Noncommutative effective field theories and the large $N$ correspondence

Abstract

We integrate the notion of an effective field theory, as described by Costello, with the framework of noncommutative symplectic geometry introduced by Kontsevich; providing a definition for the renormalization group flow in noncommutative geometry that is defined through the use of ribbon graphs. As in the commutative case, the resulting noncommutative effective field theories are in one-to-one correspondence with local interaction functionals. We explain how in this setting, the large $N$ correspondence discovered by 't Hooft appears as a relation between noncommutative and commutative effective field theories. As an example, we apply this framework to study a noncommutative analogue of Chern-Simons theory.