Skew Invariant Theory of Symplectic Groups, Pluri-Hodge Groups and 3-Manifold Invariants

Abstract

This article deals with a number of topics which are, somewhat surprisingly, related. Firstly, the fundamental theorem of skew invariant theory for the symplectic group giving the generators and relations of symplectic invariants is established. The relations are the so called P_n relations which appear in the study of certain 3-manifold invariants. Next a class of cohomology groups are introduced, called Pluri-Hodge groups (somewhat in keeping with the notion of pluri-canonical groups). These are Dolbeault groups on a complex manifold X with values in tensor powers of sheaves of holomorphic forms of various degrees. By Riemann-Roch one shows that knowledge of the Pluri-Hodge groups gives precise formulae for all Chern numbers of the manifold. When X is holomorphic symplectic the Pluri-Hodge groups form representations of Sp(g) where g counts the number of tensor products. Still with X holomorphic symplectic, the Pluri-Hodge groups are vectors in the vector space H_g(X) of states in the topological field theory of Rozansky and Witten for a 3-manifold with boundary a Riemann surface of genus g. A formulation of the Murakami-Ohtsuki invariants, due to Sawon, allows us to show that quotients of the spaces of trivalent graphs B_g carry representations of the symplectic group. There is a weight system W_X: B_g -> H_g(X) and we show that W_X preserves the various symplectic group actions.