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

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.

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