A symplectic formula of generalized Casson invariants

Abstract

Suppose Y is an integer homology 3-sphere, Taubes [Tau90] proved that the number of irreducible critical orbits of the perturbed Chern-Simons functional on Y , counted with signs, is equal to the algebraic intersection number of two character varieties associated with Heegaard splittings when the structure group is SU(2). Taubes' result established a relationship between gauge theory and the Casson invariant. This article proves an analogous identification result for SU(n) generalized Casson invariants. As a special case, we show that the SU(3) Casson invariant of Boden-Herald [BH98] can be equivalently calculated by taking an appropriate intersection number of Lagrangian submanifolds.