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…

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…

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…

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…

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.…

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…

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 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…

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…

The talk will be a survey of Higgs bundles, their moduli spaces and the associated fibration structure from a historical, and personal, point of view.

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…