action in this particular sector are manifestly O(d, d,R)

invariant. At first, this truncation seems unphysical. However, recall that in the full theory, the

internal coordinates Y i are only a fraction of the entire space. In addition, the truncation of

the spacetime theory only considers massless Kaluza-Klein modes of the spacetime fields, hence

getting rid of their dual winding modes. Thus, the theory loses all memory of the topological

properties of the torus, and in this respect, truncating the worldsheet theory to zero center-of-

mass-momentum and zero winding, explicitly reflects the truncation of the target space theory

when performing dimensional reduction. It is worth mentioning that instead of this truncation,

one could set the momenta pi 0 = ki, ki ∈ Z, which would yield an O(d, d,Z) invariance of the

sigma model. However, there seems to be no physical justification for doing so.

Now that we have identified the appropriate truncation to make the string action O(d, d,R)

invariant, we need to check its consistency. The truncated (LM = 0) is

STruncted =1

2α′

∫dτ∑n 6=0

(inyMn ∂τy−nM − u−nNn − e−nHn

), (3.28)

with the truncated Virasoro constraints

Nn = −1

2ηMN

(∑k 6=0

k(n− k)yMk yNn−k

),

Hn = −1

2HMN

(∑k 6=0

k(n− k)yMk yNn−k

).

(3.29)

If we recast the action 3.28 in a local form, we obtain

29

STruncted =1

4πα′

∫d2σ

(∂σY

M∂τY M − u∂σY M∂σY M − eHMN∂σYM∂σY

N), (3.30)

which is the generalized (without gauge fixing) version of the action 3.6, showing that the

procedure presented in [10] works only, at the classical level, if we truncate the string action.

Following the procedure shown in section 3.1, it is straightforward to see that the equations of

motion for yMn in the action 3.28 indeed coincide with those obtained from the action 3.19 for

pi 0 = 0 and Li = 0.

The zero modes yi0 and yi 0, in contrast, require a more careful treatment. As we mentioned

before, yi 0 never appears in the action, and since we set pi 0 = 0, the only term containing yi0vanished. The absence of these, gives the gauge symmetry Y M → Y M + fM(τ). While for

yi 0 this does not present any problems because yi 0 is just a redundancy of the non-local field

redefinition, yi0 does have non-trivial equations of motion in the original theory. To address this

issue, let us invert the definition of the canonical momentum 3.16, and we find the Hamiltonian

equation of motion for yi0 (with pi 0 = 0 and redefining 2πα′Pi = ∂σYi)

∂τyi0(τ) = V i(τ), (3.31)

where V i is the upper component of the O(d, d) vector

V M(τ) =

(V i(τ)

Vi(τ)

)=

1

2π

∫ 2π

0

dσ(u ∂σY

M + eHMN∂σY N

). (3.32)

Given a solution to the equations derived from 3.30 (that leave the zero modes completely

undetermined), we may integrate 3.31, which yields

yi0(τ) = yi +W i(τ), (3.33)

where yi is a constant and W i(τ) is the upper component of the vector

WM(τ) =

(W i(τ)

Wi(t)

)=

∫ τ

0

dτ ′V M(τ ′). (3.34)

In turn, given a solution 3.33, we can transform it under O(d, d) with the matrix given by 3.26,

and we get

y′i0 (τ) = Aijyi0(τ) +BijWi(τ), (3.35)

obeying ∂τy′i0 = V ′i that is the equation of motion arising from V ′M = ΩM

NVN . We can use the

30

fact that yi 0 is completely arbitrary and has no physical meaning in the original sigma model,

to fix it to

yi 0(τ) = ¯yi + Wi(τ). (3.36)

This allows us to write everything in terms of O(d, d) covariant objects

Y M0 = yM +WM(τ), (3.37)

that is now consistent with the transformation law

Y ′M(σ, τ) = ΩMNY

N(σ, τ). (3.38)

3.3 Coupling the string

The discussion in previous sections has established a better understanding of O(d, d) invariance

of the sigma model, and has shown that the truncation of the theory is consistent. Our task

now will be to couple the worldsheet action to the dynamical spacetime fields. Recall that

the spacetime action upon dimensional reduction from n + d to n dimensions is O(d, d,R),

n-dimensional diffeomorphism and gauge invariant. In this section we will see whether the

dimensional reduction of the spacetime theory is compatible with the O(d, d) invariant sigma

model that we found in the previous section. We will consider the index split X µ = (Xµ, Y i) with

µ = 0, ..., n− 1 external indices, and i = 1, ..., d internal indices. The computations presented in

this section are based on Ref. [24]. The final action that we find coincides with the results from

Refs. [9, 12].

The n+ d dimensional non-linear sigma model is given by

SString = − 1

4πα′

∫d2σ√−h(hαβGµν(X)∂αX

µ∂βXν + εαβBµν(X)∂αX

µ∂βXν). (3.39)

Following the steps carried out in the previous sections of this chapter, we may express the

action 3.39 in the Hamiltonian formalism

SString =1

2πα′

∫d2σ

(2πα′Pµ∂τX

µ − uN − eH), (3.40)

where

31

2πα′Pµ =1

eGµν

(∂τX

ν − u ∂σX ν)

+ Bµν ∂σXν . (3.41)

After using the reduction ansatze 2.33 for the spacetime fields, truncating the spacetime fields

to the massless Kaluza-Klein modes (which amounts to taking the lower dimensional fields to

be independent of Y i) and expanding, the O(d, d) invariant Virasoro constraints and the term

Pµ∂τXµ can be expressed as

Pµ∂τXµ = Pµ∂τX

µ + Pi∂τYi, N = 2πα′Pµ∂σX

µ + 2πα′Pi∂σYi = 2πα′Pµ∂σX

µ +1

2ZMZM ,

H =1

2

[GµνΠµΠν − 2GµλBλνΠµ∂σX

ν + (Gµν +GλρBλµBρν)∂σXµ∂σXν

+HMN(ZM +AMµ ∂σXµ)(ZN +ANν ∂σXν)],

(3.42)

where we use the O(d, d) vector ZM defined in the previous sections and we have defined

Πµ = 2πα′Pµ −AMµ ZM , Bµν = Bµν +1

2AMµ Aν M . (3.43)

Since O(d, d) transformations only act on the internal sector of the theory, there is no reason

to keep the terms that contain Xµ and Pµ in the Hamiltonian formalism. We can recover the

Lagrangian form of these terms by integrating out the momenta Pµ by using their equations

2πα′Pµ =1

eGµν X

ν + Bµν ∂σXν +AMµ ZM , with Xµ = ∂τXµ − u ∂σXµ. (3.44)

Substituting this into the Hamiltonian action gives the action in the mixed form

SString =1

2πα′

∫d2σ

[1

2eGµν X

µXν +(Bµν ∂σXν +AMµ ZM

)Xµ − e

2Gµν ∂σX

µ∂σXν

+2πα′Pi∂τYi − u

2ZMZM −

e

2HMN

(ZM +AMµ ∂σXµ

) (ZN +ANν ∂σXν

)].

(3.45)

Here, note that

1

2eGµν X

µXν − e

2Gµν ∂σX

µ∂σXν = −1

2

√−hhαβGµν ∂αX

µ∂βXν ,

Bµν Xµ∂σX

ν = −1

2εαβBµν ∂αX

µ∂βXν ,

(3.46)

which makes worldsheet diffeomorphism invariance in the n dimensional sector manifest.

Let us now turn to the construction of an O(d, d) invariant action using the concepts that

we have studied so far. The procedure is very similar to the case with the constant background

32

fields, but there are some issues that we need to treat carefully, such as gauge and diffeomorphism

invariance. We will consider here the spacetime fields to be dynamical, given that ultimately,

we want to have an action from which we can get equations for the worldsheet and spacetime

fields. For the sake of simplicity, in this section we will only focus on the equations of motion for

the worldsheet fields. We will get to the equations of motion of the spacetime fields, once we go

to the cosmological case. We begin with the non-local field redefinition for the internal sector

2πα′Pi = ∂σYi,

setting pi 0 = 0 once we consider the fields Yi to be periodic, i.e Yi(σ + 2π, τ) = Yi(σ, τ). With

this redefinition, we have ZM = ∂σYM , which yields

∫d2σPi∂τY

i =1

2

∫d2σ∂σY

M∂τY M .

This allows us to write the action 3.45 in an O(d, d) covariant form. To this end, it is convenient

to define a new derivative Dα as it is done in reference [5]

DαYM = ∂αY

M +AMµ ∂αXµ, (3.47)

so that we can write the O(d, d) invariant action

SString =− 1

4πα′

∫d2σ√−h[hαβGµν ∂αX

µ∂βXν + εαβ

(Bµν ∂αX

µ∂βXν +AMµ DαY M∂βX

µ)]

+1

4πα′

∫d2σ

[DσY

MDτY M − uDσYMDσY M − eHMNDσY

MDσYN],

(3.48)

which exactly coincides with the results from Refs. [9, 12].

Given that the reduced spacetime action is invaritant under n dimensional diffeomorphisms,

gauge and O(d, d) transformations, we need to check that this fundamental symmetries are still

present when we couple the string. n+d dimensional spacetime diffeomorphisms xµ → xµ−ξµ(x)

imply on the worldsheet fields the transformation X µ → X µ − ξµ(X). Following the steps

presented in section 2.4 for dimensional reduction, we obtain the transformations

δξXµ = −ξµ(X),

δλYi = −λi(X),

(3.49)

where we have ignored theGL(d) rotation Λ because it is related toO(d, d) transformations and it

is not local in Xµ. Unlike in the analysis for the spacetime theory, here we will consider separately

the gauge and diffeomorphism transformations. We will denote n dimensional diffeomorpshism

33

variations with δξ and gague transformations with δg. Sometimes we will make the distinction

between the gauge parameters ζ and λ by writing δζ and δλ. Since we are considering dynamical

spacetime fields, they transform as presented in section 2.4 under these transformations, but

they will acquire an additional contribution for diffeomorphisms, as we will see next. Under n

dimensional diffeomorphisms, the fields of the action 3.48 transform as

δξXµ = −ξµ(X),

δξYM = 0,

δξGµν = LξGµν(X) + δξXρ∂ρGµν(X),

δξBµν = LξBµν(X) + δξXρ∂ρBµν(X),

δξAMµ = LξAMµ (X) + δξXρ∂ρAMµ (X),

δξHMN = ξρ∂ρHMN + δξXρ∂ρHMN ,

(3.50)

where we added the contribution δξXρ∂ρϕ(X) for ϕ = (Gµν , Bµν ,AMµ ,HMN). This transforma-

tion comes from the dependence on Xµ of these fields. Using equation 3.50, it is straightforward

to check n dimensional diffeomorphism invariance, by realizing that

δξ(DαYM) = 0.

On the other hand, gauge invariance requires more attention. First, it is important to note

that even though we know how Y i transforms, we do not know how Yi transforms. If we want

to preserve O(d, d) and gauge invariance, the most natural transformation that we can imagine

for Yi is δλYi = −λi(X). This implies δλYM = −λM(X). In order to check this, we can derive

this transformation from the definition of Pi, namely

2πα′Pi =1

egij(DτY

j − uDσYj)

+ bij DσYj − Aµ i∂σXµ, (3.51)

whose gauge transformation yields

2πα′δλPi = −∂σλi,

after noticing that Dα is a gauge covariant derivative, so that

δλ(DαYi) = 0.

This transformation is consistent and implies δλpi 0 = 0, which preserves the chosen space of

solutions with pi 0 = 0. With these considerations, the fields in the action 3.48 transform under

gauge transformations as

34

δgXµ = 0,

δgYM = −λM ,

δgGµν = 0,

δgBµν = 2 ∂[µζν] + ∂[νλMAµ]M ,

δgAMµ = ∂µλM ,

δgHMN = 0.

(3.52)

It can easily be shown that the action 3.48 is invariant under these transformations. Everything

except for

εαβ(Bµν ∂αX

µ∂βXν +AMµ DαY M∂βX

µ)

is manifestly gauge invariant. In order to check gauge invariance of this part of the action,

consider the variation

εαβδλAMµ DαY M∂βXµ = εαβ∂αY M∂βλ

M + εαβ∂νλMAµM∂αXµ∂βX

ν .

The first term is a total derivative, and the second term cancels the δλBµν contribution. δζBµν ,

on the other hand, vanishes because it is a total derivative.

There is still one important issue that needs to be discussed, namely the equations of motion

of the zero mode yi0 and their consistency with gauge transformations. We proceed similarly

as before, and invert equation 3.51 to find the equations of motion for yi0 for pi 0 = 0 using the

non-local field redefinition yields

∂τyi0 = V i(τ), (3.53)

where this time

V M =

(V i

Vi

)=

1

2π

∫ 2π

0

dσ[uDσY

M + eHMNDσY N −AMµ ∂τXµ]. (3.54)

A gauge transformation of V i gives

δλVi = − 1

2π

∫ 2π

0

dσ∂µλi∂τX

µ = − 1

2π

∫ 2π

0

dσ∂τλi, (3.55)

ensuring gauge invariance of equation 3.53, given that

δλyi0 =

1

2π

∫ 2π

0

dσ δλYi = − 1

2π

∫ 2π

0

dσ λi.

35

Similarly, as discussed previously, since yi 0 is a redundancy of the non-local field redefinition,

and hence arbitrary, we can write the O(d, d) covariant equation

∂τYM0 = V M , (3.56)

that can be solved by integration as

Y M0 = yM +WM(τ), with WM(τ) =

∫ τ

0

dτ ′V M(τ ′), (3.57)

which again, is consistent with

Y ′M(σ, τ) = ΩMNY

N(σ, τ).

Let us turn our attention to discussing the equations of motion that arise from the O(d, d)

invariant sigma model. First, to simplify notation and some of the computations, we will do this

in conformal gauge. The action 3.48 in conformal gauge is

SString =− 1

4πα′

∫d2σ

[Gµν ∂αX

µ∂αXν + εαβ(Bµν ∂αX

µ∂βXν +AMµ DαY M∂βX

µ)]

+1

4πα′

∫d2σ

[DσY

MDτY M −HMNDσYMDσY

N].

(3.58)

Varying this action with respect to Y M gives the equations of motion

∂σ(DτY

M −HMNDσYN)

= 0, (3.59)

where we have used for the variation of the action δY S

δ(DαYM) = ∂αδY

M ,

expanded the term εαβAMµ DαY M∂βXµ and integrated by parts to express everything in terms

of covariant derivatives.

Recall that by considering pi 0 = 0, yi0 does not appear in the action, introducing the gauge

symmetry δfYM(σ, τ) = fM(τ). This means that he equations of motion 3.59 do not determine

the dynamics of the zero mode Y M0 (τ). It is worth stressing that the presence of this gauge

symmetry in this theory, as in the case for constant spacetime fields, is due to the fact that the

spacetime fields do not depend on the coordinates Y i. On the other hand, equation 3.59 tells

us that the quantity inside the brackets must be a τ dependent function CM(τ) that depends

on our gauge choice for fM . Nonetheless, an arbitrary gauge choice for fM does not provide a

36

solution to the equations of motion of the original sigma model unless equation 3.53 is satisfied.

It turns out that the only gauge choice that yields 3.53 upon integrating with respect to σ is

CM = 0, i.e

DτYM −HMNDσY

N = 0,

which corresponds to the covariant self-duality relations

DαYM = εα

βHMNDβY N . (3.60)

Similarly, to find the equations of motion for Xµ we vary the action 3.58 with respect to Xµ

and find

Gµν ∂α∂αXν +Gµν Γνλρ ∂αX

λ∂αXρ − 1

2εαβ(Hµνλ ∂αX

ν∂βXλ + 2FMµν ∂αXν∂βY M

)− 1

2∂µHMNDσY

MDσYN = 0,

(3.61)

where we have used the equations of motion 3.59 for Y M . We can restore 2 dimensional Lorentz

invariance in 3.61, by using the covariant self-duality relations 3.60. This leaves us with

Gµν ∂α∂αXν +Gµν Γνλρ ∂αX

λ∂αXρ − 1

2εαβ(Hµνλ ∂αX

ν∂βXλ + 2FMµν ∂αXν∂βY M

)− 1

4∂µHMNDαY

MDαY N = 0.(3.62)

Along the construction of the O(d, d) invariant worldsheet action 3.48, we have made a non-

local field redefinition, which explicitly eliminates manifestly worldsheet diffeomorphism invari-

ance. This symmetry is crucial in string theory, and as such, a consistent worldsheet theory

should incorporate it. Even though we recovered 2D Lorentz invariance in the equations of

motion 3.62 using the self-duality relations, the theory has to have this symmetry, i.e it has

to be invariant under diffeomorphisms (or 2D Lorentz transformations, one considering a flat

worldsheet) off-shell.

Recall that the action of the n+ d dimensional sigma model in the Hamiltonian formalism is

given by

SString =1

2πα′

∫d2σ

(2πα′Pµ∂τX

µ − uN − eH). (3.63)

The canonical fields in this formalism (X µ, Pµ) transform under gauge transformations related

37

to the Virasoro constraints H and N as [24]

δHXµ = α ∂σX

µ + ε HµN Z

N ,

δHPµ = ∂σ

[α Pµ +

1

2πα′ε HµN Z

N

]− 1

4πα′ε ∂µHMN Z

M ZN ,(3.64)

with infinitesimal gauge parameters α generated by N , and ε generated by H. We have written

these transformations, for compactness, in therms of the O(n+ d, n+ d) objects

HMN =

(G− BG−1B

)µν

(BG−1

)µ

ν

−(G−1B

)µν Gµν

ZM =

(∂σX

µ

2πα′ Pµ

), (3.65)

where the indices M can be raised and lowered with the O(n + d, n + d) metric ηMN . The

Hamiltonian gauge fields e and u, in order to guarantee invariance of the action under these

transformations, transform as

δHe = ∂τ ε− [u, ε]− [e, α] , δHu = ∂τα− [u, α]− [e, ε] , (3.66)

where the bracket [·, ·] is the Lie bracket given by

[α1, α2] := α1 ∂σα2 − α2 ∂σα1 . (3.67)

The gauge parameters α and ε are related to the worldsheet geometric vector field ωα associated

to the standard worldsheet diffeomorphism transformations in the Lagrangian formalism by

ε = e ωτ , α = ωσ + uωτ . (3.68)

One of the crucial steps in the construction of the action is to split X µ = (Xµ, Y i) and

Pµ = (Pµ, Pi), to subsequently recover the Lagrangian form of the external coordinates Xµ. For

this reason, we will consider worldsheet diffeomorphims for these coordinates in the Lagrangian

formalism, namely

δLXµ = ωα∂αX

µ. (3.69)

Nonetheless, for the internal sector it is more convenient to perform the transformations in

the Hamiltonian formalism, due to locality and O(d, d) invariance, which in the Lagrangian

formulation would be lost if the self-duality relations are not imposed. Upon doing the split, we

obtain the transformations for the internal canonical pair

38

δY i = α ∂σYi + ε Hi

N ZN ,

δPi = ∂σ

[αPi +

1

2πα′ε HiN Z

N

].

(3.70)

The next step in the construction of the action is to perform the non-local field redefinition

2πα′ Pi = ∂σYi. This redefinition consistently yields the transformation for the coordinates and

their duals

δY i = α ∂σYi + ε Hi

N ZN ,

δYi = α ∂σYi + ε HiN ZN ,

(3.71)

which after using the Kaluza-Klein decomposition of the spacetime fields, can be recast into the

O(d, d) covariant form

δY M = α ∂σYM + ε

[HMN DσY N − e−1AMµ (∂τX

µ − u ∂σXµ)]. (3.72)

Using the relation 3.68 between the gauge parameters and the vector field ωα, we can bring this

expression to the more convenient form

δY M = ωα∂αYM − ωτ

[DτY

M − uDσYM − eHMN DσY N

]. (3.73)

The action 3.48 is invariant under worldsheet diffeomorphisms acting as

δωXµ = ωα∂αX

µ ,

δωhαβ = ∇αωβ +∇βωα ,

δωYM = ωα∂αY

M − ωτ[DτY

M − uDσYM − eHMN DσY N

],

(3.74)

and the transformations of the gauge fields e and u are determined from δωhαβ, and their

definition i.e

e = − 1√−hh00

, u = −h01

h00. (3.75)

In this thesis we will prove that the action is invariant under 2D Lorentz transformations.

A more formal discussion about diffeomorphism and conformal invariance of the action can be

found in Ref. [24]. Choosing conformal gauge and specializing to 2D Lorentz transformations,

39

which amount to choosing ωα to be

ωα(σ) = ωαβ σβ, (3.76)

where ωαβ is infinitesimal and antisymmetric. The worldsheet fields transform as

δωXµ = ωαβ σβ ∂αX

µ,

δωYM = ωαβ σβ ∂αY

M − ωτβ σβ DM ,(3.77)

where we have defined the duality vector DM ≡ DτYM −HMN DσY N . 2D Lorentz invariance

of the external sector of the theory is straightforward to check. The internal sector, on the other

hand, requires more attention.

Before turning to the internal sector, let us briefly comment on the mixed part of the action

∫d2σεαβ AMµ DαY M∂βX

µ. (3.78)

A 2D Lorentz transformation of this term can be brought to the form

∫d2σεαβ

(∂γω

γAMα ∂βY M + ∂βωγAMγ ∂αY M + ∂αω

γAMβ ∂γY M

)=

1

3

∫d2σεαβ∂[γω

γAMα ∂β]Y M ,

(3.79)

where we have used the abbreviation ωα = ωαβ σβ, and introduced the pull-back of the gauge

fields AMα = AMµ ∂αXµ, which transforms as a worldsheet one form, namely δωAMα = LωAMα ,

where L is the worldsheet Lie derivative. Note that here we are discussing 2D Lorentz invariance,

but adopting the notation used for diffeomorphisms. This poses no problems, given that Lorentz

transformations can be regarded as a special type of diffeomorphisms, where the parameters are

given by equation 3.76. Expression 3.79 vanishes due to the fact that it contains the antisym-

metrization of three wordlsheet indices. The internal sector of the action can be written more

compactly in terms of the duality vector as

Sinternal =1

4πα′

∫d2σDσY

MDM . (3.80)

This part of the action is invariant under 2D Lorentz transformations. This can explicitly be

proved after a tedious computation using the transformations

40

δω(DαYM) = Lω(DαY

M)− ∂α(ωτ DM) ,

δωDM = ωα∂αD

M − ωτ[∂τD

M −HMN ∂σDN

],

(3.81)

thus concluding that the theory has the fundamental symmetries that the original non-linear

sigma model has.

41

Chapter 4

Applications

The construction of the O(d, d) invariant worldsheet action offers interesting applications. On

the one hand, one could be tempted to use this O(d, d) invariant theory to perform O(d, d)

covariant perturbation theory, and even conjecture that one could compute O(d, d) covariant

spacetime theories taking this worldsheet theory as a starting point. On the other hand, recall

that the spacetime theory, in the general setting, requires a Green Schwarz (GS)-like mechanism

in order to guarantee O(d, d) invariance. For this reason, it is worth investigating whether such

structures arise from this novel worldsheet theory upon quantization. There are some hints that

this is indeed the case. The need for the GS mechanism in heterotic string theory comes from

the chiral nature of the two families of fermions contained in the worldsheet theory. Different

symmetries in heterotic string theory give rise to classically conserved currents associated to these

fermions, and due to the chiral nature of the theory half of them vanish. Nonetheless, one finds

non-vanishing quantum correlation functions of the classically vanishing currents. Similarly, in

this worldsheet theory which we have developed, we are dealing with chiral fields. The equations

of motion of the fields Y M can be written in light-cone coordinates as

∂σ∂−YML = 0, ∂σ∂+Y

MR = 0, (4.1)

where we have assumed vanishing gauge fields AMµ , a constant HMN , used the projectors Π±MN =12

(ηMN ±HMN), and defined Y ML(R) = Π±MNY

N . These are the equations of motion for left-

and right-moving chiral bosons [25]. The fact that making O(d, d) invariance manifest implies

chiral fields strongly suggests the presence of an anomaly in the theory, as it is the case for

chiral fermions in the heterotic case. This section is mainly dedicated to finding the anomaly

of the string action, and see whether it can be canceled by means of a GS deformation of the

external B-field, as in heterotic string theory, and the target space action to first order in α′ of

the bosonic string.

42

Additionally, in the introduction, we discussed the potential cosmological applications of

O(d, d) invariance. After discussing the anomaly we will specialize to a cosmological spacetime

theory, and then see the effects of this choice on the worldsheet. To finalize, we will see that

the anomaly is of the trivial form in the cosmological case. This means that the anomaly can

be canceled by adding a local counterterm, instead of the GS mechanism. This result is to be

expected given that in cosmology, as we will see, there is no external B-field.

4.1 The anomaly

In order to investigate the anomaly we will consider a simplified setting: we will only analyze

the internal sector of the theory, we will set the gauge fields AMµ to zero, and we will choose

conformal gauge. The action that we consider is then

SY =1

4πα′

∫d2σ

(∂σY

M ∂τY M −HMN ∂σYM ∂σY

N). (4.2)

In section 2.4 we learned that in order to obtain the same structures present in the GS

mechanism, we needed to go to a frame formalism, in which we found the deformation of H and

the non-standard transformation of the B-field. In this section we will do something similar,

however, we will make a different gauge choice. We will choose the generalized metric H to be

HMN = EMA(x)hAB EN

B(x), (4.3)

where hAB is the constant SO(d)×SO(d) metric [24]. Let us clarify that the procedure is exactly

the same as the one followed in section 2.4, with the exception of the choice of the flat metric.

The choice 4.3, however, has no effect on the O(d, d) metric in this formulation, i.e

ηMN = EMA ηAB EN

B, (4.4)

and the flat indices are, as before, raised and lowered with ηAB and ηAB. We can flatten the

O(d, d) indices of the coordinates Y M as

Y M = EAM(X)Y A, (4.5)

and their derivatives take the form

∂αYM = EA

M(∂αY

A −WαBA Y B

)≡ EA

M DαY A, (4.6)

where we have introduced the covariant derivative Dα, and the pull-back of the Maurer-Cartan

43

form1, in this case for SO(d)× SO(d)

WαAB ≡ EA

M∂αEMB = WµA

B∂αXµ. (4.7)

The vielbeins transform under infinitesimal SO(d)× SO(d) transformations as

δλEMA(x) = −λBA(x)EM

B(x) , δλEAM(x) = λA

B(x)EBM(x) , (4.8)

for an antisymmetric infinitesimal gauge parameter λAB. The Maurer-Cartan form transforms

as a connection, i.e

δλWαAB = −DαλAB. (4.9)

The coordinates Y A, on the other hand, transform like

δλYA = λAB(X)Y B, (4.10)

and their covariant derivatives as

δλDαY A = λAB(X)DαY B. (4.11)

In this formulation it is possible to write the action 4.2 as

SY =1

4πα′

∫d2σ

(DσY A DτYA − hAB DσY A DσY B

), (4.12)

which is invariant under local SO(d)×SO(d) transformations, when simultaneously transforming

Y A and WµAB, according to equations 4.10 and 4.9. If we vary this action with respect to the

fields Y A, we obtain the equations of motion

Dσ(DτY A − hAB DσY B

)= 0. (4.13)

In order to find an anomaly for this theory, we need to find a conserved current in the classical

theory which does not survive the quantization procedure. Even though local SO(d) × SO(d)

invariance of the action does not yield a physical current as a rigid symmetry would using

Noether’s procedure, the action variational principle implies that a general variation of the

action 4.2, upon using the field equations 4.13, vanishes i.e

∫d2σ

δS

δWαAB

DαλAB = − 1

4πα′

∫d2σ λAB DαJ α

AB = 0. (4.14)

1Recall that WµAB is antisymmetric in (A,B)

44

The explicit expressions for J αAB are

J τAB = −Y[ADσYB] ,

J σAB = −Y[A

(DτYB] − 2hB]C DσY C

).

(4.15)

The current-like objects J αAB, as we will show next, are not conserved. This is due to the

structure of the Maurer-Cartan form, which contains tensors as well as connections. In order to

elucidate this point, it is helpful to use the notation introduced in Ref. [26]. To this end, let us

define the projectors

ΠA±B =

1

2

(δAB ± hAB

), (4.16)

with the following properties

Π2± = Π± , Π+Π− = 0 , 1 = Π+ + Π− , Tr Π± = d. (4.17)

These projectors allow us to decompose a vector V A as

V A = ΠA+B V

B, V A = ΠA−B V

B, (4.18)

where the indices A and A correspond to the (d, 0) and (0, d) representations of SO(d)× SO(d)

respectively. Using this technology, we find that the naive conservation law 4.14 implies only

projected conservation of J α, i.e

ΠAC+ ΠBD

+ (DαJ α)CD = 0,

ΠAC− ΠBD

− (DαJ α)CD = 0,(4.19)

which can be verified explicitly by using the field equations 4.13, and noting that Wµ decomposes

as

WµAB = QµAB + PµAB = QµAB +QµAB + PµAB − PµBA , (4.20)

with

QµAB := (Π+WµΠ+)AB + (Π−WµΠ−)AB = QµAB +QµAB,

PµAB := (Π+WµΠ−)AB + (Π−WµΠ+)AB = PµAB + PµAB .(4.21)

45

In this equation Qµ and Pµ correspond to the composite connection and tensor field from section

2.4 respectively, but for SO(d)× SO(d) instead of GL(d)×GL(d).

Even though equation 4.19 is not a conventional conservation law, setting the Maurer-Cartan

form to zero, which is equivalent to considering a free theory, we find the well defined SO(d)×SO(d) conserved currents

jτAB = −Y[A∂σYB] , jσAB = −Y[A

(∂τ − 2 ∂σ

)YB] ,

jτAB

= −Y[A∂σYB] , jσAB

= −Y[A

(∂τ + 2 ∂σ

)YB] ,

(4.22)

which obey the standard conservation law

∂αjαAB = 0,

∂αjαAB

= 0.(4.23)

In order to make the chiral nature of the theory manifest, let us write the currents 4.22 in

light-cone coordinates

jAB+ = Y [A(∂+ − 2 ∂−

)Y B] , jAB− = Y [A∂−Y

B] ,

jAB+ = −Y [A∂+YB] , jAB− = Y [A

(2 ∂+ − ∂−

)Y B] .

(4.24)

It is possible to gauge fix the zero mode yA0 (τ) of Y A to zero by making use of the gauge symmetry

δfYA = fA(τ). This gauge choice implies, together with the field equations 4.13 for a vanishing

Wµ, the chiral relations

∂−YA = 0,

∂+YA = 0,

(4.25)

which are the equivalent to the self-duality relations in a free theory with flat indices. With

these relations we find, on-shell and gauge fixing yA0 to zero, that

jAB− ∼ 0, jAB+ ∼ 0. (4.26)

This is similar to the fermionic case in heterotic string theory, where half of the fermionic

currents vanish classically, but the quantum correlation functions (schematically) 〈j(x)j(y)〉 of

the classically vanishing currents do not. The similarity between the O(d, d) invariant worldsheet

46

theory and the heterotic case regarding the vanishing of classically conserved currents, and the

need for the GS mechanism in the spacetime theory, suggest that the SO(d)×SO(d) currents 4.24

are potentially anomalous. One could try to compute the correlation functions of the currents

using canonical quantization and then evaluate whether the Ward identities are anomalous as

it is proved in Ref. [24]. This approach is advantageous in that one does not need to employ

any regularization to perform the computations. Nonetheless, it is not easy to interpret the

corresponding effective action in order to compute its anomalous transformation. Additionally,

this approach relies on the mode expansion of Y A and since we are working with closed strings (or

on the cylinder), 2D Lorentz transformations are ill-defined for the Fourier modes of Y A because

the worldsheet-coordinate dependence of these transformations does not respect the periodicity

of Y A. In order to overcome these difficulties we will work in 2D Minkowski space rather than on

the cylinder. This choice is justified by the fact that anomalies do not care about the topology of

the space, and we will seek the anomaly from the perspective of a non-gauge-invariant effective

action.

Let us now turn to the quantum theory, and the search for the anomaly. We begin by rewriting

the action 4.12 in terms of the projectors 4.16, and expanding the Maurer-Cartan form

SY =1

2πα′

∫d2σ

[(DσY A + PA

σ BYB)(D−YA + P−ACY

C)

+(DσY A + PA

σ BYB) (D+YA + P+ACY

C)],

(4.27)

with the new covariant derivative Dα, such that

DαY A = ∂σYA +QAB

α YB, (4.28)

and similar for the over barred indices. Note that we have written the action in terms of the Pull-

backs Qα = Qµ ∂αXµ, Pα = Pµ ∂αX

µ, and we have used the light-cone coordinate derivatives

D± = 12

(Dτ ±Dσ). Recall that σ ∈ R, and not σ ∈ [0, 2π] as before, since we are doing the

computations in 2D MInkowski space. The QABµ is antisymmetric in the gauge group indices,

i.e

QABµ = −QBA

µ , (4.29)

similar for the over barred indices, and they are SO(d) connections, namely they transform as

δλQABµ = −DµλAB. (4.30)

On the other hand, the tensor fields PµAB transform as

47

δλPµAB = λAC PµCB + λB

C PµAC . (4.31)

The action 4.27 has an interesting structure. If we set the tensor fields Pα to zero, the action

reduces to the sum of the actions for left- and right-moving chiral bosons coupled to a gauge

field Qα, very similar to the fermions of heterotic string theory. For this reason and for the sake

of simplicity let us consider, for the moment, vanishing tensor fields Pα. The action then reads

SY =1

2πα′

∫d2σ

[DσY AD−YA +DσY AD+YA

]. (4.32)

Quantizing both, left- and right-moving, Floreanini-Jackiw (FJ) or chiral bosons is essentially

identical so we will only present here how we proceed for the left-moving fields. The action for

these fields is

SFJ =1

2πα′

∫d2σDσY AD−YA

=1

4πα′

∫d2σ gαβDαY ADβYA,

(4.33)

where to get to the second line we defined the metric gαβ as

gαβ =

(0 1

1 0

), α = (−, σ). (4.34)

With this notation and inserting the definition of the covariant derivatives yields

SFJ = S0 + S3 + S4, (4.35)

where

S0 =1

4πα′

∫d2σ gαβ ∂αY

A ∂βYA, S3 = − 1

2πα′

∫d2σ gαβ QAB

α YA∂βYB,

S4 =1

4πα′

∫d2σ gαβ QCA

α QβCB YAYB.

(4.36)

Our task will now be to find the one loop effective action emerging from integrating out the

chiral bosons of the action 4.35 in order to evaluate whether it is invariant under SO(d)×SO(d)

transformations. If the effective action fails to be invariant, as we will now prove, there is an

SO(d)× SO(d) anomaly. We define the effective action for the composite connections as

48

eiW [Q] = Z−1

∫DY eiSFJ [Y,Q], (4.37)

where we have chosen to normalize the effective action using the free partition function

Z =

∫DY eiS0[Y ]. (4.38)

Splitting SFJ in equation 4.37 as indicated in 4.36 leads to

eiW [Q] = 〈ei(S3+S4)〉, (4.39)

where 〈...〉 denotes the expectation value with respect to the free theory. We will work per-

turbatively, and we will consider up to one loop contributions to the effective action. Taylor

expanding the expectation value in equation 4.39 leads to

〈ei(S3+S4)〉 = 〈1 + iS3 + iS4 +1

2(iS3 + iS4)2 + ...〉. (4.40)

We will focus on the quadratic piece of the effective action, namely the terms that only involve

two composite connections, which is given by

W (2)[Q] = 〈S4〉+i

2〈S2

3〉conn. ∼QABα Q

CDβ

+ , (4.41)

where the solid lines correspond to the Y A propagators and only connected diagrams are con-

sidered, given that the effective action can be seen as the generator for connected diagrams.

In order to evaluate the expectation values of equation 4.41 we need to find the explicit

expression for the propagators of the free theory S0. These can be found by starting from

0 =

∫DY

δ

δY A(σ)

[eiS0 Y B(σ′)

], (4.42)

which leads to

−σ〈YA(σ)Y B(σ′)〉 = −i2πα′ δAB δ(2)(σ − σ′), with −σ ≡ gαβ∂α∂β, (4.43)

and after going to Fourier space and reorganizing, we obtain the propagator

49

〈YA(σ)Y B(σ′)〉 =α′ δA

B

2π

∫d2k

i eik(σ−σ′)

k2−σ + iε

=α′ δA

B

4π

∫d2k

i eik(σ−σ′)

k−kσ + iε,

(4.44)

where k2−σ ≡ gαβkαkβ = 2k−kσ, and in the second line we let ε absorb a factor of 1/2. Using

equation 4.41 in combination with these propagators, we obtain the quadratic piece of the

effective action

W (2)[Q] = − i4

∫d2p gαγ gβδ QAB

γ (p) Παβ QABδ (−p), (4.45)

with

QABα (σ) =

∫d2p eikσQAB

α (p), (4.46)

and, from the diagrams in equation 4.40, we compute the vaccuum polarization

Παβ(p) = −1

2

∫dk− dkσ

(2kα − pα) (2kβ − pβ)− 2gαβ [k−kσ + (p− − k−) (pσ − kσ)]

kσ (pσ − kσ)(k− + iε

kσ

)(k− − p− − iε

pσ−kσ

) . (4.47)

The integral with respect to k− can be regularized by including a factor eiδk−kσ [24]. This

regulator, however, is not Lorentz invariant, but as we will see, this will not affect the anomaly.

The integral with respect to kσ, on the other hand, has an infrarred divergence in the form of

∫ |pσ |0

dkσkσ

. (4.48)

For this reason we will regulate this integral by setting an infrared cutoff µ, namely

∫ |pσ |0

dkσ →∫ |pσ |µ

dkσ. (4.49)

Performing the integrals using these regulators leads to the one loop effective action

50

W (2)[Q] = −π2

∫d2p

−2QAB

− (p)pσp−Q−AB(−p)

+QAB− (p)

pσp−

log (|pσ|/µ) Q−AB(−p)

+QABσ (p)

p−pσ

log (|pσ|/µ) QσAB(−p)

−2QABσ (p) log (|pσ|/µ) Q−AB(−p)

.

(4.50)

It is straightforward to check that, upon performing a gauge transformation to lowest order

in Qα of the connection (in momentum space), i.e δλQABα (p) = −ipα λAB(p), the last three lines

cancel, while the first one does not. For this reason this term will determine the anomaly. We

can rewrite this term, by noting that pσ = p+ − p−, as

∫d2pQAB

− (p)pσp−Q−AB(−p) =

∫d2p

[QAB− (p)

p+

p−Q−AB(−p)−QAB

− (p)Q−AB(−p)]. (4.51)

None of the above terms is gauge invariant, and their combination is not gauge invariant either.

However, while the first term is non-local, the second one is local. If it is possible to add a local

counterterm to the effective action that cancels the anomaly, the anomaly is said to be irrelevant,

otherwise, it is called relevant [27,28]. With this in mind, it is obvious that the second term in

equation 4.51 is an irrelevant contribution. Thus, in order to compute the anomaly we will only

consider the first term. The gauge variation of the effective action to lowest order in Qα is given

by

δλW(2)[Q] = −2πi

∫d2p λAB(p) p+ Q−AB(−p) +O(Q2). (4.52)

This seems to indicate that the theory is anomalous. We can try to add a local counterterm to

cancel this anomaly (as we did above in 4.51). To this end, notice that the gauge transforma-

tion is Lorentz invariant meaning that we must write the most general Lorentz invariant local

counterterm, which is

∆W (2)[Q] = α

∫d2pQAB

+ (p)Q−AB(−p). (4.53)

The transformation including this counterterm reads

δλ(W (2)[Q] + ∆W (2)[Q]

)= −i

∫d2p λAB [(2π − α)p+Q−AB(−p) + αp−Q+AB(−p)] , (4.54)

51

thus proving that this is indeed a relevant anomaly. We will choose α = −π in order to have a

purely parity violating anomalous variation, and we further define

W(2)eff [Q] = W (2)[Q]− π

∫d2pQAB

+ (p)Q−AB(−p), (4.55)

whose gauge variation to lowest order in Qα is

δλW(2)eff [Q] = −iπ

∫d2p λAB(p) [p+Q−AB(−p)− p−Q+AB(−p)]

=1

4π

∫d2σ λAB [∂+Q−AB − ∂−Q+AB]

=1

8π

∫d2σ λAB εαβ ∂αQβAB,

(4.56)

where in the last line we recover the 2D Minkowski indices α = (τ, σ), and we used the totally

antisymmetric tensor in light-cone coordinates ε+− = 2. Performing a completely analogous

procedure for the right-moving chiral bosons, one finds that the anomaly acquires an additional

contribution, namely

δλ,λW(2)eff [Q, Q] =

1

8π

∫d2σ

[λAB εαβ ∂αQβAB − λAB εαβ ∂αQβAB

]. (4.57)

Up to this point we have neglected the tensor field Pµ, which means that the result above

may change once we include it. Nevertheless, from its gauge transformation

δλPµAB = λAC PµCB + λB

C PµAC , (4.58)

it is clear that including it would have no effect on the result 4.57, given that the term containing

it in the effective action would be of the form

∫d2pPAB

α Gαβ(p)PβAB, (4.59)

and its gauge transformation, if non-vanishing, would be of quadratic order in background fields,

hence not contributing to this (linear) order. With this in mind, the final result for the gauge

transformation of the effective action, to linear order in background fields is

δλ,λW(2)eff [Q, Q, P ] =

1

8π

∫d2σ

[λAB εαβ ∂αQβAB − λAB εαβ ∂αQβAB

]=

1

8π

∫d2σ εαβ ∂αX

µ ∂βXν Tr

(∂[µλQν] − ∂[µλ Qν]

),

(4.60)

52

where to arrive at the last line we integrated by parts and wrote the index contraction as a

trace, hence getting rid of the minus sign. This anomalous variation obeys the Wess-Zumino

consistency (or integrability) conditions [29], i.e

[δλ1 , δλ2 ]W(2)eff = δ[λ1,λ2]W

(2)eff , (4.61)

which indicates that this is the full anomaly, and it does not receive any higher order contribu-

tions.

The anomaly implies that gauge-equivalent background fields (from the target space per-

spective) lead to inequivalent worldsheet quantum theories. This is not acceptable. By simple

inspection of equation 4.60, the interaction term of the worldsheet scalars with the B-field has

a similar structure:

SB = − 1

4πα′

∫d2σ εαβ Bµν ∂αX

µ ∂βXν . (4.62)

Thus, in the same spirit of the GS mechanism of heterotic string theory, we make the external

B-field acquire the following non-standard transformation under SO(d)× SO(d):

δλ,λBµν =1

2α′ Tr

(∂[µλQν]

)− 1

2α′ Tr

(∂[µλ Qν]

), (4.63)

which cancels the anomaly, and exactly matches with the findings in Ref. [6], presented in

equation 2.67.

4.2 A cosmological spacetime

We argued in the introduction of this thesis that O(d, d) invariance offers interesting potential

applications in cosmology. In this section we will specialize to a cosmological spacetime, which

we will probe with a string using the O(d, d) invariant action found in the previous chapter. In

cosmology the non-trivial dynamics of the fields is dictated only by cosmic time. This means that

the spacetime fields of the low energy effective action are independent of all spacelike coordinates

in a cosmological setting. We learned from dimensional reduction of the target space that after

truncating the theory, which amounts to declaring the fields to be independent of the internal

coordinates, we lose all information about the topology of the internal space. This means that

the internal coordinates may or may not be compact, and a sufficient condition for O(d, d)

invariance of the spacetime theory is that the fields have to be independent of the internal

coordinates, or more technically, that the space has to have d abelian isometries. This is indeed

the case in cosmology, where there are no compact coordinates, and the fields only depend on

time. Therefore, the O(d, d) invariant worldsheet action found in the previous chapter, where the

53

associated truncation is necessary, is consistent with the cosmological setting. We had previously

considered an n+d dimensional spacetime action, which we reduced to an n dimensional theory.

Here, we begin with a 1 + d dimensional theory with the index split xµ = (t, yi), i = 1, ..., d and

the ansatze for the spacetime fields are

Gµν =

(−n2 + Ai0gijA

j0 Ak0gkj

gikAk0 gij

),

Bµν =

(0 Ai 0 − Ak0bik

−Ak 0 + Ak0bjk bij

),

(4.64)

where we have used that in one dimension there are no 2-forms or antisymmetrization of indices

and n(t) is a lapse function. It is also worth mentioning that the one dimensional 1-forms do

not have a field strength in one dimension. This means that they are not dynamical.

The gauge and diffeomorphism parameters are

ξ0 = ξ(t),

ξi = Λik y

k + λi(t),

ζi = Λik yk + λi(t).

(4.65)

We can simplify things by using the fact that Ai0 and Ai 0 do not have a dynamical term in the

spacetime action and only examine the simpler reduction ansatz

Gµν =

(−n2(t) 0

0 gij(t)

),

Bµν =

(0 0

0 bij(t)

).

(4.66)

The resulting reduced spacetime action is

I0[S,Φ, n] =1

2κ2

∫dtne−Φ

(−(DΦ)2 − 1

8tr(DS)2

), (4.67)

where Φ = Φ(t) is the shifted dilaton and we have introduced the covariant derivative D and

the matrix S defined as

D ≡ 1

n(t)∂t, SMN ≡ ηMPHPN , (4.68)

and we have adapted to the notation presented in reference [30]. S transforms under O(d, d) in

matrix notation as

54

S → Ω−1SΩ, Ω ∈ O(d, d),

indicating that the lower dimensional symmetries of this action areO(d, d,R) and time reparametriza-

tion invariance for the transformation t→ t− ξ(t). The former is obvious by using the cyclicity

property of the trace. We could be tempted to think that the action is also invariant under the

general linear group of 2d dimensions GL(2d,R). Nevertheless, the generalized metric H is not

an arbitrary symmetric matrix; it belongs to the group O(d, d), thus the O(d, d) transforma-

tions are the most general transformations that preserve the structure of H. This reduces the

symmetry group to O(d, d).

Diffeomorphism invariance can be checked by noting that the fields transform as

δξn = ∂t(ξn),

δξHMN = ξ∂tHMN ,

δξSMN = ξ∂tSMN ,

δξΦ = ξ∂tΦ.

(4.69)

The covariant derivative D ensures that the derivative D of a scalar is a scalar as well and it

follows the Leibniz rule and satisfies the usual integration by parts [3]

∫dtnB(DA) =

∫dt∂t(AB)−

∫dtn(DB)A. (4.70)

Now that we have checked that the action has the desired symmetries for the situation we

want to study, we can proceed and find the equations of motion for these fields. Varying 4.67with

respect to the different fields, we obtain:

δΦI0 =

∫dtne−ΦδΦEΦ, (4.71a)

δnI0 =

∫dtne−Φ δn

nEn, (4.71b)

δSI0 =

∫dtne−Φtr(δ SFS). (4.71c)

The equations of motion for the dilaton are

EΦ ≡1

2κ2

(2D2Φ− (DΦ)2 +

1

8tr(DS)2

). (4.72)

The computation to find equation 4.72 is straightforward. In contrast, to find the equations of

motion for the lapse function n(t) we need to be somewhat careful because n(t) transforms as

55

a density. Note that in equation 4.71b we multiplied the variation δn by a factor of 1n, which

makes the quantity δnn

a scalar. This ensures that En is also a scalar. This yields for n(t) the

equation

En ≡1

2κ2

((DΦ)2 +

1

8tr(DS)2

). (4.73)

In order to compute the equations of motion for the field S, first note that S2 = 1. This

imposes the constraint δ S = −S(δ S)S. This in turn, implies that the vanishing of FS does

not correspond to the equations of motion of S. We can write δ S in terms of an unconstrained

variation δK [3]

δ S =1

2(δK − SδK S) , (4.74)

which, if we plug into equation 4.71c gives

δSI0 =

∫dtne−Φtr(δKES), ES =

1

2(FS − SFS S) . (4.75)

Since δK is unconstrained, the vanishing of ES corresponds to the equations of motion for

the field S. It can be quickly verified that tr(δ SFS) = tr(δ SES) by using the constraint

δ S = −S(δ S)S. This leads to

ES ≡1

8κ2

(D2 S + S(DS)2 −DΦDS

). (4.76)

Note that all these equations are O(d, d) invariant. The only object that is affected by these

transformations in the equations for Φ and n is the trace term which we know is invariant. The

equations for S are invariant as well by noting that all terms transform in the same manner.

4.3 The cosmological worldsheet action

Similarly to the procedure followed in section 3.3, we make the index split for the coordinates

X µ = (X0, Y i) with i = 1, ..., d. The O(d, d) invariant non-linear sigma model action for a string

using the cosmological ansatz 4.66 is

SString =1

4πα′

∫d2σ√−hhαβn2(X0)∂αX

0∂βX0

+1

4πα′

∫d2σ

(∂σY

M∂τY M − u ∂σY M∂σY M − eHMN(X0)∂σYM∂σY

N).

(4.77)

56

Time reparametrizations t→ t− ξ(t) have the effect on the worldsheet X0 → X0− ξ(X0). This

implies the transformations under time reparametrization of the fields in the action 4.77

δξX0 = −ξ,

δξYM = 0,

δξn = ∂0(ξn) + δξX0∂0n,

δξHMN = ξ∂0HMN + δξX0∂0HMN ,

(4.78)

It is straightforward to check that the action is invariant under these and O(d, d) transfor-

mations.

Now that we understand that the non-local field redefinition is consistent for a theory in which

we focus our attention on a sector that does not include winding and center of mass momenta

along the internal coordinates, it is worth checking whether in this cosmological case the O(d, d)

invariant action reproduces the same equations of motion as the usual cosmological non-linear

sigma model without any field redefinition. This will allows us to see that they are equivalent

(of course, modulo the zero mode discussion in section 3.2). The discussion for the coordinates

Y i can be found in section 3.3. It is more interesting to discuss the equations of motion of the

timelike coordinates X0, because the spacetime fields in this case depend on X0. In order to

simplify the computations, we will work in conformal gauge. The original sigma model for a

string in a background determined by the cosmological ansatz in conformal gauge is given by

SString = − 1

4πα′

∫d2σ[−n2(X0)∂αX

0∂αX0 + gij(X0) ∂αY

i∂αY j

+ εαβbij(X0) ∂αY

i∂βYj].

(4.79)

Varying this action with respect to X0 gives the equations of motion

∂α∂αX0 +

1

n∂0n ∂αX

0∂αX0 +1

2n2∂0gij ∂αY

i∂αY j +1

2n2εαβ∂0bij ∂αY

i∂βYj = 0. (4.80)

In contrast, the O(d, d) invariant sigma model action in conformal gauge is

SO(d,d) =1

4πα′

∫d2σ n2(X0)∂αX

0∂αX0

+1

4πα′

∫d2σ

(∂σY

M∂τY M −HMN(X0)∂σYM∂σY

N),

(4.81)

whose variation with respect to X0 gives the O(d, d) invariant equations of motion

∂α∂αX0 +

1

n∂0n ∂αX

0∂αX0 +1

2n2∂0HMN ∂σY

M∂σYN = 0. (4.82)

57

We can vary the action 4.81 with respect to Y M to obtain the self-duality relations for Y i and

Yi. Doing this variation, the equations of motion for Y M read

∂σ∂τY M = ∂σ(HMN(X0)∂σY

N). (4.83)

Given that n and HMN do not depend on the internal coordinates Y i, the action still has the

gauge symmetry

Y M → Y M + fM(τ).

Integrating equation 4.83 and gauge fixing τ -dependent functions arising after integration using

the gauge symmetry, we obtain the self-duality relations

∂τY M = HMN(X0)∂σYN ,

∂σYi = gik ∂τYk + bik ∂σY

k.(4.84)

Expanding equation 4.82, using the second line of equation 4.84 to eliminate terms involving Yi

and noting that gik gjl ∂0gkl = −∂0gij, we get the original equations of motion 4.80, thus, proving

that both actions are equivalent. Notice that we followed exactly the same procedure carried

out in section 3.1, which is only possible if the spacetime fields do not depend on the internal

coordinates.

One may wonder what happens with the cosmological worldsheet action at the quantum

level. Notice that in cosmology we do not have an external B-field, thus if the quantum theory

for the cosmological O(d, d) invariant sigma model is anomalous, we face serious problems.

Furthermore, this would imply an inconsistency with the cosmological target space theory, since

it is O(d, d) invariant to all orders in α′, and no special anomaly cancellation mechanism is

required. Quantization of the the sigma model in the cosmological setting is exactly the same as

for the more general case with n+ d dimensions, given that the number of internal coordinates

is identical, and the quantization procedure relies only on this sector. Let us recall that, for the

more general case, the anomaly is given by

δλ,λW(2)eff [Q, Q, P ] =

1

8π

∫d2σ εαβ ∂αX

µ ∂βXν Tr

(∂[µλQν] − ∂[µλ Qν]

), (4.85)

where we have already introduced the 2D Lorentz invariant counterterm, and set the constant

α = −π. Fortunately in cosmology, given that the only external coordinate is the timelike

coordinate X0, we have µ = ν = 0, and the anomaly vanishes identically. Hence, the anomaly

in the comoslogical case is irrelevant, due to the inclusion of the 2D Lorentz invariant local

counterterm.

58

Chapter 5

Discussion and conclusions

In this thesis we have found that in the interest of having an O(d, d,R) invariant sigma model

action for the bosonic string, we need to consider a truncation of the worldsheet fields where

the string has zero center-of-mass momentum and no winding around the internal space. This

truncation reflects the situation in the target space theory. For the low energy effective field

theory O(d, d,R) symmetry arises after performing dimensional reduction, where one has to

truncate the theory by only considering massless Kaluza-Klein modes, which means that the

spacetime fields do not probe the internal space, and the topology of the space and the dynamics

of the string in this sector are irrelevant. Once we identified the consistent truncation, we

turned to the construction of the manifestly O(d, d,R) invariant action for a string in a general

background based on Tseytlin’s approach of redefining the internal canonical momenta. The

truncation, in combination with the dimensional reduction procedure followed for the target

space theory, allowed us to construct a manifestly O(d, d,R) invariant worldsheet action that

includes all target space fields that survive the truncation of the target space theory. We found

that the worldsheet action has all the symmetries that the target space theory has, thus allowing

us to have a consistent O(d, d,R) invariant coupled system. Moreover, we proved that the

worldsheet action is 2D Lorentz invariant, albeit not manifestly so.

The construction of the worldsheet action allowed us to show that the theory is anomalous,

giving us a worldsheet-theoretical justification for the GS mechanism in the bosonic target

space theory to first order in α′. We restricted ourselves to analyzing the worldsheet internal

sector of the O(d, d,R) invariant sigma model in a simplified setting without Kaluza-Klein gauge

fields given that these have no effect on the non-standard transformation of the B-field, and in a

vielbein formulation based on the coset O(d, d,R)/SO(d)×SO(d). In this setting, we found that

the effective action for the composite SO(d)× SO(d) connections, arising from integrating out

the worldsheet chiral bosons, is not invariant under the gauge group SO(d)× SO(d), indicating

the presence of an anomaly. Fortunately, this anomaly could be canceled by a non-standard

transformation of the external B-field under the gauge transformations, resembling the GS

59

mechanism needed in the bosonic target space theory.

One of the motivating aspects for this project was the potential application of such a coupled

O(d, d) invariant model to cosmology. We found that the O(d, d) invariant worldsheet action is

consistent with the cosmological setting, because the worldsheet theory does not care about the

topology of the internal space, hence a sufficient condition for O(d, d,R) invariance is that the

target space fields be independent of the internal coordinates, as is the case in cosmology. For this

reason, it was possible to use the O(d, d) invariant sigma model in a cosmological setting, allowing

us to find the equations of motion and letting us compare the equations of motion of both models,

leading to the conclusion that the standard sigma model and the novel O(d, d) covariant action

are physically equivalent, modulo the truncation necessary to guarantee O(d, d,R) invariance.

Additionally, we saw that, in the cosmological case, the anomaly is irrelevant, since it is possible

to cancel it by the inclusion of a local counterterm. This is to be expected, given that in

cosmology there is no external B-field, and the cosmological spacetime action does not require

the GS mechanism to higher orders in α′.

These developments seem to suggest that there is a closer connection between bosonic string

theory and heterotic string theory, potentially a novel duality. Additionally, the construction

of the O(d, d) invariant worldsheet action, in combination with the GS mechanism, offer the

opportunity to investigate the computations of beta functions in an O(d, d) covariant fashion,

similar to the results in Refs. [31–33], and especially in the cosmological setting, in combination

with the classification of all α′ corrections relevant to cosmology in Ref. [3].

60

Bibliography

[1] G. Veneziano. Scale factor duality for classical and quantum strings. Physics Letters B,

265(3):287 – 294, 1991.

[2] Ashoke Sen. o(d) ⊗ o(d) symmetry of the space of cosmological solutions in string theory,

scale factor duality, and two-dimensional black holes. Physics Letters B, 271(3):295 – 300,

1991.

[3] Olaf Hohm and Barton Zwiebach. Duality Invariant Cosmology to all Orders in α′. 2019.

[4] Heliudson Bernardo, Robert Brandenberger, and Guilherme Franzmann. O(d, d) covariant

String Cosmology to all orders in α′. 2019.

[5] Jnanadeva Maharana and John H. Schwarz. Noncompact symmetries in string theory. Nucl.

Phys., B390:3–32, 1993.

[6] Camille Eloy, Olaf Hohm, and Henning Samtleben. Green-Schwarz Mechanism for String

Dualities. Phys. Rev. Lett., 124(9):091601, 2020.

[7] Michael B. Green and John H. Schwarz. Anomaly Cancellation in Supersymmetric D=10

Gauge Theory and Superstring Theory. Phys. Lett. B, 149:117–122, 1984.

[8] M. Gasperini and G. Veneziano. O(d,d) covariant string cosmology. Phys. Lett., B277:256–

264, 1992.

[9] John H. Schwarz and Ashoke Sen. Duality symmetries of 4-D heterotic strings. Phys. Lett.

B, 312:105–114, 1993.

[10] A.A. Tseytlin. Duality symmetric formulation of string world sheet dynamics. Physical

letters B, 242(2):163–174, 1990.

[11] A.A. Tseytlin. Duality symmetric closed string theory and interacting chiral scalars. Nuclear

Physics B, 350(3):395–440, 1991.

[12] Chris D. A. Blair. Doubled strings, negative strings and null waves. JHEP, 11:042, 2016.

61

[13] Chris D.A. Blair, Emanuel Malek, and Alasdair J. Routh. An O(D,D) invariant Hamilto-

nian action for the superstring. Class. Quant. Grav., 31(20):205011, 2014.

[14] Joseph Polchinski. String Theory, volume 1. Cambridge University Press, 1998.

[15] Amit Giveon, Massimo Porrati, and Eliezer Rabinovici. Target space duality in string

theory. Phys. Rept., 244:77–202, 1994.

[16] John Schwarz Katrin Becker, Melanie Becker. String Theory and M-Theory: A Modern

Introduction. Cambridge University Press, 2006.

[17] E. Cremmer, B. Julia, Hong Lu, and C.N. Pope. Dualization of dualities. 1. Nucl. Phys. B,

523:73–144, 1998.

[18] Olaf Hohm, Ashoke Sen, and Barton Zwiebach. Heterotic Effective Action and Duality

Symmetries Revisited. JHEP, 02:079, 2015.

[19] Camille Eloy, Olaf Hohm, and Henning Samtleben. Duality Invariance and Higher Deriva-

tives. Phys. Rev. D, 101(12):126018, 2020.

[20] R.R. Metsaev and A.A. Tseytlin. Order α′ (two-loop) equivalence of the string equations

of motion and the σ-model weyl invariance conditions: Dependence on the dilaton and the

antisymmetric tensor. Nuclear Physics B, 293:385 – 419, 1987.

[21] W. Siegel. Two vierbein formalism for string inspired axionic gravity. Phys. Rev. D,

47:5453–5459, 1993.

[22] W. Siegel. Superspace duality in low-energy superstrings. Phys. Rev. D, 48:2826–2837,

1993.

[23] Olaf Hohm, Chris Hull, and Barton Zwiebach. Generalized metric formulation of double

field theory. JHEP, 08:008, 2010.

[24] Roberto Bonezzi, Felipe Diaz-Jaramillo, and Olaf Hohm. Old Dualities and New Anomalies.

8 2020.

[25] R. Floreanini and R. Jackiw. Selfdual Fields as Charge Density Solitons. Phys. Rev. Lett.,

59:1873, 1987.

[26] Olaf Hohm and Barton Zwiebach. On the Riemann Tensor in Double Field Theory. JHEP,

05:126, 2012.

[27] Adel Bilal. Lectures on Anomalies. 2 2008.

62

[28] Fiorenzo Bastianelli and Peter van Nieuwenhuizen. Path Integrals and Anomalies in Curved

Space. Cambridge University Press, 2006.

[29] J. Wess and B. Zumino. Consequences of anomalous Ward identities. Phys. Lett. B, 37:95–

97, 1971.

[30] Olaf Hohm and Barton Zwiebach. T-duality Constraints on Higher Derivatives Revisited.

JHEP, 04:101, 2016.

[31] David S Berman and Neil B Copland. The String partition function in Hull’s doubled

formalism. Phys. Lett. B, 649:325–333, 2007.

[32] David S. Berman, Neil B. Copland, and Daniel C. Thompson. Background Field Equations

for the Duality Symmetric String. Nucl. Phys. B, 791:175–191, 2008.

[33] David S. Berman and Daniel C. Thompson. Duality Symmetric Strings, Dilatons and O(d,d)

Effective Actions. Phys. Lett. B, 662:279–284, 2008.

63

Declaration of authorship

I hereby certify that the master’s thesis that I am submitting is entirely my own original work,

except where otherwise stated. I am aware of the University’s regulations concerning plagiarism,

including those regulations concerning disciplinary actions that may result from plagiarism. Any

use of the works of any other author, in any form, is properly acknowledged at their point of

use.

Felipe Dıaz-Jaramillo

64