Conference: Hitchin 70  

Schedule

Programme

The moduli stack of G-Higgs bundles on a curve is the cotangent bundle of the stack of principal G-bundles. Hitchin showed that applying invariant polynomials to the Higgs field gives an integrable system. Motivated by the Langlands program we consider the following generalisation: take the cotangent bundle of the stack of torsors over a non-constant group scheme. Do we still get an integrable system? We conjecture and in many cases prove that this is the case for so-called parahoric group schemes. A special case of this includes strongly parabolic Higgs bundles. There is also an interpretation in terms of equivariant Higgs bundles. This is joint work with Masoud Kamgarpour and Rohith Varma.

I shall explain how a Hitchin-Teichmüller component for SL(∞,ℝ) can be parametrized by holomorphic differentials, and I shall discuss properties of the corresponding Hitchin's map.

In 1987 Hitchin discovered a new family of algebraic integrable systems, solvable by spectral curve methods. One novelty was that the base curve was of arbitrary genus. Later on it was understood how to extend Hitchin's viewpoint, allowing poles in the Higgs fields, and thus incorporating many of the known classical integrable systems, which occur as meromorphic Hitchin systems when the base curve has genus zero. However, in a different 1987 paper, Hitchin also proved that the total space of his integrable system admits a hyperkahler metric and (combined with work of Donaldson, Corlette and Simpson) this shows that the differentiable manifold underlying the total space of the integrable system has a simple description as a character variety

Hom(π₁(Σ),G)/G

of representations of the fundamental group of the base curve Σ into the structure group G. This misses the main cases of interest classically, but it turns out there is an extension. In work with Biquard from 2004 Hitchin's hyperkahler story was extended to the meromorphic case, upgrading the speakers holomorphic symplectic quotient approach from 1999. Using the irregular Riemann--Hilbert correspondence the total space of such integrable systems then has a simple explicit description in terms of monodromy and Stokes data, generalising the character varieties. The construction of such “wild character varieties”, as algebraic symplectic varieties, was recently completed in work with D. Yamakawa. The main aim of this talk is to describe some simple examples of wild character varieties including some cases of complex dimension 2, familiar in the theory of Painleve equations, although their structure as new examples of complete hyperkahler manifolds (gravitational instantons) is perhaps less well-known. The language of quasi-Hamiltonian geometry will be used and we will see how this leads to relations to quivers, Catalan numbers and triangulations, and in particular how simple examples of gluing wild boundary conditions for Stokes data leads to duplicial algebras in the sense of Loday.

Stable generalized complex structures are a class of structures that are not too far from being symplectic and are characterised by displaying generic behaviour. Examples of manifolds possessing these structures abound, including several manifold that do not have complex or symplectic structures. We will introduce the notion of complex divisor on a real manifold M and show it induces a corresponding Lie algrebroid over M. Stable structures induce such a complex divisor on the underlying manifold and we will show that these structures are in fact equivalent to symplectic structures on the corresponding Lie algebroid. This allows us to use symplectic techniques to study stable structures. We prove, for example, that the deformation space is unobstructed, give a normal form for the structure in a neighbourhood of the singular locus as well as in a neighbourhood of a Lagrangian brane. We also produce obstructions for the existence of stable generalized complex structures on a manifold equipped with a complex divisor. This is joint work with Marco Gualtieri.

In this talk I will discuss a parameterization of n(2g-2) connected components of the SO₀(n,n+1)-Higgs bundle moduli space. We will see how this parameterization generalizes both Hitchin's parameterization of the Hitchin component as a vector space of holomorphic differentials of degree 2, 4,…, 2n and Hitchin's parameterization of the nonzero Toledo invariant components of the PSL(2,ℝ)=SO₀(1,2)-Higgs bundle moduli space as vector bundles over certain symmetric products of the Riemann surface. Time permitting, we will give a connected count of the SO₀(p,p+1)-Higgs bundles moduli space and discuss the Zariski closure of these the representations in each connected component.

I review the geometry of heterotic string compactifications leading to supersymmetric gauge theories in 4 and 3 dimensions. The data of these compatifications are specified by a quadruple (X, V, ∇^I, H), where X is a 6 or 7 dimensional manifold, V is a vector bundle over X with a Yang Mills connection with satisfies instanton constraints, ∇^I is an instanton connection on TX, and H is a 3-form field on X defined in terms of the 2-form B field and the Chern-Simons forms for the bundle connection and the connection ∇^I. We recast the constraints imposed by supersymmetry on the geometry of these compactifications snd by the anomaly cancelation condition for H, in terms of an extension bundle Q over X (which is, topologically, of the form Q=T*X⊕P⊕TX for certain principal bundle P on X) which admits a connection which squares to zero. We show that the tangent space of the moduli space is then given in terms of the first cohomology group with values in Q. We discuss the fact that all our results can be reproduced from a superpotential. We find a Kähler metric on the moduli space which is a natural inner product on the moduli space, with a Kähler potential taking a remarkably simple form, and as in type II special geometry, it is quasi-topological. The heterotic geometry we define here may open a new door into understanding heterotic geometry. Time permitting, I will discuss dualities in the context of these theories.

In generalized geometry, the condition of having zero torsion for a metric connection does not determine the connection uniquely. Due to this special feature of the theory, curvature quantities are often difficult to tackle. In this lecture I will talk about the interplay between this new phenomenon and the notion of T-duality. Towards the end, I will mention some applications to complex hermitian geometry.

We discuss the problem of counting components of the moduli space of SO(p,q)-Higgs bundles for p>2 and q>p+1. In particular, we exhibit as a new feature the existence of exotic connected components, which cannot be distinguished neither by (refined) topological invariants of underlying bundles nor by explicit parametrization. The presence of singularities in the moduli space means that Bott-Morse theory does not apply directly. Instead our proof uses a complete description of the Biylanicki-Birula stratification of the exotic components.

This talk is based on joint work with M. Aparicio, S. Bradlow, B. Collier, O. García-Prada and A. Oliveira.

A key aspect of Poisson geometry is the ability to represent a Poisson structure as the image of a symplectic structure under a surjective submersion, called a symplectic realization. Among symplectic realizations of a given Poisson structure, there is a canonical one endowed with a much richer structure: the Weinstein symplectic groupoid. I will describe work with Michael Bailey in which we solve a problem posed by M. Crainic in 2004: What is the analogue of the Weinstein symplectic groupoid for a generalized complex structure? We show that there is a natural holomorphic symplectic groupoid which solves the problem, but to see this one must work in the appropriate stack-theoretic context. Along the way we develop some interesting new tools, such as groupoid Courant reduction, which have independent interest.

We will overview some conjectures on the mixed Hodge structure of character varieties in the framework of non-abelian Hodge theory on a Riemann surface. Then we introduce and study toric analogues of these spaces, in particular we prove that the toric character variety retracts to its core, the zero fiber of the toric Hitchin map, that its cohomology is Hodge-Tate and satisfies curious Hard Lefschetz, as well as the purity conjecture. We will indicate how these shed light on the P=W conjecture in the toric case as well as for general character varieties. This is based on joint work with Nick Proudfoot.

The geometric data classifying an irregular singularity of a (vector valued) o.d.e. is well known: a formal normal form, and Stokes data, consisting in the generic case basically of a family of n-1 upper triangular matrices and n-1 lower triangular matrices, the Stokes data, as well as a formal normal form at the singularity. Likewise, generically, for regular singularities one has the monodromy data, as well as formal normal forms at the singularity. These data, while similar, are not the same, and one would like some picture which holds relatively uniformly in the family. More precisely, let us consider, on a neighbourhood of the origin in ℂ,

y'=(A(ε,x)/p_ε(x))⋅y,

where

p_ε(x)=x^k+1+ε_k-1x^k-1+…+ε₁x+ε₀,

is a polynomial, and ε_i are small complex parameters, with ε=0 corresponding to an irregular singularity. The n×n matrix A(ε,x) is holomorphic in x, ε, on a neighbourhood of the origin. Its leading order term A₀ is supposed to be diagonalisable, with distinct eigenvalues, which, changing variables if necessary, we can assume have distinct real part. One would like monodromy data for the family. The answer starts with the introduction of an extra variable t:

ẏ=A(ε,x)⋅y,
ẋ=p_ε(x),

and proceeds from there. Joint work with Christiane Rousseau.

This will be a report on current progress on understanding the asymptotic geometry of the natural hyperKaehler metric on the Hitchin moduli space. The goal is to compare with the predictions and conjectural picture made by Gaiotto, Moore and Neitzke. Joint work with Swoboda, Weiss and Witt.

I will explain how the notion of parabolic Higgs bundle introduced by Simpson can be generalized to the context of local systems on punctured Riemann surfaces with structure group a real reductive Lie group. The monodromy around the punctures corresponds to a choice of parabolic structure given by the Kostant-Sekiguchi correspondence. Interesting features appear when the parabolic weight belongs to the boundary of a Weyl alcove: then the Higgs bundles are a generalization of the parahoric bundles studied by Balaji and Sedhadri. Our approach includes some new ingredients which may also be relevant in the case of parabolic principal bundles (without Higgs field), such as the relation between the parabolic degree and the Tits distance. This is joint work with Olivier Biquard and Oscar García-Prada.

Associated to any Lie algebroid is a pair of "characteristic varieties" that encode the singularities of its anchor map. The first is a coisotropic subvariety in the cotangent bundle; it is the singular support of various natural differential complexes. The second is a subvariety of the Lie algebroid itself, and is responsible for the localization of equivariant integrals. I will explain how various classical results can be seen through this lens, and describe some new applications to the local and global classification of Poisson varieties.

During the talk we shall visit different problems that can be studied through Higgs bundles. In particular, we shall look at how spectral data can be used to understand characteristic classes of SO(p,q)-Higgs bundles, how fibre products lead to correspondences between Hitchin fibrations, and how finite group actions can be used to construct branes in the moduli space of Higgs bundles. Some of the work in progress presented here is in collaboration with David Baraglia, Steve Bradlow, and Sebastian Heller.

Classical Teichmüller space describes the space of conformal structures on a given topological surface S. It plays an important role in several areas of mathematics as well as in theoretical physics.

Higher Teichmüller theory generalizes several aspects of classical Teichmüller theory to the context of Lie groups of higher rank, such as the symplectic group PSp(2n;ℝ) or the special linear group PSL(n;ℝ). So far, two families of higher Teichmüller spaces are known. The Hitchin component, which is defined when the Lie group is a split real forms, and the space of maximal representations, which is defined for Lie groups of Hermitian type. Interestingly, both families are linked with various notions of positivity in Lie groups.

In this talk I will give an introduction to higher Teichmüller theory, introduce new positive structures on Lie groups and discuss the (partly conjectural) relation between the two.