Title: Derived categories and Rozansky-Witten invariants

Lecture 1: The geometry of Rozansky-Witten invariants

Lecture 2: Derived categories and Rozansky-Witten weight systems

Lecture 3: The Rozansky-Witten TQFT

Abstract: 

In 1996 Rozansky and Witten described a new family of (2+1)-dimensional topological quantum field theories, quite different from the now familiar Chern-Simons theories. Instead of starting from a compact Lie group, one starts with a hyperk\"ahler manifold X^{4n}; the partition function (a topological invariant) for a closed 3-manifold M is then expressed as an integral over the space of all maps from M to X. Further analysis shows that these invariants amount to evaluations of the universal finite-type invariant of Le, Murakami and Ohtsuki, using weight systems derived purely from the hyperk\"ahler manifold X.

I will explain the geometrical origin of these weight systems and then describe (joint work with Simon Willerton and Justin Sawon) a precise analogy between hyperk\"ahler manifolds and Lie algebras, the connections with Vassiliev theory, and the rigorous construction of the TQFT arising from X. The flavour of the theory is appealingly algebro-geometrical: whereas constructions of Chern-Simons theory start from the category of representations of a quantum group, Rozansky-Witten theory turns out to be based on the derived category (don't panic!) of coherent sheaves on X.

Some references:

Roberts "Rozansky-Witten theory", math.QA/0112209

Gelfand and Manin, "Homological algebra" (Springer)

Thomas "Derived categories for the working mathematician", math.AG/0001044

Rozansky and Witten "Hyperkaehler geometry and invariants of 3-manifolds",
hep-th/9612216

Hitchin and Sawon, "Curvature and characteristic numbers of hyperkaehler
manifolds", math.DG/9908114

Kapranov "Rozansky-Witten invariants via Atiyah classes", alg-geom/9704009

Le, Murakami and Ohtsuki "On a universal invariant of 3-manifolds",
q-alg/9512002