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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.

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.

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).

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.