Noncommutative effective field theories and the large $N$ correspondence

TL;DR

This paper develops a framework combining effective field theories with noncommutative symplectic geometry, establishing a large N correspondence and applying it to noncommutative Chern-Simons theory.

Contribution

It introduces a definition of renormalization group flow in noncommutative geometry using ribbon graphs, linking noncommutative and commutative effective theories.

Findings

Established a one-to-one correspondence between noncommutative and local interaction effective theories.

Demonstrated the large N correspondence as a relation between noncommutative and commutative theories.

Applied the framework to a noncommutative analogue of Chern-Simons theory.

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.