# Recent developments in Higgs theory

October 22–27, 2018

### Roman Bezrukavnikov

###### Hitchin fibration and positive characteristic

I will review works by myself with Braverman and Travkin, by T.-S. Chen and X. Zhu and by M. Groechenig establishing geometric Langlands duality in the (much easier) framework of crystalline D-modules in positive characteristic. Asymptotics of the $$p$$-Hitchin map plays an important role in some of these constructions. I will also discuss the aspect of duality related to t-structures on the categories of coherent sheaves and D-modules, including indications of relation to Bridgeland stabilities.

###### Exotic components of surface group representation varieties and their Higgs bundle avatars

Moduli spaces of Higgs bundles on a Riemann surface correspond to representation varieties for the surface fundamental group. For representations into complex semisimple Lie groups, the connected components of these spaces are labeled by obvious topological invariants. This is no longer true if one restricts to real forms of the complex groups. Factors other than the obvious invariants lead to the existence of extra "exotic" components which can have special significance. Instances of such exotic components occur in so-called "higher Teichmüller" theory, where they are attributable to one of two distinct mechanisms. Recent Higgs bundle results for the groups $$SO(p,q)$$ reveal new examples outside the scope of these two mechanisms. This talk will describe the new $$SO(p,q)$$ results.

### Marc Burger

###### On the real spectrum compactification of Higher Teichmüller spaces

For a finitely generated group $$H$$ and a real algebraic semisimple group $$G$$ we define $$RChar(H,G)$$, the real spectrum compactification of the character variety $$Char(H,G)$$ of $$H$$ in $$G$$ and relate it to the Parreau compactification $$PChar(H,G)$$ obtained by projectivizing Weyl chamber valued length functions. As an application we deduce that the closure of a connected component $$C$$ in $$PChar(H,G)$$ is arcwise connected in a controlled way. We specialize then to the case where $$H$$ is a compact surface group, $$G=SL(n,\mathbb{R})$$ and $$C=Hit(n)$$ the Hitchin component. We show that length functions in the Parreau boundary of $$Hit(n)$$ all come from representations of $$H$$ over non-archimedean real closed fields that are positive in the sense of Fock-Goncharov. In fact this characterizes completely such length functions. When $$G=SL(2,\mathbb{R})$$, $$PChar(H,G)$$ is the Thurston boundary and the action of the mapping class group $$Mod$$ on it has complicated dynamics. We show that for $$G=SL(3,\mathbb{R})$$, there is a non-empty open set of discontinuity for the action of $$Mod$$ in the Parreau boundary and produce explicit examples of representations of $$H$$ into $$SL(3,R(X))$$, whose Weyl chamber valued length function falls into that open set. Incidentally, this produces actions of $$H$$ on the building associated to $$SL(3, R(X))$$, where $$X$$ has valuation -1, that are displacing in the sense of Delzant-Guichard-Labourie Mozes.

This is joint work with Alessandra Iozzi, Anne Parreau and Beatrice Pozzetti.

### Brian Collier

###### Conformal limits and stratifications

Both the Higgs bundle moduli space and the moduli space of flat connections have a natural stratification induced by a $$\mathbb{C}^*$$ action. In both of these stratifications, each strata is a holomorphic fibration over a connected component of complex variations of Hodge structure. While the nonabelian Hodge correspondence provides a homeomorphism between Higgs bundles and flat connections, this homeomorphism does not preserve the respective strata. The closed strata on the Higgs bundle side is the image of the Hitchin section (the Hitchin component) and the closed strata in the space of flat connections is the space of opers. In this talk we will show how many of the relationships between opers and the Hitchin component extend to general strata. This is based on joint work with Richard Wentworth.

###### Pluriharmonic maps of infinite energy

I will discuss the Jost-Zuo-Mochizuki theory of infinite energy pluriharmonic maps into manifolds with strong negative curvature in the sense of Siu and I will give some applications when the target is the Teichmüller space.

### Ben Davison

###### Higher genus Yangians and Higgs bundles

In genus one, nonAbelian Hodge theory provides a link between the Borel-Moore homology of the stack of degree zero semistable Higgs bundles and the Borel-Moore homology of the stack of commuting matrices. This latter cohomology (summed over all rank) is known to be a degeneration of the Yangian algebra studied by Maulik-Okounkov and (separately) Schiffmann-Vasserot. I will discuss the extension of this story to higher genus, and in particular the construction and fundamental properties of a "genus $$g>1$$ Yangian".

### George Dimitrov

###### Non-commutative counting invariants and Kanev's construction

Roughly, these invariants are sets of triangulated subcategories in a given triangulated category and their quotients. I will talk about the definition, some additional structures, and functoriality. Counting and intersection of the elements in these sets will be illustrated. I will discuss configurations of non-commutative genus zero curves in a category and will exemplify their incidence correspondences. The latter give opportunity to contemplate attaching Prym-Tyurin-Kanev varieties to holomorphic family of categories imitating the so called Kanev's construction. This is a joint work with L. Katzarkov.

### Alexander Efimov

###### Categorical Hitchin system

In this talk I will construct a non-commutative "lift" of the Hitchin integrable system (in type A), in which the Hitchin hamiltonians arise as higher order Hochschild cohomology of the derived category of the underlying smooth projective curve.

### Hélène Esnault

###### Higgs bundles and $$F$$-isocrystals

Based on earlier work with Michael Groechenig aiming at proving Carlos Simpson’s integrality conjecture, we view Deligne’s conjecture on the counting of irreducible $$\ell$$-adic sheaves up to scalar coming from the ground field on a smooth projective curve over a finite field as a problem on the nilpotent cone (Hitchin $$q$$-fibre). The counting itself has been proved by H. Yu by automorphic methods. It works well in rank $$1$$ (and has been worked out by Efstathia Katsigianni, a PhD student in Berlin), and in higher rank it raises natural questions on this fibre (and slightly more). The talk is based on discussions, which are still in their infancy, with Tomoyuki Abe, Marco D’Addezio, and Michael Groechenig.

### Philippe Eyssidieux

###### Orbifold Kähler groups related to Mapping class groups

The talk will survey a work in progress with Louis Funar using TQFT which constructs new Kähler groups with an interesting representation theory and relates the construction to classical examples.

### Michael Finkelberg

###### Conjectural Higgs branches of Sicilian quantum field theories

According to G. Moore and Y. Tachikawa, the conjectural Higgs branches of Sicilian quantum field theories should give rise to a 2d TQFT with values in complex symplectic manifolds. A rigorous mathematical construction of this TQFT was given recently by Ginzburg-Kazhdan and by T. Arakawa. I will talk about a construction in terms of ring objects in the equivariant derived Satake category. This is a joint work with A. Braverman and H. Nakajima.

###### Higgs bundles and cotangent bundle to the "new" higher Teichmüller space

We will study the cotangent bundle to the version of the Theichmüller space defined in the previous talk and show that it is in a certain sense analogous to Higgs bundles. Namely, nonzero cotangent vectors are associated to spectral curves, which are not holomorphic, but just Lagrangian. We also construct an analogue of the Hitchin map from flat $$\lambda$$-connections to the cotangent bundle. The still opened question of isomorphism between the "old" and the "new" Teichmüller spaces amounts to follow from local bijectivity of this map.

###### Arakelov-Milnor inequalities and maximal variations of Hodge structure

We consider the moduli space of $$G$$-Higgs bundles over a compact Riemann surface $$X$$, where $$G$$ is a real semisimple Lie group. By the non-abelian Hodge correspondence this is homeomorphic to the moduli space of representations of the fundamental group of $$X$$ in $$G$$. We are interested in the fixed point subvarieties under the action of $$\mathbb{C}^*$$, obtained by rescalling the Higgs field. The fixed points are known as Hodge bundles and correspond to variations of Hodge structure. They also correspond to critical subvarieties of a Morse function on the moduli space of Higgs bundles, known as the Hitchin functional. We show that one can define in this context an invariant that generalizes the Toledo invariant in the case where $$G$$ is of Hermitian type. Moreover, there are bounds on this invariant similar to the Milnor–Wood inequalities of the Hermitian case. These bounds also generalize the Arakelov inequalities of classical Hodge bundles arising from families of varieties over a compact Riemann surface. We explore the case where this invariant is maximal, and show that there is a rigidity phenomenon, relating to Fuchsian representations, and which in particular allows the identification of higher Teichmüller spaces.

### Tamas Hausel

###### Intersection of mirror branes on Higgs moduli spaces

I will discuss a computational approach for the semiclassical limit of mirror symmetry for Higgs moduli spaces for Langlands dual groups, by comparing the equivariant indices of intersections of mirror branes.

### Alessandra Iozzi

###### Higher Teichmüller theory and geodesic currents

We start by recalling features of classical Teichmüller space of a topological surface. We explain the notion of "higher Teichmüller" by introducing the notion of maximal representations and of Hitchin component and we see how some of the classical features, such as the Thurston compactification, generalize to the "higher Teichmüller" setting. In the process we explain a structure theorem for geodesic currents. This is joint work with M. Burger, A. Parreau and B. Pozzetti.

### Bruno Klingler

###### Tame topology and Hodge theory

The idea of tame topology was introduced by Grothendieck and developed by model theorists under the name "o-minimal structures". I will explain how tame topology can be applied in Hodge theory to give for instance a new proof of the algebraicity of the components of the Hodge locus, a celebrated result of Cattani-Deligne-Kaplan (joint work with Bakker and Tsimerman). I will also outline possible applications to non-abelian Hodge theory.

### Andrey Levin

###### Higgs theory and Arakelov geometry

I shall discus possible modification of the Higgs theory in the direction to the Arakelov geometry using the conception of numbers — polynomials correspondence. I try to present some evidences to existence number theoretical partner of the Higgs field.

### Takuro Mochizuki

###### Periodic monopoles and difference modules

One of the main themes in complex geometry is to obtain a correspondence between differential geometric objects and algebro-geometric objects. For instance, it is notable that Simpson proved the equivalence of irreducible tame harmonic bundles, stable parabolic bundles with logarithmic connections and stable parabolic bundles with logarithmic Higgs fields on compact punctured Riemann surfaces.

In this talk, we shall explain an equivalence between singular periodic monopoles of GCK type and stable parabolic difference modules. It is a variant of Simpson's theorem in the context of periodic monopoles.

### Nikita Nekrasov

###### Towards the proof of the NRS conjecture

Nekrasov-Rosly-Shatashvili conjecture (2011) relates generating functions of equivariant generalized Donaldson invariants of $$\mathbb{CP}^2$$ to the symplectic geometry of the moduli space of rank 2 local systems on curves with punctures (more specifically, the varieties of opers play a prominent role). We report on the progress in the case of genus 0 curves with 4 punctures, and the generalization of the NRS conjecture to the higher rank case. Based on the joint work with Saebyeok Jeong.

### Pranav Pandit

###### Spectral Networks

Spectral networks are certain decorated graphs drawn on a Riemann surface. I will introduce spectral networks and discuss their role in a conjectural picture relating points in the base of the Hitchin system, harmonic maps, and stability structures on certain Fukaya-type categories. This is based on various joint projects with Fabian Haiden, Ludmil Katzarkov, Maxim Kontsevich, Alexander Noll, and Carlos Simpson.

### Aleksandar Petkov

###### Center manifold theory for the Yang–Mills and the Yang–Mills–Higgs flows

Center manifold theory provides a powerful method of analysis of local behaviour of a dynamical system near a non-hyperbolic equilibrium. In the finite-dimensional case it serves for a dimensional reduction of the system. Analogously, in the infinite-dimensional case (e.g. PDEs), it provides, under some conditions, a finite-dimensional reduction of the system.

Recent work of Haiden, Kontsevich, Katzarkov and Pandit suggests a new categorical approach to the theory of center manifolds of PDEs (categorical center manifold). In this talk we explore the existence of a (local) center manifold, from analytical point of view, for the Donaldson's heat flow of Yang–Mills and Yang–Mills–Higgs functionals.

### Helge Ruddat

###### Smoothing toroidal crossing varieties

Gross-Hacking-Keel provided insight on how Cluster varieties relate to maximal degenerations. My talk is about smoothing maximal degenerate varieties. I explain the proof of a new result on smoothing toroidal and normal crossing varieties by constructing certain kinds of log structures that enjoy Hodge to de Rham degeneration for the Danilov differential forms. We generalize work by Friedman and work by Gross and Siebert. This is a joint project with Simon Felten and Matej Filip.

### Masahiko Saito

###### Moduli spaces of parabolic Higgs bundles and parabolic connections on curves and Integrable systems

We will start by reviewing algebraic constructions of moduli spaces of stable parabolic Higgs bundles and parabolic connections on a smooth curve via GIT. Due to the works of Maruyama and Yokogawa, and Inaba, Iwasaki and Saito, one can show that the moduli spaces are smooth quasiprojective algebraic schemes with natural holomorphic symplectic structures.

We will also explain about Riemann-Hilbert correspondence from moduli spaces of parabolic connections to moduli spaces of monodromy representations. Moreover, we will explain how we can obtain differential equations with Painlevé property including Classical 8 types from isomonodromic flows defined in universal family of moduli spaces of parabolic connections.

In the later part of the talk, we will give an explicit description of moduli spaces of parabolic Higgs bundles and parabolic connections by apparent singularities and their duals (a joint work of S. Szabo). We will explain the relation between the geometry of moduli spaces and spectral curves.

TBA

### Artan Sheshmani

###### Atiyah Class and Sheaf counting on local Calabi-Yau fourfolds

We discuss Donaldson-Thomas (DT) invariants of torsion sheaves with 2 dimensional support on a smooth projective surface in an ambient non-compact Calabi Yau fourfold given by the total space of a rank 2 bundle on the surface. We prove that in certain cases, when the rank 2 bundle is chosen appropriately, the universal truncated Atiyah class of these codimension 2 sheaves reduces to one, defined over the moduli space of such sheaves realized as torsion codimension 1 sheaves in a noncompact divisor (threefold) embedded in the ambient fourfold. Such reduction property of universal Atiyah class enables us to relate our fourfold DT theory to a reduced DT theory of a threefold and subsequently then to the moduli spaces of sheaves on the base surface. We finally make predictions about modularity of such fourfold invariants when the base surface is an elliptic K3. This is joint work with Emanuel Diaconescu and Shing-Tung Yau.

### Carlos Simpson

###### Construction of GL Higgs bundles on the moduli of bundles over a genus 2 curve — the even case

This is continuing joint work with R. Donagi and T. Pantev. Their idea is to apply the nonabelian Hodge correspondence to construct local systems over open subsets of the moduli space of stable bundles, as predicted by the Geometric Langlands Correspondence. Here, we look at the case of rank 2 bundles of even degree on a compact genus 2 curve. The moduli space is $$\mathbb{P}^3$$. For the Higgs bundle obtained by pushforward from a general fiber of the Hitchin fibration, we show how to adjust the parabolic structure to obtain vanishing of the first and second parabolic Chern classes. There are some differences with respect to the odd degree case that we treated earlier. Currently open for this construction is the question of verifying the eigensheaf property for the Hecke correspondence between the even and odd moduli spaces.

### Thomas Sutherland

###### From Higgs bundles to local systems: a (non)abelian perspective

In this talk we study how natural coordinate systems on moduli spaces of local systems arise from the geometry of diffeomorphic moduli spaces of Higgs bundles. Here we focus on the smallest-dimensional examples of moduli spaces of $$SL(2,\mathbb{C})$$ Higgs bundles on the Riemann sphere with (possibly irregular) singularities, which are in bijection with the Painleve equations giving isomonodromic deformations of flat connections associated to such Higgs bundles via the non-abelian Hodge correspondence. In particular we consider how Fock-Goncharov coordinates coming from the holonomy of an abelianized connection on the spectral curve solve a Riemann-Hilbert problem determined by the geometry of the Hitchin integrable system. Further we observe how traces of the holonomy of the non-abelian connection on the Riemann sphere coincide with certain theta functions in the sense of Gross-Siebert whose heuristic definition is based on the Hitchin fibration.

### Andras Szenes

###### An enumerative approach to the $$P=W$$ conjecture

The $$P=W$$ conjecture may be formulated as the multiplicativity of the perverse filtration on the cohomology of the moduli spaces of Higgs bundles. In joint work with Chiarello and Hausel, I developed an enumerative approach to this conjecture, based on an idea of de Cataldo and Migliorini. Our calculation uses equivariant intergration, residue calculus and difference equations, and yields the result for $$\mathrm{rank}=2$$.

### Alexander Thomas

###### Punctual Hilbert schemes and stable Higgs bundles — a new approach to higher Teichmüller theory

The space of stable Higgs bundles, or Hitchin's component, is the main object of study in higher Teichmüller theory. I will present a new, purely geometric approach to this theory. By means of the punctual Hilbert scheme of the plane, a new geometric structure on a surface is defined, generalizing the complex structure, whose moduli space is conjecturally isomorphic to Hitchin's component. Joint work with Vladimir Fock.

### Alan Thompson

###### Pseudolattices and Homological Mirror Symmetry

I will present recent joint work with Andrew Harder on the classification of a certain type of pseudolattice. These pseudolattices arise naturally as the numerical Grothendieck groups associated to the bounded derived categories of coherent sheaves on del Pezzo surfaces and thus, unsurprisingly, their classification parallels that of the del Pezzo surfaces. However, we will also see that one may interpret this classification as a classification of a certain type of genus one Lefschetz fibration, through the framework of homological mirror symmetry.

###### The Tate construction as a crystal

Let $$X$$ be a scheme over a perfect field of characteristic $$p$$. It is known that the functor which takes a vector bundle $$E$$ over $$X$$ to the Tate cohomology complex of the order $$p$$ cyclic group with coefficients in the $$p$$-th tensor power of $$E$$ extends to a triangulated functor from the derived category of quasi-coherent sheaves on $$X$$ to itself. We will show how this functor lifts to a functor from the derived category of quasi-coherent sheaves on $$X$$ to the (appropriately defined) derived category of crystals on $$X$$. This is a jont work with Alexander Petrov.

### Richard Wentworth

###### Harmonic maps, pleated surfaces, and the asymptotic structure of the Hitchin moduli space

Recent work has given a precise description of the large scale behavior of solutions to the Hitchin equations in terms of certain limiting configurations. I will explain how these correspond in a precise way, via harmonic maps, to Bonahon's parametrization of pleated surfaces in hyperbolic 3-space by transverse and bending cocycles for a geodesic lamination. The result gives a geometric interpretation of the asymptotics of Hitchin's integrable system. This is joint work with Andreas Ott, Jan Swobod, and Michael Wolf.

### Graeme Wilkin

###### Representations of the Heisenberg algebra on a singular Morse complex

Many examples of moduli spaces (such as Higgs bundles and quiver varieties) arise as symplectic quotients of singular spaces. I will describe my previous work to develop a Morse theory in this setting, and how one can explicitly construct spaces of flow lines via the Hecke correspondence. For single vertex quivers with complete relation , such as the ADHM quiver, I will show that the Morse function is perfect and that the associated Morse complex admits a representation of a finite-dimensional Heisenberg algebra. In the special case of Hilbert schemes of points on $$\mathbb{C}^2$$ the limit of this construction gives a Morse-theoretic description of Nakajima's representations of the Heisenberg algebra.