abstracts moduli-2024

Let X be a toric hyperkaehler manifold, i.e. a complete hyperkaehler manifold of dimension 4n with T^n-symmetry. If T^n is the maximal torus of a compact Lie group and the Weyl group W of K acts on X, we can form the singular hyperkaehler manifold X/W and try to resolve the singularities of its twistor space Z/W to obtain a new complete hyperkaehler manifold. The ur-example is X=(R^3xS^1)^n with K=U(n) and W=S_n. The resolution of Z/W in this case gives the moduli space of charge n SU(2)-monopoles on R^3. In my talk I will explain how to resolve the twistor space of X/W for arbitrary X, and why, provided the volume growth of X is r^3n, the resolved hyperkaehler manifold should be described as a moduli space of solutions to Lie(K)-valued Nahm’s equations. This is joint work with Lorenzo Foscolo.

With Gonçalo Oliveira and Rosa Sena-Dias, we study the deformed Hermitian-Yang-Mills equation on the full flag manifold, both in rank one and in higher rank.

Almost all gravitational instantons that are hyperkahler (aka tesserons) can be realized as moduli spaces of monopoles. We establish this realization and use it to describe the parameter space of such spaces.
Presentation

I will describe progress towards the construction of complete Calabi-Yau metrics on the complement of ample, simple normal crossings anti-canonical divisors. This talk will discuss joint works with Y. Li, and F. Tong and S.-T. Yau.

I will discuss the construction of large families of Yang-Mills instantons, i.e. solutions of the anti-self-duality equations of 4-dimensional gauge theory, on gravitational instantons with ALF (asymptotically locally flat) asymptotics. The latter are certain complete non-co