The Mathematical Colloquium: Ezra Getzer

Mathematical Colloquium with Ezra Getzer.

Abstract: Deformation theory goes back to the work of Kodaira and Spencer, who studied deformations of complex manifolds and holomorphic vector bundles. Deligne reformulated their work In the language of groupoids. (For many purposes, it suffices to think of a groupoid as a space with a group action.)

Consider instead the problem of studying deformations of flat vector bundles (to avoid the technical demands of complex geometry). A flat connection is (locally) an n x n matrix of one-forms A, and in this context, the equation of Kodaira and Spencer becomes the Maurer-Cartan equation dA+A^2=0. The space of solutions of this equation carries an action of the gauge group of invertible n x n matrices g of functions, by the formula

A^g = g^{-1} A g + g^{-1} dg

Topologists associate to a groupoid a topological space (its nerve): in our case, the components of this space are the gauge equivalence classes of flat connections, and the fundamental group of a component is the group of gauge symmetries of a flat connection (a finite-dimensional Lie group).

There is a parallel deformation theory for associative algebras. Let V be a vector space. The objects of the groupoid are associative products on V: this is acted on by the “gauge group” GL(V) of all invertible linear maps from V to V. According to Deligne, we should really think of this as a “two-groupoid”: the gauge transformations themselves have symmetries, namely the inner automorphisms.

The components of this (two-)groupoid are the equivalence classes of products on V, but now the fundamental group of a component is the outer automorphism group of the product. Furthermore, the second homotopy group is the centre of the algebra V. (Indeed, the second homotopy group of a space is a representation of the fundamental group.)

In this talk, I will outline a new approach to this two-groupoid, using differential forms on cubical sets.

Everyone is welcome!