Toward Realistic Intersecting D-Brane Models We provide a pedagogical introduction to a recently studied class of phenomenologically interesting string models, known as Intersecting D-Brane Models. The gauge fields of the Standard-Model are localized on D-branes wrapping certain compact cycles on an underlying geometry, whose intersections can give rise to chiral fermions. We address the basic issues and also provide an overview of the recent activity in this field. This article is intended to serve non-experts with explanations of the fundamental aspects, and also to provide some orientation for both experts and nonexperts in this active field of string phenomenology.

F-Theory and AdS_3/CFT_2 We construct supersymmetric AdS3 solutions in F-theory, that is Type IIB supergravity with varying axio-dilaton, which are holographically dual to 2d N = (0, 4) superconformal field theories with small superconformal algebra. In F-theory these arise from D3-branes wrapped on curves in the base of an elliptically fibered Calabi-Yau threefold Y3 and correspond to self-dual strings in the 6d N = (1, 0) theory obtained from F-theory on Y3. The non-trivial fibration over the wrapped curves implies a varying coupling of the N = 4 Super-Yang-Mills theory on the D3-branes. We compute the holographic central charges and show that these agree with the field theory and with the anomalies of self-dual strings in 6d. We complement our analysis with a discussion of the dual M-theory solutions and a comparison of the central charges.

N = 1 supersymmetric indices and the four-dimensional A-model We compute the supersymmetric partition function of N = 1 supersymmetric gauge theories with an R-symmetry on M4 ∼= Mg,p × S1, a principal elliptic fiber bundle of degree p over a genus-g Riemann surface, Σg. Equivalently, we compute the generalized supersymmetric index IMg,p, with the supersymmetric three-manifold Mg,p as the spatial slice. The ordinary N = 1 supersymmetric index on the round three-sphere is recovered as a special case. We approach this computation from the point of view of a topological A-model for the abelianized gauge fields on the base Σg. This A-model—or A-twisted two-dimensional N = (2, 2) gauge theory—encodes all the information about the generalized indices, which are viewed as expectations values of some canonically-defined surface defects wrapped on T2 inside Σg × T2. Being defined by compactification on the torus, the A-model also enjoys natural modular properties, governed by the four-dimensional ’t Hooft anomalies. As an application of our results, we provide new tests of Seiberg duality. We also present a new evaluation formula for the three-sphere index as a sum over two-dimensional vacua.

Mirror Symmetry Constructions The term “mirror symmetry” is used to refer to a wide array of phenomena in mathematics and physics, and there is no consensus as to its precise definition. In general, it refers to a correspondence that maps objects of a certain type— manifolds, for example, or polynomials— to objects of a possibly different type in such a way that the “A-model” of the first object is exchanged with the “B-model” of its image. The phrases “A-model” and “B-model” originate in physics, and the various definitions of mirror symmetry arise from different ideas about the mathematical data that these physical notions are supposed to capture. The Calabi-Yau A-model, for example, encodes deformations of the K¨ahler structure of a Calabi-Yau manifold, while the Calabi-Yau Bmodel encodes deformations of its complex structure. There is also a Landau-Ginzburg A-model and B-model, which are associated to a polynomial rather than a manifold, and which are somewhat less geometric in nature. The versions of mirror symmetry that we will consider in this course are: • The Batyrev construction, which interchanges the Calabi-Yau Amodel of a manifold and the Calabi-Yau B-model of its mirror manifold; • The Hori-Vafa construction, which interchanges the Calabi-Yau (or, more generally, semi-Fano) A-model of a manifold and the Landau-Ginzburg B-model of its mirror polynomial; • The Berglund-H¨ubsch-Krawitz construction, which interchanges the Landau-Ginzburg A-model of a polynomial and the LandauGinzburg B-model of its mirror polynomial. In each case, mirror symmetry is a conjectural equivalence between the sets of data encoded by the two models. In full generality it remains a conjecture, but many cases are known to hold. The CalabiYau/Calabi-Yau mirror symmetry, for example, has been proven whenever the Calabi-Yau manifold X is a complete intersection in a toric variety, and in some cases when X is a complete intersection in a more general GIT quotient. We should note that, in these notes, mirror symmetry will only be discussed as an interchange of cohomology groups (or “state spaces”) on the A-side and B-side. At least in the Calabi-Yau case, however, both the A-model and the B-model are understood to capture much more data than these vector spaces alone. The Calabi-Yau A-model, for example, can be encoded in terms of Gromov-Witten theory. The structure of the notes is as follows. In Chapter 0, we will review the fundamentals of toric geometry, which are necessary to explain the Batyrev construction. Chapters 1, 2, and 3 develop the three forms of mirror symmetry outlined above. The Appendix reviews the basics of Chen-Ruan cohomology, a cohomology theory for orbifolds that is needed in order to define the state spaces of the Calabi-Yau A- and B-model, and that also provides a useful parallel to the definition of the states spaces in Landau-Ginzburg theory.

The Witten equation, mirror symmetry and quantum singularity theory For any non-degenerate, quasi-homogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a corresponding cohomological field theory associated to the singularity. This theory is analogous to Gromov-Witten theory and generalizes the theory of r-spin curves, which corresponds to the simple singularity Ar−1. We also resolve two outstanding conjectures of Witten. The first conjecture is that ADE-singularities are self-dual; and the second conjecture is that the total potential functions of ADE-singularities satisfy corresponding ADE-integrable hierarchies. Other cases of integrable hierarchies are also discussed.

Symplectic Gromov-Witten invariants Originally Gromov-Witten (GW-) invariants belonged to the realm of symplectic rather than algebraic geometry. For a smooth projective variety X, GW-invariants “count” algebraic curves with certain incidence conditions, but in a rather refined way. Salient features are (1) in unobstructed situations, i.e. if the relevant moduli spaces of algebraic curves are smooth of the expected dimension (“expected” by looking at the Riemann-Roch theorem), one obtains the number that one would naively expect from algebraic geometry. A typical such example is the number of plane rational curves of degree d passing through 3d−1 generic closed points, which is in fact a finite number. (2) GW-invariants are constant under (smooth projective) deformations of the variety. For the original definition one deforms X as almost complex manifold and replaces algebraic by pseudo-holomorphic curves (i.e. holomorphic with respect to the almost complex structure). For a generic choice of almost complex structure on X the relevant moduli spaces of pseudo-holomorphic curves are oriented manifolds of the expected dimension, and GW-invariants can be defined by naive counting. Not every almost complex structure J is admitted though, but (for compactness results) only those that are tamed by a symplectic form ω, which by definition means ω(v, Jv) > 0 for any nonzero tangent vector v ∈ TX. In the algebraic case, if J is sufficiently close to the integrable structure, ω may be chosen as pull-back of the Fubini-Study form. It turns out that GW-invariants really depend only on (the deformation class of) the symplectic structure, hence are symplectic in nature. Since in the original definition singular curves are basically neglected, GW-invariants were bound to projective manifolds with numerically effective anticanonical bundle. More recently the situation has changed with the advent of a beautiful, purely algebraic and completely general theory of GW-invariants based on an idea of Li and Tian [Be1] [BeFa], [LiTi1]. This development is surveyed in [Be2]. Due to the independent effort of many there is now also a completely general definition of symplectic GW-invariants available [FkOn] [LiTi2] [Ru2] [Si1]. The purpose of the present paper is to supplement Behrend’s contribution to this volume by the symplectic point of view. We will also sketch the author’s more recent proof of equivalence of symplectic and algebraic GW-invariants for projective manifolds. While it is perfect to have a purely algebraic theory, I believe that the symplectic point of view is still rewarding, even if one is not interested in symplectic questions: Apart from the aesthetic appeal, which the interplay between geometric and algebraic methods usually has, it is sometimes easier and more instructive to use symplectic techniques (if only as preparation for an algebraic treatment). In [Si3, Prop. 1.1] I gave an example of GW-invariants of certain projective bundles, that are much better accessible by symplectic techniques. I also find the properties of GW-invariants, most prominently deformation invariance, intuitively more apparent from the symplectic side, cf. also Section 4.2 (but this might be a matter of taste). More philosophically, the symplectic nature of enumerative invariants in algebraic geometry should mean something, especially in view of their appearance in 2 mirror symmmetry. Finally, it is important to establish algebraic techniques for the computation of symplectic invariants. In fact, a closed formula for GW-invariants, holding in even the most degenerate situations, can be easily derived from the definition, cf. [Si2]. The formula involves only Fulton’s canonical class of the moduli space and the Chern class of a virtual bundle. Gromov-Witten (GW) invariants have a rather interesting and involved history, with connections to gauge theory, quantum field theory, symplectic geometry and algebraic geometry. One referee encouraged me to include some remarks on this. I would like to point out that I concentrate only on the history of defining these invariants rather than the many interesting applications and computations. The story begins with Gromov’s seminal paper of 1985 [Gv]. In this paper Gromov laid the foundations for a theory of (pseudo-) holomorphic curves in almost complex manifolds. Of course, a notion of holomorphic maps between almost complex manifolds existed already for a long time. Gromov’s points were however that (1) while there might not exist higher dimensional almost complex submanifolds or holomorphic functions even locally, there are always many local holomorphic curves (2) the local theory of curves in almost complex manifolds largely parallels the theory in the integrable case, i.e. on Cn with the standard complex structure (Riemann removable singularities theorem, isolatedness of singular points and intersections, identity theorem) (3) to get good global properties one should require the existence of a “taming ” symplectic form ω (a closed, non-degenerate two-form) with ω(v, Jv) > 0 for any nonzero tangent vector v (J the almost complex structure). In fact, in the tamed setting, Gromov proves a compactness result for spaces of pseudo-holomorphic curves in a fixed homology class. At first sight the requirement of a taming symplectic form seems to be merely a technical one. However, Gromov turned this around and observed that given a symplectic manifold (M, ω), the space of almost complex structures tamed by ω is always nonempty and connected. With the ideas of gauge theory just having come up, Gromov studied moduli spaces of pseudo-holomorphic curves in some simple cases for generic tamed almost complex structures. One such case was pseudo-holomorphic curves homologous to IP1 × {pt} on IP1 × T with T an n-dimensional (compact) complex torus. He shows that for any almost complex structure on IP1 × T tamed by the product symplectic structure there exists such a pseudo-holomorphic curves. In nowadays terms he shows that the associated GW-invariant is nonzero. This can then be used to prove his famous squeezing theorem: The symplectic ball of radius r can not be symplectically embedded into the cylinder B2 R × Cn for R < r. Several more applications of pseudo-holomorphic curves to the global structure of symplectic manifolds were already given in Gromov’s paper, and many more have been given in the meantime. The probably most striking one is however due to Floer [Fl]. He interpreted the Cauchy-Riemann equation of pseudo-holomorphic curves as flow lines of a functional on a space of maps from the circle S1 to the manifold. He can then do Morse theory on this space of maps. The homology of the associated Morse complex is the celebrated (symplectic) Floer homology, which has been used 3 to solve the Arnold conjecture on fixed points of nondegenerate Hamiltonian symplectomorphisms. I mention Floer’s work also because it is in the (rather extended) introduction to [Fl] that a (quantum) product structure on the cohomology of a symplectic manifold makes its first appearance (and is worked out for IPn ). As we now (almost) know [RuTi2] [PiSaSc] this agrees with the product structure defined via GW-invariants, i.e. quantum cohomology. An entirely different, albeit related, development took place in physics. Witten [Wi1] observed from Floer’s instanton homology, a homology theory developed by Floer in analogy to the symplectic case for gauge theory on three manifolds, that one can formulate supersymmetric gauge theory on closed four-manifolds, provided one changes the definition of the fields in an appropriate way (“twisting procedure”). The result is a physical theory that reproduces Donaldson’s polynomial invariants as correlation functions. Because the latter are (differential-) topological invariants, the twisted theory is referred to as topological quantum field theory. In [Wi2] Witten applied the twisting procedure to non-linear sigma models instead of gauge theory. Such a theory is modelled on maps from a Riemann surface to a closed, almost complex manifold. The classical extrema of the action functional are then pseudoholomorphic maps. The correlation functions of the theory are physical analogs of GW-invariants. Witten was the first to observe much of the rich algebraic structure that one expects for these correlation functions from degenerations of Riemann surfaces [Wi3]. It is a curious fact that while simple versions of GW-invariants were used as a tool in symplectic topology, and the technical prerequisites for a systematic treatment along the lines of Donaldson theory were all available (notably through the work of McDuff, the compactness theorem by Gromov, Pansu, Parker/Wolfson and Ye), it was only in 1993 that Ruan tied up the loose ends [Ru1] and defined symplectic invariants based on moduli spaces of pseudo-holomorphic (rational) curves, mostly for positive symplectic manifolds. It was quickly pointed out to him that one of his invariants was the mathematical analogue of correlation functions in Witten’s topological sigma model. At the end of 1993 the breakthrough in the mathematical development of GW-invariants and their relations was achieved by Ruan and Tian in the important paper [RuTi1]. Apart from special cases (complex homogeneous manifolds), up until recently the methods of Ruan and Tian were the only available to make precise sense of GW-invariants for a large class of manifolds (semi-positive) including Fano and Calabi-Yau manifolds, and to establish relations between them, notably associativity of the quantum product and the WDVV equation. And many of the deeper developments in GW-theory used these methods, including Taubes’ relationship between GW-invariants and Seiberg-Witten invariants of symplectic four-manifolds [Ta], as well as Givental’s proof of the mirror conjecture for the quintic via equivariant GW-invariants [Gi]. For the case of positive symplectic manifolds proofs for the gluing theorem for two rational pseudo-holomorphic curves, which is the reason for associativity of the quantum product, were also given by different methods in the PhD thesis of G. Liu [Lu] and in the lecture notes [McSa]. Early in 1994 Kontsevich and Manin advanced the theory in a different direction 4 [KoMa]: Rather than proving the relations among GW-invariants, they formulated them as axioms and investigated their formal behaviour. They introduced a rather big compactification of the moduli space of maps from a Riemann surface by “stable maps” (cf. Def. 1.1 below). With this choice all relations coming from degenerations of domains can be formulated in a rather regular and neat way. In the algebraic setting spaces of stable maps have projective algebraic coarse moduli spaces [FuPa]; fine moduli spaces exist in the category of Deligne-Mumford stacks [BeMa]. Another plus is the regular combinatorial structure that allows to employ methods of graph theory to compute GW-invariants in certain cases. No suggestion was made however of how to address the problem of degeneracy of moduli spaces, that in the Ruan/Tian approach applied to projective algebraic manifolds forces the use of general almost complex structures rather than the integrable one. This problem was only solved in the more recent references given above, first in the algebraic and finally in the symplectic category, by constructing virtual fundamental classes on spaces of stable maps. Here is an outline of the paper: We start in the first chapter with a simple model case to discuss both the traditional approach and the basic ideas of [Si1]. Chapter 2 is devoted to the most technical part of my approach, the construction of a Banach orbifold containing the moduli space of pseudo-holomorphic curves. The ambient Banach orbifold will be used in Chapter 3 to construct the virtual fundamental class on the moduli space. The fourth chapter discusses the properties of GW-invariants, that one obtains easily from the virtual fundamental class. We follow here the same framework as in [Be2], so a comparison is easily possible. A fairly detailed sketch of the equivalence with the algebraic definition is given in the last chapter. The proof shows that the obstruction theory chosen in the algebraic context is natural also from the symplectic point of view. For this chapter we assume some understanding of the algebraic definition. After this survey had been finished, the author received a similar survey by Li and Tian [LiTi3], in which they also announce a proof of equivalence of symplectic and algebraic Gromov-Witten invariants. A little warning is in order: The symplectic definition of GW-invariants is more involved than the algebraic one. Modulo checking the axioms and the formal apparatus needed to do things properly, the latter can be given a rather concise treatment, cf. [Si2]. But as long as symplectic GW-invariants are based on pseudo-holomorphic curves, even to find local embeddings of the moduli space into finite dimensional manifolds (“Kuranishi model”) means a considerable amount of technical work. In this survey I tried to emphasize ideas and the reasons for doing things in a particular way, but at the same time keep the presentation as non-technical as possible. While we do not assume any knowledge of symplectic geometry or GW-theory, the ideal reader would have some basic acquaintance with the traditional approach, e.g. from [McSa]. Whoever feels uneasy with symplectic manifolds is invited to replace the word “symplectic” by “Kahler”.

THE WITTEN EQUATION AND ITS VIRTUAL FUNDAMENTAL CYCLE We study a system of nonlinear elliptic PDEs associated with a quasi-homogeneous polynomial. These equations were proposed by Witten as the replacement for the CauchyRiemann equation in the singularity (Landau-Ginzburg) setting. We introduce a perturbation to the equation and construct a virtual cycle for the moduli space of its solutions. Then, we study the wall-crossing of the deformation of the virtual cycle under perturbation and match it to classical Picard-Lefschetz theory. An extended virtual cycle is obtained for the original equation. Finally, we prove that the extended virtual cycle satisfies a set of axioms similar to those of Gromov-Witten theory and r-spin theory.

Timelike duality, M′-theory and an exotic form of the Englert solution Through timelike dualities, one can generate exotic versions of M-theory with different spacetime signatures. These are the M∗-theory with signature (9, 2, −), the M′- theory, with signature (6, 5, +) and the theories with reversed signatures (1, 10, −), (2, 9, +) and (5, 6, −). In (s, t, ±), s is the number of space directions, t the number of time directions, and ± refers to the sign of the kinetic term of the 3 form. The only irreducible pseudo-riemannian manifolds admitting absolute parallelism are, besides Lie groups, the seven-sphere S7 ≡ SO(8)/SO(7) and its pseudo-riemannian version S3,4 ≡ SO(4, 4)/SO(3, 4). [There is also the complexification SO(8, C)/SO(7, C), but it is of dimension too high for our considerations.] The seven-sphere S7 ≡ S 7,0 has been found to play an important role in 11-dimensional supergravity, both through the Freund-Rubin solution and the Englert solution that uses its remarkable parallelizability to turn on non trivial internal fluxes. The spacetime manifold is in both cases AdS4×S7. We show that S3,4 enjoys a similar role in M′-theory and construct the exotic form AdS4×S 3,4 of the Englert solution, with non zero internal fluxes turned on. There is no analogous solution in M∗-theory.

Edward Witten: QUANTUM BACKGROUND INDEPENDENCE IN STRING THEORY Not only in physical string theories, but also in some highly simplified situations, background independence has been difficult to understand. It is argued that the “holomorphic anomaly” of Bershadsky, Cecotti, Ooguri, and Vafa gives a fundamental explanation of some of the problems. Moreover, their anomaly equation can be interpreted in terms of a rather peculiar quantum version of background independence: in systems afflicted by the anomaly, background independence does not hold order by order in perturbation theory, but the exact partition function as a function of the coupling constants has a background independent interpretation as a state in an auxiliary quantum Hilbert space. The significance of this auxiliary space is otherwise unknown.

Mirror Symmetry, Mirror Map and Applications to Calabi-Yau Hypersurfaces Mirror Symmetry, Picard-Fuchs equations and instanton corrected Yukawa couplings are discussed within the framework of toric geometry. It allows to establish mirror symmetry of Calabi-Yau spaces for which the mirror manifold had been unavailable in previous constructions. Mirror maps and Yukawa couplings are explicitly given for several examples with two and three moduli.

Mirror symmetry and supersymmetry on SU(4)-structure backgrounds We revisit the backgrounds of type IIB on manifolds with SU(4)-structure and discuss two sets of solutions arising from internal geometries that are complex and symplectic respectively. Both can be realized in terms of generalized complex geometry. We identify a map which relates the complex and symplectic supersymmetric systems. In the semi-flat torus bundle setting this map corresponds to T-duality and suggest a way of extending the mirror transform to non-K¨ahler geometries.

On mirror symmetry for Calabi-Yau fourfolds with three-form cohomology We study the action of mirror symmetry on two-dimensional N = (2, 2) effective theories obtained by compactifying Type IIA string theory on Calabi-Yau fourfolds. Our focus is on fourfold geometries with non-trivial three-form cohomology. The couplings of the massless zero-modes arising by expanding in these forms depend both on the complex structure deformations and the K¨ahler structure deformations of the Calabi-Yau fourfold. We argue that two holomorphic functions of the deformation moduli capture this information. These are exchanged under mirror symmetry, which allows us to derive them at the large complex structure and large volume point. We discuss the application of the resulting explicit expression to F-theory compactifications and their weak string coupling limit. In the latter orientifold settings we demonstrate compatibility with mirror symmetry of Calabi-Yau threefolds at large complex structure. As a byproduct we find an interesting relation of no-scale like conditions on K¨ahler potentials to the existence of chiral and twisted-chiral descriptions in two dimensions.

Branes And Quantization The problem of quantizing a symplectic manifold (M, ω) can be formulated in terms of the A-model of a complexification of M. This leads to an interesting new perspective on quantization. From this point of view, the Hilbert space obtained by quantization of (M, ω) is the space of (Bcc, B′) strings, where Bcc and B′ are two A-branes; B′ is an ordinary Lagrangian A-brane, and Bcc is a space-filling coisotropic A-brane. B′ is supported on M, and the choice of ω is encoded in the choice of Bcc. As an example, we describe from this point of view the representations of the group SL(2, R). Another application is to Chern-Simons gauge theory.

Ooguri-Vafa Invariants and Off-shell Superpotentials of Type II/F-theory compactification In this paper, we make a further step of [1] and calculate off-shell superpotential of two Calabi-Yau manifold with three parameters by integrating the period of subsystem. We also obtain the Ooguri-Vafa invariants with open mirror symmetry.

D-brane Superpotentials and Ooguri-Vafa Invariants of Compact Calabi-Yau Threefolds We calculate the D-brane superpotentials for two non-Fermat type compact Calabi-Yau manifolds which are the hypersurface of degree 14 in the weighed projective space P(1,1,2,3,7) and the hypersurface of degree 8 in the weighed projective space P(1,1,1,2,3) in type II string theory respectively. By constructing the open-closed mirror maps, we also compute the Ooguri-Vafa invariants, which are related to the open Gromov-Witten invariants.

Cohomology of Heisenberg-Virasoro conformal algebra  The notion of Lie conformal algebra, introduced by Kac in [5], encodes an axiomatic description of the operator product expansion of chiral fields in conformal field theory. In a more general context, a Lie conformal algebra is just an algebra in the pseudotensor category [1]. Closely related to vertex algebras, Lie conformal algebras have many applications in other areas of algebras and integrable systems. In particular, they give us powerful tools for the study of infinite-dimensional Lie (super)algebras and associative algebras (and their representations), satisfying the sole locality property [7]. The main examples of such Lie algebras are those based on the punctured complex plane, such as the Virasoro algebra and the loop Lie algebras [4]. In addition, Lie conformal algebras resemble Lie algebras in many ways [6, 9, 10, 12, 13]. A general cohomology theory of conformal algebras with coefficients in an arbitrary conformal module was developed in [2], where explicit computations of cohomologies for the Virasoro conformal algebra and current conformal algebra were given. The low-dimensional cohomologies of the general Lie conformal algebras gcN were studied in [8]. The cohomologies of the W(2, 2)-type conformal algebra with trivial coefficients were completely determined in [11]. In this paper, we study the cohomology of the Heisenberg-Virasoro conformal algebra, which was introduced in [10] as a Lie conformal algebra associated with the twisted Heisenberg-Virasoro Lie algebra.

On Instanton Superpotentials, Calabi-Yau Geometry, and Fibrations In this paper we explore contributions to non-perturbative superpotentials arising from instantons wrapping effective divisors in smooth Calabi-Yau four-folds. We concentrate on the case of manifolds constructed as complete intersections in products of projective spaces (CICYs) or generalizations thereof (gCICYs). We systematically investigate the structure of the cone of effective (algebraic) divisors in the four-fold geometries and employ the same tools recently developed in [1] to construct more general instanton geometries than have previously been considered in the literature. We provide examples of instanton configurations on Calabi-Yau manifolds that are elliptically and K3-fibered and explore their consequences in the context of string dualities. The examples discussed include manifolds containing infinite families of divisors with arithmetic genus, χ(D, OD) = 1 and superpotentials exhibiting modular symmetry.

Introduction to Seiberg-Witten Theory and its Stringy Origin We give an elementary introduction to the recent solution of N = 2 supersymmetric Yang-Mills theory. In addition, we review how it can be re-derived from string duality.

A MATHEMATICAL THEORY OF WITTEN'S GAUGED LINEAR SIGMA MODEL Abstract. We construct a mathematical theory of Witten’s Gauged Linear Sigma Model (GLSM). Our theory applies to a wide range of examples, including many cases with non-Abelian gauge group. Both the Gromov-Witten theory of a Calabi-Yau complete intersection X and the Landau-Ginzburg dual (FJRW-theory) of X can be expressed as gauged linear sigma models. Furthermore, the Landau-Ginzburg/Calabi-Yau correspondence can be interpreted as a variation of the moment map or a deformation of GIT in the GLSM. This paper focuses primarily on the algebraic theory, while a companion article [FJR16] will treat the analytic theory.    

Review of M(atrix)-Theory, Type IIB Matrix Model and Matrix String Theory A review of M-(atrix) theory (the BFFS matrix quantum mechanics), type IIB matrix model (the IKKT matrix model) and Matrix String Theory (the DVV matrix gauge theory) is presented.

Mirror symmetry, D-brane superpotentials and Ooguri–Vafa invariants of Calabi–Yau manifolds The D-brane superpotential is very important in the low energy effective theory. As the generating function of all disk instantons from the worldsheet point of view, it plays a crucial role in deriving some important properties of the compact Calabi–Yau manifolds. By using the generalized GKZ hypergeometric system, we will calculate the D-brane superpotentials of two non-Fermat type compact Calabi–Yau hypersurfaces in toric varieties, respectively. Then according to the mirror symmetry, we obtain the A-model superpotentials and the Ooguri–Vafa invariants for the mirror Calabi–Yau manifolds.

Special geometry of local Calabi-Yau manifolds and superpotentials from holomorphic matrix models We analyse the (rigid) special geometry of a class of local Calabi-Yau manifolds given by hypersurfaces in C4 as W0(x)2 + f0(x) + v2 + w2 + z2 = 0, that arise in the study of the largeN duals of four-dimensional N = 1 supersymmetric SU(N) Yang-Mills theories with adjoint field Φ and superpotential W(Φ). The special geometry relations are deduced from the planar limit of the corresponding holomorphic matrix model. The set of cycles is split into a bulk sector, for which we obtain the standard rigid special geometry relations, and a set of relative cycles, that come from the non-compactness of the manifold, for which we find cut-off dependent corrections to the usual special geometry relations. The (cutoff independent) prepotential is identified with the free energy of the holomorphic matrix model in the planar limit. On the way, we clarify various subtleties pertaining to the saddle point approximation of the holomorphic matrix model. A formula for the superpotential of IIB string theory with background fluxes on these local Calabi-Yau manifolds is proposed that is based on pairings similar to the ones of relative cohomology.

Graded 3-Calabi-Yau algebras as Ore extensions of 2-Calabi-Yau algebras We study a class of graded algebras obtained from Ore extensions of graded Calabi-Yau algebras of dimension 2. It is proved that these algebras are graded Calabi-Yau and graded coherent. The superpotentials associated to these graded Calabi-Yau algebras are also constructed.

Moment maps, monodromy and mirror manifolds Via considerations of symplectic reduction, monodromy, mirror symmetry and Chern-Simons functionals, a conjecture is proposed on the existence of special Lagrangians in the hamiltonian deformation class of a given Lagrangian submanifold of a Calabi-Yau manifold. It involves a stability condition for graded Lagrangians, and can be proved for the simple case of T2.

Calabi-Yau Manifolds, Hermitian Yang-Mills Instantons and Mirror Symmetry We address the issue why Calabi-Yau manifolds exist with a mirror pair. We observe that the irreducible spinor representation of the Lorentz group Spin(6) requires us to consider the vector spaces of two-forms and four-forms on an equal footing. The doubling of the two-form vector space due to the Hodge duality doubles the variety of six-dimensional spin manifolds. We explore how the doubling is related to the mirror symmetry of Calabi-Yau manifolds. Via the gauge theory formulation of six-dimensional Riemannian manifolds, we show that the curvature tensor of a Calabi-Yau manifold satisfies the Hermitian Yang-Mills equations on the Calabi-Yau manifold. Therefore the mirror symmetry of Calabi-Yau manifolds can be recast as the mirror pair of Hermitian Yang-Mills instantons. We discuss the mirror symmetry from the gauge theory perspective.

Kähler potential of heterotic orbifolds with multiple Kähler moduli String theory is possibly the prime candidate for an UV-complete and unified description of particle physics and cosmology. In order to make contact with our observable 4D universe, expecting N = 1 supersymmetry at some scale, the six extra dimensions predicted by the theory must be compactified on Calabi-Yau manifolds or orbifolds. Below the compactification scale, the features of the resulting SUGRA theory are entirely determined by the geometry and topology of the compactification space. Thus, one expects to identify a compactification space that yields the standard models of particle physics and cosmology. Toroidal orbifold compactifications of the E8×E8 heterotic string1 have shown to lead to large sets of models that, beyond reproducing properties of the MSSM, provide plausible solutions to problems of particle physics [5, 6, 7, 8, 9, 10]. However, to study the cosmological consequences of this scheme one has still to overcome the hurdles posed by moduli stabilization. Bulk moduli are complex scalar fields parametrizing the size and shape of the compact space; they have a flat potential at tree level in the effective theory of these models, leading thus to undesirable properties, such as unobserved fifth forces. Efforts to solve this issue have been made [11, 12], revealing that it is still unclear whether orbifold compactifications can lead to stable 4D vacua. Addressing moduli stabilization requires knowing the details of the associated effective field theory. In N = 1 preserving heterotic orbifolds, one then needs to determine the superpotential, the gauge kinetic functions and the K¨ahler potential. The former two have been computed2 even considering some non-perturbative contributions [17, 18, 19, 20, 21]. The K¨ahler potential is less known because, among other things, it is not protected by non-renormalization theorems [22], although it has been explicitly obtained at leading order in some cases [23, 24].

Edward Witten: More On Gauge Theory And Geometric Langlands The geometric Langlands correspondence was described some years ago in terms of S-duality of N = 4 super Yang-Mills theory. Some additional matters relevant to this story are described here. The main goal is to explain directly why an A-brane of a certain simple kind can be an eigenbrane for the action of ’t Hooft operators. To set the stage, we review some facts about Higgs bundles and the Hitchin fibration. We consider only the simplest examples, in which many technical questions can be avoided.

EM-Duality And The Geometric Langlands Program The geometric Langlands program can be described in a natural way by compactifying on a Riemann surface C a twisted version of N = 4 super Yang-Mills theory in four dimensions. The key ingredients are electric-magnetic duality of gauge theory, mirror symmetry of sigma-models, branes, Wilson and ’t Hooft operators, and topological field theory. Seemingly esoteric notions of the geometric Langlands program, such as Hecke eigensheaves and D-modules, arise naturally from the physics.

Edward Frenkel, Edward Witten: Geometric Endoscopy and Mirror Symmetry The geometric Langlands correspondence has been interpreted as the mirror symmetry of the Hitchin fibrations for two dual reductive groups. This mirror symmetry, in turn, reduces to T–duality on the generic Hitchin fibers, which are smooth tori. In this paper we study what happens when the Hitchin fibers on the B-model side develop orbifold singularities. These singularities correspond to local systems with finite groups of automorphisms. In the classical Langlands Program local systems of this type are called endoscopic. They play an important role in the theory of automorphic representations, in particular, in the stabilization of the trace formula. Our goal is to use the mirror symmetry of the Hitchin fibrations to expose the special role played by these local systems in the geometric theory. The study of the categories of A-branes on the dual Hitchin fibers allows us to uncover some interesting phenomena associated with the endoscopy in the geometric Langlands correspondence. We then follow our predictions back to the classical theory of automorphic functions. This enables us to test and confirm them. The geometry we use is similar to that which is exploited in recent work by B.-C. Ngˆo, a fact which could be significant for understanding the trace formula.

Stringy Gravity: Solving the Dark Problems at ‘short’ distance Dictated by Symmetry Principle, string theory predicts not General Relativity but its own gravity which assumes the entire closed string massless sector to be geometric and thus gravitational. In terms of R/(MG), i.e. the dimensionless radial variable normalized by mass, Stringy Gravity agrees with General Relativity toward infinity, but modifies it at short distance. At far short distance, gravitational force can be even repulsive. These may solve the dark matter and energy problems, as they arise essentially from small R/(MG) observations: long distance divided by much heavier mass. We address the pertinent differential geometry for Stringy Gravity, stringy Equivalence Principle, stringy geodesics and the minimal coupling to the Standard Model. We highlight the notion of ‘doubled-yet-gauged’ coordinate system, in which a gauge orbit corresponds to a single physical point and proper distance is defined between two gauge orbits by a path integral.