The Batalin–Vilkovisky Formalism in Noncommutative Effective Field Theory

We address the treatment of gauge theories within the framework that is formed from combining the machinery of noncommutative symplectic geometry, as introduced by Kontsevich, with Costello’s approach to effective gauge field theories within the Batalin–Vilkovisky formalism; discussing the problem of quantization in this context; and identifying the relevant cohomology theory controlling this process. We explain how the resulting noncommutative effective gauge field theories produce classes in a compactification of the moduli space of Riemann surfaces, when we pass to the large length scale limit. Within this setting, the large N correspondence of ’t Hooft—describing a connection between open string theories and gauge theories—appears as a relation between the noncommutative and commutative geometries. We use this correspondence to investigate and ultimately quantize a noncommutative analogue of Chern–Simons theory.