08.06.2009 (Monday)

Gauge theory on G2 manifolds and bundles over Calabi-Yau moduli spaces

String Theory & Geometry Seminar Simon Donaldson (Imperial College)

13:00 IC
room IMS seminar room

In the first part of the talk I will review differential geometric background involving associative submanifolds of 7 manifolds with G2 holonomy and an analogue of the Yang Mills instanton equation for connections on bundles over such a manifold. Then I will describe joint work with Ed Segal in which, assuming these objects have suitable formal properties, we define holomorphic bundles over moduli spaces of Calabi-Yau 3-folds which can be viewed as the complexification of Floer theory. In the last part of the talk I will consider in more detail the problem of establishing the foundations required for the theory. This involves pertubations of the equations and the question of how to count solutions at infinity, which are pairs consisting of a connection and an associative submanifold. The counting problem leads to a nonlinear generalisation of the spectral flow for eigenvalues of Dirac operators.