The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles - IMS

The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles
(07 Jul - 29 Aug 2014)

~ Abstracts ~

Gravitating vortices and coupled Kahler-Yang-Mills equations
Luis Álvarez-Cónsul, Institute of Mathematical Sciences, ICMAT, Spain

After explaining the relation between vortices on a Riemann surface and instantons on a Kähler surface via dimensional reduction, as well as more recent non-abelian generalizations, we go on to study gravitating vortices and their relation to certain coupled equations for Kähler metrics and Yang-Mills connections. Joint work with M. Garcia-Fernandez and O. Garcia-Prada.

The Hitchin-Witten connection and quantum Chern-Simons theory for complex gauge groups
Jørgen Ellegaard Andersen, Aarhus University, Denmark

In this talk we will construct the 2 dimensional part of the Quantum Chern-Simons theory for the complex gauge group SL(n,C). We will do this by applying geometric quantization of the moduli space of flat SL(n,C) connections on a closed oriented surface. This will devolve to the construction of the Hitchin-Witten connection, which is a mapping class group invariant projectively flat connection in an infinite rank vector bundle over Teichmüller space of the surface. We get in this way a construction of the quantum representations of the mapping class groups of the quantum Chern-Simons theory for the complex gauge group SL(n,C).

Hitchin pairs on a singular curve
Pabitra Barik, Institute of Mathematical Sciences, India

In this talk, we present the moduli problem of rank $2$ torsion free Hitchin pairs of fixed Euler characteristic $\chi$ on a reducible nodal curve. We describe the moduli space of the Hitchin pairs. We define the analogue of the classical Hitchin map and describe the geometry of general Hitchin fibre.

Symplectic quot scheme
Indranil Biswas, Tata Institute of Fundamental Research, India

We construct a symplectic analog of the usual Quot scheme parametrizing torsion quotients of a vector bundle over a curve. Its' properties are investigated.

Wild character varieties and wild mapping class groups
Philip Boalch, École Normale Supérieure and CNRS, France

The wild character varieties are a new class of symplectic/Poisson varieties that generalise the complex character varieties of Riemann surfaces. They were first defined analytically in 1999 and more recently there is a purely algebraic approach generalising the quasi-Hamiltonian framework. I'll describe the main features of this story, including the link to meromorphic Higgs bundles, and the natural generalisations of the notions of "Riemann surface" and "mapping class group" that it leads to.

List of lectures (approx. 40-45 min each)

1) motivation, background and examples
2) wild nonabelian Hodge theory on curves
3) wild character varieties and Stokes local systems
4) wild mapping class groups

Higgs bundles, connections and quivers
Philip Boalch, École Normale Supérieure and CNRS, France

In string theory one is apparently supposed to replace a (Feynman) graph by a (Riemann) surface, to pass from a perturbative picture to a nonperturbative one. In the theory of hyperkahler manifolds there is a class of examples attached to graphs (and some data on the graph)--- the Nakajima quiver varieties, and a class of examples attached to Riemann surfaces (and some data on the surface, to specify the boundary conditions)---the wild Hitchin spaces.

I will talk about these "nonperturbative" hyperkahler manifolds attached to surfaces, and how in some cases they are related to graphs. This yields a new theory of "multiplicative quiver varieties", and enables us to extend work of Okamoto and Crawley-Boevey to see the appearance of many non-affine Kac-Moody Weyl groups and root systems in the theory of connections/Higgs bundles on Riemann surfaces (in contrast to the usual, local, understanding of affine Kac-Moody algebras, in terms of loop algebras).

Higgs bundles, spectral data, and isomorphisms among low dimensional Lie groups.
Steven Bradlow, University of Illinois at Urbana Champaign, USA

We will explore some interesting relations among Higgs bundles, from the point of view of spectral data, that result from special isomorphisms among low dimensional Lie algebras and Lie groups.

Stacky compactifications of minimal resolutions of singularities of type A_k and gauge theory
Ugo Bruzzo, Scuola Internazionale Superiore di Studi Avanzati, Italy

We use stacky compactifications of minimal resolutions of singularities of type A_k (ALE spaces) and a theory of framed sheaves on projective stacks to give rigorous definitions of partition functions for supersymmetric gauge theories on ALE spaces. We show that in dimension 2 the moduli functor for framed sheaves on projective stacks is representable, and is represent by a scheme, for which it is possible to compute the obstruction to smoothness. For the case of the stacky compactifications of A_k ALE spaces, that scheme contains a dense open subvariety which is a moduli spaces for instantons on the ALE spaces with fixed Chern classes and holonomy at infinity.

The resulting partition functions show nice factorization properties with respect to the toric structure of the ALE spaces and it is possible to prove Nekrasov's conjecture for them; in the vanishing limit of the equivariant parameter they define suitable pre-potentials.

Joint work with F. Sala, M. Pedrini and R. Szabo

Geometry of compactified Jacobians
Sebastian Casalaina-Martin, University of Colorado at Boulder, USA

There is a close connection between the geometry of a smooth curve, and the geometry of its Jacobian. In this talk I will discuss joint work with J. Kass and F. Viviani where we investigate the connection between the geometry of a stable curve and its compactified Jacobians, constructed by Caporaso, Oda-Seshadri, Pandharipande, and Simpson. Specifically, we describe the singularities of the spaces, as well as singularities of their theta divisors in the integral case. Applications to the birational geometry of universal compactified Jacobians will also be discussed.

Introduction to stacks II, Definitions and basic examples
Sebastian Casalaina-Martin, University of Colorado at Boulder, USA

Continuing from the first lecture, I will give a brief overview of the definition of an algebraic stack, using various examples for illustration.

Introduction to stacks I, Moduli spaces and motivation
Sebastian Casalaina-Martin, University of Colorado at Boulder, USA

Since their introduction in Deligne and Mumford's work on the moduli space of curves, algebraic stacks have played an important role in the theory of moduli. In this lecture I will discuss several moduli problems, and motivate some of the background that goes into the definition of a stack.

Higgs bundles and fixed points
Brian Collier, University of Illinois , USA

For G a real semisimple Lie group, we will study fixed points of a roots of unity action on the moduli space of G-Higgs bundles. For certain real groups we will classify these fixed points and give families of examples in the Hitchin component for any split real form and also in certain non-Hitchin maximal components for $SP(4,\mathbb{R}).$ If time permits, the relation between fixed points and the harmonic metric solving the Hitchin equations will be discussed.

Donagi-Markman cubic for the G-generalised Hitchin system
Peter Dalakov, Bulgarian Academy of Sciences, Bulgaria

Donagi and Markman have shown that the infinitesimal period map for an algebraic (or analytic) completely integrable Hamiltonian system is encoded in a section of the third symmetric power of the cotangent bundle to the base of the system. For the Hitchin system, this cubic is given by a formula of Balduzzi and Pantev. I will discuss a recent work with U.Bruzzo (IJM, vol 25, #2) where we show that the Balduzzi-Pantev formula also holds along the maximal rank symplectic leaves of the G-generalised Hitchin system.

Essential dimension of the moduli stack of vector bundles
Ajneet Dhillon, The University of Western Ontario, Canada

I will start with a general overview of essential dimension. The powerful genericity theorem of Brosnan-Reichstein-Vistoli will be discussed and applied to the moduli stack of curves. In the final part of the talk, which is joint work with I. Biswas and N. Hoffmann, we discuss the essential dimension of the moduli stack vector bundles over a curve.

3d-3d correspondence, lens spaces, and quantizations of Teichmuller theory
Tudor Dimofte, Institute for Advanced Study, USA

I will review aspects of the "d-3d correspondence" in physics, which associates to a 3-manifold M and a positive integer n a 3d supersymmetric quantum field theory T[M,n]. Various observables in the theory T[M,n] compute classical, quantum, and categorical invariants of SL(n,C) local systems on M. Some new examples include partition functions of T[M,n] on L(k,p) lens spaces, which coincide with partition functions of SL(n,C) Chern-Simons theory at level k on M itself. For k>1 these partition functions provide new quantizations of Teichmuller theory (on the boundary of M). I will also mention some early hints of categorification for SL(n,C) Chern-Simons partition functions that come from studying T[M,n].

Polygons, polynomials, fences, and flows
David Dumas, University of Illinois at Chicago, USA

In a recent paper with Michael Wolf, we use affine differential geometry to construct a homeomorphism between the moduli space of polynomial cubic differentials on the complex plane and the space of projective equivalence classes of convex polygons in RP^2. In this talk I will briefly recall the results of this project, and then focus primarily on discussing various connections, questions, and conjectures suggested by this work. These include an interpretation of our main theorem in terms of a family of rank-3 meromorphic Higgs bundles, a conjectural relation with the Stokes phenomenon, and a question about the Poisson geometry of the space of twisted polygons.

Quantization of spectral curves of Higgs bundles via a B-model topological recursion
Olivia Dumitrescu, University of Hannover, Germany

These lectures are aimed at serving as an introduction to the recent circle of ideas and mathematical works on quantum curves.

What is the mirror dual of Catalan numbers? We start with asking this naïve question. The answer we propose is the Hermite differential equation. Surprisingly, we can construct its exact WKB solution by Catalan numbers and their higher genus analogue. In this process, we encounter the Eynard-Orantin topological recursion, which is a B-model TQFT.

Why Catalan numbers? Do they have anything to do with Higgs bundles? Yes, of course! The geometric framework of the above story is the quantization of spectral curves of Higgs bundles via the generalized Eynard-Orantin theory. We give a mathematical introduction to this powerful technique. The simplest genus 0 cases recover classical equations: the Airy differential equation and its WKB analysis via the Witten-Kontsevich theory, and the Gauss hypergeometric equation and its WKB analysis via the Seiberg-Witten theory. Indeed, the point of view of spectral curves of Higgs bundles provides a simple way of understanding the Witten-Kontsevich theory for the intersection numbers of the moduli of stable curves. This is because the Witten-Kontsevich theory is equivalent to the local theory of a spectral covering near its simple ramification point.

What are the motivations for "quantum" curves? Gukov and Sulkowski proposed a quantization of the "A-polynomials" of knot theory using the Eynard-Orantin topological recursion, and conjectured that the quantum curve should generate the D-module characterizing the Jones polynomials. In search of a rigorous mathematical framework for quantum curves, we have discovered a quantization mechanism for spectral curves of Higgs bundles, by generalizing the Eynard-Orantin theory. We present a mathematical theory based on Hilbert schemes of points of log Calabi-Yau surfaces. The final outcome is a concrete formula for the exact WKB analysis of quantum curves. Its relation to non-Abelian Hodge correspondence and geometric Langlands correspondence will be discussed.

The lectures are based on our joint work with Motohico Mulase, Vincent Bouchard, Bertrand Eynard, Sergey Shadrin, Piotr Sulkowski, and others.

Stability data, irregular connections and tropical curves
Mario Garcia Fernandez, École Polytechnique Fédérale de Lausanne, Switzerland

I will give an overview of recent joint work with S. Filippini and J. Stoppa, in which we construct isomonodromic families of irregular meromorphic connections \nabla(Z) on P1, with values in the derivations of a class of infinite dimensional Poisson algebras. Our main results concern the limits of the families as we vary a scaling parameter R. In the R \to 0 'conformal limit' we recover a semi-classical version of the connections introduced by Bridgeland and Toledano Laredo (and so the Joyce holomorphic generating functions for DT invariants). In the R \to \infty 'large complex structure limit' the families relate to tropical curves in the plane and tropical/GW invariants. The connections \nabla(Z) are a rough but rigorous approximation to the (mostly conjectural) four-dimensional tt*-connections introduced by Gaiotto-Moore-Neitzke.

Geometry of character varieties of abelian groups
Carlos Florentino, Universidade de Lisboa, Portugal

The description of the space of commuting elements in a compact Lie group is an interesting algebro-geometric problem with applications in Mathematical Physics, notably in Supersymmetric Yang Mills theories. When the Lie group is complex reductive, this space is the character variety of a free abelian group. Let K be a compact Lie group (not necessarily connected) and G be its complexification. We consider, more generally, an arbitrary finitely generated abelian group A, and show that the conjugation orbit space Hom(A,K)/K is a strong deformation retract of the character variety Hom(A,G)/G (this can also be shown for nilpotent groups replacing A, and possibly more generally). As a Corollary, when G is connected and semisimple, we obtain necessary and sufficient conditions for Hom(A,G)/G to be irreducible. This relates to G-Higgs bundles over abelian varieties, to intriguing problems on irreducibility of the variety of k-tuples of n by n commuting matrices, and to the Hilbert scheme of n points on C^k.

Abelianisation of local system
Vladimir Fock, University of Strasbourg, France

Hitchin integrable system is a map which takes a Higgs bundle over a Riemann surface to a collection of homomorphic differentials on the surface and a point of a Prymian of a certain ramified cover of it.

We suggest a construction analogous to the inverse of the Hitchin's one namely an explicit parametrisation of the moduli of decorated flat connections on a punctured surface (instead of Higgs bundles) by a Prymian of the cover.

Involutions of Higgs bundle moduli spaces
Oscar García-Prada, Instituto de Ciencias Matemáticas, Spain

We consider the moduli space of G-Higgs bundles over a compact Riemann surface X, where G is a complex semisimple Lie group, and study various involutions of the moduli space involving involutions in G and X, as well as twisting with Z(G)-bundles, where Z(G) is the centre of G. We describe the fixed points and study their relations with representations of the fundamental group of X in G. We will also comment on the relation of the fixed points with Langlands duality and mirror symmetry for Higgs bundles.

Higgs bundles for the non-compact dual of the special orthogonal group
Peter Gothen, Faculdade de Ciencias da Universidade do Porto, Portugal

We use Higgs bundles to study the character variety for representations of a surface group in the non-compact dual of the special orthogonal group. This talk is based on joint work with Steve Bradlow and Oscar Garcia-Prada.

Introduction to Higgs bundles
Olivier Guichard, Université de Strasbourg, France

These lectures will focus on the "Non Abelian Hodge Theory" (NAHT) which relates the topology (here the representations of the fundamental group) of a compact Kähler manifold and its holomorphic structure (via Higgs bundle, i.e. holomorphic vector bundles with an additional structure).

The relevant definitions and properties will first be presented: smooth, flat and holomorphic vector bundles, connections, pseudo-connections, hermitian structures, etc. The mentioned theory (NAHT) is sometimes called the Hitchin-Kobayashi correspondence and establishes a bijection between the (simple) flat bundles and the (stable and of degree 0) Higgs bundles on a Kähler manifold.

Both directions in this correspondence amount to solving an partial differential equation associated with a functional: the energy functional (from flat to Higgs) and the Yang-Mills functional (from Higgs to flat). The minimizing flows of those functionals (i.e. the heat flow and the Yang-Mills flow) will then enable us to prove existence theorems for the PDEs of concern.

A series of companion problem sessions will be the opportunity to gain familiarity with the introduced objects and also to go beyond the program of the lectures.

Arithmetic of wild character varieties
Tamás Hausel, École Polytechnique Fédérale de Lausanne, Switzerland

I explain some conjectures, originating both in arithmetic and physics, on the mixed Hodge polynomials and perverse Hodge polynomials of tame character varieties and moduli of parabolic Higgs bundles on Riemann surfaces. We will then study the arithmetic of one class of Boalch's wild character varieties using the character theory of Yokonuma-Hecke algebras, and point out the relationship of the point count, and a natural conjecture on their mixed Hodge polynomials to the tame case. Finally I will discuss the twisted case, where torus knots through their HOMFLY homology will appear naturally in our formulas. This is joint work with Martin Mereb and Michael Wong.

A compactness theorem for the Seiberg-Witten equations with multiple spinors
Andriy Haydys, Universität Bielefeld, Germany

This is a joint project with Th. Walpuski. Motivated by higher dimensional gauge theory, we consider the compactness problem for the Seiberg-Witten equations with multiple spinors in dimension three. We show that a sequence of solutions of the Seiberg-Witten equations has a subsequence converging to a Fueter section, which is a non-linear version of a harmonic spinor.

On the intersection form on moduli spaces of Higgs bundles
Jochen Heinloth, Universität Duisburg-Essen, Germany

T. Hausel and F. Rodriguez-Villegas conjectured that the intersection form on the moduli space of PGL_n Higgs bundles vanishes if rank and degree are coprime.
In this talk I will report on recent progress towards this conjecture.

Stratifications for moduli of sheaves and quiver representations
Victoria Hoskins, Freie Universität Berlin, Germany

Many moduli spaces are constructed via geometric invariant theory and in this talk we focus on moduli of coherent sheaves and moduli of quiver representations. In both cases, we compare two stratifications: a stratification coming from the GIT construction and a Harder-Narasimhan stratification. We see that for quivers both stratifications coincide, but this is not quite true for sheaves. We explain why this is the case and what can be done to rectify this. If time permits, we will relate these stratifications for sheaves and quiver representations via a construction of çlvarez-C-nsul and King.

Toric GIT
David Donghoon Hyeon, Pohang University of Science and Technology, Korea

Geometric Invariant Theory (GIT) is a tool for constructing quotient varieties. Worked out by D. Mumford in the 1960's, it has been used heavily for constructing various moduli spaces.

Toric varieties can be described completely combinatorially, and they naturally arise as GIT quotients. I will explain basics of GIT and toric varieties, and how the nuts of bolts of GIT can be described combinatorially in the toric setting.

Derived categories of functors and Fourier--Mukai transform for quiver sheaves
Marcos Jardim, Universidade Estadual de Campinas, Brazil

Let C be small category and A an arbitrary category. Consider the category C(A) whose objects are functors from C in A and whose morphisms are natural transformations. Given a functor F : A -- B one obtains an induced functor F_C : C(A) --> C(B). If A and B are abelian categories, we have that C(A) and C(B) are also abelian, and one has two functors R(F_C) : D(C(A)) --> D(C(B)) and (RF)_C : C(D(A)) --> C (D(B)). The goals of this paper are 1) to find a relationship between D(C(A)) and C(D(A)); 2) to relate the functors R(F_ C) and (RF)_C. As an application, we prove a version of Mukai's Theorem for quiver sheaves.

Higgs bundle categories and variations of stability structures
Ludmil Katzarkov, University of California, Irvine, USA

In this talk we look from a new perspective on the classical by now Hitchin system. We will introduce the notion of variations of stability structures.

Minimal surfaces and the complex geometry of Hitchin components
François Labourie, Université Paris-Sud, France

Teichmüller theory is a fascinating interplay between complex geometry and topology: Teichmüller space is both a space of complex structures on a surface $S$ and a connected component $\chi(\mathsf{PSL}(2,\mathbb R))$ of representations of $\pi_1(S)$ in $\mathsf{PSL}(2,\mathbb R)$. For a general (simple real split) group $\mathsf{G}$, a similar connected component $\chi(\mathsf G)$, called the {\em Hitchin component} and its Thurstonian dynamical geometry has attracted a lot of attention recently. Yet, the complex interpretation of the Hitchin component is largely unknown (although conjectured). In this talk, I will explain the complex interpretation of Hitchin components for rank 2 groups as a space of pairs $(J,q)$ where $J$ is a complex structure on $S$ and $q$ a holomorphic differential of an order depending on the group $\mathsf G$.

Topology of moduli spaces of representations
Sean Lawton, University of Texas-Pan American, USA

In this talk we will give a general definition for the moduli space of Lie group valued representations of a discrete group, give examples, and briefly discuss why these spaces are interesting. Then we will present recent theorems concerning the topology of these spaces.

Cubic differentials and limits of convex $RP^2$ strucures under neck pinches
John Loftin, Rutgers University, USA

Given a closed oriented surface $S$ of genus at least 2, fix for now $\Sigma$ a Riemann surface structure on $S$. The Hitchin component of the space of representations of $\pi_1S$ into a simple split real Lie group $G$ with trivial center is shown via a Higgs bundle construction to be diffeomorphic to a vector space of sections of various powers of the canonical bundle over $\Sigma$. The Hitchin component of $G=PSL(3,R)$ is parametrized by the space $H^0(\Sigma,K^2) \oplus H^0(\Sigma,K^3)$ of pairs of quadratic and cubic differentials. For each such Hitchin representation $\rho$ into $PSL(3,R)$, there is a bounded convex domain $\Omega$ in $R^2\subset RP^2$ so that $\rho(\pi_1)$ acts discretely and properly discontinuously on $\Omega$. This construction introduces canonical real projective coordinates on $S$ to make it a convex $RP^2$ surface.

In this case (and in all rank 2 cases, as shown by Labourie), we can ignore the quadratic differential by allowing $\Sigma$ to vary in Teichm\"uller space. In other words, the space of Hitchin representations (or convex $RP^2$ structures) is parametrized by $(\Sigma,U)$ for $\Sigma$ a Riemann surface structure on $S$ and $U$ a cubic differential. These provide natural complex coordinates on the moduli space of convex $RP^2$ structures. We are interested in the boundary behavior as $\Sigma$ degenerates under a neck pinch. In these cases, we can produce from the residue of a regular cubic differential near the neck pinch the structure of the end of the $RP^2$ structure. Our new result is the converse of this statement (start with an appropriate end to an $RP^2$ surface and produce cubic differential with the correct residue). We also will explain some work in progress on degenerating families of $RP^2$ surfaces. The proofs depend on recent techniques developed by Benoist-Hulin and Dumas-Wolf.

Hodge polynomials of the SL(2,C)-character variety of an elliptic curve with two marked points
Marina Logares, Instituto de Ciencias Matemáticas, Spain

We compute Hodge polynomials for the moduli space of representations of an elliptic curve with two marked points into SL(2,C). When we fix the conjugacy classes of the representations around the marked points to be diagonal, the character variety is diffeomorphic to the moduli space of parabolic Higgs bundles for which the Betti numbers are known. In that case we can recover some of the Hodge numbers of the character variety. This is joint work with Vicente Mu-oz.

A dilogarithm identity on the moduli space of curves
Feng Luo, Rutgers University, USA

We establish an identity for closed hyperbolic surfaces whose terms depend on the dilogarithms of the lengths of simple closed geodesics in all 3-holed spheres and 1-holed tori in the surface. This is a joint work with Ser Peow Tan.

Conformal blocks and Kummer surfaces
Johan Martens, The University of Edinburgh, Scotland

Originating in statistical mechanics, bundles of conformal blocks have in the last few decades increasingly become useful in geometry and topology. I will sketch a brief overview of some such developments, all of which are concerned with Òlarge level asymptoticsÓ. I will then try to convince the audience that also sporadic low-level behaviour can be interesting, by focusing on the (WZW) conformal blocks for SU(2) at level 4, in particular for Riemann surfaces of genus 2. This is joint ongoing work with T. Baier and M. Bolognesi.

E-polynomials of SL(2,C)-character varieties
Javier Martinez, Universidad Complutense de Madrid, Spain

The G-character variety of a Riemann surface of genus g is the moduli space parametrizing representations of its fundamental group into G. Twisted character varieties consist of representations of the fundamental group of a punctured surface with fixed holonomy around the puncture. We will describe these varieties and we will show how to compute their E-polynomials for G=SL(2,C) and arbitrary genus, using a geometric technique based on stratifications and fibrations of the moduli space (joint work with V.Muñoz).

Noncommutative K3 surfaces and moduli spaces of sheaves
Sukhendu Mehrotra, Chennai Mathematical Institute, India

Let X be a K3 surface, and X(n) its Hilbert scheme of points. Beauville showed that X(n) is a holomorphic, symplectic manifold with a 21 dimensional family of deformations. The general such deformation is, however, not the Hilbert scheme on any K3. Mukai showed that a similar statements are true for moduli spaces of stable sheaves on X.

Let M be the component of the moduli space of (marked) holomorphic symplectic manifolds containing Hilbert schemes of K3s. In joint work with Eyal Markman, we show how to naturally associate to a "K3 category" to every point in a dense open set of M. This talk discusses this result, together with applications and further questions.

Mixed twistor structure and GKZ-hypergeometric systems
Takuro Mochizuki, Kyoto University, Japan

Mixed twistor D-modules are D-modules with mixed twistor structure. Here, the concept of twistor structure is a generalization of that of Hodge structure, introduced by C. Simpson in his study on harmonic bundles. As in the Hodge case, various operations for D-modules are enhanced to those for mixed twistor D-modules. It implies that some important mathematical objects are naturally equipped with mixed twistor structure. For example, the D-modules associated to families of Laurent polynomials, called the GKZ hypergeometric systems, are naturally equipped with mixed twistor structure. In this talk, after a review of the theory of mixed twistor structure, we shall report our work in progress on potential applications of the mixed twistor structure of the GKZ hypergeometric systems in the study of quantum D-modules.

Quantum Cohomology and conformal blocks divisor
Swarnava Mukhopadhyay, University of Maryland, USA

Conformal blocks are vector bundles on moduli space of curves with marked points that arise naturally in rational conformal field theory. They also arise as global sections of ample line bundles on moduli stack of parabolic G-bundles on a smooth curve. Recent work of Fakhruddin has refocused our attention on conformal blocks and changed our perspective on the birational geometry of the moduli space of genus zero curves with marked points. Conformal blocks give rise to a very interesting family of numerically effective divisors and hence relate to interesting questions on nef cones of moduli spaces of curves. I will describe joint work with Prakash Belkale and Angela Gibney where we study these divisors. I will focus on our vanishing theorems, new symmetries and non-vanishing properties of these divisors (quantum cohomology of Grassmannians is one of our main tools).

Quantization of spectral curves of Higgs bundles via a B-model topological recursion
Motohico Mulase, University of California, Davis, USA

These lectures are aimed at serving as an introduction to the recent circle of ideas and mathematical works on quantum curves.

What is the mirror dual of Catalan numbers? We start with asking this naïve question. The answer we propose is the Hermite differential equation. Surprisingly, we can construct its exact WKB solution by Catalan numbers and their higher genus analogue. In this process, we encounter the Eynard-Orantin topological recursion, which is a B-model TQFT.

Why Catalan numbers? Do they have anything to do with Higgs bundles? Yes, of course! The geometric framework of the above story is the quantization of spectral curves of Higgs bundles via the generalized Eynard-Orantin theory. We give a mathematical introduction to this powerful technique. The simplest genus 0 cases recover classical equations: the Airy differential equation and its WKB analysis via the Witten-Kontsevich theory, and the Gauss hypergeometric equation and its WKB analysis via the Seiberg-Witten theory. Indeed, the point of view of spectral curves of Higgs bundles provides a simple way of understanding the Witten-Kontsevich theory for the intersection numbers of the moduli of stable curves. This is because the Witten-Kontsevich theory is equivalent to the local theory of a spectral covering near its simple ramification point.

What are the motivations for "quantum" curves? Gukov and Su |kowski proposed a quantization of the "A-polynomials" of knot theory using the Eynard-Orantin topological recursion, and conjectured that the quantum curve should generate the D-module characterizing the Jones polynomials. In search of a rigorous mathematical framework for quantum curves, we have discovered a quantization mechanism for spectral curves of Higgs bundles, by generalizing the Eynard-Orantin theory. We present a mathematical theory based on Hilbert schemes of points of log Calabi-Yau surfaces. The final outcome is a concrete formula for the exact WKB analysis of quantum curves. Its relation to non-Abelian Hodge correspondence and geometric Langlands correspondence will be discussed.

The lectures are based on our joint work with Olivia Dumitrescu, Vincent Bouchard, Bertrand Eynard, Sergey Shadrin, Piotr Sulkowski, and others.

Components of moduli spaces of Higgs bundles: an intrinsic approach
Andre Gama Oliveira, Universidade de Trás-os-Montes e Alto Douro, Portugal

Let G be any reductive Lie group and M_G be the moduli space of G-Higgs bundles over a closed Riemann surface X. We give an abstract approach (i.e. without specifying G) to the general techniques used to determine the simplest topological invariant of M_G, namely its connected components. Non-abelian Hodge theory says then that the same information will hold for the space of surface group representations in G. This is joint work in progress with O. García-Prada and P. Gothen.n.

Character varieties and nested Hilbert schemes
Tony Pantev, University of Pennsylvania, USA

I will describe two geometric setups which relate Higgs bundles on curves to Calabi-Yau threefolds. In the first setup the Hitchin system with poles is identified with a Calabi-Yau integrable system, and in the second setup the moduli of parabolic Higgs bundles identified with a moduli of perverse coherent sheaves on a Calabi-Yau threefold. I will discuss the dependence of these identifications on the particular geometric model for the Calabi-Yau geometry and the effect the choice of a model has on the description of stability. The lectures will describe joint works with W.-Y. Chuang, D.-E. Diaconescu and R. Donagi.

Fivebranes and 4-manifolds
Pavel Putrov, California Institute of Technology, USA

In my talk I will consider a correspondence between 4 manifolds and 2d N=(0,2) supersymmetric quantum field theories. I will conjecture some new results about the Vafa-Witten partition function (the generating function of Euler characteristics of instanton moduli spaces) associated to a 4-manifold. This talk is based on a joint work with A. Gadde and S. Gukov.

Quantum vortices on closed surfaces and L^2-invariants
Nuno Romao, University of Gottingen, Germany

I will describe joint work with M. Boekstedt and C. Wegner aiming at uncovering fundamental features of N=(2,2) supersymmetric quantum mechanics on moduli spaces of vortices on compact Riemann surfaces. My focus in this talk will be on the (topological A-twisted) supersymmetric Abelian Higgs model coupling to local systems, for both linear and nonlinear targets; the corresponding ground states can be investigated by means of the theory of L^2-invariants applied to the natural Kaehler metrics on the universal covers of the moduli spaces. I shall explain why the quanta of such Abelian gauge theories can nontrivially realize non-Abelian statistics in particular examples, and motivate a precise conjecture regarding the nonlinear superposition of ground states for rank one.

Representation varieties in real algebraic geometry
Florent Schaffhauser, Universidad de Los Andes, Colombia

The GL(n;C) representation variety associated to a compact Riemann surface X has various real structures of interest, the most immediate of which being perhaps that which fixes representations with values in GL(n; R). When the Riemann surface X itself has a real structure, a new real structure of the GL(n;C) representation variety arises, whose fixed points can be interpreted nicely in terms of representations of the fundamental group of X. In this introductory talk, we study in detail that example.

An introduction to spectral data for Higgs bundles
Laura Schaposnik, University of Illinois at Urbana Champaign, USA

During the first lecture we will define Higgs bundles for complex groups, and construct Higgs bundles for real forms as fixed points of involutions acting on the moduli space of complex pairs. Then, in the second talk we will review how spectral data has been defined for complex Higgs bundles, and extend the idea to real Higgs bundles via the action involutions.

A-branes and B-branes in the moduli space of Higgs bundles
Laura Schaposnik, University of Illinois at Urbana Champaign, USA

Through the hyperkähler structure of the moduli space of Higgs bundles for complex groups we shall construct three natural involutions whose fixed points in the moduli space give branes in the A-model and B-model with respect to each of the complex structures. Then, by means of spectral data and K-theory, we shall look at these branes and their topological invariants, and propose what the dual branes should be under Langlands duality.

Limits and bubbling sets for the Yang-Mills flow on Kaehler manifolds
Ben Sibley, Max-Planck-Institut für Mathematik, Germany

In this talk I will discuss convergence of the Yang-Mills flow for holomorphic vector bundles on Kähler manifolds. In the case that the bundle is stable, it is well known by work of Donaldson and Simpson that this flow converges at infinity to an Hermitian-Einstein connection on the original bundle. If the bundle is unstable one still has a notion of a limit that arises from applying Uhlenbeck compactness to the flow in this setting. The limiting data consists of a singular connection (to which a sequence along the flow converges in a suitable sense) defined on a reflexive sheaf, which is smoothly isomorphic to the original bundle away from a certain singular set of small Hausdorff measure. The connection, together with its associated singular set is called an Uhlenbeck limit, and is not a priori unique, i.e. might depend on the sequence chosen. Around twenty years ago, Bando and Siu conjectured that this should indeed be the case. Namely, that the limit should be canonically determined by complex- geometric properties of the initial data, that is, the Kaehler class of the manifold and the holomorphic structure of the bundle. I will explain the solution to this conjecture, which is naturally broken up into two parts: First the proof of uniqueness of the connection, and second of the singular set. I will emphasise the latter, which is joint work with Richard Wentworth.

Deformations of twisted harmonic maps and variations of the Morse function
Marco Spinaci, Université Joseph Fourier, France

Consider a semisimple representation of a Kähler group to a reductive algebraic group. Infinitesimal deformations of this representation may be represented by adequate cocycles; we study the problem of how the (non-unique) harmonic equivariant map varies with respect to such a deformation. We give explicit constructions up to the second order. This allows us to compute the first and second derivatives of the energy Morse function and to generalize some results due to Hitchin for smooth points of the moduli space of representations of surface groups. To the first order, critical points are complex variations of Hodge structure. To the second order, on the one hand the Morse function is strictly plurisubharmonic; on the other hand, we prove Hitchin's formula for the Morse indices. Finally, we present some ideas of how this formula could help in proving the following conjecture: The representations of cocompact complex hyperbolic lattice in a Lie group of Hermitian type with maximal Toledo invariant are superrigid.

Nahm transform for parabolic connections and Higgs bundles on the Riemann sphere
Szilárd Szabó, Hungarian Academy of Sciences, Hungary

We define Nahm transform analytically and then identify it as an algebraic map with respect to both the de Rham and Dolbeault structures. The Dolbeault interpretation goes through spectral data, and we explain how this implies that the map is a hyper-Kaehler isometry.

Stable sheaves with twisted sections and the Vafa-Witten equations on smooth projective surfaces
Yuuji Tanaka, National Cheng Kung University, Taiwan

This talk describes a Hitchin-Kobayashi style correspondence for the Vafa-Witten equations on smooth projective surfaces.

The Vafa-Witten equations are a set of gauge theoretic equations introduced by Vafa and Witten in 90's for the study of S-duality conjecture of N=4 supersymmetric Yang-Mills theory. Recently, these were picked up also by Haydys and Witten in the context of "categorification" of Khovanov homology.

The correspondence we mention is a one to one correspondence between the existence of a solution to the equations on a locally-free sheaf E on a projective surface X; and a suitable notion of stability for a pair consisting of a locally free sheaf E and a section of a sheaf of endomorphisms of E twisted by the canonical line bundle of X.

This turns out to be a special case of results obtained by Alvarez-Consul and Garcia-Prada as the case of a twisted quiver bundle with one vertex and one arrow, whose head and tail coincide, and with twisting sheaf the anti-canonical bundle.

In this talk, we sketch an alternative proof, which uses a Mehta-Ramanathan style argument originally developed by Donaldson for the Hermitian-Einstein problem, referring to its relation to the Hitchin equations on Riemann surfaces.

Hitchin-Kobayashi correspondence for vortices on non-compact Riemann surfaces
Sushmita Venugopalan, Tata Institute of Fundamental Research, India

A Hitchin-Kobayashi correspondence relates stable holomorphic pairs (a holomorphic section on a Hermitian vector bundle and a holomorphic section on it) with the zeros of the vortex equation. This set up can be generalized - the vector bundle can be replaced by a fiber bundle, whose fibers are a Kaehler manifold with Hamiltonian action of a compact Lie group K. In this talk I present a Hitchin-Kobayashi correspondence for K-vortices on certain Riemann surfaces with inifinite volume. In the particular case of affine vortices - i.e. when the Riemann surface is the complex line, our result has applications in gauged Gromov-Witten theory.

Holography principle and Moishezon twistor spaces
Misha Verbitsky, Higher School of Economics, Russia

Let Tw(M) be a twistor space of a hyperkaehler or ASD 4-manifold M, and S the rational curve in Tw(M) obtained as a fiber of the projection to M. I prove "a holography principle": any meromorphic function defined in a neighbourhood U of S can be extended to a meromorphic function on Tw(M), and any section of a holomorphic line bundle can be extended from U to Tw(M). This is used to define the notion of a Moishezon twistor space. I prove that the twistor spaces of hyperkahler manifolds obtained by finite-dimensional hyperkahler reduction (such as Nakajima's quiver varieties) are always Moishezon.

Ends of the moduli space of Higgs bundles
Frederik Witt, University of Münster, Germany

Hitchin's existence theorem asserts that a stable Higgs bundle of rank $2$ carries a unitary connection satisfying Hitchin's self-duality equation. In this talk we discuss a new proof, via gluing methods, for elements in the ends of the Higgs bundle moduli space and identify a dense open subset of the boundary of the compactification of this moduli space.

Geometry of branes, duality and quantization
Siye Wu, The University of Hong Kong, Hong Kong

Branes are extended objects that define the boundary conditions of sigma models. These lectures cover the geometry of branes, their roles in duality and mirror symmetry, especially in Hitchin's moduli space, and the relation to quantization.

A desingularization of the moduli space of Higgs bundles over a curve
Sang-Bum Yoo, Pohang University of Science and Technology, Korea

Let $X$ be a smooth projective curve $X$ of genus $g\geq 3$. Let $\mathbf{M}$ be the moduli space of semistable rank $2$ Higgs bundles over $X$ with trivial determinant. Let $\mathbf{K}$ be Kirwan's desingularization of $\mathbf{M}$. A smooth variety $\mathbf{K}_\epsilon$ can be obtained after two blow-downs of $\mathbf{K}$. But there is no known information on the moduli theoretic meaning of $\mathbf{K}_\epsilon$. The purpose of this work is to explain the meaning precisely. In this talk, we will explain two results: (a) A construction of a new desingularization $\mathbf{S}$ of $\mathbf{M}$ as an analogous one of Seshadri's desingularization of the moduli space of semistable rank $2$ vector bundles over $X$ with trivial determinant, and (b) $\mathbf{K}_\epsilon\cong\mathbf{S}$.

Quiver invariants and orientifolds
Matthew Young, The University of Hong Kong, Hong Kong

We study self-dual representations of a quiver. These differ from ordinary representations by having classical groups, instead of general linear groups, as symmetries. We compute counting functions for the stack of semistable self-dual representations and describe their wall-crossing under changes in stability. The observed behaviour is consistent with formulas appearing in the physics literature on BPS state counting in the presence of an orientifold. Our primary tool is a representation of the Hall algebra which is a mathematical model for the space of BPS states in an orientifold theory.

GIT Characterization of the Harder-Narasimhan filtration for finite dimensional representations of quivers
Alfonso Zamora-Saiz, Instituto Superior Técnico, Portugal

In a moduli problem where we use Geometric Invariant Theory to take the quotient to get a moduli space, an unstable object gives a GIT unstable point in certain parameter space. To a GIT unstable point, Kempf associates a "maximally destabilizing" 1-parameter subgroup, and this induces a filtration of the object. We show that this filtration coincides with the Harder-Narasimhan filtration for finite dimensional representations of a finite quiver on vector spaces, using the construction of a moduli space for these objects given by King. This work is a continuation of previous work joint with T. Gomez and I. Sols, where we show a similar correspondence for moduli problems related to sheaves with additional structure, now in the different moduli problem of quiver representations.

The gradient heat flow of Higgs pairs
Xi Zhang, University of Science and Technology of China, China

In this talk, we consider generalized Hermitian-Einstein type equations and related heat flow on holomorphic vector bundles with extra structures, and introduce our (joint with Li JiaYu) recent work on the limiting behavior of the gradient heat flow of Higgs pair.

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