D-Branes on Calabi-Yau Spaces and Their Mirrors We study the boundary states of D-branes wrapped around supersymmetric cycles in a general Calabi-Yau manifold. In particular, we show how the geometric data on the cycles are encoded in the boundary states. As an application, we analyze how the mirror symmetry transforms D-branes, and we verify that it is consistent with the conjectured periodicity and the monodromy of the Ramond-Ramond field configuration on a Calabi-Yau manifold. This also enables us to study open string worldsheet instanton corrections and relate them to closed string instanton counting. The cases when the mirror symmetry is realized as T-duality are also discussed.

T-Branes at the Limits of Geometry Singular limits of 6D F-theory compactifications are often captured by T-branes, namely a non-abelian configuration of intersecting 7-branes with a nilpotent matrix of normal deformations. The long distance approximation of such 7-branes is a Hitchin-like system in which simple and irregular poles emerge at marked points of the geometry. When multiple matter fields localize at the same point in the geometry, the associated Higgs field can exhibit irregular behavior, namely poles of order greater than one. This provides a geometric mechanism to engineer wild Higgs bundles. Physical constraints such as anomaly cancellation and consistent coupling to gravity also limit the order of such poles. Using this geometric formulation, we unify seemingly different wild Hitchin systems in a single framework in which orders of poles become adjustable parameters dictated by tuning gauge singlet moduli of the F-theory model.

Dirichlet branes, homological mirror symmetry, and stability We discuss some mathematical conjectures which have come out of the study of Dirichlet branes in superstring theory, focusing on the case of supersymmetric branes in Calabi-Yau compactification. This has led to the formulation of a notion of stability for objects in a derived category, contact with Kontsevich’s homological mirror symmetry conjecture, and “physics proofs” for many of the subsequent conjectures based on it, such as the representation of Calabi-Yau monodromy by autoequivalences of the derived category.

Clay Mathematics Monographs: MIRROR SYMMETRY Since the 1980s, there has been an extremely rich interaction between mathematics and physics. Viewed against the backdrop of relations between these two fields throughout the history of science, that may not appear to be so surprising. For example, throughout most of their history the two subjects were not clearly distinguished. For much of the 1900s, however, physics and mathematics developed to a great extent independently and, except for relatively rare and not-so-deep interconnections, the two fields went their separate ways. With the appreciation of the importance of Yang–Mills gauge theories in describing the physics of particle interactions, and with the appreciation of its importance in the mathematics of vector bundles, renewed interaction between the two fields began to take place. For example, the importance of instantons and monopoles came to be appreciated from both the physical and mathematical points of view. With the discovery of supersymmetry and its logical completion to superstring theory, a vast arena of interaction opened up between physics and mathematics and continues today at a very deep level for both fields. Fundamental questions in one field often turn out to be fundamental questions in the other field as well. But even today mathematicians and physicists often find it difficult to discuss their work and interact with each other. The reason for this appears to be twofold. First, the languages used in the two fields are rather different. This problem is gradually being resolved as we recognize the need to become “bilingual.” The second and more serious problem is that the established scientific methods in the two fields do not converge. Whereas mathematics places emphasis on rigorous foundations and the interplay of various structures, to a physicist the relevant aspects are physical clarity and physical interconnection of ideas, even if they come at the cost of some mathematical rigor. This can lead to friction between mathematicians and physicists. While mathematicians respect xv xvi INTRODUCTION physicists for their intuition, they sometimes do not fully trust how those results were obtained and so they erect their own rigorous foundations as a substitute for the physical reasoning leading to those results. At the same time, physicists, who now appreciate the importance of modern mathematics as a powerful tool for theoretical physics, feel that attempts to build on a more rigorous foundation, while noble, will distract them from their real goal of understanding nature. Thus we are at a delicate point in the history of the interaction of these two fields: While both fields desperately need each other, the relationship seems at times to be a dysfunctional codependence rather than a happy marriage! The aim of this book is to develop an aspect of this interplay known as “mirror symmetry” from both physical and mathematical perspectives, in order to further interaction between the two fields. With this goal in mind, almost half of the book includes introductory mathematics and physics material, while we try to emphasize the interconnection between the two areas. Unfortunately, however, the book also reflects the present status, namely, we find two distinct approaches to understanding mirror symmetry, without a clear connection between physical and mathematical methods of proof. Even the notion of what one means by “proof” of mirror symmetry differs between the two fields. Mirror symmetry is an example of a general phenomenon known as duality, which occurs when two seemingly different physical systems are isomorphic in a non-trivial way. The non-triviality of this isomorphism involves the fact that quantum corrections must be taken into account. Mathematically, a good analogy is the Fourier transform, where local concepts such as products are equivalent to convolution products, requiring integration over the whole space. Thus it is difficult to understand such isomorphisms in the classical context. In particular, under such an isomorphism, certain complicated quantities involving quantum corrections in one system get mapped to simple classical questions in the other. Thus, finding such dualities leads to solving complicated physical questions in terms of simple ones in the dual theory. Precisely for this reason the discovery of duality symmetries has revolutionized our understanding of quantum theories and string theory. It is fair to say that we do not have a deep understanding of the reason for the prevalence of duality symmetries in physics. Nor do we have a proof of why a duality should exist in any given case. Most of the arguments in A HISTORY OF MIRROR SYMMETRY favor of duality symmetries involve checking consequences and seeing that they are indeed satisfied in a non-trivial way. Because there have been so many non-trivial checks, we have no doubts about their validity, but that does not mean we have a deep understanding of the inner workings of duality symmetries. The only heuristic explanation of dualities we know of is the “scarcity of rich structures,” and consistent quantum theories are indeed rather rich. So different ways of coming up with similar quantum systems end up being equivalent! There is, however, one exception to this rule, mirror symmetry; for we have a reasonably clear picture of how it works. Moreover, a mathematical framework to rigorize many of the statements arising from the physics picture has also been constructed, and the subject is in a rather mature state of development. It is our hope that by elaborating aspects of this beautiful duality to both physicists and mathematicians, we can inspire further clarifications of this duality, which may also serve as a model for a deeper understanding of other dualities and interconnections between physics and mathematics. A History of Mirror Symmetry The history of the development of mirror symmetry is a very complicated one. Here we give a brief account of it, without any claim to completeness. The origin of the idea can be traced back to a simple observation of [154], [223] that string theory propagation on a target space that is a circle of radius R is equivalent to string propagation on a circle of radius 1/R (in some natural units). This has become known as T-duality. Upon the emergence of Calabi–Yau manifolds as interesting geometries for string propagation [41], a more intensive study of the corresponding string theories was initiated. It was soon appreciated that N = 2 supersymmetry on the worldsheet is a key organizing principle for the study of the corresponding string theories. It was noticed by [71] and [173] that given an N = 2 worldsheet theory, it is not possible to uniquely reconstruct a corresponding Calabi–Yau manifold. Instead there was a twofold ambiguity. In other words, it was seen that there could be pairs of Calabi–Yau manifolds that lead to the same underlying worldsheet theory, and it was conjectured that perhaps this was a general feature of all Calabi–Yau manifolds. Such pairs did not even have to have the same cohomology dimensions. In fact, the Hodge numbers hp,q for one of xvii xviii INTRODUCTION them was mapped to hd−p,q for the mirror, where d is the complex dimension of the Calabi–Yau manifold. Moreover, it was seen that the instantoncorrected cohomology ring (i.e., quantum cohomology ring) for one is related to a classical computation for the mirror. Phenomenological evidence for this conjecture was found in [42], where a search through a large class of Calabi–Yau threefolds showed a high degree of symmetry for the number of Calabi–Yaus with Euler numbers that differ by sign, as is predicted by the mirror conjecture. Non-trivial examples of mirror pairs were constructed in [123], using the relation between Calabi–Yau manifolds and Landau– Ginzburg models [107], [189], [124]. It was shown in [45] that one could use these mirror pairs to compute the instanton corrections for one Calabi– Yau manifold in terms of the variations of Hodge structure for the mirror. The instanton corrections involve certain questions of enumerative geometry; roughly speaking, one needs to know how many holomorphic maps exist from the two-sphere to the Calabi–Yau for any fixed choice of homology class for the two-cycle image. The notion of topological strings was introduced in [262] where it abstracted from the full worldsheet theory only the holomorphic maps to the target. It was noted in [245] and [264] that mirror symmetry descends to a statement of the equivalence of two topological theories. It is this latter statement that is often taken to be the definition of the mirror conjecture in the mathematics literature. In [16] and [17] it was suggested that one could use toric geometry to propose a large class of mirror pairs. In [265] linear sigma models were introduced, which gave a simple description of a string propagating on a Calabi–Yau, for which toric geometry was rather natural. In [267] it was shown how to define topological strings on Riemann surfaces with boundaries and what data is needed to determine the boundary condition (the choice of the boundary condition is what we now call the choice of a D-brane and was first introduced in [67]). In [24] and [25], it was shown how one can use mirror symmetry to count holomorphic maps from higher genus curves to Calabi–Yau threefolds. In [164] a conjecture was made about mirror symmetry as a statement about the equivalence of the derived category and the Fukaya category. In [163] it was shown how one can use localization ideas to compute the “number” of rational curves directly. It was shown in [108, 109] and [180, 181, 182, 183] how one may refine this program to find a more effective method for computation of the number of THE ORGANIZATION OF THIS BOOK xix rational curves. Moreover, it was shown that this agrees with the predictions of the number of rational curves based on mirror symmetry (this is what is now understood to be the “mathematical proof of mirror symmetry”). In [234] it was shown, based on how mirror symmetry acts on D0-branes, that Calabi–Yau mirror pairs are geometrically related: One is the moduli of some special Lagrangian submanifold (equipped with a flat bundle) of the other. In [246] the implications of mirror symmetry for topological strings in the context of branes was sketched. In [114] the integrality property of topological string amplitudes was discovered and connected to the physical question of counting of certain solitons. In [135] a proof of mirror symmetry was presented based on T-duality applied to the linear sigma model. Work on mirror symmetry continues with major developments in the context of topological strings on Riemann surfaces with boundaries, which is beyond the scope of the present book.

Topological AdS/CFT We define a holographic dual to the Donaldson-Witten topological twist of N = 2 gauge theories on a Riemannian four-manifold. This is described by a class of asymptotically locally hyperbolic solutions to N = 4 gauged supergravity in five dimensions, with the four-manifold as conformal boundary. Under AdS/CFT, minus the logarithm of the partition function of the gauge theory is identified with the holographically renormalized supergravity action. We show that the latter is independent of the metric on the boundary four-manifold, as required for a topological theory. Supersymmetric solutions in the bulk satisfy first order differential equations for a twisted Sp(1) structure, which extends the quaternionic K¨ahler structure that exists on any Riemannian four-manifold boundary. We comment on applications and extensions, including generalizations to other topological twists.

U-dualities in Type II string theories and M-theory String theory is arguably the most developed candidate for a theory of everything. It appeared as an attempt to describe strong interactions and dualities in scattering amplitudes. Soon it was rediscovered as a possible theory of quantum gravity [6]. It was realised that the spectrum of a closed string contains excitations of spin 2 which were then identified with gravitons, which caused the significant transition in the understanding of strings from simply tubes between quarks to the most elementary constituents of matter. This resulted in intense studying of fundamental strings and led to discovery of five different consistent superstring theories that live in 10 dimension: Type I, Type IIA and IIB, SO(32) and E8 heterotic strings. These theories differ by gauge symmetries, set of fields, boundary conditions and realisation of supersymmetry. The situation appeared to be very strange: after years of looking for a theory of everything one eventually ends up with five of them having no way to choose the correct one. The way out of this trouble was tightly connected to the problem of extra dimensions in string theories. Almost one hundred years before these events T. Kaluza and F. Klein suggested one could consider the Maxwell field Aµ as a part of 5-dimensional metric. Assuming, that the fifth dimension is compact with very small radius of compactification they showed that General Relativity on such a background is equivalent to the 4-dimensional theory of electromagnetic field interacting with gravity. The same idea can be used to get rid of extra 6 dimensions of string theories. For example one can choose a 6-dimensional torus T6 as an internal space. Since the torus is flat it preserves reparametrisation invariance of the worldsheet and Virasoro algebra, that is local. An amazing feature of Type IIA and Type IIB string theories is that compactified on T1 they become equivalent on quantum level [7–9]. This is a particular case of the so-called T-duality that is the oldest known duality in string theory [10, 11]. It relates two heterotic string theories to each other as well. T-duality is a perturbative symmetry and can be seen manifestly in the spectrum of a closed string living on a background with compact directions. An example of a non-perturbative symmetry is provided by S-duality of Type IIB string theory in 10 dimensions, that is SL(2, Z). In addition, S-duality relates heterotic SO(32) strings to Type I strings. Finally, type IIA theory in the strong coupling regime behaves as an 11- dimensional theory whose low-energy limit is captured by 11-dimensional supergravity. The same supergravity being compactified on a unit interval I = [0, 1] leads to the low-energy limit of E8 heterotic theory. The net of dualities that unifies all five string theories gives a hint that there should exist a mother theory that gives all string theories in various limits and lives in 11 dimensions. Such theory is commonly referred to as M-theory and, although it has not been understood in great details, a lot of is already known about its structure. M-theory describes dynamics of 2- and 5-dimensional membranes (the so-called M2- and M5-branes) and reduces to 11-dimensional supergravity in its low-energy limit. Being compactified on a circle S1 M-theory is equivalent to Type IIA string theory. A fundamental string then is associated to an M2-brane wrapped around the circle. The other objects of Type IIA string theory like D2, D4 branes for example appear from the fundamental objects of M-theory in a similar way [12–14]. On the other hand M-theory compactified on a torus T2 gives rise to Type IIB string theory compactified on a circle S1. S-duality symmetry SL(2, Z) of Type IIB theory becomes transparent in this picture and is just the modular group of the 2- dimensional torus. Together S- and T-dualities are combined into a non-perturbative set of symmetries of M-theory that is called U-duality [15]. These dualities provide a powerful instrument for studying string compactifications, moduli stabilization, properties of string backgrounds, and were intensively studied for many years (for review see [11, 16–18]). However, the partition function of a superstring is not manifestly invariant under these transformations. In [19–21] the formulation of the worldsheet action for a string where T-duality of a background is manifest was proposed. The idea was to consider combinations of coordinates of a closed string X = X+ + X− and X˜ = X+ − X− as independent variables. Then O(d, d) T-duality symmetry becomes manifest if the action is rewritten in terms of 2d extended coordinates X = (X, X˜). The Buscher procedure, described in details in further sections, gives a well defined algorithm for gauging the isometry, integrating out gauge fields and obtaining the T-dual sigma-model. This leads to the notion of the so-called generalised metric that puts the space-time metric and the gauge fields on an equal footing and allows one to consider diffeomorphisms and gauge transformations as a part of more general transformations of extended space. The duality invariant approach on which the thesis is focused, is an incredibly fascinating construction. Among other applications, the most intriguing feature of this approach is that both non-geometric and geometric backgrounds of string theory become geometric in terms of the extended space. Although geometry of the extended space is still a mystery and very little is known about its structure, one already sees useful applications such as gauged supergravities, studying non-geometric fluxes, SU(3) structures, global properties of backgrounds and many others. Good pedagogical reviews of this approach and its applications can be found in [22, 23].

Quantisation conditions of the quantum Hitchin system and the real geometric Langlands correspondence We are going to propose a natural quantisation condition for the Hitchin system, and explain how it can be reformulated in terms of a function Y(a, t). The function Y(a, t) relevant for this task is found to be the generating function for the variety of opers within the space of all local systems as predicted in [6, 9]. However, the condition on Y expressing the quantisation condition turns out to be different from the types of conditions considered in [1]. Our derivation is essentially complete for Hitchin systems associated to the Lie algebra sl2 in genus 0 and 1, which may be called the Gaudin and elliptic Calogero-Moser models assciated to the group SL(2, C). It reduces to a conjecture of E. Frenkel [11] for g > 1, as will be discussed below. Reformulating the quantisation conditions in terms of Y can be done using the Separation of Variables (SOV) method pioneered by Sklyanin [12]. This method may be seen as a more concrete procedure to construct the geometric Langlands correspondence relating opers to Dmodules (eigenvalue equations), as was pointed out in [11]. In our case it will be found that the SOV method relates single-valued solutions of the eigenvalue equations to opers having Fuchsian holonomy. The classification of opers or equivalently projective structures on C with Fuchsian holonomy has been studied in [13]. Using complex Fenchel-Nielsen coordinates we will reformulate this description in terms of the generating function for the variety of opers. From the point of view of the geometric Langlands correspondence we obtain a correspondence between a special class of real opers, opers with Fuchsian holonomy which is in particular real, and D-modules admitting single-valued solutions. We expect that a generalisation to more general local system with real holonomy will exist. We propose to call such correspondences the real geometric Langlands correspondence.

Supersymmetric Yang-Mills theory as higher Chern-Simons theory We observe that the string field theory actions for the topological sigma models describe higher or categorified Chern-Simons theories. These theories yield dynamical equations for connective structures on higher principal bundles. As a special case, we consider holomorphic higher Chern-Simons theory on the ambitwistor space of four-dimensional space-time. In particular, we propose a higher ambitwistor space action functional for maximally supersymmetric Yang-Mills theory.

Supersymmetric branes and instantons on curved spaces We discuss non-linear instantons in supersymmetric field theories on curved spaces arising from D-branes. Focusing on D3-branes and four-dimensional field theories, we derive the supersymmetry conditions and show the intimate relation between the instanton solutions and the non-linearly realized supersymmetries of the field theory. We demonstrate that field theories with non-linearly realized supersymmetries are coupled to supergravity backgrounds in a similar fashion as those with linearly realized supersymmetries, and provide details on how to derive such couplings from a type II perspective.

New nilpotent N = 2 superfields We propose new off-shell models for spontaneously broken local N = 2 supersymmetry, in which the supergravity multiplet couples to nilpotent Goldstino superfields that contain either a gauge one-form or a gauge two-form in addition to spin-1/2 Goldstone fermions and auxiliary fields. In the case of N = 2 Poincar´e supersymmetry, we elaborate on the concept of twisted chiral superfields and present a nilpotent N = 2 superfield that underlies the cubic nilpotency conditions given in arXiv:1707.03414 in terms of constrained N = 1 superfields.

Heterotic Instanton Superpotentials from Complete Intersection Calabi-Yau Manifolds We study Pfaffians that appear in non-perturbative superpotential terms arising from worldsheet instantons in heterotic theories. A result by Beasley and Witten shows that these instanton contributions cancel among curves within a given homology class for Calabi-Yau manifolds that can be described as hypersurfaces or complete intersections in projective or toric ambient spaces. We provide a prescription that identifies all P1 curves in certain homology classes of complete intersection Calabi-Yau manifolds in products of projective spaces (CICYs) and cross-check our results by a comparison with the genus zero Gromov-Witten invariants. We then use this construction to study instanton superpotentials on those manifolds and their quotients. We identify a non-toric quotient of a non-favorable CICY with a single genus zero curve in a certain homology class, so that a cancellation `a la Beasley-Witten is not possible. In another example, we study a non-toric quotient of a favorable CICY and check that the superpotential still vanishes. From this and related examples, we conjecture that the Beasley-Witten cancellation result can be extended to toric and non-toric quotients of CICYs, but can be avoided if the CICY is non-favorable.

Anti-D3 branes and moduli in non-linear supergravity Anti-D3 branes and non-perturbative effects in flux compactifications spontaneously break supersymmetry and stabilise moduli in a metastable de Sitter vacua. The low energy 4D effective field theory description for such models would be a supergravity theory with non-linearly realised supersymmetry. Guided by string theory modular symmetry, we compute this non-linear supergravity theory, including dependence on all bulk moduli. Using either a constrained chiral superfield or a constrained vector field, the uplifting contribution to the scalar potential from the anti-D3 brane can be parameterised either as an F-term or Fayet-Iliopoulos D-term. Using again the modular symmetry, we show that 4D non-linear supergravities that descend from string theory have an enhanced protection from quantum corrections by non-renormalisation theorems. The superpotential giving rise to metastable de Sitter vacua is robust against perturbative string-loop and α′ corrections.

The Gauge Group Ambiguity of the Standard Model of Physics There is an ambiguity in the gauge group of the Standard Model. The group is G = SU(3) × SU(2) × U(1)/Γ, where Γ is a subgroup of Z6 which cannot be determined by current experiments. We describe how the electric, magnetic and dyonic line operators of the theory depend on the choice of Γ. We also explain how the periodicity of the theta angles, associated to each factor of G, differ.

D-Branes on the Quintic We study D-branes on the quintic CY by combining results from several directions: general results on holomorphic curves and vector bundles, stringy geometry and mirror symmetry, and the boundary states in Gepner models recently constructed by Recknagel and Schomerus, to begin sketching a picture of D-branes in the stringy regime. We also make first steps towards computing superpotentials on the D-brane world-volumes.

String-Theory Fibre Inflation and α-Attractors Fibre inflation is a specific string theory construction based on the Large Volume Scenario that produces an inflationary plateau. We outline its relation to α-attractor models for inflation, with the cosmological sector originating from certain string theory corrections leading to α = 2 and α = 1/2. Above a certain field range, the steepening effect of higher-order corrections leads first to the breakdown of single-field slow-roll and after that to the onset of 2-field dynamics: the overall volume of the extra dimensions starts to participate in the effective dynamics. Finally, we propose effective supergravity models of fibre inflation based on an D3 uplift term with a nilpotent superfield. Specific moduli dependent D3 induced geometries lead to cosmological fibre models but have in addition a de Sitter minimum exit. These supergravity models motivated by fibre inflation are relatively simple, stabilize the axions and disentangle the Hubble parameter from supersymmetry breaking.

BPS Objects in D = 7 Supergravity and their M-Theory Origin We study several different types of BPS flows within minimal N = 1, D = 7 supergravity with SU(2) gauge group and non-vanishing topological mass. After reviewing some known domain wall solutions involving only the metric and the R+ scalar field, we move to considering more general flows involving a “dyonic” profile for the 3-form gauge potential. In this context, we consider flows featuring a Mkw3 as well as an AdS3 slicing, write down the corresponding flow equations, and integrate them analytically to obtain many examples of asymptotically AdS7 solutions in presence of a running 3-form. Furthermore, we move to adding the possibility of non-vanishing vector fields, find the new corresponding flows and integrate them numerically. Finally, we discuss the eleven-dimensional interpretation of the aforementioned solutions as effective descriptions of M2 − M5 bound states.

E8 instantons on type-A ALE spaces and supersymmetric field theories We consider the 6d superconformal field theory realized on M5-branes probing the E8 end-of-theworld brane on the deformed and resolved C2/Zk singularity. We give an explicit algorithm which determines, for arbitrary holonomy at infinity, the 6d quiver gauge theory on the tensor branch, the type-A class S description of the T2 compactification, and the star-shaped quiver obtained as the mirror of the T3 compactification.

A Generalized Construction of Calabi-Yau Models and Mirror Symmetry We extend the construction of Calabi-Yau manifolds to hypersurfaces in nonFano toric varieties, requiring the use of certain Laurent defining polynomials, and explore the phases of the corresponding gauged linear sigma models. The associated non-reflexive and non-convex polytopes provide a generalization of Batyrev’s original work, allowing us to construct novel pairs of mirror models. We showcase our proposal for this generalization by examining Calabi-Yau hypersurfaces in Hirzebruch n-folds, focusing on n = 3, 4 sequences, and outline the more general class of so-defined geometries.

K3 Surfaces and String Duality The primary purpose of these lecture notes is to explore the moduli space of type IIA, type IIB, and heterotic string compactified on a K3 surface. The main tool which is invoked is that of string duality. K3 surfaces provide a fascinating arena for string compactification as they are not trivial spaces but are sufficiently simple for one to be able to analyze most of their properties in detail. They also make an almost ubiquitous appearance in the common statements concerning string duality. We review the necessary facts concerning the classical geometry of K3 surfaces that will be needed and then we review “old string theory” on K3 surfaces in terms of conformal field theory. The type IIA string, the type IIB string, the E8 × E8 heterotic string, and Spin(32)/Z2 heterotic string on a K3 surface are then each analyzed in turn. The discussion is biased in favour of purely geometric notions concerning the K3 surface itself. Compactification of superstring theory Superstring theories and M theory, at present the best candidate quantum theories which unify gravity, Yang-Mills fields and matter, are directly formulated in ten and eleven space-time dimensions. To obtain a candidate theory of our four dimensional universe, one must find a solution of one of these theories whose low energy physics is well described by a four dimensional effective field theory (EFT), containing the well established Standard Model of particle physics (SM) coupled to Einstein’s general relativity. The standard paradigm for finding such solutions is compactification, along the lines originally proposed by Kaluza and Klein in the context of higher dimensional general relativity. One postulates that the underlying D-dimensional space-time is a product of four-dimensional Minkowski space-time, with a D − 4-dimensional compact and small Riemannian manifold K. One then finds that low energy physics effectively averages over K, leading to a four dimensional EFT whose field content and Lagrangian are determined in terms of the topology and geometry of K. Of the huge body of prior work on this subject, the part most relevant for string/M theory is supergravity compactification, as in the limit of low energies, small curvatures and weak coupling, the various string theories and M theory reduce to ten and eleven dimensional supergravity theories. Many of the qualitative features of string/M theory compactification, and a good deal of what is known quantitatively, can be understood simply in terms of compactification of these field theories, with the addition of a few crucial ingredients from string/M theory. Thus, most of this article will restrict attention to this case, leaving many “stringy” topics to the articles on conformal field theory, topological string theory and so on. We also largely restrict attention to compactifications based on Ricci flat compact spaces. There is an equally important class in which K has positive curvature; these lead to anti-de Sitter space-times and are discussed in the articles on AdS/CFT. After a general review, we begin with compactification of the heterotic string on a three complex dimensional Calabi-Yau manifold. This was the first construction which led convincingly to the SM, and remains one of the most important examples. We then survey the various families of compactifications to higher dimensions, with an eye on the relations between these compacti- fications which follow from superstring duality. We then discuss some of the phenomena which arise in the regimes of large curvature and strong coupling. In the final section, we bring these ideas together in a survey of the various known four dimensional constructions.

Dynamics of M-Theory Cosmology A complete global analysis of spatially–flat, four–dimensional cosmologies derived from the type IIA string and M–theory effective actions is presented. A non–trivial Ramond–Ramond sector is included. The governing equations are written as a dynamical system. Asymptotically, the form fields are dynamically negligible, but play a crucial rˆole in determining the possible intermediate behaviour of the solutions (i.e. the nature of the equilibrium points). The only past-attracting solution (source in the system) may be interpreted in the eleven–dimensional setting in terms of flat space. This source is unstable to the introduction of spatial curvature.  

Superstring Cosmology Aspects of superstring cosmology are reviewed with an emphasis on the cosmological implications of duality symmetries in the theory. The string effective actions are summarized and toroidal compactification to four dimensions reviewed. Global symmetries that arise in the compactification are discussed and the duality relationships between the string effective actions are then highlighted. Higher–dimensional Kasner cosmologies are presented and interpreted in both string and Einstein frames, and then given in dimensionally reduced forms. String cosmologies containing both non–trivial Neveu–Schwarz/Neveu–Schwarz and Ramond–Ramond fields are derived by employing the global symmetries of the effective actions. Anisotropic and inhomogeneous cosmologies in four–dimensions are also developed. The review concludes with a detailed analysis of the pre–big bang inflationary scenario. The generation of primordial spectra of cosmological perturbations in such a scenario is discussed. Possible future directions offered in the Hoˇrava–Witten theory are outlined.

Supergravity Brane Cosmologies Solitonic brane cosmologies are found where the world-volume is curved due to the evolution of the dilaton field on the brane. In many cases, these may be related to the solitonic Dp- and M5-branes of string and M-theory. An eleven-dimensional interpretation of the D8-brane cosmology of the massive type IIA theory is discussed in terms of compactification on a torus bundle. Braneworlds are also found in Horava-Witten theory compactified on a Calabi-Yau three-fold. The possibility of dilaton-driven inflation on the brane is discussed.

D-branes on Calabi–Yau Manifolds and Superpotentials We show how to compute terms in an expansion of the world-volume superpotential for fairly general D-branes on the quintic Calabi-Yau using linear sigma model techniques, and show in examples that this superpotential captures the geometry and obstruction theory of bundles and sheaves on this Calabi-Yau.

Mirror Symmetry, D-branes and Counting Holomorphic Discs We consider a class of special Lagrangian subspaces of Calabi-Yau manifolds and identify their mirrors, using the recent derivation of mirror symmetry, as certain holomorphic varieties of the mirror geometry. This transforms the counting of holomorphic disc instantons ending on the Lagrangian submanifold to the classical Abel-Jacobi map on the mirror. We recover some results already anticipated as well as obtain some highly non-trivial new predictions.

D-Brane Stability and Monodromy We review the idea of Π-stability for B-type D-branes on a Calabi–Yau manifold. It is shown that the octahedral axiom from the theory of derived categories is an essential ingredient in the study of stability. Various examples in the context of the quintic Calabi–Yau threefold are studied and we plot the lines of marginal stability in several cases. We derive the conjecture of Kontsevich, Horja and Morrison for the derived category version of monodromy around a “conifold” point. Finally, we propose an application of these ideas to the study of supersymmetry breaking.

D-branes on Stringy Calabi–Yau Manifolds We argue that D-branes corresponding to rational B boundary states in a Gepner model can be understood as fractional branes in the Landau–Ginzburg orbifold phase of the linear sigma model description. Combining this idea with the generalized McKay correspondence allows us to identify these states with coherent sheaves, and to calculate their K-theory classes in the large volume limit, without needing to invoke mirror symmetry. We check this identification against the mirror symmetry results for the example of the Calabi–Yau hypersurface in WIP 1 , 1 , 2 , 2 , 2 .

Orbifold Resolution by D-Branes We study topological properties of the D-brane resolution of three-dimensional orbifold singularities, C3/Γ, for finite abelian groups Γ. The D-brane vacuum moduli space is shown to fill out the background spacetime with Fayet–Iliopoulos parameters controlling the size of the blow-ups. This D-brane vacuum moduli space can be classically described by a gauged linear sigma model, which is shown to be non-generic in a manner that projects out non-geometric regions in its phase diagram, as anticipated from a number of perspectives.

Worldsheet approaches to D-branes on supersymmetric cycles We consider D-branes wrapped around supersymmetric cycles of CalabiYau manifolds from the viewpoint of N = 2 Landau-Ginzburg models with boundary as well as by consideration of boundary states in the corresponding Gepner models. The Landau-Ginzburg approach enables us to provide a target space interpretation for the boundary states. The boundary states are obtained by applying Cardy’s procedure to combinations of characters in the Gepner models which are invariant under spectral flow. We are able to relate the two descriptions using the common discrete symmetries of the two descriptions. We thus provide an extension to the boundary, the bulk correspondence between Landau-Ginzburg orbifolds and the corresponding Gepner models.

TOWARDS MIRROR SYMMETRY AS DUALITY FOR TWO DIMENSIONAL ABELIAN GAUGE THEORIES Superconformal sigma models with Calabi–Yau target spaces described as complete intersection subvarieties in toric varieties can be obtained as the low-energy limit of certain abelian gauge theories in two dimensions. We formulate mirror symmetry for this class of Calabi–Yau spaces as a duality in the abelian gauge theory, giving the explicit mapping relating the two Lagrangians. The duality relates inequivalent theories which lead to isomorphic theories in the low-energy limit. This formulation suggests that mirror symmetry could be derived using abelian duality. The application of duality in this context is complicated by the presence of nontrivial dynamics and the absence of a global symmetry. We propose a way to overcome these obstacles, leading to a more symmetric Lagrangian. The argument, however, fails to produce a derivation of the conjecture.

D-Branes And Mirror Symmetry We study (2, 2) supersymmetric field theories on two-dimensional worldsheet with boundaries. We determine D-branes (boundary conditions and boundary interactions) that preserve half of the bulk supercharges in nonlinear sigma models, gauged linear sigma models, and Landau-Ginzburg models. We identify a mechanism for brane creation in LG theories and provide a new derivation of a link between soliton numbers of the massive theories and R-charges of vacua at the UV fixed point. Moreover we identify Lagrangian submanifolds that arise as the mirror of certain D-branes wrapped around holomorphic cycles of K¨ahler manifolds. In the case of Fano varieties this leads to the explanation of Helix structure of the collection of exceptional bundles and soliton numbers, through Picard-Lefshetz theory applied to the mirror LG theory. Furthermore using the LG realization of minimal models we find a purely geometric realization of Verlinde Algebra for SU(2) level k as intersection numbers of D-branes. This also leads to a direct computation of modular transformation matrix and provides a geometric interpretation for its role in diagonalizing the Fusion algebra.