Aspects of Vacuum Stability in String Theory with Broken Supersymmetry

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Aspects of Vacuum Stability in String

Theory with Broken Supersymmetry

Author:

Ivano BA S I L E

Supervisor:

Prof. Augusto SA G N O T T I

A thesis submitted in fulfillment of the requirements for the Master’s Degree

in

Theoretical Physics Department of Physics

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Abstract

University of Pisa Department of Physics

Master’s Degree

Aspects of Vacuum Stability in String Theory with Broken Supersymmetry

by Ivano BA S I L E

In this Thesis we address the issue of perturbative stability of non–supersymmetric string vacua with AdS_×S spacetime backgrounds. To this end, we build a framework which generalizes the Breitenlohner–Freedman criterion for perturbative vacuum stability in Field Theory and rests on the diagonalization of certain asymptotic mixing matrices of field fluctuations. After an introductory overview, which we present in Chapter2and is meant to highlight some current frontiers of String Theory, in Chapter3we review the key features of perturbative closed–string spectra and of their orientifold descendants in ten dimensions. We also explain how Supersymmetry breaking brings along, in the low–energy field theory, runaway potentials that destabilize an initial Minkowski vacuum. In Chapter4we then consider a class of AdS×S vacua that recently emerged, where the effect of tadpoles is compensated by suitable fluxes in AdS and/or in the internal spheres. In Chapter5we introduce the Breitenlohner–Freedman bound, computing it for various types of free fields in AdS, and in Chapter6we discuss the simplified asymptotic analysis of mixed systems of fluctuations. We then apply this method to the string vacua described in Chapter4, diagonalizing the resulting asymptotic mass and mixing matrices, in order to compare the corresponding eigenvalues to the bounds. The resulting conditions for vacuum stability in different regimes of string couplings are then explored, and we identify two perturbatively stable vacua: an AdS3×S7orientifold vacuum and an AdS7×S3 heterotic vacuum. Chapter7contains some concluding remarks.

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Acknowledgements

I started working on this Thesis at the beginning of April, when I also met Prof. J. Mourad of U. Paris 7, who was visiting my advisor, Prof. A. Sagnotti, at Scuola Normale Superiore. I am grateful to Prof. Mourad for discussions and clarifications on the material presented here.

I would also like to express my gratitude to Prof. Sagnotti, for his guidance and patience in addressing my doubts on this fascinating, yet intricate, subject.

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To my parents, for their endless support and their, sometimes

amusing, concerns.

To my brothers, for being the best!

To my friends, for their wholehearted honesty and their

perspectives on life.

To Carolina, for her loving and caring presence regardless of

circumstances. So hey, let’s celebrate.

To Malvina, for our strength contests.

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1

Introduction

String Theory [1] is, among many other things, an attempt to solve the long–standing theoretical problem of quantum gravity [2]. It was initially conceived as a candidate model to describe strong interactions: the discovery of the Veneziano amplitude [3], with its remarkable “planar duality” soon ascribed to string scattering, sparkled some interest in quantum models of relativistic extended objects. The model soon clashed with deep inelastic scattering, since it exhibited an exponential decrease at high–energies and fixed scattering angle [4], and surely it was not at all obvious that it would have anything to do with gravity. Evidence in this respect, however, piled up as more in–depth investigations were carried out. Surely, together with tachyons and other massless gauge and matter particles, it soon became evident that the spectrum of a free closed string contains massless spin–2 particles, which hint to fundamental excitations of a quantized version of the gravitational field. Moreover, while fundamental extended objects provide a natural way to smooth out the singular high–energy behavior of gravity, String Theory actually requires that gravity be present for its inner consistency. This very fact, when it was properly acknowledged, drove many physicists to regard it as an unprecedented candidate for a unified theory.

Up to the mid 1980’s, String Theory was not regarded as a mainstream approach to unify gravity with the other interactions at the quantum level. It was natural to resist its allures after its early difficulties and its serendipitous emergence, and moreover mathematical dangers were lurking in various corners, which appeared to threaten its consistency in one way or another. Remarkably, the theory withstood all such consistency checks, and the Green–Schwarz mechanism [5] not only guaranteed its consistency but it also added new lessons on the deep subject of gauge anomalies. In the following years many new developments were found to be deeply connected to one another, unveiling a fascinating web of perspectives on gravity and Quantum Field Theory.

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String Theory helped to attain new insights on generalizations of the so–called electric–magnetic duality [6] and new non–perturbative methods for Quantum Field Theory, thus opening new perspectives in vastly different fields such as Condensed Matter Physics and Quantum Information Theory. There is however a downside, since String Theory appears today a remarkably difficult subject. As a result, despite strenuous efforts that have stretched over four decades by now, we still lack a satisfactory understanding of its foundations. There is a tempting analogy with an unexplored island, whose coastline has been mapped and studied to some extent, but whose internal regions remain unscathed by any available approach to the regimes of strong quantum fluctuations and gravitational effects. To some extent, this is also true for Quantum Field Theory (and indeed this is not the only parallel that one can draw between the two frameworks), but for String Theory the only available tools that go beyond perturbation theory are still, to a large extent, duality symmetries of its low–energy manifestations. They have been successfully applied to a number of relevant circumstances, and the resulting complete picture, commonly dubbed M–theory [7], which rests heavily on the low–energy Supergravity [8], encompasses all ten–dimensional supersymmetric strings. Remarkably, however, it also connects them to the eleven–dimensional Supergravity of [9], a fact that makes its deep meaning tantalizingly elusive.

A big asset of String Theory is certainly the breath of new ideas and ways of looking at Physics that it appears to provide, despite our limited means of exploring it in depth. These have included, so far, new techniques in phenomenological model building, but also insights into the microscopic description of black holes and beautiful geometric realisations of gauge theories, of their symmetries and of the breaking thereof [1]. Some authors have thus dared to stress that the relevance of String Theory as a fruitful framework in Theoretical Physics appears largely vindicated, regardless of its eventual validity as a description of our physical Universe.

This Thesis is motivated, nonetheless, by the compelling need to try and come closer to the roots of the subject. It focuses on the issue of Supersymmetry breaking in String Theory, which is relatively simple, in some manifestations, and yet both theoretically deep and potentially of relevance for Physics. A description of the Fundamental Interactions that goes beyond the Standard Model, formulated within

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Chapter 1. Introduction 7

String Theory, has to take into account the breaking of Supersymmetry (SUSY for short) [10]1 for clear phenomenological reasons: to put it bluntly, we do not see signs of Supersymmetry, which in its simplest manifestations would require equal numbers of Bose and Fermi particles of all masses, at the energy scales currently explored in particle accelerators. It spontaneous breaking would be the natural recipe to hide it while maintaining some of its virtues, and similar ideas that are central to the Standard Model were recently vindicated at the CERN LHC. However, our understanding of SUSY breaking in Quantum Field Theory is still incomplete, while when approaching the problem at a more fundamental level one is readily confronted with some conundrums. The breaking of SUSY, which could very well occur around the string scale or even at the Planck scale, appears to cause havoc in the whole framework of String Theory insofar as we understand it at present, since it tends to destabilize the vacuum. There is, among others, one such mechanism that stands out for its relative simplicity, and can therefore serve an ideal entry point into the subject. It naturally comes about in String Theory and is commonly termed “Brane Supersymmetry Breaking” (BSB for short) [12]. It arises in orientifold vacua [13] that, as such, involve the simplest types of available dynamical objects, called BPS branes and orientifolds [14], which appear in the theory and can populate spacetime while preserving, individually, portions of the initial SUSY. Branes and orientifolds can give rise to a host of vacuum configurations, including some relatively simple ones that preserve a number of supersymmetries. However, if some of them preserve incompatible portions of SUSY this results in its breaking at the string scale, with a low–energy physics that may resemble, in principle, that of our Universe. This, however, brings one readily to the frontier of String Theory as we can currently conceive it. One should keep in mind, in this respect, the main question that inspires our search, which is related to the possibly fundamental rôle of SUSY in String Theory. A number of people have advocated it over the years, since String Theory appears unstable and out of control in the absence of SUSY, but this lesson has never been quantified in a satisfactory manner. For instance, bosonic string models and also some non–supersymmetric fermionic ones contain tachyons in their spectra, but problems appear to be lingering around more subtly also for tachyon–free models, due to the

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potential emergence of non–perturbative instabilities [15]. Supersymmetry, even in a non–linear phase, might play an important rôle in this respect.

A general feature of String Theory is the apparent multitude of available vac-uum states that it inherits from Einstein gravity coupled to matter. This is the so–called landscape problem [1]. Each of these vacuum states determines, in principle, the low–energy physics one is after, so that the vastity of options appears disconcert-ing. What is lacking, as of now, is a criterion to select a stable subsector of vacua that can be physically realized. For this reason, anthropic arguments have been employed in this context, translating the issue of uniqueness into one of typicality, and often attempting to look at statistical properties of the landscape. We shall not develop the discussion in this direction. Rather, we shall try to push the frontier a bit further, trying to understand how somewhat more realistic spacetimes can be realized by stable string vacua in a context where SUSY is broken, at least within a perturbative approach.

As we have anticipated, a main problem with this setting is indeed that such spacetimes, along with all the other degrees of freedom that specify the low–energy physics, appear generically unstable: there are mechanisms which drive the vacuum to modify itself. These instabilities present themselves mainly in two varieties: perturbative, whenever the vacuum is unstable against small fluctuations, and non–perturbative, when they involve quantum tunneling phenomena that can be approached in a semi–classical setting. In this Thesis we develop a formalism that provides convenient ways to investigate the perturbative stability of vacuum states AdS field theories of the type that arise in low–energy descriptions of AdS string vacua with broken Supersymmetry, simplifying to some extent the available methods. The resulting techniques, which rest heavily on the classic work of Breitenlohner and Freedman [17], are then applied to the problem of SUSY–breaking: more specifically, we study the regimes of stability of a class of non–supersymmetric string vacua recently found in [18]. The main attractive feature of these relatively simple vacua is that they provide, potentially, a handle to deepen our understanding of unexplored regimes of String Theory, because their parameter spaces include regions where the physics is expected to be well–approximated by the low–energy theories that we can investigate, as well as others involving more extreme regimes.

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2

Basics of String Theory

Let us now move on to give a short overview of the basic aspects of String Theory that we shall need in this thesis. More in–depth discussions of the material covered in this section can be found in [1]1.

A natural starting point for the construction of a quantum theory of relativistic strings is the question of how to formulate the dynamics. To this end, one can proceed by analogy with the more familiar case of a point particle, described via a worldline τ 7→ xµ(τ)in spacetime. For a string, the corresponding object is a worldsheet in D–dimensional spacetime, described by an embedding

(σ, τ)7→ Xµ(σ, τ). (2.1)

The worldsheet coordinates (σ, τ), henceforth abbreviated to σα, α = 1, 2, are a convenient way to parametrize a string moving and fluctuating in spacetime, but they also introduce some redundancy from the outset. As in the case of the point particle, the physics of a string should be independent of the specific parametrization chosen, which brings forth the invariance under worldsheet reparametrization.

2.1

T H E N A M B U

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G O T O A C T I O N

The Nambu–Goto action [20]

SNG= −T Z d2σ s −det ∂Xµ ∂σα ∂Xν ∂σβ ηµν (2.2)

provides a description with a clearcut geometric meaning. It computes the area of the worldsheet swept by the string as it moves in spacetime. Too see this, it is convenient

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to look at the tensor γαβ ≡ ∂Xµ ∂σα ∂Xν ∂σβ ηµν, (2.3)

which is the metric tensor induced by pullback on the worldsheet. Taking into account the Lorentzian signature of γαβ, rewriting (2.2) as

SNG =−T Z

d2σ p

−det γ (2.4)

makes its geometric nature manifest, along with its reparametrization invariance. Notice that the action is also manifestly Poincaré invariant, by construction. The constant T is identified with the string tension, as can be seen for example by evaluating the action on a static string configuration. This quantity is often written in the form

T= 1

2πα0 , (2.5)

where α0is termed Regge slope for historical reasons, and it is linked to the string length scale ls. Indeed, by dimensional analysis2, one can write α0 ∝ ls2. For convenience, one can choose σ in the range[0, π], which means that closed strings should satisfy the periodicity conditions X(σ+π, τ) =X(σ, τ). Analogously, an open string coordinate allows two different types of boundary conditions, depending on whether their endpoints are fixed (Dirichlet boundary conditions) or ∂σX = 0 at the endpoints (Neumann boundary conditions).

2.2

T H E P O LYA K O V A C T I O N

Despite its natural geometric interpretation, the Nambu–Goto action is not well suited to carry out a quantization procedure. The square root makes the Hamiltonian formulation and subsequent canonical quantization cumbersome. Fortunately, there is an easy way out: one can simplify the form of the Nambu–Goto action, at the price of adding further redundancies in the description. The trick is to work with an independent worldsheet metric gαβ, and to write an action that couples the embeddings Xµ_{with the worldsheet metric in the canonical way in which scalar fields couple to}

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2.2. The Polyakov Action 11

FI G U R E 2 . 1 : Schematic representation of how a string sweeps out a two–dimensional surface in spacetime, in contrast to a particle’s one–dimensional worldline. Taken from K. Becker, M. Becker and J. H. Schwarz, “String theory and M-theory: A modern introduction”.

gravity. The resulting action, written using the shorthand notation ∂α ≡∂/∂σα, is then

SP =− 1 4πα0 Z d2σp−g gαβ∂αXµ∂βX ν ηµν. (2.6)

This is usually called the Polyakov action, since Polyakov carried out its detailed quantization [21], although it was originally proposed in [22]. It has the form of a sigma–model. Such actions are found in many places, including low–energy effective descriptions of condensed matter systems and QCD. It can be checked to coincide with the Nambu–Goto action on–shell, when the equations of motion of gαβ, which plays the rôle of a Lagrange multiplier, are imposed. Both descriptions are reparametrization invariant, but in the case of the Polyakov action an additional gauge invariance appears: the invariance under Weyl transformations, rescalings of the metric by a positive function

gαβ →Ω 2₍

σ)gαβ. (2.7)

Weyl invariance, together with reparametrization invariance, suffices to eliminate gαβ altogether, since it can be locally identified with the 2D Minkowski metric ηαβ.

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action is unique to two dimensions. It is a feature of strings which, in the quantum theory, has far–reaching consequences. It is what ultimately leads to the appearance of low–energy supergravity descriptions and string interactions, while also fixing the number of spacetime dimensions (which is the reason why we did not specify it in the beginning). Preserving Weyl invariance through quantization gives strong consistency conditions.

FI G U R E 2 . 2 : A Weyl rescaling acting on a surface. From the point of view of a Weyl invariant theory, these two manifolds are

indistinguish-able. Taken from [19].

Thanks to reparametrization and Weyl invariance, the Polyakov action can be considerably simplified. Using reparametrization invariance, one can fix the metric to be conformally flat, and subsequently Weyl invariance reduces it to the flat metric. In detail: gαβ reparam. −−−−→Ω2₍ σ)ηαβ Weyl −−→ηαβ. (2.8)

The fact that Weyl rescalings can locally reduce the metric to a flat one can also be seen noting that, in two dimensions, the Riemann tensor is proportional to the Ricci scalar. However, the possibility of doing this globally is restricted to worldsheet manifolds with vanishing Euler characteristic χ, as the worldsheet Ricci scalar integrates to χ by the Gauss–Bonnet theorem. This means that when different worldsheet topologies are considered there are complications in the gauge fixing procedure, and the simplest option is to concentrate the curvature at special points. This has important consequences on the nature of string perturbation theory, where higher loop corrections correspond to worldsheets with higher genera. For the time being we restrict to cases where this topological obstruction is absent. With this gauge

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2.3. Quantization 13

fixing the action (2.6), using the Minkowski metric both in spacetime and on the worldsheet to raise and lower indices and perform contractions, reads

SP= − 1 4πα0

Z

d2σ ∂αX·∂αX , (2.9)

which can be thought of as the action for D massless free scalar fields living on the worldsheet. Indeed, the field equations for Xµ_{are simply free wave equations in flat} two–dimensional spacetime,

∂α∂

α_Xµ₌_{0 ,} _(2.10)

and can be solved, for example, resorting to lightcone coordinates σ±≡σ±τ. These coordinates, along with their spacetime counterparts, are also useful for quantization, even though the procedure can be also carried out in a manifestly covariant way. But we are missing something: the simplicity of (2.9) hides the fact that the equation of motion for gαβstill has to be satisfied. Having fixed a gauge such that gαβ =ηαβ means that this equation turns into a constraint, namely the vanishing of the energy– momentum tensor obtained varying (2.6) with respect to the metric tensor

Tαβ ≡ − 2 T 1 √ −g δSP[g, X] δgαβ =∂αX·∂βX− 1 2gαβg ρλ ∂ρX·∂λX=0 , (2.11)

where the normalization is chosen for later convenience.

2.3

Q U A N T I Z A T I O N

Without delving into all the technical details of the calculations, let us simply state the main results of applying canonical quantization of the string. The mode expansions arising from solutions of the wave equations (2.10) with suitable boundary conditions feature an infinite set of right–mode and left–mode coefficients{αµn, ˜αµn}, with n∈Z. Explicitly, the closed string mode expansion is given, in terms of these coefficients, by

Xµ₌ _xµ₊_2α0_pµ τ+i r α0 2 _n

∑

₆₌₀ αµn n e −2in(τ−σ)₊ ˜α µ n n e −2in(τ+σ) ! . (2.12)

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by the boundary conditions in terms of the first. This can be seen by the presence of only one set of oscillator, e.g.{αn}, in the mode expansion for Neumann boundary conditions, which reads

Xµ₌ _xµ₊_2α0_pµ τ+i√2α0

∑

n6=0 αµn n e −inτ_cos₍_nσ₎_. _(2.13)

In addition, for closed strings the zero mode coefficients α₀µ, ˜αµ₀ are identified with

αµ₀ = ˜αµ₀ =√α0/2pµ. (2.14)

On the other hand, for open strings

αµ₀ =√2α0pµ. (2.15)

This notation allows us to compactly express the Fourier modes of the constraints Tαβ =0 as the set of conditions Ln =Ln=0, n≥0, where for n∈Z

Ln= 1

2

∑

_m αn−m·αm, Ln= 1

2

∑

_m αn−m·αm (2.16) are oscillator sums which play an important rôle in the quantum theory, which we now come to describe. In the quantum theory, specified by equal–time canonical commutation relations of Xµ_{and the associated canonical momentum}Π

µ,

[Xµ₍

σ, τ),Πν(σ0, τ)] =iδµνδ(σ−σ0), (2.17) the coefficients αµnare promoted to operators which, up to a rescaling by 1/√n, play a rôle analogous to that of ladder operators for the quantum harmonic oscillator: they allow to move between different excitation levels of the string. This can be most clearly seen in lightcone quantization, where one only looks at the physical excitations of the string. These are the transverse excitations, described by Xj_where, say, j=1, ... , D₋2. The remaining components, expressed in lightcone coordinates as X± = (X0±XD−1)/√2), are specified by the constraints. One can observe that transverse string excitations indeed arrange in levels akin to those of the quantum

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harmonic oscillator, as the corresponding operators αjn, ˜αjnsatisfy the appropriate commutation relations

[αi_n, αjm] = [˜αin, ˜α j

m] =n δijδm+n, 0 (2.18)

derived by (2.17). One can then proceed to build the Hilbert space of the theory, starting with the vacuum and constructing higher excitations. The operator α0 = ˜α0 corresponds to the center–of–mass momentum of the string, which serves as an additional label for states. The constraints coming from the decomposition of (2.11) into oscillator modes, which as mentioned have the form Ln= Ln =0(n>0), have to imposed on states to pick out the Hilbert space of physical states. In particular, one of the n=0 constraints, the one corresponding to L0−L0, imposes level matching, i.e. the equality of the number of right–moving and left–moving excitations, also called the levels N= D−2

∑

i=0 n

∑

>0 αi₋_nαi_n, N˜ = D−2

∑

i=0 n

∑

>0 ˜αi₋_n˜αi_n. (2.19) They behave as number operators on the Hilbert space, much like the operator a†a in the case of the quantum harmonic oscillator. On the other hand, the constraint corresponding to L0and L0, carefully treated in the quantum theory, gives an explicit formula for the mass M2₌₋_p

µpµof the string excitations. For concreteness, for the closed string the mass formula is given by

M2 = 4 α0(N−1) = 4 α0( ˜ N₋1) = 2 α0(N+ ˜ N−2), (2.20)

while for the open string

M2= 1

α0(N−1), (2.21)

and the level matching condition is absent because left–moving and right–moving oscillators are not independent. It should be stressed that the mass formula (2.20) comes from a proper implementation of the L0, L0constraints on physical states, which involves the introduction of an overall additive constant3. This constant is

3_{The introduction of an additive constant a in the constraint}₍_L

0−a)|physi =0, imposed on physical

states, originates from the need to specify the order in which ladder operators appear in L0(and

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then derived in terms of the space–time dimension, which is subsequently fixed by the requirement of preserving Lorentz invariance at the quantum level. The most concise way of deriving this result makes use of zeta function regularization, which in this context amounts to substitutions of the form

∞

∑

n=1

(n+α)→ −6α(α−1) +1

12 , (2.22)

which originate from a formal analytic continuation of the sum using a generalized ζ function. This regularization method is consistent with Weyl and Lorentz invariance, but despite giving the correct result it can be somewhat unconvincing due to lack of rigor. On the other hand, a more rigorous approach involves the study of the worldsheet theory with the powerful algebraic tools of Conformal Field Theory (CFT). In particular, one finds that the correct constant reflects an all–important quantity in the worldsheet CFT, the central charge c.

2.3.1 The Virasoro Algebra

As we briefly mentioned, the operators (2.16), as well as their supersymmetric generalizations, are central objects in String Theory. Their utmost importance stems from the properties of the algebra they generate. In the quantum theory, then, Lnand Lnare termed Virasoro generators, and they satisfy commutations relations of the form

[Lm, Ln] = (m−n)Lm+n+ c 12m(m 2₋₁₎ δm+n, 0, (2.23) [Lm, Ln] = (m−n)Lm+n+ ˜c 12m(m 2 −1)δm+n, 0, (2.24)

as well as[Ln, Lm] =0. Thus, they generate two commuting Virasoro algebras with central charges c , ˜c. This is in contrast to the classical algebra, which does not feature this central extension. Indeed, (2.23) can be taken as a definition of central charge, an ubiquitous quantity which appears in a number of observables of interest in Conformal Field Theory. In particular, it reflects the number of degrees of freedom in a given model. For example, a free scalar contributes 1 to the central charge.

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non–vanishing of the (expectation value of the) trace of the energy–momentum tensor. Indeed, there is a prescription such that the anomaly is specified by the central charge c= ˜c and the worldsheet Ricci scalar R,

hTα αi =

c

12 R . (2.25)

In worldsheet theories the vanishing of the total central charge vanishes is feasible, because Fadeev–Popov ghosts, which originate from the gauge fixing procedure, provide a negative contribution to the central charge of₋26. In particular, in bosonic String Theory the total central charge is D−26. This fixes the critical dimension D=26 of the theory4.

2.3.2 Open Strings: Chan–Paton Factors

For open strings there is an additional subtlety: one can independently impose Dirichlet or Neumann boundary conditions in each direction, and this changes spectra and other features of the quantum theory. Despite the initial lack of interest towards Dirichlet boundary conditions, which violate Lorentz invariance, a key insight was to realize that these conditions actually correspond to open strings ending on other dynamical extended objects, termed D–branes, where the D stands for Dirichlet [1,14]. D–branes are dynamical objects, solitons in the perturbation expansion of String Theory, but in contrast to strings they are characterized by a tensionO(1/gs), which becomes infinitely large in the perturbative regime and confers them the aspect of hyperplanes in typical situations. They carry tension and R–R charges, and they interact. In the perturbative regime their interaction is mediated by exchange of closed strings, and in the low–energy spacetime theory branes appear indeed as solitons, i.e. non–trivial solutions of the classical equations of motion. Perhaps one of the most attractive features of D–branes is that the low–energy physics of open strings attached to branes describes transverse fluctuations of branes in spacetime, as well as gauge theories living on the brane’s worldvolume. Gauge invariance and the Higgs 4_{It should be stressed that working in the critical dimension is not the only way of having a consistent}

quantum theory, but it has the important feature of bringing along flat space–time backgrounds. Different types of worldsheet CFT can feature different contributions to the central charge: a typical example is the so–called linear dilaton CFT, which describes the dynamics in a curved cosmological background.

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mechanisms acquire, in this picture, a geometric description that relies exclusively on brane configurations, such as intersections and coincidence.

FI G U R E 2 . 3 : A schematic representation of D–branes, with open string ending on them. Taken from K. Becker, M. Becker and J. H. Schwarz, “String theory and M-theory: A modern

introduc-tion”.

To be more concrete regarding how geometric properties of D–brane configurations affect the physics of their low–energy excitations, that is open strings ending on them, consider a stack of N coincident D–branes. In order to specify configurations of strings ending on these branes, each endpoint has to carry an integer label j=1, ... , N specifying on which brane it ends. This label is called a Chan-Paton factor [23]. Given that each field associated to open strings, and massless gauge fields in particular, carries two of such labels, it is natural to suspect that there is a matrix–like character to these fields. Indeed there is, and computations involving the low–energy physics of coincident branes turn out to lead to U(N)Yang–Mills theory on the worldvolume of the stack. In addition to providing a nice geometric picture for the emergence of gauge invariance, the Higgs mechanism follows as well from the natural idea of separating the branes: strings stretching between separated branes acquire a mass proportional to the separation. The idea of Chan–Paton factors also ties in to the original desire of using strings to model the strong interaction, as degrees of freedom on the strings’ endpoints can be associated with quarks, and the strings with confining flux tubes.

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2.4. The Free Spectrum 19

Another geometric interpretation, stemming from the spacetime perspective, is that Chan–Paton factors provide reflection coefficients for closed strings with respect to boundaries specified by extended objects in spacetime. In general, Chan-Paton factor build classical gauge groups, filling the three families U(n), O(n)and USp(2n), but the latter two cases involve, in this space–time picture, orientifolds, additional non–dynamical mirrors that reverse the orientation of closed strings. We shall return to their rôle later on.

2.4

T H E F R E E S P E C T R U M

The resulting spectrum for the free string contains the first hints that quantum relativistic strings can accommodate, and indeed require, gravity and gauge theories in a single framework. One can indeed show that, while the perturbative vacuum is tachyonic for the bosonic string (this problem is absent in the superstring) the first excited states

αi₋₁˜αj₋₁|0, ˜0i (closed string),

αi₋₁|0i (open string), (2.26)

which have to be massless to allow for Lorentz invariance, organize in irreducible multiplets corresponding to single particle states with the same quantum numbers as particles, living in spacetime, starting from massless modes: a spin–1 gauge field (for the open string) and a scalarΦ, called the dilaton, an antisymmetric two–tensor Bµν and a symmetric two–tensor Gµν(for the closed string). In other words, excitations of the string produce photons and, most importantly, gravitons. As famously argued by Deser, Feynman and Weinberg, a field theory of a massless spin–2 field recon-structs General Relativity (possibly with higher derivative corrections) when one introduces interactions consistent with gauge invariance, which becomes full–fledged diffeomorphism invariance in the non–linear theory.

Another, stronger piece of evidence that gravity emerges from strings comes from investigating the low–energy effective dynamics which governs these massless modes. This is done primarily in two ways, either matching the low–energy limit of scattering

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amplitudes to Field Theory or looking at strings in the presence of coherent states of the massless string modes5or, at the worldsheet level, studying the sigma–model with background fields in curved spacetime

S=₋ 1 4πα0 Z d2σ Gµνgαβ∂αXµ∂βXν+eαβBµν∂αXµ∂βXν+α0Ric(g)Φ . (2.27)

The conditions for cancellation of the Weyl anomaly [1,21], which as we anticipated plays a prominent rôle in the story, can be interpreted in terms of equations of motions originating from an effective theory for the fields corresponding to the massless string modes. In both cases the result is the same: the massless modes of the bosonic string are governed by a theory which involves Einstein gravity. In addition, the requirement that Minkowski spacetime be a classical solution fixes the theory to live in 26 dimensions, the so–called critical dimension of the bosonic string. For the superstring the critical dimension turns out to be 10. In the same fashion, one can show that the open string dynamics is governed by Yang–Mills theories, where the emerging non–Abelian U(n)(and also, as we have anticipated, O(n)or USp(2n)) gauge invariance has a beautiful geometric realization in terms of D–branes (or D–branes and orientifolds). Building upon the low–energy dynamics, one can look at how strings behave in the presence of coherent states of its massless modes. From the structure of string perturbation theory, as well as by symmetry arguments, the string coupling gs, along with the open string coupling go, with gs= g2_o, which weigh different worldsheet topologies in the path integral are not free parameters of the theory: they are actually given by the (exponential of the) vacuum expectation value of the dilaton,

gs=ehΦi. (2.28)

This is an important result. One of its consequences is that one can use the low– energy dynamics to investigate regimes where quantum corrections are negligible and the semi–classical gravity description is expected to be accurate. In the same fashion, the curvature length scales R set by spacetime backgrounds, solutions of the low–energy theory, identify a dimensionless parameter√α0/R that measures how well flat space string perturbation theory (or, more generally, string perturbation

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2.4. The Free Spectrum 21

theory in a fixed spacetime) can be trusted. In reality, strings and branes backreact on the spacetime geometry: gravity and its sources appear in String Theory as strings and branes interacting amongst themselves. It is not clear what the relevant degrees of freedom in strong coupling and curvature regimes are, but there are indications that some kind of democracy of dimensions emerges, where extended objects (strings, branes, ...) of various dimensions participate in the dynamics. This is opposed to the stringy perturbative regime, where branes become infinitely rigid in the limit gs→0, thus effectively decoupling from the dynamics. Nonetheless, their effect can still be seen from the perspective of strings, both in the open and closed sector. Moreover, one can go beyond the rigid approximation and incorporate transverse fluctuations of branes in the low–energy effective theory.

As we have briefly mentioned, there is an elephant in the room: the bosonic string has the issue of having a tachyonic vacuum, a state with negative squared mass. This suggests some kind of instability, in the sense that a more complete formulation of the theory in a non–perturbative fashion should perhaps show that naïve string perturbation theory is an expansion around an unstable equilibrium, and not an actual vacuum given by a minimum of some potential. There are indeed clear indications of this fact for the open–string tachyon: integrating out massive modes, one can build a modified potential associated to the tachyonic mode of the bosonic string, showing explicitly that string perturbation theory is formulated around a local maximum. More physically, open string tachyons can be understood as instabilities related to D–brane decay, with the minimum of the potential corresponding to the vanishing tension of the evaporated D–brane6. No such clear results, however, have been obtained for the closed–string tachyon. However, the superstring can consistently eliminate tachyon instabilities (even when Supersymmetry is broken, in a number of cases, at least at tree level), while enriching the spectrum with the fermions needed to account for matter dynamics.

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2.5

S U P E R S T R I N G S

In this section we give a brief overview of what happens when one includes Su-persymmetry in the worldsheet theory. The calculations proceed along the lines of what we have seen for the bosonic string, but are much more involved. See [1] for a comprehensive introduction.

Supersymmetry provides a natural starting point to introduce fermions in String Theory, while also removing the unpleasant tachyon from the spectrum. Supersym-metry is the invariance of a theory under the interchange of bosons and fermions: the SUSY transformations map one type of degrees of freedom into the other in such a way that the physics is unchanged. Field theories with SUSY are often much easier to deal with, thanks to cancellations and the emergence of consequent powerful analytical techniques (e.g. localization) to probe them in non–perturbative regimes, even with exact results. As such, SUSY is an invaluable theoretical tool which is quite desirable to have at our disposal, but at the same time it is of crucial importance to understand mechanisms of SUSY breaking relevant to the description of somewhat more realistic spacetimes and, eventually, to recover the Standard Model physics. Some steps in this direction are indeed the main aim of this Thesis.

2.5.1 Worldsheet Supersymmetry

One can either introduce SUSY in spacetime or on the worldsheet. In this Thesis we take the latter approach. The inclusion of fermion fields living on the worldsheet turns out to reproduce the spacetime Dirac equation, instead of the Klein–Gordon mass– shell condition, when the constraints are imposed on the physical states. We can thus write a supersymmetric version of the Polyakov action, the Ramond–Neveu–Schwarz (RNS) action SRNS =− 1 4πα0 Z d2σ ∂αXµ·∂αXµ+ψ µ ρα∂αψµ , (2.29)

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2.5. Superstrings 23

where the Dirac matrices ρα_{satisfy the Clifford algebra}

{ρα, ρβ} =2 ηαβ (2.30)

and are chosen to be free of factors of i: it is a Majorana representation of the Clifford algebra, and ψµ_{are thus Majorana spinors. Moreover, the fermion fields are a pairs of} Majorana–Weyl spinors that propagate independently in closed strings, consistently with the fact that we are working in two dimensions7_{. On the other hand, for open}

strings pairs of Majorana–Weyl fermions turn into one another upon reflection at the ends.

2.5.2 Quantization

From this action the quantization can be carried out much in the same manner as in the case of the bosonic string, with some key differences here and there. For example, the constraints are given by the vanishing of the energy–momentum tensor and the supercurrent, the Noether current associated to Supersymmetry transformations. Another key difference is that both the left–moving and the right–moving sectors of the worldsheet fermion fields independently allow two types of boundary conditions, called Ramond (R) and Neveu–Schwarz (NS) boundary conditions. This means that open strings have two sectors, R (which gives spacetime fermions after quantization) and NS (which gives bosons), while closed strings can have R–R, NS–NS sectors (both of which give bosons) as well as N–RS and RS–N sectors (which give fermions). The resulting mass formula contains bosonic as well as fermionic oscillators. Quantization, once again, fixes the spacetime dimension: for the superstring it is D=10. This is due to the central charge of the super–Virasoro algebra satisfied by the Fourier modes of the constraints. For instance, the (normal–ordered) Virasoro generators for the superstring read Ln= 1 2

∑

_m : αn−m·αm :+ 1 2

∑

_r r₋ n 2 : ψn−r·ψr:+δn, 0∆ , (2.31) 7_{Recall that Majorana–Weyl conditions can only be imposed in dimensions 2 (mod 8). This is related}

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where r is half–integer in the NS sector, and integer in the R sector. The normal ordering shift∆ shall be described in the next chapter, where we discuss the fermionic contribution to vacuum amplitudes.

Furthermore, in order to get a consistent quantum theory, one has to incorporate the GSO projection, which projects out of the physical Hilbert space states according to a certain parity operator named G–parity. This is required in order to preserve modular invariance for closed in the quantum theory, and in a notable subset of cases also space–time Supersymmetry. It can also conveniently remove tachyons, curing the problems of the bosonic theory. The projection is fixed in the NS sector, so this means that the two possible projections in the R sector lead to two types of superstrings, named Type IIA and Type IIB String Theory. They are supersymmetric theories in ten spacetime dimensions.

2.5.3 The Free Spectrum and Low–Energy Supergravity

The free massless spectrum of Type IIA and Type IIB String Theory, owing to the fact that the GSO projections makes the theories supersymmetric, can be arranged into SUSY multiplets. In addition, the presence of the graviton in the spectrum implies that gravitini have to be present as well. This leads one to expect that the low–energy dynamics of superstrings, governed by their massless modes, be described by Supergravity. Indeed, the free massless spectra of the two theories correspond to quanta of the following fields, where the notation C(n)refers to n–form gauge fields with field strengths F(n+1)=dC(n).

• Type IIA: Φ, Bµν, Gµν, C(1), C(3)+fermions; • Type IIB: Φ, Bµν, Gµν, C(0), C(2), C(4)+fermions.

In other words, the sector containing the graviton, dilaton and the 2–form B is the same, namely the NS–NS sector, while the so–called Ramond–Ramond fields correspond to forms with field strengths of even (IIA) or odd (IIB) degrees. In addition, the field strength F(5)_{associated to the 4–form C}(4)_{has to satisfy a self–duality requirement} that cannot be easily implemented in a Lagrangian formulation. These fields, together with the corresponding low–energy effective actions, build up precisely Type IIA

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2.5. Superstrings 25

and Type IIB Supergravity, which have the unique ten–dimensional actions with

N = 2 Supersymmetry (which means 32 SUSY generators, or supercharges, in a four–dimensional sense).

2.5.4 The Web of 10D–11D Dualities

In similar ways the other types of ten–dimensional string theories can be built on grounds of consistency. It turns out that there are five perturbative supersymmetric formulations, each of which appears to describe a corner of the metaphorical coastline of M–theory. Some of the non–perturbative understanding we have gained is based on the web of dualities [7], which relate different regimes of these five theories in a way that, in some circumstances, hard computations in one setting translate into easy ones in another, and viceversa. These dualities also relate these five string theories to eleven–dimensional Supergravity, which is the unique Supergravity theory in the maximal dimension. It is believed to arise as a low–energy limit of M–theory, without constraints on the strength of the string coupling. The main dualities, in ten and eleven dimensions, which allow one to unify the pictures we have of String Theory and its uniqueness are neatly illustrated in fig.2.4.

FI G U R E 2 . 4 : The 10D–11D web of dualities, which relates the five ten–dimensional supersymmetric string theories, together with eleven–

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In the next chapter we explore in more detail how the structure of superstrings arises from consistency conditions. As we mentioned, Weyl invariance of the two– dimensional worldsheet theory, together with Poincaré invariance and Supersym-metry, place extremely stringent consistency requirements on the physics of strings, so much so that it appears that there is a unique theory which satisfies these condi-tions in the presence of Supersymmetry. Indeed, the strength of these requirements can already be seen in one–loop computations, by looking at the one–loop vacuum amplitudes, given by integrals of partition functions which encode a great deal of information. In the next chapter we shall focus on vacuum amplitudes, which contain a structure rich enough to reproduce the above results. Moreover, they provide an interesting way of looking at the phenomenon of Supersymmetry breaking in String Theory.

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3

(Super)string Vacuum Amplitudes

In this chapter we take a more in–depth look at the structure of superstring theories, resorting to the one–loop vacuum amplitudes, in the formalism of [25]. The strong constraints generated by Weyl invariance and modular invariance determine the structure of these amplitudes, from which one can read off the free spectra of the resulting theories. Furthermore, these constraints can be used to study how various types of projections affect these spectra. Geometrically, some of these deformations, the ones involving open strings, correspond to the addition of extended objects, such as branes and orientifolds, to the vacuum. A key feature of vacuum amplitudes, which we shall make wide use of, is that they have a sufficiently rich mathematical structure to completely characterize various superstring theories at the perturbative level, despite their being only generating functions for integer multiplicities. The subtlety is that these quantities (namely the one–loop corrections to the vacuum energy) emerge from integrals over inequivalent worldsheet geometries, and as such there are strong constraints that the integrand and the integration domain have to satisfy. For a more comprehensive review of the material covered in this section, see [25].

3.1

VA C U U M A M P L I T U D E S I N Q U A N T U M F I E L D T H E O R Y

Before discussing vacuum amplitudes in String Theory, let us briefly review their computation in Field Theory. Our discussion of string amplitudes in the following sections will rest on the simpler results that we present in this section.

In Quantum Field Theory, the vacuum–to–vacuum amplitude can the defined via the effective action using the Euclidean functional integral. In the simplest cases the vacuum energyΓ is defined, in this context, as the effective action evaluated on vanishing classical field configurations. In perturbation theory, this is related to summing vacuum bubble diagrams, with no external legs. In the functional integral

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formulation, this simply corresponds to performing the Euclidean functional integral without any external source or classical field background. Using a schematic notation, denoting all the fields in a given QFT by ϕ and the Euclidean action by SE, this means that the vacuum energy is given by

e−Γ=

Z

Dϕ e−SE, (3.1)

with the usual, implicit, assumptions about field configurations vanishing sufficiently fast at infinity. In typical covariant, local field theories the action can be split into the sum of a free, kinetic term and interaction terms, which can be treated in various approximations. The one–loop approximation, owing to the fact that the vacuum energy is given by putting classical fields and sources to zero, only depends on the structure of the free part of the theory. This means that the information contained therein is apparently of little use in understanding the underlying Physics. This, as we shall see, is in striking contrast to String Theory, where the one–loop vacuum amplitudes severely constrain the spectrum of the theory and completely determine the perturbative excitations.

In the one–loop approximation, the integral is approximated using the saddle point method around the vanishing field configuration. This results in a functional determinant involving the quadratic operator

S(_E2)(x, y)_≡ δ

2_S E

δϕ(x)δϕ(y), (3.2)

whose form depends on the nature of the fields in the theory. In a typical QFT, the free part of the action is the sum of free actions for the individual fields, so that the vacuum energy also decomposes into a corresponding sum. Using this fact, we can restrict our attention to the cases of free scalar and fermion fields in D dimensions. The vacuum energy is only sensitive to masses and physical polarizations, so that the computations can be straightforwardly generalized to other kinds of bosonic and fermionic fields. For a scalar field of mass M the one–loop vacuum energy is thus

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3.1. Vacuum Amplitudes in Quantum Field Theory 29

given by the formal expressions

Γ= 1 2Tr log(∆+M 2_{) =}₋1 2 Z ∞ e dt t e −tM2 Tr e−t∆. (3.3)

Here we made use of the (formal) identity

log(det A) =₋ Z ∞ e dt t Tr e −tA_, _(3.4)

where e can be thought of as an ultraviolet cutoff. Its presence is necessary because functional determinants of operators are usually ill–defined. Ratios of functional determinants are typically better objects to deal with. Another strategy, which is also used, is to define functional determinants using analytic continuations of regulated expressions, which make sense in the limit where the regulator is removed. In zeta regularization, for example, this usually leads to the appearance of a renormalization scale µ, which is one of many manifestations of how renormalization is deeply linked with the mathematics, as well as the physics, of Field Theory. In (3.3), which is akin to the Heat Kernel regularization procedure, this fact is reflected by the presence of the UV cutoff e. In addition to these ultraviolet subtleties, the trace in (3.3) leads to an infrared divergence as well. This is associated with working in a space with infinite volume. The most direct way of extracting a meaningful quantity out of this is to start from a finite large volume, as is usually done in Statistical Physics, and then to look at the vacuum energy densityΓ/Vol in the limit of infinite spacetime volume. We shall thus place the theory in a box of volume V, looking at the asymptotics in the infinite V limit, where the discretized momentum eigenstates approach a continuum.

The upshot of this discussion is that one can express the regulated vacuum energy of a scalar field by evaluating the trace using momentum eigenstates (strictly speaking, they are not actual states in the Hilbert space, but the usual framework of rigged Hilbert spaces provides a way of accommodating such situations). The expression (3.3) thus becomes Γ=₋V 2 Z ∞ e dt t e− tM2Z dDp (2π)D e− tp2 ₌₋ V 2(4π)D/2 Z ∞ e dt tD/2+1 e− tM2_, _(3.5)

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of mass M has the same vacuum energy, apart from an overall sign and a different numerical factor which counts the physical polarizations. This means that, in general, one can write

Γ=₋ V 2(4π)D/2 Z ∞ e dt tD/2+1Str e−tM2, (3.6)

where the supertrace operation Str sums over signed polarizations, i.e. with a minus sign for fermions, and M2is promoted to a mass matrix acting on the space of field components. The lowercase tr refers to a trace on this space. Physically, the variable of integration t can be thought of as the Schwinger parameter describing the proper time of the loop’s worldline. The UV divergence due to integrating down to small values of t thus corresponds to arbitrarily small loops, or alternatively arbitrarily high momentum running through the loop as expected.

3.2

VA C U U M A M P L I T U D E S F O R T H E B O S O N I C S T R I N G

Eq. (3.6) expresses the one–loop vacuum energy in a compact form which can be readily applied to any field theory, but in String Theory there are subtle differences which make a straightforward application of (3.6) incorrect. The reasons behind this lie behind the elimination of UV divergences in String Theory, and can be ultimately traced to the extended nature of strings. In this section we look at vacuum amplitudes in the bosonic string, which provides the simplest setting to understand these key differences, as well as a stepping stone towards getting to superstring vacuum amplitudes.

3.2.1 Closed Strings

Let us first try to naïvely apply (3.6) to the closed bosonic string. The more thorough way to derive the vacuum amplitude uses the framework of Conformal Field Theory, which makes the modifications we shall need manifest from the outset. In particular, it shows how the vacuum energy in String Theory corresponds to the vacuum amplitude of the worldsheet theory, which may not be obvious given that in String Theory there are both a worldsheet and a space–time perspective. However, a naïve approach

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3.2. Vacuum Amplitudes for the Bosonic String 31

to the computation, along with more detailed scrutiny of the worldsheet vacuum amplitude, will be sufficient to arrive at the correct result. In applying (3.6) to the bosonic string, it is useful to recall the mass formula, expressed as

M2 = 2

α0(L0+L0−2). (3.7)

This result turns the the trace into a sum over oscillator levels for each transverse mode, where each oscillator brings forth an infinite ladder of levels to sum over, analogously to what happens for the partition function of a free Bose gas. But here there is the additional condition of level matching. This is readily implemented in the trace by inserting, at each level n, ˜n, a Kronecker δ in its integral representation

δn, ˜n= Z 1 2 −1 2 ds e2πi(n−˜n)s, (3.8)

which can be recast in operator form inside the trace. Recalling that D=26 for the (critical) bosonic string, the resulting vacuum amplitude reads

Γ=− V 2(4π)13 Z 1₂ −1 2 ds Z ∞ e dt t14tr e−α20(L0+L0−2)te2πi(N−N˜)s . (3.9)

A more elegant form obtains defining the complex parameter

τ=τ1+iτ2≡ s+i t

α0π (3.10)

and letting

q=e2πiτ. (3.11)

In terms of these quantities, (3.9) becomes

Γ=− V 2(4π2_α0₎13 Z 1 2 −1 2 dτ1 Z ∞ e dτ2 τ₂14tr q L0−1_qL0−1_. _(3.12)

Left–moving and right–moving oscillators are decoupled once one has introduced the δ, and therefore one can perform the traces looking at each individual factor, say

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the one containing q. We have tr qL0−1 ₌ 1 q ∞

∏

n=1 trnqL0 !24 , (3.13)

where the trace trn, over a single transverse polarization and fixed Fourier mode n, is given by trnqL0 ₌_{tr q}na†nan ₌ ∞

∑

k=1 qkn = 1 1−qn. (3.14) Substituting these results into (3.12) yields a result that can be expressed in terms of the Dedekind η function

η(τ)≡ q 1 24 ∞

∏

n=1 (1₋qn). (3.15)

The Dedekind η function has nice properties under so–called modular transformations, as described shortly. This language makes the modular invariance, the invariance under such transformations, manifest: its presence is a crucial consequence of remnants of diffeomorphism and Weyl invariance, appear in string perturbation theory. Indeed, the mistake made following the naïve field–theoretic approach is failing to take into account modular invariance, as we shall see shortly. In terms of this function, the vacuum amplitude reads

Γ =− V

2(4π2_α0₎13T , (3.16) where the torus amplitude, defined stripping the overall factor away from Γ for convenience, reads T = Z strip d2τ τ₂2 1 τ₂12|η(τ)|48 . (3.17) There are various subtleties hidden in (3.17), and we are now in a position to address them. The first thing that should ring a bell is the name, torus amplitude. It corresponds to the diagrammatic picture of the string worldsheet associated to the one–loop vacuum amplitude: at one–loop level, Feynman diagrams of closed strings propagating in spacetime are worldsheets with the topology of a torus. The justification for this comes from the functional integral formulation of string perturbation theory, which involves the Polyakov action and the string coupling gs = ehΦi. In order to compute the functional integral one has to integrate over the embedding fields, as well as the worldsheet metric, but the presence of Weyl

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and diffeomorphism invariance has to be treated via gauge fixing. The resulting expressions, however, retain a global remnant of this gauge invariance, and it turns out that it corresponds to conformal transformations which can be defined globally. In the case of a worldsheet with the topology of a torus, they make up the modular group SL(2,Z)/Z2, which acts on the torus’ modular parameter τ as

τ7→ aτ+b

cτ+d. (3.18)

A useful characterization rests on the generating transformations

S : τ7→ −1/τ , (3.19)

T : τ 7→τ+1 . (3.20)

This is because τ has an intrinsic, geometric meaning: it parametrizes the inequivalent tori from the point of view of the residual gauge invariance. In other words, it parametrizes the moduli space of the (inequivalent complex structures of the) torus, defining it as the quotientC/(Z×τZ). Indeed, a second subtlety is that both the measure d2τ/τ₂2and the integrand are modular invariant, owing to the properties of the η function. Of course, this is no coincidence: it follows from diffeomorphism and Weyl invariance of the worldsheet theory, the latter being, as we have seen, a distinctive property of strings.

The upshot of this is that modular invariance forces us to restrict the integration region to the fundamental domainF of the torus, which is depicted in fig.3.1. Using modular transformations, generated by S and T, the fundamental domain can be moved anywhere in the upper–half plane. The most striking consequence is that with the choice of fig.3.1the region that gave rise to a UV divergence in the integral is now absent. This is a manifestation of how extended objects can tame the ultraviolet behaviour. Another way of looking at this is that, despite having a free spectrum made by an infinite tower of massive particles, free strings have additional features which are not captured in a naïve field–theoretic approach. This is the mistake that we mentioned at the beginning of this section: a close inspection of (3.17) shows that the integrand is modular invariant, so that the integration in the strip[₋1

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FI G U R E 3 . 1 : The fundamental domain of the torus, parametrized by the modular parameter τ. Each point corresponds to an inequivalent torus. Hence, the functional integral instructs us to integrate over this region. There actually is a further identification along the imaginary axis, which gives the so–called toothpaste region. Notice the crucial absence of the UV region. Taken from K. Becker, M. Becker and J. H. Schwarz, “String theory and M-theory: A modern introduction”.

overcounts infinitely many times the contribution of each torus. This is an efficient way of finding the correct result that the torus amplitude is expressed as an integral over the moduli space

T = Z F d2τ τ₂2 1 τ₂12|η(τ)|48 . (3.21) In addition to the measure and the integrand, the combination τ₂1/2|η(τ)2|, of which the 24–th power is the integrand of the above torus amplitude, is also modular invariant, owing to the transformation properties

η(τ+1) =e

iπ

12_η(_τ), _η(−1/τ) =√−iτ η(_τ). (3.22)

It corresponds to the contribution of a single transverse string coordinate. This language allows for a convenient way of understanding the structure of vacuum amplitudes and, consequently, of the free spectra they encode, as shall be more clearly seen in the context of the superstring.

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Other Topologies

If we allow non–oriented strings (this can be accomplished by a suitable projection on the Hilbert space), there are other worldsheet topologies to consider, in order to compute the full vacuum amplitude. To see this, let us recall a few facts about the topology of surfaces. A topological invariant that partially classifies surfaces is the Euler characteristic χ. For manifolds equipped with a metric tensor, χ can be computed in terms of local quantities (curvature) by integrating over the manifold: this is the content of the Gauss–Bonnet theorem. For surfaces with boundary, there is an additional integral of the geodesic curvature along the boundary.

The topology of two–dimensional surfaces can be characterized by three invariants: the number h of handles, the number b of boundaries, and the number c of crosscaps. In terms of these invariants, the Euler characteristic is given by

χ=2−2h−b−c . (3.23)

The reason why we are interested in the Euler characteristic is that the worldsheet action term which describes the coupling to the dilaton,

Z

d2σRic(g)Φ , (3.24) precisely corresponds to χ for constant dilaton profiles. But this also means that the sum over worldsheet topologies in the Polyakov functional integral weighs each topology by a factor gχ

s. This shows that the one–loop contribution is given by worldsheet topologies with vanishing Euler characteristic. There are four surfaces of this type: the torus, which we discussed above, but also

• The annulus (h=0, b=2, c=0); • The Klein bottle (h=0, b=0, c=2); • The Möbius strip (h=0, b=1, c=1).

These topologies, much in the same way as the torus, can be defined by suitable identifications of a rectangular region in the plane. Moreover, they can also be

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constructed in terms of their doubly covering tori, which aptly gives us a way to write down the associated vacuum amplitudes using appropriate modular parameters. These three topologies, along with the fundamental polygons which describe the covering tori, are depicted in fig.3.2.

FI G U R E 3 . 2 : The string worldsheet topologies (excluding the torus) which contribute to the one–loop vacuum energy, and the correspond-ing fundamental polygons. From the point of view of open strcorrespond-ings, they can be seen as providing boundary conditions which feature boundaries or crosscaps. Physically, this is interpreted as the presence

of D–branes or orientifolds. Taken from [25].

In terms of modular parameters, the covering tori of the annulus and the Klein bottle have modular parameters τ=iτ2/2 and τ=2iτ2, respectively, with τ2 ∈R, while the Möbius strip is obtained from a covering torus with modular parameter τ=iτ2/2+1/2 which has non–vanishing real part. However, to get to these surfaces from their covering tori one must perform identifications with respect to certain involutions. The result is that the modular group of the covering torus is broken after identification, and the moduli spaces for the resulting integrals have to change accordingly.

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The Klein Bottle Amplitude: the Orientifold Projection

In the case of the Klein bottle, the modular parameter of the covering torus is of the form 2iτ2, with τ2 ∈R. Furthermore, the identification breaks the modular group

in such a way that the fundamental region of integration_F_Kis the whole positive imaginary axis, i.e. we have to integrate over all values t≥0. This has far–reaching consequences that we shall get to shortly. The other subtlety when dealing with the Klein bottle is the lack of orientation, which means that the corresponding strings have to be non–oriented, namely we should identify the left–moving and right–moving sectors of the spectrum. A convenient way to implement this projection, owing to the orthogonality of the left–moving and right–moving sectors, is to insert the worldsheet parity operatorΩ : σ _{7→ −}σ inside the trace, while also dividing the overall amplitude (including the torus contribution) by 2 in order not to overcount. It should be noted that this projection, at the level of the spectrum, has to keep the even–parity states to have consistent interactions. Were this not the case, odd–parity states could produce even states by scattering. Putting these ingredients together, the Klein bottle amplitude reads

K = 1₂ Z FK dτ2 τ₂2 1 τ₂12tr qL0−1_qL0−1Ω , (3.25)

which can be evaluated to give

K = 1 2 Z ∞ 0 dτ2 τ₂2 1 τ₂12η(2iτ2)24. (3.26)

The dependence on the modular parameter 2iτ2in the integrand is thus manifest. A convenient check that the number of states in the free spectrum is consistent with the even–parity projection can be made expanding the integrand in a Laurent series in q, q, and adding the resulting contribution to the level–matched terms in the torus amplitude, which also depend on(qq)n_{. The coefficients then count the number of} states at each level. The torus amplitude by itself, which is the only contribution to the