Curved Beta-Gamma Systems and Quantum Koszul Resolution

We consider the partition function of beta-gamma systems in curved space of the type discussed by Nekrasov and Witten. We show how the Koszul resolution theorem can be applied to the computation of the partition functions and to characters of these systems and find a prescription to enforce the hypotheses of the theorem at the path integral level. We illustrate the technique in a few examples: a simple 2-dimensional target space, the N-dimensional conifold, and a superconifold. Our method can also be applied to the Pure Spinor constraints of superstrings.