Vortex Partition Functions, Wall Crossing and Equivariant Gromov–Witten Invariants

Topological String Partition Functions as Equivariant Indices

In this work we exploit the relationship with certain equivariant genera of isntanton moduli spaces to study the string partition functions of some local Calabi-Yau geometries, in particular, the Gopakumar-Vafa conjecture for them [8]. Gromov-Witten invariants are in general rational numbers. However as conjectured by Gopakumar and Vafa [8] using M-theory, the generating series of GromovWitten ...

Equivariant Cohomology and Wall Crossing Formulas in Seiberg-Witten Theory

We use localization formulas in the theory of equivariant cohomology to rederive the wall crossing formulas of Li-Liu [7] and Okonek-Teleman [8] for Seiberg-Witten invariants. One of the difficulties in the study of Donaldson invariants or Seiberg-Witten invariants for closed oriented 4-manifold with b2 = 1 is that one has to deal with reducible solutions. There have been a lot of work in this ...

Paradan’s wall crossing formula for partition functions and Khovanski-Pukhlikov differential operator

Equivariant -invariants and -forms

Let a Lie group G act isometrically on an odd dimensional Riemannian manifold M and let D be a G-equivariant operator of Dirac type on M. We construct a formal power series X (D) in X 2 g, which can be interpreted as a universal-form for families of Dirac operators induced by D over brations with bres isometric to M and with structure group G. For a Killing eld X 2 g without zeroes, rX (D) is t...

Area Dependence in Gauged Gromov-witten Theory

We study the variation of the moduli space of symplectic vortices on a fixed holomorphic curve with respect to the area form. For compact, convex varieties we define symplectic vortex invariants and prove a wall-crossing formula for them. As an application, we prove a vortex version of the abelianization conjecture of Bertram, Ciocan-Fontanine, and Kim [4], which related GromovWitten invariants...