Gauge theoretical Gromov-Witten invariants and virtual fundamental classes

Abstract

This article is an expanded version of talks which the authors have given in Oberwolfach, Bochum, and at the Fano Conference in Torino. In these talks we explained the main results of our papers "Gauge theoretical equivariant Gromov-Witten invariants and the full Seiberg-Witten invariants of ruled surfaces" and "Comparing virtual fundamental classes: Gauge theoretical Gromov-Witten invariants for toric varieties". We have also included new results, e. g. the material concerning flag varieties, Quot spaces over P1, and the generalized quiver representations. The common theme is the construction of gauge theoretical Gromov- Witten type invariants of arbitrary genus associated with certain symplec- tic factorization problems with additional symmetry, and the computation of these invariants in terms of complex geometric objects.