A symplectic formula of generalized Casson invariants

TL;DR

This paper extends Taubes' relationship between gauge theory and the Casson invariant from SU(2) to SU(n), showing that generalized SU(n) Casson invariants can be computed via intersection theory of character varieties.

Contribution

It proves an analogous intersection formula for SU(n) generalized Casson invariants, generalizing Taubes' SU(2) result to higher rank groups.

Findings

Established an intersection number formula for SU(n) Casson invariants.

Demonstrated the SU(3) case aligns with Boden-Herald's invariant calculation.

Connected gauge theory critical points with character variety intersections.

Abstract

Suppose Y is an integer homology 3-sphere, Taubes 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 can be equivalently calculated by taking an appropriate intersection number of Lagrangian submanifolds.