This is one rambling paragraph previously at symplectic geometry. We should also have equivariant localization per se. Hopefully this entry will be cleaned up later. It obviously has its origin from the informal discussion here
Jacobi’s elimination of nodes and Witten’s “Two dimensional gauge theories revisited.”
Conservation laws arising from symmetries have been formalized as moment maps by Kirillov, Kostant and Souriau in late 1960s. Elimination of nodes procedure has been made rigorous by Marsden and Weinstein and, independently, by Meyer, as symplectic reduction (symplectic quotient construction).
In the early 1980s Mumford observed that symplectic quotients are closely related to geometric invariant theory quotients and that many moduli spaces important in algebraic geometry and in mathematical physics can be realized as symplectic quotients. Atiyah and Bott used this point of view in “The moment map and equivariant cohomology” to construct cohomology classes of moduli spaces of flat connections on Riemann surfaces. In “Two dimensional gauge theories revisited” Witten conjectured a method for computing the intersection pairings of cohomology classes of symplectic quotients. The work of Atiyah and Bott and Witten’s conjecture stimulated a large research effort to understand the topology of symplectic quotients in terms of the equivariant cohomology of the original spaces. Witten’s conjecture was proved by Jeffrey and Kirwan several years later.