2009-11-13
Andrei Caldararu (Univ. Wisconsin)
A conjecture of Duflo and the Ext algebra of branes
The Duflo theorem is a statement in Lie theory which allows us to
compute the ring structure of the center of the universal enveloping algebra of
a finite-dimensional Lie algebra. A categorical version of it was used by Maxim
Kontsevich to give a spectacular proof of the so-called "Theorem on complex
manifolds," which computes the multiplicative structure of Hochschild
cohomology of a complex manifold in terms of the algebra of polyvector fields.
In Lie theory there are also more general Duflo-type statements (mostly
conjectural), which study the case of a pair (Lie algebra, Lie subalgebra). I
will explain how these translate into conjectures about the multiplicative
structure of the Ext-algebra of the structure sheaf of a complex submanifold of
a complex manifold, and how from this interaction we can hope to gain new
insights into both algebraic geometry and Lie theory. (Based on discussions
with Damien Callaque.)