Course overview -- Wintersemester 2014/2015 (Archive)

We study a set of measures that represent immersed submanifolds.Our main result is a set of stability conditions that include the Euler-Lagrange equations, but are strictly more general.

I will explain how the supersymmetric disk partition function Z of gauged linear sigma models relates to the central charge of objects in the category of B-branes of a Calabi-Yau (CY). The advantage of this approach is that Z provides an expression at every point in the quantum corrected moduli space of the CY. The B-branes in these models are realized naturally as matrix factorizations, equivariant under the gauge group. I will explain how to relate them to more familiar objects such as coherent sheaves on the CY and show examples, if time alllows.

Many integrable systems can be formulated as a so-called Lax equation. In this talk, we will review the up to now well-known construction which relates such integrable systems to algebraic geometry. If time permits, we also discuss some further directions due to Donagi, McDaniel-Smolinsky and others leading to decomposition of spectral covers and Prym-Tyurin varieties.

​A theorem of Bumsig Kim (1999) says that the quantum cohomology ring of a full flag manifold (i.e. ​generic ​adjoint orbit of a compact Lie group) ​is determined by​ a certain integrable system, the open ​ Toda lattice , which is ​​canonically associated to the Lie group. In my talk I will present ​this result in some more detail and then I will explain how one can extend it to the context of affine Kac-Moody flag manifolds. The quantum cohomology ring is this time determined by another integrable system, the periodic Toda lattice. This has been observed by Martin Guest and Takashi Otofuji (2001) for some particular flag manifolds. Extensions of their result have been obtained recently by Leonardo Mihalcea and myself in a joint work. They will be outlined in the talk. ​

Abstract: In the first part we recall the definition of the symplectic structure on nilpotent Lie groups. We apply the information to the Heisenberg Lie group and its quotients. The goal is to find first integrals for the geodesic flow.

In my talk I will discuss a family of matrix models, which describes the generating functions of intersection numbers on moduli spaces both for open and closed Riemann surfaces. Linear (Virasoro\W-constraints) and bilinear (KP\MKP integrable hierarchies) equations follow from the matrix model representation.

As discovered by Donagi and Markman, the existence of Lagrangian structure on a holomorphic family of abelian varieties (of appropriate dimension) depends on the vanishing of a certain local obstruction. In particular, the infinitesimal period map for the family must be a section of the third symmetric power of the cotangent bundle to the base of the family. I will discuss recent work with U.Bruzzo (IJM, vol.25 (2), 2014 ) where we compute the Donagi-Markman cubic for the generalised Hitchin system. In particular, we show that the Balduzzi-Pantev formula holds along the maximal rank symplectic leaves.