Mathematical Physics

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 $$\Hom(\pi_1(\Sigma), G)/G$$ of representations of the fundamental group of the base curve $\Sigma$ 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, generalizing the author's construction in the untwisted case (2002-2014). For example, by hyperkahler rotation, the wild character varieties all thus admit special Lagrangian fibrations. 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. The new results to be discussed are joint work with R. Paluba and/or D. Yamakawa.

Cohomological Hall algebras associated with preprojective algebras of quivers play a preeminent role in geometric representation theory and mathematical physics. In the present talk, I will introduce and describe CoHAs associated with the stack of Higgs sheaves on a smooth projective curve. Moreover, I will address the connections with representation theory and gauge theory. (This is a joint work with Olivier Schiffmann.)

The Witten d-bar equation is a generalization of the parametrized holomorphic curve equation associated to a holomorphic function (superpotential) on a Kahler manifold X. It plays a central role in the work of Gaiotto-Moore-Witten on the "algebra of the infrared". The talk will explain an "intrinsic" point of view on the equation as a condition on a real surface S embedded into X (i.e., not involving any parametrization of S). This is possible if S is not a holomorphic curve in the usual sense.

We construct an action of the Weyl group on the affine closure of the cotangent bundle on G/U. The construction involves Hamiltonian reduction with respect to the `universal centralizer' and an interesting Lagrangian variety, the Miura variety. A closely related construction produces symplectic manifolds which play a role in `Sicilian theories' and whose existence was conjectured by Moore and Tachikawa. Some of these constructions may be reinterpreted, via the Geometric Satake, in terms of the affine grassmannian.

The index of rigidity was introduced by Katz as the Euler characteristic of de Rham cohomology of End-connection of a meromorphic connection on curve. As its name suggests, the index valuates the rigidity of the connection on curve. Especially, in P^1 case, this index makes a significant contribution together with middle convolution. Namely Katz showed that regular singular connection on P^1 can be reduced to a rank 1 connection by middle convolution if and only if the index of rigidity is 2. After that, the work of Crawley-Boevey gave an interpretation of the index of rigidity and the Katz' algorithm from the theory of root system. Namely, he gave a realization of moduli spaces of regular singular connections on a trivial bundle as quiver varieties. In this setting the index of rigidity can be naturally computed by the Euler form of quiver, and the Katz algorithm can be understood as a special example of the theory of Weyl group orbits of positive roots of the quiver. I will give an overview of this story with a generalization to the case of irregular singular connections. Moreover, I will introduce an algebraic curve associated to a linear differential equation on Riemann surface as an analogy of the spectral curve of Higgs bundle. And compare some indices of singularities of differential equation and its associated curve, Milnor numbers and Komatsu-Malgrange irregularities. Finally as a corollary of this comparison of local indices, I will give a comparison between cohomology of the curve and de Rham cohomology of the differential equation and show the coincidence of the index of rigidity and the Euler characteristic of the associated curve.

In this talk I will explain how a perverse filtration on the Kontsevich-Soibelman cohomological Hall algebra enables us to define the Lie algebra of BPS states associated to a smooth algebra with potential. I will then explain what this means for character varieties, and in particular, how to build the "genus g Kac-Moody Lie algebra" out of the cohomology of representations of the fundamental group of a surface.

In their paper, "On the motivic class of the stack of bundles", Behrend and Dhillon were able to derive a formula for the class of a stack of vector bundles on a curve in a completion of the K-ring of varieties. Later, Mozgovoy and Schiffmann performed a similar computation in order to obtain the number of points over a finite field in the moduli space of twisted Higgs bundles. We will briefly introduce motivic classes. Then, following Mozgovoy and Schiffmann's argument, we will outline an approach for computing motivic classes for the moduli stack of vector bundles with connections on a curve. This is a work in progress with Roman Fedorov and Yan Soibelman.

I will discuss in the framework of the P=W conjecture, how one can conjecture formulas for the perverse Hirzebruch y-genus of Higgs moduli spaces. The form of the conjecture raises the possibility that they can be obtained as the partition function of a 2D TQFT.

3d field theories with N=2 supersymmetry play a special role in the evolving web of connections between geometry and physics originating in the 6d (2,0) theory. Specifically, these 3d theories are associated to 3-manifolds M, and their vacuum structure captures the geometry of local systems on M. (Sometimes M arises as a cobordism between two surfaces C, C', in which case the 3d theories encode some functorial relation between the geometry of Hitchin systems on C and C'.) I would like to explain some of the mathematics of 3d N=2 theories. In particular, I would like to explain how Hilbert spaces in these theories arise as Dolbeault cohomology of certain moduli spaces of bundles. One application is a homological interpretation of the "pentagon relation" relating flips of triangulation on a surface.

I will define a generalization of the classical Laplace transform for D-modules on the projective line to parabolic harmonic bundles with finitely many logarithmic singularities with regular residues and one irregular singularity, and show some of its properties. The construction involves on the analytic side L2-cohomology, and it has algebraic de Rham and Dolbeault interpretations using certain elementary modifications of complexes. We establish stationary phase formulas, in patricular a transformation rule for the parabolic weights. In the regular semi-simple case we show that the transformation is a hyper-Kaehler isometry.