Comparing virtual fundamental classes: Gauge theoretical Gromov-Witten invariants for toric varieties

In general, a Kobayashi-Hitchin correspondence establishes an isomorphism between a moduli space of stable algebraic geometric objects and a moduli space of solutions of a certain (generalized) Hermite-Einstein equation. We believe that, for a large class of moduli problems, this correspondence respects the virtual fundamental classes defined in the two categories. We prove this statement in an interesting case, namely the moduli problem associated with the symplectic factorization problem which yields the complete toric varieties. Therefore, our main objects are the moduli spaces of "twisted" linear sigma models associated with this symplectic factorization problem. We describe these moduli spaces in both gauge theoretic and algebraic geometric frameworks and we identify the correponding virtual fundamental classes.