Scuola di Dottorato in Scienza ed Alta Tecnologia
Indirizzo in Fisica ed Astrofisica
Dbranes and NonPerturbative Quantum Field Theory: Stringy Instantons and Strongly Coupled Spintronics
Daniele Musso
Universitá degli Studi di Torino Dipartimento di Fisica Teorica
Relatore Commissione
Prof. Alberto Lerda Prof. Matteo Bertolini
CoRelatore Prof. Marco Billó
Dr. Aldo Cotrone Prof. Silvia Penati
Controrelatori
Prof. Riccardo Argurio
Prof. Nick Evans
Dipartimento di Fisica Teorica
Facoltà di Scienze Matematiche, Fisiche e Naturali
Università degli studi di Torino
March 16th, 2012
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To my parents, my grandmother and Claudio
Contents
 0 Abstract
 1 Preamble
 2 Introduction

I Stringy Instanton Calculus
 3 Instanton Preliminaries
 4 Dbrane Instantons

5 Stringy Instantons
 5.1 Motivations
 5.2 Stringy instanton salient features
 5.3 Stringy instantons in theories
 5.4 Description of the stringy instanton model
 5.5 Moduli action
 5.6 BRST structure of the moduli action
 5.7 String scale and renormalization

5.8 SU conformal case
 5.8.1 Localization limit
 5.8.2 Details of the partition function integral computation
 5.8.3 Explicit computations for the smallest instanton numbers
 5.8.4 Last step: the computation of the exotic nonperturbative prepotential
 5.8.5 Resumming exotic contributions
 5.8.6 Comment to the SU conformal case
 5.9 SU notconformal case
 5.10 Final Comments and Future Developments

II Holographic Superconductors
 6 Holographic Techniques
 7 Minimal Holographic Description of a Superconductor

8 Holographic Spintronics
 8.1 Mixed spinelectric conductivities and spintronics
 8.2 Superconductor with two fermion species
 8.3 Inhomogeneous superconductivity

8.4 Holographic unbalanced superconductor: Dual gravity setup
 8.4.1 Backreacted bulk dynamics
 8.4.2 Boundary conditions
 8.4.3 Normal phase
 8.4.4 A criterion for instability and hair formation
 8.4.5 ChandrasekharClogston bound at weakcoupling
 8.4.6 ChandrasekharClogston bound at strongcoupling
 8.4.7 The condensate
 8.4.8 A look to the “unbalanced” gravitational solutions
 8.5 Fluctuations
 8.6 Conductivities
 8.7 Nonhomogeneous phases?
 8.8 String embeddings and UV completion
 8.9 General models and holographic fit
 9 Future Directions

III Appendices
 A Graviton
 B ’t Hooft Symbols
 C ADHM Projector
 D Shape of the ChanPaton Orientifold Matrix
 E Details on the Dinstanton Computations
 F Bulk Massive Scalar Field
 G Clues for /CFT Correspondence
 H MeissnerOchsenfeld Effect
 I Probe Approximation
 J The nearhorizon geometry of RN black hole at low temperature.
 K Green’s Functions and Linear Response
 L Onsager’s Reciprocity Relation
 M Josephson Effect
 N Temperature Gradient, Heat Flow, Electric Fields and the Metric
 O Holographic Renormalization of our Model
 P Thermal Conductivity
 Brief Conclusion
 Acknowledgements
Chapter 0 Abstract
The nonperturbative dynamics of quantum field theories is studied using theoretical tools inspired by string formalism. Two main lines are developed: the analysis of stringy instantons in a class of fourdimensional gauge theories and the holographic study of the minimal model for a strongly coupled unbalanced superconductor.
The field theory instanton calculus admits a natural and efficient description in terms of Dbrane models. In addition, the string viewpoint offers the possibility of generalizing the ordinary instanton configurations. Even though such generalized, or stringy, instantons would be absent in a purely fieldtheoretical, lowenergy treatment, we demonstrate that they do alter the IR effective description of the brane dynamics by introducing contributions related to the string scale . In the first part of this thesis we compute explicitly the stringy instanton corrections to the effective prepotential in a class of quiver gauge theories.
In the second part of the thesis, we present a detailed analysis of the minimal holographic setup yielding an effective description of a superconductor with two Abelian currents. The model contains a scalar field whose condensation produces a spontaneous symmetry breaking which describes the transition to a superfluid phase. This system has important applications in both QCD and condensed matter physics; moreover, it allows us to study mixed electricspin transport properties (i.e. spintronics) at strong coupling.
Chapter 1 Preamble
The subject of the present thesis consists in the study of the nonperturbative dynamics of quantum field theory using stringinspired theoretical tools.
The nonperturbative dynamics of quantum field theory is relevant for an extremely wide scenario of physical contexts. Indeed, it plays a crucial role in various sectors of physics ranging from the dynamics of fundamental constituents and interactions to the condensed matter panorama. The string approach offers a versatile and powerful framework in which many distinct nonperturbative aspects of quantum field theory can be accommodated and studied. As the range of application is wide, also the ensemble of possible techniques is very wealthy; our treatment concentrates especially on two different lines, namely the Dbrane instanton calculus and the /CFT correspondence. They both originate from a common string environment.
In the string formalism, the dynamics is described in terms of the evolution and the interactions of some fundamental objects: the membranes. In contrast to particles and as the name intuitively suggests, membranes can have extended directions. If we agree on defining a generic brane as an object that is extended in spacetime directions, we can think of strings as branes. In addition to the strings, the set of basic objects in the string formalism comprehends also the Dbranes that are particular dimensional membranes to which the open string endpoints can be attached. Focusing the attention on the concept of membrane rather than just on strings, sets a democratic viewpoint encompassing all the fundamental constituents of the formalism, which, not so democratically, we will still indicate as “string formalism”.
The Dbranes, or Dbranes for short, are the pivotal ingredient that allows us to explore the nonperturbative aspects of quantum field theory of interest here. An essential feature of Dbranes is their relation and interactions with strings. Historically they have been introduced as surfaces on which the open string endpoints can lie; they actually define Dirichlet (from which the ‘‘D’’ of Dbrane) boundary conditions for the strings^{1}^{1}1More precisely, we have Dirichlet boundary conditions (i.e. constrained) for the directions that are transverse to the Dbrane and Neumann (i.e. free) boundary condition along the Dbrane itself.. It is possible to argue that the open strings attached to a brane offer a description of the dynamics of the brane itself, hence Dbranes and open strings are closely related. Open strings are objects with tension; an open string connected to a Dbrane can be naively thought of as a quantum “tension fluctuation” or excitation of the Dbrane itself.
In the lowenergy spectrum of closed strings there is a massless mode which can be identified with the graviton, i.e. the spin quantum mediating gravitational interactions. Since the Dbranes are objects possessing a rest intrinsic energy, they interact gravitationally and it is therefore straightforward to expect that the Dbranes can source and absorb closed strings.
The relation with both open and closed strings puts the Dbranes in a central position for the developments we are going to analyze throughout the thesis. Depicting a naive image which will be clarified and made precise in the following treatment, the open strings attached to a Dbrane are described at lowenergy by means of a gauge theory whose base manifold coincides with the dimensional hypervolume spanned by the Dbrane in its timeevolution^{2}^{2}2Henceforth referred to as the dimensional worldvolume of the Dbrane.. Instead, the closed strings propagating at lowenergy in the whole ambient spacetime containing the Dbranes are described at an effective level by supergravity or even classical gravity models with extended solutions.
If we concentrate on the physics of closed strings in the proximity of a Dbrane, we can wonder if the closedstring/gravitational behavior could account as well for the whole dynamics of the brane itself. At a sketchy and lowenergy level, we could hope that looking at the local spacetime deformations induced by the presence of a Dbrane we could recover full information on the Dbrane dynamics. Since, as we have just mentioned, the brane dynamics is already encoded in the physics of the open string modes attached to it, we are here speculating about an open/closed string connection relying crucially on the Dbrane physics. Such idea lies at the heart of the /CFT correspondence and holographic models in general.
We can also consider a different aspect of Dbrane dynamics which leads us to the stringy instanton calculus. Since, in appropriate lowenergy regimes, the open strings “living on a brane” are welldescribed by a quantum supersymmetric gauge field theory, it is natural to ask whether such effective field theory emerges with all its perturbative and nonperturbative content. The proper answer is that from the analysis of Dbrane models we do not only recover all the standard perturbative and nonperturbative features of the lowenergy quantum field theory but, in addition, the string framework makes it possible to study some significant generalized configurations. Such generalizations are outside of the reach of an IR (i.e. lowenergy) purely field theoretical approach and we henceforth refer to them as stringy or exotic. In particular, our focus is on the stringy instantonic configurations and the modifications that they induce in the couplings of the lowenergy effective field theory. As we will see, although the origin of the exotic effects is intrinsically related to the stringy nature of the model, they do produce important modifications to the lowenergy quantum field theory.
1.1 Historical and “philosophical” note
Historically, the string formalism has been firstly introduced to describe strong interactions, specifically as a model for meson scattering^{3}^{3}3Two synthetic historical accounts about the early steps of string models can be found in the introductory chapters of [1] and [2].. The string description of meson scattering is particularly suitable to account for the  duality of meson scattering where the two Mandelstam channels presenting the same amplitude value are naturally encoded in a single open string amplitude. Moreover, also the proliferation of mesons is describable in terms of spinning string states possessing a precise interconnection between the rest energy and the spin ,
(1.1) 
where represents a constant shift while is the celebrated Regge slope. From relation (1.1) we can observe that is inversely related to the string tension; indeed, keeping fixed the energy we have greater angular momentum if is increased. We can then expect that, increasing , we increase the length of the spinning string and then, at fixed energy, this is equivalent to reducing its tension^{4}^{4}4The tension corresponds to the energy density on the worldsheet spanned by the string: as such it is related to the “linear energy density” along the string. Here “increase the length of the string” means “let the length increase”; since we keep the energy fixed the increase in length does not correspond to a mechanical stretch of the string (i.e. there is no “work” done on the string)..
In the context of the strong interaction, the later introduction of a renormalizable quantum field theory, namely the QCD, overcame the string description, at least in the highenergy perturbative regime. Before long, the recognition in the closed string spectrum of a spin gravitonlike particle interacting democratically^{5}^{5}5See Appendix A for a brief argument. with all the objects possessing mass gave a tremendous new momentum to the theoretical research in this field. Indeed, this discovery opened the doors to a completely new description of gravity hopefully admitting a consistent quantum treatment.
Another greatly interesting feature of the string formalism is the possibility of comprehending various, and maybe all, kinds of fundamental interactions in one single theory. This aspect of unification of all fundamental interactions produced widespread interest and even radical enthusiasm leading some people to call string theory the “theory of everything”. Without spending much time to discuss this point, it is generally possible to assume a different and more moderate perspective. As the frequent use of the name “string formalism” instead of “string theory” suggests, we want to adopt an instrumental attitude towards the stringy mathematical tools and techniques; these are extremely fertile and insightful in describing many important physical systems. In the part of the present thesis regarding the holographic approaches to study strongly coupled systems, in some cases we will even adopt a phenomenological effective approach, meaning that the string inspired models we study describe macroscopic physical properties without the pretension of accounting precisely for the microscopic features of the system under analysis. Already at the bottomup level we will observe how insightful a string inspired model can be in shedding light on the strongly coupled dynamics of quantum field theories. The moderate attitude does not deny the conceptual and philosophical fascination of aiming to a unified theory of everything, it simply focuses on the operative purpose of studying and exploiting the string formalism as deeply as possible before, or even independently, of how the dispute on whether we deal with the theory of everything or not will be settled.
1.2 Purpose and Original Content
This thesis is aimed at producing a text which could result as clear as possible also to a partial or even a “localized” reading. The body of the text is indeed divided in many paragraphs containing various footnotes and references both to the numerous appendices and to papers in the literature; this fragmentation is deliberate considering that an exiting Ph.D. student feels the moral duty to be as useful as possible to the doctorate students following him.
The treated subjects require quite massive introductions. At the outset we underline that the original content of the thesis is contained mainly in the chapters:
1.3 Disclaimer
The thesis was publicly defended on March the 16th 2012. All the content and in particular the bibliographic references are referred to that date. In the meantime, there have been further developments in the field which do not appear here.
Chapter 2 Introduction
2.1 Strings, Branes and Gauge Theories
When regarding the string formalism as a candidate description of the fundamental interactions, various important questions arise. One among the most significant points consists in how the physics that we have already experimentally tested could admit a stringy description, or rather, how can it be embedded in a string model. Specifically, we are interested in the way in which General Relativity and the quantum field theory describing the electroweak and strong interactions (i.e. the Standard Model) can appear in the string context.
Let us remind ourselves that the characteristic energy scale of strings is Planck’s scale () which is much higher than all the scales directly probed in human particle physics experiments so far^{1}^{1}1This is strictly true when the string model under consideration does not include compactified directions. The compactification scale can “lower” the string scale (i.e. the scale at which stringy effects have to be taken into account) many orders of magnitude below Plank’s scale; as it is natural to expect thinking of KaluzaKlein modes, the larger the compact directions, the lower the string scale. For details see for instance [3].. This can be naively thought of as a consequence of the fact that the string degrees of freedom appear as a quantum description (among other things) of gravity. The string scale has therefore a close relationship with the scale at which gravity becomes sensitive to quantum corrections, i.e. Planck’s scale. The physical theories in which we are confident, i.e. the theories that passed many experimental tests, must therefore emerge in the lowenergy (with respect to Planck’s scale) regime of string theory. The string formalism aims to furnish the unifying UV completion of General Relativity and the Standard Model, hence the lowenergy limit of string theory is required to reproduce them, both at the perturbative and the nonperturbative levels.
Studying the lowenergy limit of a string model is equivalent to analyzing it at an energy level which is small with respect to the characteristic energy needed to excite the strings. The string excitation energy is measured by the tension, therefore the lowenergy limit can be taken considering the infinite tension limit. Indeed, the string tension appears as an overall factor which scales the string action; in a wouldbe stringfieldtheory^{2}^{2}2With stringfieldtheory is usually meant the secondquantized formulation of a string model. Note that the existence of such a formulation is still an open question in all the string models studied so far, including the bosonic string. in which the exponential of minus the action weights the probability amplitude of a possible evolution (i.e. the amplitude associated to a path in a path integral formulation), the string excitation “cost” scales as the action and then according to the tension .
As already mentioned, the string tension is expressed in terms of the dimensionful constant ; in natural units , we have:
(2.1) 
Moreover, it is straightforward to define a characteristic string length . Actually, the worldsheet is the twodimensional surface spanned by the string in its evolution and then the tension has the dimensions of an inverse area, that is an inverse squared length. In natural units, we define:
(2.2) 
Notice that is not to be thought of as the string length tout court. In fact, a worldsheet amplitude can be regarded as describing different classical string propagations. Consider for instance a rectangular plane worldsheet much longer than wide. It can either represent the propagation of a long string on a short path or the propagation of a short string on a long path. Since, as we will see shortly, the action measures the worldsheet proper area, in this simple rectangular case, the characteristic length we defined is related to the geometric mean of the two sides of the rectangular worldsheet.
The lowenergy regime of a string model as the infinite stringtension limit implies, through (2.2), that the original extended strings whose tension diverges become effectively pointlike, . It is pretty reasonable that the physics of relativistic pointlike objects can be described with quantum field theory; this is indeed the case and the naive expectation can be precisely tested.
In the perturbative analysis of string dynamics one considers the scattering amplitudes of the lowest lying string vibrational modes, namely the massless ones. Actually, the massive modes correspond to excited and so more energetic vibrational modes of the strings which are then strongly suppressed in the lowenergy limit. Within the string formalism, the lowenergy limit is achieved performing the computations at finite and then considering the results in the infinite tension limit . The results obtained in this fashion are to be compared to the scattering amplitudes computed for the corresponding^{3}^{3}3The correspondence is set by the identification of particles and string vibrational modes presenting the same quantum numbers. massless particles in quantum field theory. The string computations yield, in general, the field theoretical results with additional corrections; these corrections are weighted by positive powers of the string constant . Such corrections vanish in the limit. At the perturbative level, the field theory for the set of massless particles corresponding to the massless string modes, can be regarded as an effective lowenergy description of the corresponding string model.
Let us show an explicit example, namely the scattering amplitude of three nonAbelian massless vectors^{4}^{4}4I.e. massless vectors corresponding to gauge bosons of a nonAbelian gauge theory.. As shown in detailed in e.g. [4], the string computation of this amplitude returns:
(2.3) 
where and are respectively the momenta and polarization vectors of the three vector modes labeled by the index . For the sake of compactness, we have dropped the color structure and the corresponding factor from the amplitude (2.3); we also defined . Equation (2.3) coincides with the YangMills theory result for three gauge bosons up to first order in the momenta. It is evident that in the infinitetension limit the additional stringy contribution vanishes and we recover precisely the field theoretical result.
The complete perturbative test of the consistency of the lowenergy field theory description for a string model is a wide topic. Furthermore, one can start considering many different string models and ask whether some features of field theories such as masses or potentials can arise in the lowenergy regime of appropriate string configurations. Leaving this important aspect somewhat aside, our interest would be instead directed towards the nonperturbative side of the story.
Gauge theories and their supersymmetric versions have a rich vacuum structure. We can study perturbatively these theories around different classical field configurations that, at least locally, minimize the action. The perturbative agreement between string formalism and its lowenergy field theory description holds also around a nontrivial vacuum. Of course, this can be tested explicitly and, even more importantly, we have to understand how the nontrivial background itself could arise from a string model.
This relevant question has been first addressed as soon as the Dbranes were discovered. Actually, Dbranes can be seen as nonperturbative objects in string theory, meaning that they have a solitonic nature in the string context. A nice feature of these extended objects is that, even though they have a nonperturbative origin, their dynamics is describable in terms of perturbative modes as we will see accurately in Section 2.2. One can expect this observing, for instance, that it is perfectly legitimate to consider small oscillations, i.e. fluctuations, of a field around a nontrivial vacuum in which the field itself assumes a big VEV. The perturbative picture of lowenergy Dbrane dynamics is realized by open strings whose endpoints live on the Dbranes and by closed string modes emitted and absorbed by the Dbranes themselves.
One could ask why the membranes such Dbranes do not shrink to pointlike objects themselves in the limit. The answer is related to topology because Dbranes are solitonic objects representing topologically nontrivial ground states of the string theory. The lowenergy limit of a topologicallynon trivial sector is the topologically nontrivial vacuum; this vacuum configuration can contain extended objects or extended field configurations. In the lowenergy picture, the strings describe the small fluctuations around the topologically nontrivial vacuum represented by the Dbranes at rest. Hence we can guess that their “solitonic character” would not be spoiled by the limit, and “naively” the expectation is true. In other terms, the Dbranes are topologically protected. To be slightly more precise, we must say that Dbrane models, containing the appropriate kinds and number of Dbranes, give indeed rise to setups whose lowenergy dynamics could be encoded in a supersymmetric gauge theory with all its perturbative and nonperturbative features.
2.2 Perturbative description of the Dbranes
In the present section we indulge on how supergravity and gauge theories emerge in the lowenergy description of the closed and open string sector respectively in the presence of Dbranes. As a necessary introductory step, we must concentrate first on the quantum description of string dynamics. Later we will focus on the lowenergy limit.
2.2.1 Quantum treatment of strings
From a classical point of view, a propagating string sweeps the twodimensional worldsheet embedded into spacetime. In order to associate an action to a particular string evolution, we generalize what is standard in particle dynamics. Actually, considering a relativistic particle, we associate to its propagation an action that measures the proper length of the worldline representing the particle evolution in spacetime. The proper length is invariant with respect to reparametrizations of the worldline, in accordance with the relativistic coordinate invariance requirement. Another important characteristic of the proper length is its additivity: namely, the action of the composition of two wordlines sharing an endpoint is given by the sum of the values of the action associated to the component paths.
Inspired by the classical relativistic particle, we are straightforwardly led to think that the classical evolution of the string is encoded in the worldsheet with minimal proper area, being this expressed by the following action
(2.4) 
This functional is usually referred to as the NambuGoto action. As usual, we indicated the tension with , the worldsheet with , the two worldsheet coordinates with where , the spacetime coordinates with , the spacetime metric with and the corresponding induced worldsheet metric with . Notice that the variational study of (2.4) has to be performed choosing suitable boundary conditions.
The relativistic string is endowed with a new crucial feature with respect to the relativistic particle: the presence of “internal” freedom. Actually, a particle is just a pointlike object without internal characteristics whereas a string can oscillate. Since the string length is usually of the order of Plank’s length,
(2.5) 
its oscillations are clearly a quantum effect. The study of the internal modes of the string brings us to the question of the string quantization.
The detail of string quantization is far beyond the purpose of this introductory part; for a thorough treatment of the topic we refer to the literature (see for instance [1, 4]). The NambuGoto action (2.4) is not suitable for a quantum treatment because the presence of the square root in the integrand makes the quantization process cumbersome. Indeed, for the sake of treating the string at the quantum level one actually considers the action
(2.6) 
that is classically equivalent to the NambuGoto action where is the metric on the worldsheet; notice that is here regarded as a dynamical field. The action (2.6) is usually referred to as the BrinkDi VecchiaHoweDeserZuminoPolyakov action and putting onshell we recover (2.4) (see for instance [4]). Choosing appropriate boundary conditions, one studies the equations of motion descending from the variational analysis of the action. The oscillatory modes of such string solutions describe the profile in spacetime of the string itself. More precisely, any point of the string is mapped into spacetime by the socalled embedding functions . Notice that we are actually embedding the worldsheet spanned by the two ’s into spacetime . These embedding functions can be regarded as a collection of scalar fields living on the worldsheet whose Fourier modes are promoted to operators in a worldsheetFock space. The spacetime quantum dynamics of the string is encoded in the quantum field theory defined on the worldsheet. This crucial point is both natural and surprising. Its naturalness descends from the fact that we are actually generalizing straightforwardly the approach which is standard for relativistic particles; its novelty originates from the fact that scattering amplitudes for string processes in spacetime are obtained computing matrix elements of the quantum field theory living on the worldsheet.
2.2.2 Supersymmetry and superstrings
So far we have considered only bosonic string modes. A crucial ingredient in developing string theory and defining the Dbranes is represented by supersymmetry. It relates bosonic and fermionic degrees of freedom and it can be thought of as an extension of the standard Poincaré invariance^{5}^{5}5Supersymmetry can be expressed as a generalized Poincaré invariance on an extended spacetime comprehending also fermionic (i.e. Grassmann) directions.. In a supersymmetric theory any bosonic mode has a corresponding fermionic partner. To promote the bosonic string model (2.6) to a superstring (i.e. supersymmetric string) model one possibility is to introduce supersymmetry in the worldsheet theory. At the level of the action we add to (2.6) the fermion term
(2.7) 
where the are a collection of Majorana spinors where is the ambient spacetime dimensionality. The matrices satisfy the bidimensional worldsheet Clifford algebra.
Once that the worldsheet model is supersymmetric, in order to obtain a supersymmetric string theory also from the ambient spacetime viewpoint a careful analysis is required. At first, anomaly cancellation implies that a superstring theory can only be consistent for . In addition, Gliozzi, Sherk and Olive found a way of projecting the superstring spectrum to render it actually supersymmetric^{6}^{6}6The GSO projection can be regarded as a consequence of oneloop modular invariance requirement, see for instance [5].. Depending on the relative chirality choice between left and right fermion modes on the closed strings^{7}^{7}7Strings can be open ore closed. Closedstring vibrations admit both “clockwise” and “counterclockwise” running wave solutions around the string., we have two possible GSO projections leading to two consistent string theories usually referred to as Type IIA and Type IIB.
2.2.3 String scattering amplitudes and vertex operators
The building blocks of the perturbative analysis are the string scattering amplitudes. These are classified in accordance with the topology of the worldsheet ; indeed, the number of “handles” of a worldsheet topology generalizes the number of loops of a Feynman diagram in particle theories.
The asymptotic string states participating in a scattering process, are encoded in localized operators (the socalled vertex operators) defined on the worldsheet. Any external or asymptotic string state can be associated to a puncture on the worldsheet; the latter is then a bidimensional punctured Riemann surface. The punctures (or vertices), where we insert the vertex operators, correspond to the external legs of particle diagrams.
Since in a string diagram the punctures are associated to the emission/absorption of a state in the string spectrum, it is quite natural to expect that, in accordance with the viewpoint of the secondquantized field theory living on the worldsheet, they are represented by operators. Without entering into further detail, this is indeed the case^{8}^{8}8We recommend to look at [1] for a deeper analysis.. The vertex operators carry the quantum numbers of the string state they create/annihilate. Let us underline that the secondquantized treatment of the worldsheet field theory corresponds to a firstquantized picture of string in spacetime. More precisely, in studying string scattering at the firstquantized level, the fundamental object is the worldsheet, i.e. the string trajectory, which is specified a priori. In this sense, we consider string fluctuations propagating on a given worldsheet “background”. In a secondquantized picture for the strings, the worldsheet would be dynamically determined and the fundamental object would be the stringFock space.
A scattering amplitude represents a matrix element between asymptotic states which, by definition, involve a string propagation for infinite time. To be neat, think about the point (i.e. propagator) amplitude for a closed string. It is obviously described by a cylindrical worldsheet extending from negative to positive infinite time. Exploiting the invariances of the theory^{9}^{9}9Namely the reparametrization and Weyl freedom, see for instance [1]. it is possible to map this infinite cylinder to a sphere with two punctures. The topology of the cylinder and the topology of the sphere with two punctures are the same. In a similar fashion, it is possible to map any “loop” and “particle” scattering diagram on a sphere or on an torus with the appropriate number of handles and the appropriate number of punctures.
Proceeding in analogy with the closed string case, the free open string point function is associated to a rectangular worldsheet extending from minus to plus infinite time; this presents the same topology of a disk. The asymptotic open string states are again represented by vertex operators, in this case they are localized on points belonging to the disk boundary.
Given a certain topology for the worldsheet , we can compute the associated point string amplitude evaluating the vacuum expectation value of the vertex operators in the framework of the conformal field theory^{10}^{10}10The tracelessness of the energymomentum tensor together with the fact that the base manifold (i.e. the worldsheet) is twodimensional implies that the Poincaré and Weyl invariance of the worldsheet field theory is promoted to full conformal invariance, look for instance [4]. living on the worldsheet,
(2.8) 
The integral could be in general quite complicated. As it will be useful in what follows, a vertex operator associated to a generic mode can be split in its operator part and the polarization part (again indicated with ), namely
(2.9) 
2.2.4 Disk and sphere diagrams in the presence of Dbranes
The tree level propagation of closed and open strings is encoded by worldsheets having the sphere and disk topology respectively. They in fact correspond to Feynman diagrams with no handles, i.e. no loops.
Computing explicitly the vacuum expectation value of a generic closed string vertex operator on the sphere we obtain zero,
(2.10) 
meaning that there is no tadpole amplitude associated to any closedstring mode . Analogously, for the generic open string mode , we can compute directly by means of conformal theory methods the vacuum expectation value of a single vertex operator on a disk. Again we obtain zero,
(2.11) 
We interpret these zero results as the absence of tadpoles for both open and closed string modes. This picture matches the idea of a trivial vacuum in which all the fields have vanishing VEV. To rephrase the point, we obtained that in a model possessing just open and closed string, the lowest scattering topologies describe the perturbative physics around the trivial vacuum.
The next step consists in introducing new actors on the stage. The new objects we consider are the Dbranes. Assuming the viewpoint of the strings, the presence of the Dbranes implies the possibility of having worldsheets with new characteristics. In addition to the sphere and the disk amplitudes already considered without the Dbranes, we can now have diagrams with different boundary conditions. For the sake of simplicity let us start introducing a single Dbrane. Due to the presence of the brane, it is possible to consider the insertion of a boundary into the closed string sphere diagram (think about a soap bubble on the surface of the sink); in other terms closed string diagrams with the topology of the disks. Such a topology accounts for the possibility of having an emission/absorption of closed strings by the Dbrane. On the boundary just introduced the brane induces boundary identifications between right and leftmoving closed string modes, [6]. As a consequence, we can have nonnull expectations even for a diagram with a single close vertex insertion,
(2.12) 
The open string counterpart consists instead in the possibility of having different boundary conditions at the two endpoints of the strings. In a configuration with two different Dbranes (for instance a D and a D with ) we can have strings stretching between them. The propagation of such a string stretching between two different kinds of Dbranes is computationally described introducing appropriate operators on the boundary of the disk diagrams. These operators are called boundary changing operators.
The expectation of a single vertex operator on a disk containing also boundary changing operators can be nonvanishing,
(2.13) 
For a pictorial explicit example corresponding to the DD case see Figure2.3.
The string scattering computations are performed in the framework of the worldsheet conformal field theory; indeed scattering amplitudes are obtained studying expectation values on the worldsheet; to have more details on the scattering computations in the presence of Dbranes see [7, 8, 9] and references therein. A selfcontained account of the actual techniques and conformal computations is beyond the purpose of the present treatment; we refer the reader to the review [10] for a throughout analysis.
2.2.5 Effective supersymmetric gauge theory on the Dbrane worldvolume
We already faced the question of defining an action on the worldsheet which is a twodimensional surface embedded into tendimensional spacetime. In order to specify an action for a Dbrane we generalize the same approach to dimensional hypersurfaces. These represent the worldvolume of the branes and on them we have bosonic fields each associated to a spacetime coordinate; we can always choose (at least locally) a convenient coordinatization (usually said “well adapted”) in which the first spacetime coordinates parametrize the Dbrane worldvolume. The remaining spacetime coordinates span the so called transverse space.
Intuitively the fields associated to the transverse coordinates represent the oscillations of the brane itself in the surrounding spacetime, while the longitudinal field can be organized in a dimensional array with . Notice that a brane breaks the entire tendimensional Poincaré invariance preserving a Poincaré subinvariance corresponding to its worldvolume. The array behaves indeed as a vector of the preserved Poincaré invariance and, being a massless mode, it is straightforwardly interpreted as a U gauge field.
Having observed the presence of a U gauge vector we can naturally accept that the bosonic part of the Dbrane action assumes the following form:
(2.14) 
where . This is the renown DiracBornInfeld action^{11}^{11}11Historically the DiracBornInfeld action was studied in the attempt of clarifying the problem of infinite Coulomb energy of pointlike particles like electrons. In the DBI theory there is an infinite tail of nonlinear terms in generalizing the usual Maxwell electrodynamics. As a result, the pointlike particles presents finite fieldstrength and finite total energy; the fields are however not smooth. (DBI). The metric is induced on the Dbrane worldvolume by the ambient metric, while is the fieldstrength associated to . The field , called dilaton, is a scalar mode emerging in the closed string spectrum and it will be introduced in the Subsection 2.2.6.
Let us observe that in the absence of the fieldstrength, (2.14) returns simply the NambuGoto action (2.4) generalized to the dimensional brane. The introduction of the DBI dependence on is naturally understandable in the framework of Tduality arguments (see for instance [11]). In the lowenergy theory we can expand the DBI action in powers of the fieldstrength and retain only the lowest order, namely the one quadratic in ; in this way we obtain the pure Maxwell Lagrangian. Similarly, the supersymmetric extension of the DBI action, at quadratic order in the derivatives, returns the SuperMaxwell gauge field theory Lagrangian. We remind the reader that the expansion of the DBI action contains kinetic and interaction terms that can be computed from the study of string scattering.
Let us concentrate on more than a single brane at a time and, more specifically, let us consider a stack of coinciding (i.e. on top of each others) Dbranes all of the same kind^{12}^{12}12The branes in the stack have the same geometrical arrangement and symmetry properties with respect to background operators as orbifolds or orientifolds (see Section 4.1).. In such a setup each open string can start and end on any brane in the stack. To account for this possibility, we can associate a label to each Dbrane so that any open string state is characterized by a couple of labels expressing the additional information concerning the branes to which it is attached. These are called ChanPaton indexes.
As observed in [6], it is possible to consider additional nondynamical degrees of freedom to the endpoints of open strings. Indeed, such an addition respects the symmetry of the theory, namely the spacetime Poincaré invariance of the D worldvolume and the worldsheet conformal invariance. Given their nondynamical character, the ChanPaton indexes can be regarded as mere labels which are preserved in free string evolution.
The ChanPaton labels running over the values have to be introduced in the space of asymptotic string states. Any state has therefore the following form:
(2.15) 
where represents the center of mass momentum of the string. Note that, before the introduction of the ChanPaton labels, the momentum furnished a complete set of quantum numbers for defining an open string state.
We can use a different representation and consider a basis for the space of ChanPaton matrices in indexes; runs over . The relation with the old basis is:
(2.16) 
For the sake of clarity, let us consider an explicit example: the strings scattering amplitude. The corresponding diagram has four external “legs” that are associated to vertex operators. Since we have introduced the ChanPaton indexes to label the states, also the vertex operators have to carry the ChanPaton structure. They therefore contain as a factor a matrix expressible in the basis. The ChanPaton factors are nondynamical and as a consequence they have to be conserved along the worldsheet boundary comprehended between two vertex operators.
The ChanPaton factor is going to describe the gauge structure of the lowenergy effective model; let us consider this point carefully. In general, our interest is concentrated on gaugeinvariant quantities, i.e. objects whose “gauge indexes” are saturated. Let us consider string amplitudes summed over all the possible ChanPaton configurations. We indicate with the ChanPaton factors corresponding to the four asymptotic states in Figure 2.4. The amplitude we want to compute results from summing all the adjacent ChanPaton indexes, leading to an overall factor
(2.17) 
Notice the important fact that the factor (2.17) is manifestly invariant with respect to the following transformation:
(2.18) 
where . We have to remember that a matrix transforms a quantum state labeled with into a quantum state labeled with ; as a quantum transformation, it is required to be unitary and we have to limit our attention to U,
(2.19) 
It is natural to interpret the two indexes of a ChanPaton matrix as belonging respectively to the fundamental and antifundamental representation of U. The lambda’s transform then in the representation that is actually the adjoint representation of U. One of the lowenergy open string excitation modes is a massless vector that, as just stated, transforms in the adjoint representation of U. In the lowenergy effective field theory, this mode plays the role of the gauge field. To realize this, one has to analyze the string scattering and check that at lowenergy the string amplitudes involving gauge vectors coincide with the amplitudes obtained by the standard effective field Lagrangian , being the nonAbelian fieldstrength.
For a stack of coinciding branes the generalized version of (2.14) is not know in a closed form. An expansion for the nonAbelian generalization of DBI action can be in principle obtained (with such an effort that usually only the first terms are computable) following specific requirements or prescriptions such as offshell supersymmetry, [12]. The choice is not unique and the literature presents several possibilities which, however, lead all to classically equivalent results (i.e. upon using the equations of motion). Our particular preference for the offshell supersymmetry requirement has a profound motivation in relation to instanton solutions; indeed, in this framework, the instantons are believed to represent solutions of the complete quantum theory and not only of its firstterms approximation.
Up to quadratic order, all the nonAbelian versions of the DBI expansion coincide with SYM theory and, specifically for the case of Dbranes, we have fourdimensional SYM theory. The gauge group is U but the U part (associated to the trace) constitutes an infraredfree Abelian subsector; at lowenergy scales this Abelian part decouples from the remaining SU part because the former becomes negligible with respect to the running nonAbelian coupling. Henceforth we will simply understand this caveat, and indicate the Dbrane stack as simply supporting an SU gauge theory. Let us write the explicit SYM action:
(2.20) 
The index labels the scalars and, in the stringy picture, is associated to the internal space directions; the index is the spinorial counterpart of so it runs on the internal space spinor components. The action (2.20) can be recovered from a systematic study of the lowenergy dynamics as performed in detail in [9].
2.2.6 Effective supergravity in the bulk
Studying the lowenergy, closed superstring spectrum we find a set of massless modes including a complex scalar called dilaton, the already mentioned spin graviton, and some totally antisymmetric fields referred to as RamondRamond forms^{13}^{13}13We do not enter into much detail here since the subject is described in any introductory string theory book. Let us mention that we have a different set of RamondRamond forms depending on the kind of superstring model we are considering, namely Type IIA or Type IIB. We will consider Type IIB whose spectrum contains all the forms where is even. We will then have (i.e. with indexes) which naturally couples with the Dbranes that are the central object for the subjects presented in the thesis.. Repeating somehow the approach we followed with open strings, we can study the scattering of lowenergy closed strings and account for their propagation and interactions by means of an effective field theory. Such an effective field theory for closedstring, massless modes is called supergravity.
In general superstring models and their effective supergravity descriptions admit extended solutions. Among these we find the Dbranes which are dimensional spatial surfaces. The worldvolume of a brane has dimensions including time and then it naturally couples to the RamondRamond field with indices. Indeed, we can regard the branes as generalizing the relativistic particle electrodynamics; there we have a zerodimensional object, i.e. the charged particle, spanning in its evolution a onedimensional manifold, the worldline. A vector field with one spacetime index couples with the particle because the worldline has a onedimensional tangent space in any of its points. The electromagnetic coupling of a charged particle is given by the following term in the action:
(2.21) 
where is the worldline, is the charge and is the tangent vector to the worldline. Let us generalize this to a Dbrane, for example for ,
(2.22) 
The integration is performed on the fourdimensional Dbrane worldvolume. The Dbrane then can emit and absorb quanta. In a supergravity picture, the presence of such a Dbrane translates to the possibility of having a source for the generalized gauge field. From now on, the coupling constant will be fixed to .
Without entering into details, let us give the relevant terms in the supergravity action (in the string frame) that are needed to describe at lowenergy the dynamics of the ambient spacetime in the presence of a stack of coinciding Dbranes:
(2.23) 
where is the spacetime metric and the associated Ricci scalar, is the dilaton and is the selfdual part of the fieldstrength associated to the RamondRamond potential with respect to which the Dbranes are charged. The indicate the omission of the other RamondRamond forms and of the form and also of the fermionic terms^{14}^{14}14It is appropriate to remind ourselves that supergravity models generalize supersymmetric ones making the supersymmetry local. In supergravity we then have again a fermionic partner for any boson of the theory..
It is possible to show that the equations of motion deriving from (2.23) admit the following solution:
(2.24)  
(2.25)  
(2.26) 
where the coordinates with are longitudinal while the with are transverse with respect to the stack of Dbranes. The coordinate is the hyperspherical radius in the transverse space where the solution (2.24) is spherically symmetric. Finally, is the string coupling constant and is the following harmonic function:
(2.27) 
being the number of branes in the stack. This solution describes a stack of parallel and coinciding branes^{15}^{15}15The case of parallel but non coinciding branes is described by the following harmonic function:
From (2.24) and (2.27) it is possible to see that computing the flux of the RR field through a hypersphere containing the branes (i.e. a hypersphere in the transverse space) we obtain
(2.29) 
that is the number of the branes (remember that we have fixed the brane coupling constant ).
The SYM theory living on the Dbranes is coupled with the dynamics of the bulk fields living in the ambient spacetime. The coupling is related to the string constant ; once we take , it is possible to consider the following expansion
(2.30) 
and effectively regard the two theories on the worldvolume and in the bulk as independent; this is usually referred as decoupling limit. As explained for instance in [13], the decoupling limit should be more precisely defined in order to maintain the physical interesting quantities finite (e.g. the “Higgs mass” of a string stretching between two separated branes, namely ). Specifically, the decoupling or Maldacena limit is given by
(2.31) 
Let us rewrite the metric (2.24) using hyperspherical coordinates in the space orthogonal to the Dbranes,
(2.32) 
where represents the elemental solid angle; in the decoupling limit (2.31) and using (2.27) we have
(2.33) 
We can define and substitute it into (2.33) obtaining:
(2.34) 
where we have used (2.31) and . Notice that contains and therefore vanishes in the Maldacena limit. We ought to define . Substituting in the metric (2.34) resulting from the Maldacena (or decoupling) limit, we obtain
(2.35) 
this metric is manifestly the product of two Einstein spaces of constant curvature: . Notice that both spaces are characterized by the same curvature radius, namely (in string units). In addition, we should note that the decoupling limit works for any value of and .
2.3 Holography and the /CFT correspondence
The /CFT correspondence is a conjectured duality between a specific gravity theory defined on an Antide Sitter space () and a corresponding conformal field theory (CFT) that can be thought of as living on the “conformal boundary” (see Section 6.1.1) of the spacetime.
The word holography comes from the combination of the two ancient Greek terms: holos meaning “whole” and grafé meaning “writing” or “painting”; it is usually referred to optical techniques which are able to reconstruct the whole threedimensional information of an image by means of a twodimensional support. The term has been adopted in the gauge/gravity correspondences framework because the gauge and gravity theories related by the correspondence are defined on spacetimes with different dimensionality. In its stronger sense, a duality is a map between two theories describing the same physics and therefore “containing” the same information; the content of the theory is just written in different terms, i.e. using different degrees of freedom. In this sense, the gauge/gravity dualities are holographic because they show that the content of the theory living in a higher dimensional spacetime is encoded holographically in the lower dimensional theory.
The first holographic hint in theoretical physics was suggested by black hole thermodynamics^{16}^{16}16The seminal papers in which the holographic principle has been proposed are [14] and [15].. The entropy of a black hole, which is related to the number of quantum states contained in the black hole volume, is proportional to the surface of the black hole horizon [16, 17]. A property of the bulk volume of a black hole is indeed related to the surface or boundary containing the same volume. Let us note that the holographic hint given by black holes comes from a context in which gravity needs to be treated at the quantum level.
At the outset of describing holography and AdS/CFT it should be understood that the topic is very wide and we must often refer to the numerous reviews present in the literature. In particular, for an introductory treatment of some fundamental ingredients such as conformal field theory and gravity we refer especially to [18, 13] and references therein.
2.3.1 String/field connections
As we have mentioned, the dynamics of Dbrane models can be described in the lowenergy regime with appropriate gauge field theories. In the lowenergy and infinite tension (or zero length limit) for the strings, the open strings themselves become effectively pointlike objects accountable for within a quantum field theory living on the worldvolume of the branes. However, this direct connection is not the only link between field and string theory. Indeed, in hindsight, we can observe that string theory was originally developed in the context of strongly coupled hadronic interactions, i.e. a context which should be also describable with a strongly coupled quantum field theory in the confining regime^{17}^{17}17Notice that sometimes this early string models are called dual models. Here the term “dual” refers to a property of hadronic scattering consisting in the equality of hadronic scatterings in the and channels for small values of and ( and are Mandelstam variables), [1]. Note that the worldsheet crossing symmetry can be seen as a first hint of open/closed string duality.. Actually there are many stringlike objects involved in this context like flux tubes and Wilson lines. Flux tubes give an effective description of the interaction between two quarks and their behavior resembles the dynamics of strings. Think for instance of the baglike potential for quarks that is related to the area spanned by the flux tube in spacetime evolution of the quark pair, [19]; this is analogous to NambuGoto action for a string propagation where the action actually measures the proper area of the worldsheet.
Even though, in the context of strong interactions, string models have been superseded by QCD, we are generally not able to employ analytical tools for the analysis of its strongly coupled and confining regime. Before the employment of /CFT inspired techniques, the main theoretical instrument to investigate the strongly coupled regime of QCD and more in general nonAbelian gauge theories was provided by numerical simulations on the lattice and effective models (such as NJL, models,…).
The intimate relation between field and string theory has been significantly boosted in the 90’s after the proposal of the /CFT Maldacena’s conjecture [20]. In its strongest version, the conjecture claims a complete equivalence between Type IIB string theory compactified on an asymptotically background and SYM (Super Yang Mills) theory living in fourdimensional flat spacetime. With we indicate the manifold obtained from the Cartesian product of fivedimensional Antide Sitter spacetime and the dimensional hypersphere. Notice that the duality relates a theory containing quantum gravity (among other interactions) to a gauge field theory without gravity. /CFT has stimulated much interest on the possibility of having gauge/gravity dualities and much effort has been tributed to this field in the last fifteen years.
One crucial point to be highlighted at the outset is that /CFT is a strong/weak duality, meaning that it relates the strongly coupled regime of one theory with the weakly regime of the other and vice versa. This feature, which renders extremely ambitious the task of finding a direct proof of the conjecture^{18}^{18}18To find a succinct account of /CFT tests look at [18]., is its most interesting practical characteristics. In fact, because of its strong/weak character the /CFT provides a powerful tool to obtain analytical results at strong coupling in field theory by means of string lowenergy and perturbative calculations. Such a possibility is particularly interesting because the theoretical methods to perform analytical computation at strong coupling in field theory are generically quite poor and, a part from numerical simulations on the lattice, the strong coupling regime has been quite often theoretically unaccessible.
2.3.2 ’t Hooft’s large limit
A very suggestive relation between nonAbelian gauge theory and string theory follows from an observation proposed by ’t Hooft in 1974. He noticed how a U YangMills theory in the large , i.e. large number of colors, admits a classification of Feynman’s diagrams according to their topological properties, [21].
Let us look to the large limit of U YangMills theory in more detail. Apparently the limit seems to lead to illdefined quantities; actually, instead of performing just the large limit, we have to act on the coupling constant as well. Consider for instance a selfenergy diagram for the gauge field that belongs to the adjoint representation of the gauge group; the gauge field is then a Hermitian matrix expressible on a basis of independent gluons. It is possible to show, [13], that the selfenergy for a gluon scales as and, in the large , limit it is then of order . Nevertheless, if we include in the analysis the coupling we notice that actually the selfenergy behavior is . It is then natural to consider the so called ’t Hooft limit, namely
(2.36) 
In this limit all the diagrams either remain finite or vanish.
In pure YangMills theory the only dimensional parameter is the QCD scale . Let us observe that in ’t Hooft’s limit the QCD scale remains constant, indeed the function equation (which defines^{19}^{19}19We can define as the scale at which runs to an infinite value. However, we ought to note that the function equation (or renormalization equation) (2.37) is derived in perturbation theory and then it is reliable only for small values of the coupling. When becomes large the perturbation scheme ceases to be justified. Keeping clear mind about this caveat, we can nevertheless retain the formal definition of the QCD scale , [22].) for pure SU YM theory is given by:
(2.37) 
where is a reference renormalization mass scale. The two sides of Eq.(2.37) scale in the same way in the ’t Hooft limit (2.36).
Following [13], in a nonAbelian YangMills theory it is possible to express the adjoint field with a doubleline notation essentially treating the adjoint representation in line with its bifundamental nature. Adopting such doubleline notation, it is possible to show that any Feynman’s diagram can be accommodated on a Riemann surface whose genus (i.e. the number of holes) is related to the features of the diagram itself, namely
(2.38) 
where corresponds to the number of loops (faces), is the number of vertices and is the number of propagators (edges).
Let us focus on the pure U gauge theory^{20}^{20}20We are concentrating just on the adjoint fields; the diagrams containing also fundamental fields result suppressed with respect to the leading contribution. An observation which is nevertheless interesting is that, in the presence of fundamental fields, the topological classifications of the diagrams has to involve also varieties with boundaries, [13].. Its Lagrangian density is
(2.39) 
where we have put in evidence the YangMills coupling constant. Any propagator is accompanied with a factor of and any interaction vertex is instead accompanied with a factor . Furthermore, any loop contributes is accompanied by a factor of ; the reason is that, since we are adopting the doubleline notation, we are considering loops associated to the fundamental representation which has indeed components. Collecting these observations in one formula, we have that the generic diagram scales as
(2.40) 
Any amplitude in the field theory can be expanded accordingly to the topology of the contributing Feynman’s diagrams,
(2.41) 
where the functions are polynomials in . In the ’t Hooft limit, the expansion is clearly dominated by the lowgenus configurations, and in particular by the planar graphs possessing the topology of a sphere. This expansion resembles precisely the topological expansion of perturbative multiloop string diagrams.
Part I Stringy Instanton Calculus
Chapter 3 Instanton Preliminaries
In this part of the thesis we devote keen attention to nonperturbative effects and particularly to instantons both in SUSY gauge field theories and in superstring theories.
The term “nonperturbative” refers to configurations whose action is proportional to a negative power of the coupling constant. In the partition function, any configuration is weighted by , therefore the nonperturbative effects are exponentially suppressed at small coupling. In this regime they are generally negligible with respect to any perturbative contribution whose weight vanishes instead as a positive power of the coupling constant. The nonperturbative physics is relevant either when we consider a strongly coupled regime or whenever the competing perturbative effects are absent. In relation to the latter case, some perturbative contributions can be forbidden, for instance, by nonrenormalization theorems induced by the supersymmetry of the theory.
In the s the theoretical physics community started to study systematically the nontrivial solutions of the classical field equations of motion of many field theories comprehending YangMills theory and its supersymmetric generalizations^{1}^{1}1As an aside curiosity, it is interesting to recall that far before the systematic study of nontrivial field solutions (or solitons) and even before the modern atomic theory was proposed, Kelvin suggested a model based on vortexes (that are a particular kind of solitons) in a fluid to represent atoms. In Kelvin’s picture, the chemical variety of atoms was explained in terms of different topological arrangements (i.e. different topological charges); the stability of atoms corresponded to the topological stability of solitons with respect to small fluctuations. As we will see, topological features are an essential property of solitons.. In the quantum field theory framework, a classical solution of the equations of motion represents a background around which the quantum fluctuations are studied. Notably, the nontrivial solutions usually mingle global and localized features. On the one side they are related to topological characteristics corresponding to global properties of the classical field configuration as a whole, on the other their energy density is nonvanishing on a finite support^{2}^{2}2It is intuitive to expect that a field configuration whose potential (i.e. static) energy density is nonvanishing everywhere has a diverging action. In a pathintegral (or partition function) formulation, a diverging action is translated into the total suppression of any amplitude involving such configuration.. Because of their localized character, the nontrivial solutions of the equations of motion are usually referred to as particlelike configurations or pseudoparticles^{3}^{3}3For an ample panoramic view on the topic of solitons and their particlelike behavior (e.g. in scattering phenomena), consult [23]. . They are nevertheless distinguished from the fundamental particle excitations arising from the perturbative quantization of the fields. In fact, as opposed to solitons, the perturbative quantum fluctuations around classical configurations emerge from the quantization of continuous deformations of the background field profile. As such they cannot, by definition, change the topology of the background itself. Indeed, the topological sectors in the field configuration space are closed (i.e. not connected with each other) with respect to continuous deformations of the fields.
Instantons constitute a prototypical example of totally localized nonperturbative field configurations of YangMills theory; the name is formed by the prefix “instant” suggesting localization also in the time direction, and the suffix “on”, usually attributed to particles. The first analysis of instantons dates back to and was performed by Belavin, Polyakov, Schwarz and Tyupkin in [24]. Notice that the localized nature also in the time direction makes it impossible to think of instantons as stable propagating particles.
3.1 Topological charge
Instantons are nontrivial classical solutions of the equations of motion of pure YangMills theory defined on fourdimensional Euclidean spacetime. They have finite action and enjoy the property of selfduality, i.e.
(3.1) 
where is the standard nonAbelian fieldstrength
(3.2) 
and is its Hodge dual,
(3.3) 
In YangMills theory the Hodge duality constitutes the nonAbelian generalization of the electromagnetic duality. We define the nonAbelian “electric” and “magnetic” fields as follows:
(3.4)  
(3.5) 
where are spatial indexes and represents the index associated to the gauge group generators which are traceless antiHermitian matrices satisfying
(3.6)  
(3.7) 
where are real structure constants. Remember that we are adopting an Euclidean metric; the upper or lower position of spacetime indexes is unimportant and the Hodge duality squares to . Euclidean electromagnetic duality transforms
(3.8)  
(3.9) 
This is in contrast with Minkowskian electromagnetic duality which introduces a minus in (3.9). Indeed, in Minkowski spacetime the Hodge dual squares to and it is impossible to define (nontrivial) selfdual or antiselfdual configurations.
Instantons correspond to Euclidean classical configurations locally minimizing the action and are therefore stable against field fluctuations. Instantons possess a finite but nonvanishing value for the action and are characterized by an integer number called topological charge or Pontryagin number or also winding number. It corresponds to the integral
(3.10) 
and its meaning will be clarified shortly. The Euclidean action of YangMills gauge theory is
(3.11) 
this, in the absence of sources, leads to the YangMills equations of motion,
(3.12) 
A field configuration that, like an instanton, has finite action must then correspond to a fieldstrength tending to zero faster than for large values of the fourdimensional Euclidean spacetime radius
(3.13) 
Explicitly, a finite value for the action requires
(3.14) 
with . In order to present such an asymptotically vanishing fieldstrength , the gauge field has to tend for large to a pure gauge plus terms vanishing faster than . A pure gauge configuration is a field configuration that can be obtained applying a gauge transformation to the trivial vacuum. Remember in fact that the fieldstrength is a gauge invariant quantity and its value on the trivial vacuum is zero. Mathematically, we then have:
(3.15) 
being a positive quantity and the matrix field representing a gauge transformation.
The integrand in the definition of the topological charge (3.10) can be expressed as a total derivative,
(3.16) 
where we have used the cyclic property of the trace to discard the term^{4}^{4}4Note that the tensor acquires a minus upon a cyclic permutation of its indexes. and the symmetry of . We apply Stoke’s theorem and compute via an integral on the ‘‘boundary’’ at infinite radius^{5}^{5}5Assuming that any field tends to a constant for , with boundary we mean a spherical shell with asymptotic radius.. From the asymptotical behavior of the fieldstrength (3.14), we have that the term in (3.16) containing is neglectable. Therefore the topological charge is given by
(3.17) 
where is the radial hypersurface element of the asymptotic threesphere. The function considered on the asymptotic defines a map from the boundary itself to the gauge group manifold. It is possible to show^{6}^{6}6Look at the appendices of [25] to have an explicit example in the case of SU gauge group. that the integer counts how many times the asymptotic “winds” around the gauge group manifold according to the map . This justifies the name “winding number” for .
Given the topological nature of the winding number , it cannot be affected by continuous deformations of the gauge field configuration. The possibility of obtaining a field configuration by continuously deforming a configuration can be regarded as an equivalence relation between and . In this framework, the gauge field configuration space splits into distinct equivalence classes (usually called topological sectors) associated to different values of the topological charge . Within a generic topological sector the action of any element of the sector has a value satisfying the inequality
(3.18) 
This is called BPS bound from the names of Bogomol’nyi, Prasad and Sommerfield who studied it for the first time. The BPS inequality (3.18) can be proven rewriting the action (3.11) as follows:
(3.19) 
Recalling (3.7), in the third passage of (3.19) we have discarded a positive quantity. Notice that in (3.18) the equality holds if and only if the field configuration corresponds to a fieldstrength that is either selfdual or antiselfdual, namely
(3.20) 
Moreover, the BPS argument implies that the configurations satisfying the selfduality condition (3.20) minimize the action within the topological sector to which they belong. Conventionally we refer to the selfdual configurations as instantons and to the antiselfdual configurations as antiinstantons^{7}^{7}7In the Mathematical community the opposite definition is usually considered.. From the definition of the topological charge (3.10) and (3.7), we have that instantons and antiinstantons have positive and negative respectively.
3.2 Vacua and tunneling amplitudes
Instantons can be interpreted as tunneling processes interpolating between different vacua of the Minkowskian formulation of the YM model, [26]. In order to illustrate this crucial point and before moving from the Euclidean to the Minkowskian formulation, it is necessary to consider the temporal gauge, namely
(3.21) 
In the temporal gauge (sometimes also referred to as Weyl gauge) it is possible to canonically quantize the theory. We have the following (Euclidean) Lagrangian and Hamiltonian densities:
(3.22)  
(3.23) 
Note that the relative signs are opposite to the usual Minkowskian expectations, indeed the terms in represent the kinetic part of the densities () and the terms in constitute the potential energy density. ^{8}^{8}8The Euclidean formulation of YM theory can be regarded as the Wickrotated (i.e. imaginary time) version of the Minkowskian version.
A general feature of field theories defined on a noncompact base manifold is that the topological considerations are strictly related to the asymptotic (i.e. at large radius) behavior of the fields. To rephrase (3.16) and (3.17), the topological charge is given by the flux integral of the Chern current
(3.24) 
through an asymptotic hypersurface. From the temporal gauge condition (3.21) we have that is directed in the direction for large . Since we deal with a local theory, we add the general assumption that the fields vanish at spatial infinity, namely
(3.25) 
Equations (3.24) and (3.25) combined together mean that the total flux of at “infinity” receives contributions only from the asymptotic regions corresponding to , that is
(3.26) 
More precisely, and represent the fluxes of “through” the spatial threedimensional manifolds corresponding to positive and negative temporal infinity respectively. We can repeat the argument connecting (3.16) to (3.17) for the threedimensional configurations at asymptotic time; namely, we can associate to the two configurations a (spatial) topological charge:
(3.27)  
(3.28) 
where represent the threedimensional hypersurface element (i.e. the volume element) oriented along the time direction. Note that the fourdimensional overall topological charge is given by
(3.29) 
The formal similarity between (3.17) and (3.27), (3.28) is evident, however a doubt could arise. Indeed, while (3.17) is defined on the asymptotic of Euclidean space, (3.27) and (3.28) are defined on the spatial manifold. The former is a compact space while the latter is not. The homotopy argument that led us to interpret as the winding number seems to be impossible for because it apparently lacks one of the essential ingredients: the compactness of the manifold on which the integral is considered. A subtle observation comes to our help. Note that we assumed in (3.25) that the gauge potential vanishes at spatial infinity. For configurations related to the trivial vacuum by a gauge transformation ,
(3.30) 
we have that at spatial infinity tends to a constant value that can be fixed to be the identity^{9}^{9}9We can discard rigid gauge rotations from our analysis without spoiling its generality.,
(3.31) 
The gauge fields and transformations assume a fixed value in the limit independently of the particular direction along which we move towards spatial infinity. In this sense we can add “the point at infinity” assigning
(3.32) 
“completing” the spatial manifold to a compact . In this sense, the integrals (3.27) expressing can be regarded as properly defined spatial winding numbers.
As a consequence of the preceding arguments, it is natural to interpret the exponentiated action of the instanton as the transition amplitude between the two configurations at temporal infinity. This transition connects field configurations presenting different spatial winding numbers . Let us remark the fact that the instanton amplitude is given by its classical action
(3.33) 
where the coupling constant appears at the denominator of the exponent. This nonperturbative feature reminds us the semiclassical WKB tunneling amplitudes. Indeed, we are interpreting the instanton as the transition amplitude through the barrier dividing distinct topological sectors. The semiclassical character of the present analysis arises from the fact that we are considering just the instanton amplitude with lowest action, i.e. only the minimal classical path in a quantum path integral.
If we consider the gauge configurations (3.30) which are obtained by applying a constanttime gauge transformation^{10}^{10}10The constanttime gauge transformations constitute a residual gauge symmetry of the temporal gauge (3.21). to the vacuum, we have that the corresponding spatial part of the fieldstrength vanishes everywhere. From equation (3.4) we have then and since the potential energy density in (3.22) is
(3.34) 
it vanishes too. The configurations with zero potential energy are degenerate with the vacuum and we henceforth refer to them as the vacua of the theory. Let us argue that an instanton solution describes a semiclassical transition amplitude connecting two such vacua. For the sake of clarity, let us stick to an explicit instanton example
(3.35) 
where is an arbitrary length scale^{11}^{11}11Further comments on the parameter as quantifying the “size” of the instanton are given in Section 3.3.. It is manifest that for large the field satisfies the requirement (3.15). Moreover, if we want to bring (3.35) into the temporal gauge we have to perform a gauge transformation that for asymptotic time (asymptotic time implies asymptotic ) will return a configuration of the form (3.30). The instanton (3.35) then connects two vacua of the theory.
Although instantons are classical solutions that exist only in the Euclidean formulation of the theory^{12}^{12}12Where they represent zeroenergy solutions; indeed selfduality implies the vanishing of the Euclidean Hamiltonian density (3.22)., they can be interpreted in Minkowski spacetime as transition amplitudes interpolating between distinct vacua related by a topologically nontrivial gauge transformation; the Euclidean derivation of instanton amplitudes can be regarded in fact as the imaginary time continuation of the theory in its Minkowski formulation. Imaginarytime methods for the computation of semiclassical realtime tunneling amplitudes are a standard technique.
3.2.1 The angle
The gauge fixing procedure in the presence of nontrivial topological sectors may generate some doubts. Take a specific gauge configuration defined on the Euclidean
(3.36) 
where has nontrivial winding on the asymptotic spacetime threesphere. A gauge fixing procedure is in general intended to remove the gauge redundancy and we could be tempted to discard (3.36) as a gauge equivalent representative of the trivial vacuum . Following the arguments in [27], we must specify that the gauge fixing procedure removes from the functional integral over the gauge field configurations those which are related by a topologically trivial gauge transformation^{13}^{13}13I.e. a transformation obtainable deforming continuously the constant gauge transformation .. In other words, fixing the gauge prevents redundancy within the various topological sectors. In fact, configurations belonging to different sectors cannot at all describe the same physical circumstance and cannot therefore be redundant. As we describe in the following, the topology has indeed phenomenological effects. Sometimes in the literature people use the terms ‘‘small’’ and ‘‘large’’ gauge transformations to denote respectively the proper gauge transformations and the topology changing ones^{14}^{14}14The same terminology has some other times a different meaning: “small” and “large” are referred to local as opposed to global (called also “rigid”) gauge transformations..
The quantum vacuum state is in general expected to be given by a functional of that is peaked on the classical vacuum. The spread of the vacuum functional is given by Heisenberg’s indeterminacy of quantum fluctuations. We can have a sketchy idea figuring a well whose bottom is the classical vacuum. However, the picture that emerged form the study of instantons is richer. The various topological sectors of YM theory can be imagined as different wells arranged in a periodic lattice whose period is measured by the elemental increment of the winding number. Instantons themselves represent transitions from one well to another. The vacuum state is sensitive to this periodic structure and therefore we have to represent the candidate fundamental state functional as follows
(3.37) 
where the component functionals are peaked around the vacuum with winding number and are coefficients. As a physical state, the quantum vacuum has to be invariant with respect to small gauge transformations; moreover, since it is stable by definition, it must be invariant with respect to the topology changing gauge transformations as well. The latter feature fixes the shape of the coefficients in (3.37) to be
(3.38) 
where is a parameter that spans a continuous onedimensional family of vacua for the YM theory. They are indeed usually called vacua. The vacua have a behavior that reminds us of Bloch waves in periodic potentials. In this respect, parametrizes the ‘‘conduction band’’ of YM vacua and is analogous to the Bloch momentum^{15}^{15}15To have further details and comments we refer the reader to [27, 28, 26, 29]..
3.3 Collective coordinates
The global features of instanton solutions like, for instance, the instanton center position, are encoded in a set of parameters usually referred to as “collective coordinates” or “moduli”. For a given value of the topological charge, the corresponding moduli space spanned by the instanton collective coordinates is denoted with and contains all the instanton solutions associated to winding number .
Let us have a direct look at the moduli of the simplest instanton example^{16}^{16}16This is the first instance of instanton studied in the original paper [24] by Belavin, Polyakov, Schwarz and Tyupkin. Indeed it is commonly referred to as BPST instanton.. Consider pure Euclidean YangMills theory with gauge group SU in Landau’s gauge, i.e. ; the index runs over the adjoint representation of the gauge group. Take the gauge transformation
(3.39) 
where we notice that the gauge adjoint space is linked to the physical space; in other words, which is a vector in the adjoint space of SU is multiplied by which instead is a spatial vector. The topological nontrivial character of the instanton emerges form such relation between space and gauge representation. Let us insert (3.39) in the instanton solution (3.35). Performing some not difficult passages, we obtain the following explicit form for the SU instanton
(3.40) 
where represents the antiselfdual ’t Hooft symbols defined in Appendix B. To go from (3.39) to (3.40) we have used the properties of the ’t Hooft symbols (see Appendix B) and we have also manually inserted the matrix