Kontsevich's Cocycle Construction and Quantization of the Loday-Quillen-Tsygan Theorem

TL;DR

This paper connects graph complexes, Calabi-Yau categories, and Kontsevich's cocycle to generalize the Loday-Quillen-Tsygan theorem, producing a commutative diagram of shifted Poisson algebras and a quantized version via Beilinson-Drinfeld algebras.

Contribution

It introduces a categorical framework that extends the Loday-Quillen-Tsygan map to A-infinity categories and constructs a quantized algebraic structure, linking geometry and gauge theories.

Findings

Established a commutative square of shifted Poisson algebras.

Generalized the Loday-Quillen-Tsygan map to A-infinity categories.

Developed a quantized version using Beilinson-Drinfeld algebras.

Abstract

We relate graph complexes, Calabi-Yau $A_\infty$-categories and Kontsevich's cocycle construction. Our main result produces a commutative square of shifted Poisson algebras; one of its edges is the Loday-Quillen-Tsygan map, generalized to $A_\infty$-categories. We describe a quantized version via Beilinson-Drinfeld algebras. The larger context is to provide categorical methods which relate enumerative geometry (as in mirror symmetry) and large $N$ gauge theories.