Hitchin Systems in Mathematics and Physics

Critical values of the integrable system correspond to singular spectral curves. In this talk we shall discuss critical points, points in the moduli space where one of the Hamiltonian vector fields vanishes. These involve torsion-free sheaves on the spectral curve instead of line bundles and a lifting to a 3-manifold which fibres over the cotangent bundle. The case of rank 2 will be described in more detail.

Quivers emerge naturally in the study of instantons on flat four-space (ADHM), its orbifolds and their deformations, called ALE space (Kronheimer-Nakajima). Pursuing this direction, we study instantons on other hyperkaehler spaces, such as ALF, ALG, and ALH spaces. Each of these cases produces instanton data that organize, respectively, into a bow (involving the Nahm equations), a sling (involving the Hitchin equations), and a monopole wall (Bogomolny equation).

Since the introduction of generalized Kahler geometry in 1984 by Gates, Hull, and Rocek in the context of two-dimensional supersymmetric sigma models, we have lacked a compelling picture of the degrees of freedom inherent in the geometry. In particular, the description of a usual Kahler structure in terms of a complex manifold together with a Kahler potential function is not available for generalized Kahler structures, despite many positive indications in the literature over the last decade. I will explain recent work showing that a generalized Kahler structure may be viewed in terms of a Morita equivalence between holomorphic Poisson manifolds; this allows us to solve the problem of existence of a generalized Kahler potential.

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.

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 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.

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.

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.

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.

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 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.