Week 08.06.2009 – 14.06.2009

In this talk I will show how the characteristic polynomial of a random unitary matrix has been successfully used in number theory to model the Riemann zeta function, and then I will present some new work re-interpreting and generalizing the previous random matrix results in a probabilistic setting. This is joint work with Paul Bourgade, Ashkan Nikeghbali and Marc Yor.

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.

Recent work on membranes of M-theory has lead to a new type of physical realization of fuzzy 2-spheres in Matrix Brane actions. In standard realizations, the fuzzy two-sphere coordinates transforming in the vector of the spherical rotation symmetry are identified with matrix variables in the adjoint representation of the unitary symmetry of the Matrix brane actions.In these new realizations, spinors of the rotational symmetry are identified with variables in bi-fundamental representations of the unitary symmetry. The vector coordinates of the fuzzy 2-sphere are recovered from bilinears in the spinors by a fuzzy version of the projection map of the Hopf fibration over the two-sphere. The outcome is the emergence of a five dimensional theory from a three dimensional one as expected from the intersection of the fundamental branes of M-Theory in ABJM quotients of eleven dimensional spacetime. The demonstration of the emergent six dimensional theory expected for the brane intersection in M-Theory without a quotient remains an open problem.