Large $N$ phenomena and quantization of the Loday-Quillen-Tsygan theorem

TL;DR

This paper introduces a BV formalism-based approach to large N limits, enabling quantization of the Loday-Quillen-Tsygan Theorem and connecting noncommutative geometry with matrix integrals and topological field theories.

Contribution

It presents a novel BV quantization framework for the Loday-Quillen-Tsygan Theorem, linking large N phenomena with noncommutative geometry and matrix models.

Findings

Derived a recurrence relation for multi-point correlation functions.

Expressed Harer-Zagier relations in noncommutative geometric terms.

Provided a quantization solution connecting moduli of branes and gauge theories.

Abstract

We offer a new approach to large $N$ limits using the Batalin-Vilkovisky formalism, both commutative and noncommutative, and we exhibit how the Loday-Quillen-Tsygan Theorem admits BV quantizations in that setting. Matrix integrals offer a key example: we demonstrate how this formalism leads to a recurrence relation that in principle allows us to compute all multi-point correlation functions. We also explain how the Harer-Zagier relations may be expressed in terms of this noncommutative geometry derived from the BV formalism. As another application, we consider the problem of quantization in the large $N$ limit and demonstrate how the Loday-Quillen-Tsygan Theorem leads us to a solution in terms of noncommutative geometry. These constructions are relevant to open topological field theories and string field theory, providing a mechanism that relates moduli of categories of branes to moduli of brane gauge theories.