Contents

LAPTH-1366/09


Applications of String Theory:

Non-perturbative Effects in Flux Compactifications

and Effective Description of Statistical Systems


Livia Ferro

LAPTH, Université de Savoie, CNRS

9, chemin de Bellevue, BP 110, 74941 Annecy le Vieux Cedex, France

livia.ferro@lapp.in2p3.fr


Abstract

In this paper, which is a revised version of the author’s PhD thesis, we analyze two different applications of string theory. In the first part, we focus on four dimensional compactifications of Type II string theories preserving supersymmetry, in presence of intersecting or magnetized D-branes. We show, through world-sheet methods, how the insertion of closed string background fluxes may modify the effective interactions on Dirichlet and Euclidean branes. In particular, we compute flux-induced fermionic masses. The generality of our results is exploited to determine the soft terms of the action on the instanton moduli space. Finally, we investigate how fluxes create new non-perturbative superpotential terms in presence of gauge and stringy instantons in SQCD-like models. The second part is devoted to the description of statistical systems through effective string models. In particular, we focus our attention on -dimensional interfaces, present in particular statistical systems defined on compact -dimensional spaces. We compute their exact partition function by resorting to standard covariant quantization of the Nambu-Goto theory, and we compare it with Monte Carlo data. Then, we propose an effective model to describe interfaces in 2 space and test it against the dimensional reduction of the Nambu-Goto description of the 2 interface.

UNIVERSITÀ DEGLI STUDI DI TORINO

Dipartimento di Fisica Teorica

DOTTORATO DI RICERCA IN FISICA FONDAMENTALE, APPLICATA E ASTROFISICA

Ciclo XXI



Applications of String Theory:

Non-perturbative Effects

in Flux Compactifications

and

Effective Description of Statistical Systems

Candidato:                                  Relatore:
       Livia Ferro                                   Prof. Marco Billò

Coordinatore:                               Controrelatore:
       Prof. Stefano Sciuto                       Prof. Angel Uranga

Anni accademici 2005/06 - 2006/07 - 2007/08

Settore scientifico disciplinare di afferenza: FIS/02

Contents

Chapter 1 Introduction

String theory presents many different interesting facets; in this thesis we want to focus our attention to two of its various applications. We will first analyse how the presence of closed string background fluxes may modify the perturbative and non-perturbative sectors of the gauge theories realized by means of particular D-brane configurations; then we will explain how the stringy formalism can be applied to the effective description of certain aspects of Lattice Gauge Theories and more general statistical systems.

As many other theoretical discoveries, string theory has a fascinating history, which goes back several decades. String theory arose indeed in the late sixties, in a different form with respect to the modern one. In that period experiments were providing an enormous proliferation of strongly interacting particles of higher spins.

Regge trajectories

Tullio Regge in 1957 introduced the complex angular momentum method [1]. In its relativistic formulation this helped to study the properties of scatterings as functions of angular momentum, after having analytically continued the scattering amplitude to the whole complex plane. The main characteristic was that the amplitude had an explicit exponential dependence on a Regge trajectory function , which enclosed the information of the angular momentum with respect to the state energy . Meanwhile regularities in the spectrum of strongly interacting particles were observed. In 1960 G. Chew and S. Frautschi [2] conjectured for them a simple dependence between their angular momentum and squared mass

(1.1)

in other words the particles were aligned on Regge trajectories which were straight lines. The constant was called Regge slope and is an additive shift. This description predicted the existence of infinitely many particle families, in function of .

The observation that the amplitudes for mesons scattering in the -channel had a perfect match with amplitudes for the -channel scattering (i.e. there was a duality between the description in terms of Regge poles or of resonances) lead to Dual Resonance Models. In 1968 Veneziano proposed his famous formula [3] which describes the scattering of four particles lying on Regge trajectories by means of the Euler Beta function

(1.2)

where is the Regge trajectory. Let us remark that this expression is explicitly - crossing symmetric. In 1970 it was argued independently by Nambu, Nielsen and Susskind [4, 5, 6] that the Veneziano dual formula could be derived from the quantum mechanics of relativistic oscillating one-dimensional objects, strings, i.e. of an infinite tower of simple harmonic oscillators. In this description the - and -channel were naturally identified with the same process; indeed a tree level open string diagram at fixed external legs is unique while in the QFT limit it can be viewed as the -channel or the -channel and the straight-line Regge trajectories were then understood as arising from a rotating relativistic string of tension proportional to . The idea was that the vibrational modes of these one-dimensional objects coincide with hadronic particles but, while particles are zero-dimensional objects, so that their classical motion is a one-dimensional line of minimal length, the string, which is a one-dimensional object, will classically describe a two-dimensional surface, the worldsheet. The natural classical action is just the area of the worldsheet.

Nambu-Goto action

Such an action was first introduced by Y. Nambu and T. Goto [13]:

(1.3)

where are the coordinates of the worldsheet and is the tension of the string, which is therefore proportional to the Regge slope. The fields give the embedding of the world-sheet in space-time (). In 1976, by means of the definition of an independent metric on the worldsheet , a first order version of Nambu-Goto action was proposed [14]

(1.4)

from which the NG action (1.3) is retrieved by integrating out . In this theory both open strings, with two distinct endpoints, and closed strings, where the endpoints make a complete loop, can be naturally considered. For the closed string, where , we can write the following mode expansion

(1.5)

where is the center of mass of the string and the momentum associated to it. In the case of open strings, the equations of motion and the boundary conditions for the fields derived from (1.4)

(1.6)
(1.7)

can be satisfied in different ways, depending on the chosen boundary conditions for each endpoint. Indeed, to solve (1.6) one can choose Neumann boundary conditions

(1.8)

or Dirichlet ones:

(1.9)

so that there can be Neumann-Neumann, Dirichlet-Dirichlet or Neumann-Dirichlet mode expansions.
The process of quantization promotes the oscillators and to annihilation and creation operators acting on a Fock space. The absence of non-physical states (or, equivalently, the cancellation of conformal anomalies) fixes the dimensions of the target space to .

The description of strong interactions based on bosonic strings was not satisfying because its spectrum contained only bosons, among which a tachyon responsible of instability. Moreover, it made many predictions that directly contradicted experimental results and could not explain all the kinematical regimes. Indeed, dual models did not incorporate the parton-like behaviour; a different theory of strong interactions was required and since 1974 Quantum Chromodynamics was recognized to give a more accurate description of experimental data in the perturbative regime. It was indeed discovered that hadrons and mesons are made by quarks and well described by an gauge theory, QCD. However, QCD is very useful to describe the behaviour of strong interactions at high energies but, since at low energies it becomes strongly coupled, calculations on items like confinement and chiral symmetry breaking are not easily performed.

The interaction between a quark and an antiquark can be instead well described with a string-like colour flux tube (i.e. an effective QCD string) stretching between them, as we will see from Chapter 7 on. For large distances , the potential goes like where is the string tension related to . The asymptotic expansion of Nambu-Goto bosonic string gives just a confining term . For this reason the presence of a string-like behaviour in some regimes of strong interactions is still believed to be correct. Strong support to this idea came from G. ’t Hooft suggestion [7] of studying gauge theories with colours in the large- limit. The diagrammatic expansion in the parameter turned out to be organized according to the genus of the diagram surface, just as an expansion of a perturbative theory with closed oriented strings. This suggested that gauge theories admit a dual representation by means of string models. The large- limit turned out to give a good qualitative description on confinement, U(1) anomalies and other dynamical items. For two-dimensional gauge theories much progress was made in the early nineties and dual string theories were found, starting with [8]. The four-dimensional case is much more complicated but in 1997 J. Maldacena succeeded to find a dual of a specific four-dimensional gauge theory [9]. He indeed found a correspondence between super Yang-Mills field theory in four space-time dimension and Type IIB string theory in a background of five-dimensional anti-de Sitter space times a five-sphere. His AdS/CFT correspondence brought back to the fore the idea of the effective QCD string. Nowadays, the comparison between results of effective QCD string and numerical lattice simulations can give some important hints and insights for consistent models.

While string theory was being supplanted by QCD, a new discovery promoted it as a good candidate for a theory of quantum gravity. Indeed, it was always in 1974 that a massless spin two excitation from the closed string sector was discovered [10], which could be interpreted as the graviton. String theory therefore evolved to a more general theory of interactions and was examined as a possible ultimate theory of nature in the quest for a unified description of Fundamental Interactions, the so-called Theory of Everything (TOE). The Nambu-Goto theory had already been extended to a supersymmetric version including fermions, the superstring theory [11, 12].

Superstrings

The fermionic worldsheet action (to be definite we will focus on Type II superstring theories) is based on the supersymmetry on the world-sheet

(1.10)

where are worldsheet Majorana spinors and the matrices provide a representation of the Clifford algebra. This action is invariant under supersymmetric tranformations. As for the pure bosonic string, one can proceed by looking for the mode expansions given by the equations of motion and boundary conditions and then promoting the respective oscillators to operators. From the quantization of world-sheet spinors two sectors arise, the Ramond (R) and the Neveu-Schwarz (NS) one.


NS-NS R-R
Type IIA
Type IIB
Table 1.1: The closed string sector of Type IIA and Type IIB theories.

In the Ramond sector of the open string the oscillators satisfy the anti-commutation relation

(1.11)

which for reduces to the Clifford algebra

(1.12)

The ground state of the Ramond sector is therefore a spacetime spinor. Enforcing the GSO projection, a supersymmetric spectrum is left. For the superstring it can be shown that the absence of non-physical states requires a 10 dimensional spacetime . As the space-time we can ”feel” is only four-dimensional, this needs a way to compactify the six spatial dimensions which are exceeding. To compare this construction with particle physics, one needs low-energy effective actions, which describe the dynamics of the massless states of the string in the field theory limit , where the string reduces to a point particle. To do this one has to find massless (or light) states and construct the effective interaction terms. (Also massive states can have impact on them, for instance if one computes a loop amplitude.) This can be performed for instance by computing string amplitudes with these states and then going to the limit .

Different kinds of consistent string theories were constructed; totally, they were five: Type I, Type IIA and Type IIB (on which we will mainly concentrate our attention), and two heterotic, E8 X E8 and SO(32). At that time it was believed that only one of these five candidates, the theory whose low energy limit after compactification would be able to match the physics observed, was the actual correct TOE. They indeed present many different characteristics. For instance Type IIA is non-chiral, whereas the other four are chiral; Type I, Type IIA and IIB contain open and closed strings, while the heterotic theories only closed strings.

In 1984 the first superstring revolution started by the discovery of M. Green and J. H. Schwarz of anomaly cancellation in type I string theory (Green-Schwarz mechanism). String theory became to be accepted as an actual candidate for the unification theory.

Approximately between 1994 and 1997 the second superstring revolution took place. It was realized that the five 10-dimensional string theories were related through a web of duality transformations, which for instance connect large and small distance scales (-duality), or strong and weak coupling constants (-duality) of different theories. A particular combination of - and -duality is called -duality. When dimensions are compactified other dualities arise. In 1995 E. Witten discovered [16] that the five 10-dimensional superstring theories were not only related between them but actually were different limits of a new 11-dimensional theory called M-theory, see Fig. (1.1). Its fundamental objects should be membranes which appear as solitons of a 11-dimensional supergravity, but its understanding is not yet precise.

Figure 1.1: The five superstring theories and the eleven dimensional supergravity are nothing else than different limits of M-theory.

This duality web required in some cases the matching of the non-perturbative spectrum of a theory with the perturbative one of the dual theory. Non-perturbative states were represented by higher-dimensional objects, branes, which play a key rôle in this respect. In particular, Dirichlet branes, or D-branes, which were been studied since 1990 and developed by J. Polchinski [17], correspond to extended objects where open strings could end (microscopic decription) but can also be viewed as soliton solutions of low energy superstring theory (macroscopic description).

D-branes

D-branes were discussed during the quest for classical solutions of the low-energy string effective action and then they became an essential element to better understand the links between the five superstring theories, as their existence is required by various duality transformations. Later, it was understood that they can be efficiently used in the construction of four-dimensional phenomenological models. A Dp-brane is an extended object with spatial dimensions, where indicates that the endpoints of the strings attached to them have Dirichlet boundary conditions. Their worldvolume action is the action of the massless open string modes embedded in a closed string background living in the bulk. It is divided into two pieces which involve respectively the NS-NS and the R-R sector. At leading order in the string coupling it reads

(1.13)

where is the Dirac-Born-Infeld (DBI) action and is the generalization of the Maxwell theory with higher derivative couplings

(1.14)

with . The Wess-Zumino (WZ) action measures the Ramond-Ramond charges of a Dp-brane and does not include the metric (so it is topologic). It involves the R-R sector of the theory

(1.15)

At low energy, i.e. at leading order in , one can retrieve the Super Yang-Mills (SYM) theory. In particular, the DBI action leads to the gauge fields and scalar kinetic terms of SYM while the WZ part to the -term.
coincident D-branes support on their worldvolume the interactions of gauge group, while gravity propagates on the whole ten-dimensional target space, the bulk. Moreover, intersecting D-branes (or branes with world-volume fluxes) permit the existence of chiral matter localized at their intersection points. The presence of tadpoles in Type II compactifications with D-branes led to the introduction of orientifold projections, i.e. transformations involving the world-sheet parity operator.

The enormous variety of possible constructions opened the way to the engineering of more and more models with semi-realistic properties.

Let us mention that D-branes entered essentially in the AdS/CFT correspondence developed by Maldacena. In these recent years many other developments have been made and many models constructed, in the search for a brane construction which could mimic the properties of Standard Model or of its minimal supersymmetric extension (MSSM). One of the main requirements is a way to break supersymmetry.

Engineering supersymmetry breaking

In four-dimensional compactifications, the supersymmetry content depends on the choice of the compactification manifold and the embedding of D-branes. There are several methods to reduce the supersymmetries in the bulk or the ones preserved by the theories living on D-branes. For instance a Calabi-Yau compactification manifold preserves supersymmetries in the bulk. D-branes can then be included in such a way to preserve . supersymmetry is desirable from the phenomenological point of view, most due to hierarchy reasons, but it should be broken at some level to retrieve the physics we observe. The search for precise supersymmetry breaking setups in string models is therefore very important. If one does not want to spoil the good soft UV behaviour of the theory, supersymmetry has to be softly broken, by adding explicit soft supersymmetry breaking terms which respect the renormalization behaviour of supersymmetric gauge theories. One of the possible terms is the introduction of gaugino masses

(1.16)

where is the gauge group index. Other ones are scalar masses , Yukawa couplings , quadratic terms in the potential for the scalar (these terms, which are allowed by the symmetries of MSSM, give rise to the -problem when the scalar is the Higgs field). Usually these possibilities arise within a so-called mediated supersymmetry breaking scheme. The supersymmetry is spontaneously broken at very high mass scales in some hidden sector; then, through the messenger sector, it is communicated to the visible one, which can be for instance the MSSM, where soft terms are produced.

In the MSSM there is no microscopic description of these soft supersymmetry breaking terms.

To reproduce such terms in string theory, one can turn on background values for field strengths

coming from the closed sector of the theory. Type IIB closed sector contains the antisymmetric tensor , coming from the NS-NS sector, and the forms, with , coming from RR one.

In the low-energy effective action the matter visible sector of MSSM is coupled to 4d supergravity and the gravitational interactions act as the messenger sector. The supersymmetry breaking terms can also be directly retrieved by computing the couplings between three-form fluxes and open string matter fields on Dp-branes through coupled to them [111] or from scattering amplitudes in closed string background. They can lead to supersymmetry breaking in the bulk by giving mass to gravitinos and/or in the open string sector via their coupling to D-branes, by generating soft supersymmetry breaking terms on the worldvolume of branes, such as gaugino masses.

During the development of the theory, it became clear that general string compactifications have hundreds of parameters, called moduli, which encode the data of the string model under consideration, such as the D-brane positions, size and shape of the manifold and so on. Each of them appears in the four-dimensional theory as a massless scalar field, giving rise to long-range interactions which are not observed and affecting the four dimensional effective action via its vacuum expectation value. Moreover they have a flat potential to all orders in perturbation theory. To stabilize them there are many possibilities. One is based on the introduction of background fluxes [28, 29, 30] in the internal dimensions, to preserve Poincaré invariance in the Minkowskian space-time.

Background flux compactifications play therefore many non-trivial rôles in phenomenological models. As already pointed out, they can create an effective potential for the moduli and break supersymmetry by generating soft supersymmetry breaking terms on D-branes.

From their start in the mid eighties with the study of heterotic string compactifications in presence of three-form H-flux [21, 22, 23], flux compactifications have enormously developed.

Other deep developments involved the non-perturbative sector of gauge theories, starting from the discovery of Yang-Mills instantons [24]. It was pointed out in 1995 [66, 67] that gauge instantons could have a realization in the frame of string theory. Systems of suitably chosen D-branes, D-instantons and Euclidean branes can indeed support the stringy description of gauge instantons. It was argued that non-perturbative effects, such as superpotentials arising from instantons and gaugino condensation, could for instance solve the problem of moduli stabilization. Moreover it was found that string theory could provide new kinds of instantons, called exotic, which still do not have a complete field theory explanation.

As we will see, the interplay between fluxes and instantons is very deep. Indeed, in presence of fluxes, non-perturbative superpotentials can be generated by instantons giving rise to new low-energy effects. Moreover, fluxes can contribute to get non-vanishing results in presence of exotic instantons by lifting fermionic zero-modes which would make vanish instanton-generated interactions.

We will discuss these topics in detail in next Chapters, where we will give general informations about the models we will consider in our computations.

In particular, we will focus on four dimensional compactifications of Type II string theories preserving supersymmetry in the presence of intersecting or magnetized D-branes, which constitute a promising scenario for phenomenological applications of string theory and realistic model building. Indeed, in these compactifications, gauge interactions similar to those of the supersymmetric extensions of the Standard Model of particle physics can be engineered using space-filling D-branes that partially or totally wrap the internal six-dimensional space. By introducing several stacks of such D-branes, one can realize adjoint gauge fields for various groups by means of the massless excitations of open strings that start and end on the same stack, while open strings stretching between different stacks provide bi-fundamental matter fields. On the other hand, from the closed string point of view, (wrapped) D-branes are sources for various fields of Type II supergravity, which acquire a non-trivial profile in the bulk. Thus the effective actions of these brane-world models describe interactions of both open string (boundary) and closed string (bulk) degrees of freedom and have the generic structure of supergravity in four dimensions coupled to vector and chiral multiplets. Several important aspects of such effective actions have been intensively investigated over the years from various points of view [18, 19, 20]. We will study, through world-sheet methods, how the insertion of background fluxes may modify effective interactions on Dirichlet and Euclidean branes and create new non-perturbative superpotential terms in presence of instantons.

1.1 Scheme of the thesis

The thesis is divided into two parts, related to very different aspects of string theory. The first one is related to string theory viewed as the candidate for the theory of everything. In particular we will drive our attention to flux compactifications and non-perturbative terms, analyzing the interplay, given by fluxes, among soft supersymmetry breaking, moduli stabilization and non-perturbative effects in the low-energy theory.

The second part of the thesis is devoted to the description of statistical systems, in particular interfaces, via the effective string, coming back to the purpose string theory was born for. After the discovery of the AdS/CFT correspondence, the interest on QCD string has been renewed. We will show how the bosonic string of Nambu-goto model in the first order formulation can mimic very well the behaviour of interfaces. To support it we will present not only the theoretical evaluation but also the comparison with precise data provided by Monte Carlo simulations.

For the detailed partial schemes see the corresponding introductions.

Chapter 2 Three-form Fluxes in compactifications

As we already stressed in Chapter 1, an important ingredient of Type II string theories compactifications preserving supersymmetry in the presence of intersecting or magnetized D-branes is the possibility of adding internal (to preserve 4d Poincaré invariance) antisymmetric fluxes both in the Neveu-Schwarz-Neveu-Schwarz and in the Ramond-Ramond sector of the bulk theory [60, 61, 62]. These fluxes bear important consequences on the low-energy effective action of the brane-worlds, such as moduli stabilization, supersymmetry breaking and also the generation of non-perturbative superpotentials.

Indeed, as is well-known [31], four-dimensional supergravity theories are specified by the choice of a gauge group , with the corresponding adjoint fields and gauge kinetic functions, by a Kähler potential and a superpotential , which are, respectively, a real and a holomorphic function of some chiral superfields . The supergravity vacuum is parametrized by the expectation values of these chiral multiplets that minimize the scalar potential

(2.1)

where is the Kähler covariant derivative of the superpotential and the () are the D-terms. Supersymmetric vacua, in particular, correspond to those solutions of the equations satisfying the D- and F-flatness conditions .

The chiral superfields of the theory comprise the fields and that parameterize the deformations of the complex and Kähler structures of the three-fold, the axion-dilaton field

(2.2)

where is the R-R scalar and the dilaton, and also some multiplets coming from the open strings attached to the D-branes. The resulting low energy supergravity model has a highly degenerate vacuum.

One way to lift (at least partially) this degeneracy is provided by the addition of internal 3-form fluxes of the bulk theory [60, 61, 62] via the generation of a superpotential [63, 28]

(2.3)

where is the holomorphic -form of the Calabi-Yau three-fold and

(2.4)

is the complex 3-form flux given in terms of the R-R and NS-NS fluxes and . The flux superpotential (2.3) depends explicitly on through and implicitly on the complex structure parameters which specify , while it does not depend on Kahler structure moduli .

Using standard supergravity methods, F-terms for the various compactification moduli can be obtained from (2.3). Insisting on unbroken supersymmetry requires the flux to be an Imaginary Self Dual 3-form of type [30], since the F-terms , , and are proportional to the , and components of the -flux respectively:

(2.5)
(2.6)
(2.7)

and only survives. So to preserve supersymmetry the flux has to be Imaginary Self Dual and with vanishing part:

(2.8)
Table 2.1: Decomposition of the ISD and IASD parts of the 3-form .

The requirement of existence of solutions to the supergravity equations of motions with fluxes imposes only [30, 32]

(2.9)

therefore for instance can break supersymmetry without destroying the solution. A consistent model including gauge and gravity would require fluxes which satisfy eq.(2.9). However, if in the setup under consideration the regime is such that the dynamical effects of gravity can be neglected (as in our model), gauge theories with ”soft” couplings with all kinds of fluxes coming from closed strings can be considered. The F-terms can also be interpreted as the “auxiliary” components of the kinetic functions for the gauge theory defined on the space-filling branes, and thus are soft supersymmetry breaking terms for the brane-world effective action. These soft terms have been computed in various scenarios of flux compactifications [33] - [38] and their effects, such as flux-induced masses for the gauginos and the gravitino, have been analyzed in various scenarios of flux compactifications relying on the structure of the bulk supergravity Lagrangian and on -symmetry considerations (see for instance the reviews [60, 61, 62] and references therein); here we derive them by a direct world-sheet analyisis.

So far the consequences of the presence of internal NS-NS or R-R flux backgrounds onto the world-volume theory of space-filling or instantonic branes have been investigated relying entirely on space-time supergravity methods [39] -[44], rather than through a string world-sheet approach111For some recent developments using world-sheet methods see Ref. [45].. A paper recently appeared with an alternative approach which does not require a microscopic description, see [46].

In this thesis we fill this gap and derive the flux induced fermionic terms of the D-brane effective actions with an explicit conformal field theory calculation of scattering amplitudes among two open string vertex operators describing the fermionic excitations at a generic brane intersection and one closed string vertex operator describing the background flux. Our world-sheet approach is quite generic and allows to obtain the flux induced couplings in a unified way for a large variety of different cases: space-filling or instantonic branes, with or without magnetization, with twisted or untwisted boundary conditions. Indeed, the scattering amplitudes we compute are generic mixed disk amplitudes, i.e. mixed open/closed string amplitudes on disks with mixed boundary conditions, similar to the ones considered in Refs. [47, 48, 49, 50, 74].

Our approach not only reproduces correctly all known results but can be applied also to cases where the supergravity methods are less obvious, like for example to study how NS-NS or R-R fluxes couple to fields with twisted boundary conditions or how they modify the action which gives the measure of integration on the moduli space of instantons. Finding the flux-induced soft terms on instantonic branes of both ordinary and exotic type is a necessary step towards the investigations of the non-perturbative aspects of flux compactifications we have mentioned above.

Indeed, in addition to fluxes, another important issue to study is the non-perturbative sector of the effective actions coming from string theory compactifications [66, 67]. Only in the last few years, concrete computational techniques have been developed to analyze non-perturbative effects using systems of branes with different boundary conditions [72, 73]. Non-perturbative effects were also recently connected to topological strings [97]. These non-perturbative contributions to the effective actions may play an important rôle in the moduli stabilization process [69, 70] and bear phenomenologically relevant implications for string theory compactifications. In the framework we are considering, non-perturbative sectors are described by configurations of D-instantons or, more generally, by wrapped Euclidean branes which may lead to the generation of a non-perturbative superpotential of the form

(2.10)

Here we have labeled the gauge group components (corresponding to different stacks of D-branes) by an index and denoted by their complexified gauge couplings. In general, the ’s depend on the axion-dilaton modulus and the Kähler parameters that describe the volumes of the cycles which are wrapped by the D-branes222The explicit dependence of on and can be derived from the Dirac-Born-Infeld action.. Furthermore, in (2.10) the exponent represents the total classical action for an instanton configuration with second Chern class with respect to the gauge component A, and are (holomorphic) functions of the chiral superfields whose particular form depends on the details of the model.

The interplay of fluxes and non-perturbative contributions, leading to a combined superpotential

(2.11)

offers new possibilities for finding supersymmetric vacua.

Indeed, the derivatives , and might now be compensated by , and [70] so that also the , and components of may become compatible with supersymmetry and help in removing the vacuum degeneracy [71].

Another option could be to arrange things in such a way to have a Minkowski vacuum with and broken supersymmetry. If the superpotential is divided into an observable and a hidden sector, with the flux-induced supersymmetry breaking happening in the latter, this could be a viable model for supersymmetry breaking mediation. If all moduli are present in , the number of equations necessary to satisfy the extremality condition for seems sufficient to obtain a complete moduli stabilization. To fully explore these, or other, possibilities, it is crucial however to develop reliable techniques to compute non-perturbative corrections to the effective action and determine the detailed structure of the non-perturbative superpotentials that can be generated, also in presence of background fluxes.

These methods not only allow to reproduce [73]-[77] the known instanton calculus of (supersymmetric) field theories [78], but can also be generalized to more exotic configurations where a field theory explanation became avalaible only recently, but it is still far from being complete [81] -[107]. The study of these exotic instanton configurations has led to interesting results in relation to moduli stabilization, (partial) supersymmetry breaking and even fermion masses and Yukawa couplings [81, 82, 91, 108] (for a recent systematic analysis see [109]). A delicate point about these stringy instantons concerns the presence of neutral anti-chiral fermionic zero-modes which completely decouple from all other instanton moduli, contrarily to what happens for the usual gauge theory instantons where they act as Lagrange multipliers for the fermionic ADHM constraints [73]. In order to get non-vanishing contributions to the effective action from such exotic instantons, it is therefore necessary to remove these anti-chiral zero modes [88, 89] or lift them by some mechanism [93, 98]. The presence of internal background fluxes may allow for such a lifting and points to the existence of an intriguing interplay among soft supersymmetry breaking, moduli stabilization, instantons and more-generally non-perturbative effects in the low-energy theory which may lead to interesting developments and applications.

If really generated, such exotic interactions could also become part of a scheme in which the supersymmetry breaking is mediated by non-perturbative soft-terms arising in the hidden sector of the theory, as recently advocated also in [105]. Nonetheless, the stringent conditions required for the non-perturbative terms to be different from zero, severely limit the freedom to engineer models which are phenomenologically viable.

To make this program more realistic, in this thesis we address the study of the generation of non-perturbative terms in presence of fluxes. In the following we will consider the interactions generated by gauge and stringy instantons in a specific setup consisting of fractional D3-branes at a singularity which engineer a quiver gauge theory with bi-fundamental matter fields. In order to simplify the treatment, still keeping the desired supergravity interpretation, this quiver theory can thought of as a local description of a Type IIB Calabi-Yau compactification on the toroidal orbifold . From this local standpoint, it is not necessary to consider global restrictions on the number and of D3-branes, which can therefore be arbitrary, nor add orientifold planes for tadpole cancelation. In such a setup we then introduce background fluxes of type and , and study the induced non-perturbative interactions in the presence of gauge and stringy instantons which we realize by means of fractional D-instantons. In this way we are able to obtain a very rich class of non-perturbative effects which range from “exotic” superpotentials terms in the effective gauge theory to non-supersymmetric multi-fermion couplings. We also show that stringy instantons in presence of -fluxes can generate non-perturbative interactions even for gauge theories. This has to be compared with the case without fluxes where an orientifold projection [88, 89] (leading to orthogonal or symplectic gauge groups) is required in order to solve the problem of the neutral fermionic zero-modes. Notice also that since the and components of the are related to the gaugino and gravitino masses (see for instance [111, 112]), the non-perturbative flux-induced interactions can be regarded as the analog of the Affleck-Dine-Seiberg (ADS) superpotentials [113] for gauge/gravity theories with soft supersymmetry breaking terms. In particular the presence of a flux has no effect on the gauge theory at a perturbative level but it generates new instanton-mediated effective interactions [96].

For the sake of simplicity most of our computations will be carried out for instantons with winding number ; however we also briefly discuss some multi-instanton effects. In particular, from a simple counting of zero-modes we find that in our quiver gauge theory an infinite tower of D-instanton corrections can contribute to the low-energy superpotential, even in the field theory limit with no fluxes, in constrast to what happens in theories with simple gauge groups where the ADS-like superpotentials are generated only by instanton with winding number . These multi-instanton effects in the quiver theories certainly deserve further analysis and investigations. For an interesting connection between matrix models and D-brane instanton calculus (and a perturbative way of computing stringy multi-instanton effects) see [110]. Results about multi-instanton processes have also appeared in Ref. [104].

More specifically, this part of the thesis is organized as follows: in next Chapter we will briefly review the notion of instantons in gauge theories and how it can be derived in the stringy side.

Chapter 4, based on the publication [64], is devoted to the computation of interaction of massless fermions in presence of closed string background fluxes. In Section 4.1 we describe in detail the world-sheet derivation of the flux induced fermionic terms of the D-brane effective action from mixed open/closed string scattering amplitudes. The explicit results for various unmagnetized or magnetized branes as well as for instantonic branes are spelled out in Section 4.2 in the case of untwisted open strings and in Section 4.3 in some case of twisted open strings. The flux-induced fermionic couplings are further analyzed for the orbifold compactification which we briefly review in Section 4.4. Later in Section 4.4.1 we compare our world-sheet results for the flux couplings on fractional D3-branes with the effective supergravity approach to the soft supersymmetry breaking terms, finding perfect agreement. In Section 4.5 we exploit the generality of our world-sheet based results to determine the soft terms of the action on the instanton moduli space.

Then in Chapter 5 we present the results of [65], where we analyse the non-perturbative side of flux compactifications. In Section 5.1 we discuss a quick method to infer the structure of the non-perturbative contributions to the effective action based on dimensional analysis and symmetry considerations. In Section 5.2 we analyze the ADHM instanton action and discuss in detail the one-instanton induced interactions in SQCD-like models without introducing -fluxes. Finally in Sections 5.3 and 5.4 we consider gauge and stringy instantons in presence of -fluxes and compute the non-perturbative interactions they produce.

Chapter 6 is devoted to summary of results, conclusions and future perspectives.

Some more technical details, such as our conventions on spinors, on the orbifold and on the flux couplings for wrapped fractional D9-branes are contained in the Appendix.

Chapter 3 Space-time Instantons in Gauge and String Theories

In this Chapter we want to briefly recall some basic facts about instantons in gauge theories and how they can be realized in string theory (many good reviews exist; see for instance [78, 79, 80]). Setups which reproduce the usual Yang-Mills instantons (i.e. gauge instantons) can be performed by means of D-brane models. As we will discuss, systems of Dp- and D(p-4)-branes in a suitably compactified target space give rise to instanton configurations of the gauge theory on the Dp’s. An important aspect of string theory realization is that new kinds of instantons can arise, which do not have an explanation on the gauge theory side yet. They are called exotic instantons and, under appropriate conditions, can actually contribute to the low-energy effective actions. Moreover, other non-perturbative effects may arise when string corrections are taken into consideration.

3.1 In gauge theory

Instantons in gauge theories, defined in Minkowski spacetime, describe tunneling processes from one vacuum to another. The simplest models which exhibit this phenomenon are the quantum mechanical point particle with a double-well potential having two vacua, or a periodic potential with infinitely many vacua. There is no classical allowed trajectory for a particle to travel from one vacuum to the other, but quantum mechanically tunneling occurs. The tunneling amplitude can be computed in the WKB approximation and is exponentially suppressed.
Sometimes it is useful to perform a Wick rotation since path integrals are more conveniently computed in Euclidean spacetime. In the Euclidean regime instantons are defined as finite action solutions to the fields equations of motion.
When a theory admits different topological sectors, in each of them a configuration of lowest finite Euclidean action can be identified. Euclidean path integral requires to keep in consideration all these configurations, where fields assume a non-trivial profile, by summing over them. The contribution of instantons to the path integral is very tiny as it turns out to be exponentially suppressed. Moreover, as we will see, when fermions are present strong selection rules appear and may eventually lead to a vanishing instanton contribution.

3.1.1 Instantons in pure Yang-Mills

Let’s take the 4-dimensional SU(N) pure Yang-Mills:

(3.1)

As we said instantons are Euclidean solutions of motion equations with finite action. The requirement of finite action implies that the field strength goes to zero faster than at infinity. This requires that the gauge field approaches a pure gauge

(3.2)

for some . Actually, there is a way to classify such fields into sectors characterized by an integer number

(3.3)

where

(3.4)

is called instanton number and corresponds to the second Chern class of the theory. By means of the Bogomoln’yi trick one can write the following bound for the action

(3.5)

which is saturated by (anti)self-dual configurations

(3.6)

The self-dual configuration is called instanton and corresponds to while yields the antiself-dual one, called anti-instanton. They satisfy the equations of motion

(3.7)

by means of the Bianchi identity. The action for an instanton, as well as for an anti-instanton, is simply:

(3.8)

If we have a -angle term

(3.9)

the classical action for an instanton number becomes

(3.10)

where is the complex gauge constant

(3.11)

The goal of the so-called instanton calculus is to evaluate correlation functions in the instanton sectors. Correlators are expressed as

(3.12)

where the field insertions can be replaced at first order by their values in the instanton background.

Moduli space and partition function

The partition function is obtained by integrating over all the possible inequivalent histories, i.e. over the inequivalent configurations of the fields. This can be traded for an integral over the so-called moduli space , which is the space of inequivalent solutions of self-dual SU(N) Yang-Mills equations. The moduli correspond to the parameters on which the gauge profile depends. For instance, in a SU(2) theory, this field assumes the following profile:

(3.13)

where is the position and the size of the instanton. These, together with the moduli associated to the gauge orientation of the instanton111Remember that the solutions of self-dual e.o.m. are equivalent for local gauge transformations but inequivalent for global ones., form the collective coordinates. We outline here a simple example, which can clarify this notion. Suppose we have only one massless field depending on a unique collective coordinate and we want to perform the path integral

(3.14)

in a one-instanton background with vanishing -term. With the saddle-point approximation, the field can be expanded around the instanton solution

(3.15)

where satisfies the self-dual equation. Therefore at first order the action reads

(3.16)

The quantum fluctuation can be written as a linear combination of the eigenfunctions of

(3.17)

where the coefficient is called zero mode and corresponds to an eigenfunction of with zero eigenvalue. It indeed represents the fluctuations which do not change the action. The path integral measure can be rewritten as

(3.18)

To perform the computation, the integral over has to be converted to an integral over the corresponding collective coordinate with a Fadeev-Popov-like method. This procedure is necessary to get a finite result from the integral, as corresponds to an eigenfunction with zero eigenvalue and does not appear in the expansion of the action. If instead a mass term is present into the action, the zero modes are said to be lifted and behave like the other ’s.

We have just used the fact that zero modes are associated to collective coordinates. When the gauge theory is not pure but the gauge fields couple to other fields, it can happen that not every zero mode is connected to a collective coordinate. Nevertheless, one continues to call moduli space the space constructed by zero modes. In general, the dimension of moduli space (i.e. the number of zero modes) can be evaluated through index theorem techniques and turns out to be

(3.19)

in the case of pure gauge theory, where the zero modes are only bosonic.

The most powerful method to solve the (anti)self-dual equations (3.6) is the ADHM construction [68]. It realizes the instanton moduli space as a hyper-Kähler quotient of a flat space by an auxiliary U(k) gauge theory. The Higgs branch of this U(k) theory, which is related to , is defined through a triplet of algebraic equations, the ADHM constraints, for each solution of which a solution to the set of equations can be built. We do not want to enter into technical details here, but we will see that the ADHM construction can be naturally embedded in a stringy setup of D-branes.

We finally remark that usually the entire moduli space can be rewritten as

(3.20)

where is the centered moduli space which defines the centered partition function.

3.1.2 Adding fermionic and scalar fields

We now want to consider the Dirac equation for a massless fermion in an (anti-)instanton background

(3.21)

where the covariant derivatives are evaluated in the (anti-)instanton background. Decomposing into its chiral and antichiral parts ( and respectively)

(3.22)

where .

One can demonstrate that has non-trivial solutions only in an instanton background, while only in an anti-instanton one. Therefore in the background of an instanton only picks up zero modes (and the reverse is true for the anti-instanton). Since the current which modifies the pure gauge equations of motion is bilinear in and , the (anti-)instanton configurations described before in the case of a pure gauge theory still remain exact solutions of this background. The non-trivial solutions of Dirac equations lead to further zero modes, fermionic ones, which obviously are Grassmann variables. The Atiyah-Singer index theorem tells us that, if the massless fermions are in the adjoint representation, the number of fermionic zero-modes is . The presence of fermionic zero-modes is a very delicate point because Grassmann variables must be in some way saturated to give a non-vanishing result in the path integral. This provides strong selection rules determining which correlation functions admit instanton corrections. In particular, for each zero mode there must be one external fermion leg, which may be given for instance by introducing fermionic mass terms or external interactions; zero modes are therefore lifted. The fermionic zero-modes can then be saturated by bringing down enough powers of the action; the correlator

(3.23)

is therefore non-vanishing only if .

A more delicate and subtle procedure should be used if scalar fields are introduced. One can indeed demonstrate that bosonic fields other than gauge ones do not lead to new zero-modes but can drastically modify the equations of motion, with an important impact on solutions. (Anti-)Instantons turn out to no longer be exact solutions of the coupled equations of motion. Nevertheless, different methods have been developed which lead to approximate solutions.

In the case of absent scalar vev’s, the equations of motions can be solved perturbatively in the gauge coupling constant and the resulting non-exact configuration is called -. The action turns out to explicitly depend on Grassmann collective coordinates (besides possible bosonic ones), meaning that the corresponding zero modes are not exact but rather quasi-zero modes.

If scalars acquire a non-vanishing vacuum expectation values, the classical action gets modified by the instanton scale size. For instance in the SU(2) case with it becomes

(3.24)

where is the scale size and the scalar vev. The term proportional to is essential to make converge the path integral on the scale size. To leading order in they can be well approximated by an ordinary instanton. These were called constrained instantons by Affleck and are examples of more general quasi-instantons as collective coordinates appear in the action and therefore some zero modes are lifted.

These considerations can be generalized to supersymmetric theories.

3.1.3 Instanton-generated superpotentials and F-terms

Instantons play a leading rôle in the understanding of non-perturbative regime of four-dimensional supersymmetric gauge theories. As shown by Affleck, Dine and Seiberg [113], instantons in SQCD with gauge group and massless flavors generate a superpotential in the case . This is not the end of the story because, even in cases where instantons do not generate such superpotential, they can deform the complex structure of the moduli space of supersymmetric vacua, which we will call , via the creation of an F-term [118], which cannot be integrated to retrieve a corresponding superpotential but is nevertheless a genuine F-term. The properties of SQCD are often listed in function of the number of flavours with respect to the number of colours :

  • : is generated but not by instantons

  • : instantons generate which lifts all flat directions on the moduli space

  • : instantons do not generate a superpotential but deform the complex structure of . It is described by an F-term which is a 4-fermion interaction on

  • : a superpotential is not generated and the moduli space is undeformed. However, there are F-terms which generate, for instance, fermions interactions, called multi-fermion F-terms. Indeed, far from the origin of the moduli space the theory has gauge instantons which generate them.

3.2 In string theory

Instanton configuration are realized in string theory by systems of D(p-4)- and Dp-branes suitably wrapped. As we already mentioned, besides the ordinary gauge instantons, other non-perturbative effects may appear in a stringy construction. There is indeed the possibility of exotic instantons, which have not a gauge field realization yet. These instantons present unbalanced fermionic zero-modes which may combine to make vanish some contributions. To lift them, one needs to drastically change the background by means of orientifold planes, deformations of the Calabi-Yau geometry or introduction of fluxes. They are very attractive because they seem to stabilize the gauge theory and can give a possible explanation for neutrino Majorana masses in the context of string phenomenology (see for instance the models constructed in [82, 83]).

3.2.1 Gauge instantons from string theory

Yang-Mills instantons have a simple realization in string theory, by systems involving D(p-4)- and Dp-branes. Witten first showed this in the maximal case [66] in Type I string theory. Let us consider, in Type IIB theory, D9-branes, which we know supporting on their world-volume a ten-dimensional U(N) supersymmetric gauge theory. Their world-volume theory contains also the couplings to the various R-R fields of the bulk. In particular it includes the term

(3.25)

which comes from the expansion of the Wess-Zumino part of the world-volume action. The 6-form field is also the same R-R form which minimally couples to the D5-branes. Therefore an instanton configuration of the gauge theory living on the D9-branes with non-zero second Chern class corresponds to units of the D5-brane charge [67]. In more detail, one can demonstrates that the mass and the charge of the D5-brane are the same of the instanton. This can obviously be extended to generic and so

(3.26)

We now list some examples. In the uncompactified case we for instance have

  • D3/D(-1) system. This model exactly gives the 4-dimensional Yang-Mills instanton.

  • D5/D1 system. The 6- plus the 2-dimensional gauge theories which are supported by this setup do not describe the 4-dimensional Yang-Mills instanton; however the D1 represents, with respect to the D5, a configuration with non-trivial in 4 of the 6 directions. The spectrum of mixed strings corresponds to the ADHM construction. The D9/D5 setup shows the same characteristics.

In the compactified case, we can have for instance

  • D9/E5 system. The D9 is completely wrapped in the 6d so that on its non-compact 4d world-volume lives a Yang-Mills theory. Its istantons are represented by 6d branes completely wrapped on (with the sam magnetization of D9) which are points in 4d. These are called euclidean branes because their world-volume directions are euclidean, indipendently of a possible Wick rotation of the non-compact part.

  • More generally, let us consider a D-brane wrapping a -cycle on , with a field strength only in the 4dimensional spacetime, and denote by the complexified gauge coupling of the resulting four-dimensional (super) Yang-Mills theory as defined in (3.11). A gauge instanton in this theory can be described in terms of a Euclidean -brane wrapping the same -cycle . The instanton induces non-perturbative interactions weighted by with being the number of instantonic branes and the action for a single instanton. We want to demonstrate that

    (3.27)

    Eq. (3.27) follows from a comparison of the world-volume action of the Euclidean E-brane with that of the wrapped D-brane [76]. To get consistency with previous sections, we move to Euclidean signature; the action of the Dp-brane is222Here we assume and take with and running in the adjoint of the gauge.

    (3.28)

    where is the D-brane tension, the dilaton, the string frame metric and the R-R -form potentials. Expanding (3.28) to quadratic order in and comparing with the standard form of the Yang-Mills action in Euclidean signature, we find that the complexified four-dimensional gauge coupling is

    (3.29)

    On the other hand the action for a Euclidean -brane wrapping is given by

    (3.30)

3.2.2 Exotic instantons

As we mentioned at the beginning, exotic instantons are instantons which arise from string constructions and do not have a field theory interpretation yet (for recent developments see [86]). They have been intensively studied in the last years as they may give contribution to neutrino Majorana masses and to moduli stabilizing terms. They can arise from many different brane setups; for instance

  • in quiver gauge theories where instanton branes are in an unoccupied node of the theory,

  • in D9/E5 systems with different magnetization,

  • in brane systems with more than 4 mixed ND directions.

Their characteristic is to present an unbalanced fermionic zero mode integration which, if not suitably handled, leads to a vanishing contribution to the instanton calculus. Unlike in gauge string instantons, these Grassmann variables no more appear in the action and therefore they must be removed, for instance with an orientifold projection [88], or lifted, for instance with bulk fluxes, as we will show later.

3.2.3 The D3/D(-1) model

We now focus our attention on a particular model, which we will study in detail in next chapters. We consider a configuration of parallel D3-branes and D(-1)-branes (or D-instantons.). For reviews see [73, 88].

A stack of D3-branes in flat space gives rise to a four-dimensional gauge theory with supersymmetry. Its field content, corresponding to the massless excitations of the open strings attached to the D3-branes, can be organized into a vector multiplet and three chiral multiplets (). These are matrices:

(3.31)

with . In superspace notation, the action of the theory is

(3.32)

where is the axion-dilaton field (3.11) and is the chiral superfield whose lowest component is the gaugino. Strings of D(-1)/D(-1) type give rise to the neutral sector. They have no longitudinal Neumann directions, so the fields arising from these strings do not have momentum and they are moduli rather than dynamical fields. They do not transform under the gauge group (and so do not couple to gauge sector) but couple to the charged sector. They are matrices, where is the instanton number, i.e. the number of D(-1)’s. The bosonic massless fields are called , where the index identifies is the space-time directions and the internal ones. There are then fermionic zero modes , (let’s assume negative chirality) which are treated independently in Euclidean space. Moreover one can introduce a triplet of auxiliary fields . The charged sector is made by strings stretching between D(-1)/D3. They have mixed boundary conditions, so the fields have no momentum. The Neveu-Schwarz sector is made up by (where negative chirality is imposed by GSO projection) and for the conjugate sector. The Ramond sector, after the GSO, leaves us with a pair of fermions , which respectively are and matrices.
If there are D3’s and D(-1), we have to introduce Chan-Paton factors in the bifundamental of . The interactions between D3/D3 fields give the usual 4d gauge theory, whereas instantons corrections are obtained by constructing the interaction terms between gauge and charged sectors and then integrating out all zero-modes (charged and neutral).

ADHM Meaning Vertex Chan-Paton

NS
centers adj.
aux.
Lagrange mult.
R partners
Lagrange mult.
Table 3.1: ADHM moduli in the neutral sector.

ADHM Meaning Vertex Chan-Paton
NS sizes
sizes
R partners


Table 3.2: ADHM moduli in the charged sector.

The moduli space is then

(3.33)

with and labeling the D and the D3 boundaries respectively. The other indices run over the following domains: ; ; ; , labeling, respectively, the vector and spinor representations of the Lorentz group and of the internal-symmetry group333See footnote 4., while . This system is described in terms of a matrix theory whose action is [78]

(3.34)

with

(3.35)

where we have also defined

(3.36)

where and are the chiral and anti-chiral blocks of the Dirac matrices in the six-dimensional internal space, and are the six vacuum expectation values of the scalar fields in the real basis. Finally,

(3.37)

is the coupling constant of the gauge theory on the D branes. The scaling dimensions of the various moduli appearing in (3.35) are listed in Tab. 3.3.

moduli