Parabolic structures appear in Mazzeo–Swoboda–Weiss–Witt’s study of the limiting configuration of the ends of the Dolbeault moduli space of Higgs bundles. Historically, parabolic structures were introduced by Seshadri and naturally appear in the nonabelian Hodge correspondence of Simpson for the group GL(n, C) over a noncompact algebraic curve. The full problem involving an arbitrary complex reductive group G is much more subtle and the generalization of a weighted flag in this case does not seem to provide an one-to-one correspondence between the Dolbeault and the Betti spaces. The correct set up for groups beyond GL(n, C) is in the language of parahoric group schemes and a nonabelian Hodge correspondence can be fully described in this case. Joint work in progress with G. Kydonakis, H. Sun and L. Zhao.

In this talk I will describe a number of different ways to view ALE hyperkähler four manifolds of type A. From one point of view we can see an analog of the Hitchin system (with affine fibres rather than projective), and by compactifying the fibres we arrive at a partial compactification which (for type A_n with n=3 or n=4) has the structure of a holomorphic symplectic manifold. In particular, we can see continuous families of complex Lagrangian submanifolds inside this space, where one family appears as fibres of the Hitchin map and the other families can be viewed as rotations of these fibres. The whole picture has many features in common with general moduli of Higgs bundles, and I will focus on these analogies during the talk. This is joint work with Rafe Mazzeo.

We will see an application of standard gluing techniques from gauge theory in order to obtain specific models of Higgs bundles lying inside certain connected components of the moduli space. The parameters involved in the procedure are the genus of the underlying Riemann surface and the holonomy of a surface group representation along the boundary components of the surface. Such gauge theoretic techniques are also used to understand the family of Teichmüller moduli spaces in the limit when one approaches a boundary point of the Deligne-Mumford compactification.

Abstract: Given a 4d N=2 supersymmetric field theory, one can obtain an associated hyperkähler (HK) manifold by considering its dimensional reduction on a circle. Such HK metrics were described in the physics literature in twistorial terms by Gaiotto-Moore-Neitzke (arXiv:0807.4723), and are an instance of ``instanton-corrected HK metrics". When we restrict to the special case of 4d N=2 theories of class S, the corresponding HK manifold is expected to be a Hitchin moduli space. The main goal of this talk is to present a mathematical treatment of the constructions of such instanton corrected HK metrics in the simpler case of ``mutually-local" instanton corrections. Namely, given the structure of an affine special Kähler (ASK) manifold together with a (compatible) variation of mutually local BPS structures, we construct an instanton corrected HK manifold. The simplifying assumption of mutual-locality allows for an explicit description of the HK structure which does not require dealing with the twistor space. For this I will mostly follow section 3 of the joint work with V. Cortés arXiv:2105.09011. I will close with some comments on possible relations between the Hitchin moduli space HK metric and the above mutually local instanton-corrected HK metrics. P.S: no physics knowledge will be required for the talk.