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.

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