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

We relate graph complexes, Calabi-Yau $A_{\mathrm{\infty }}$-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 $A_{\mathrm{\infty }}$-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.