Algebraic string topology from the neighborhood of infinity

TL;DR

This paper develops an algebraic framework for string topology operations, specifically the Goresky-Hingston coproduct, using Hochschild chains of smooth A-infinity categories with Calabi-Yau structures, and computes explicit examples for spheres.

Contribution

It introduces an algebraic analogue of the loop coproduct via Hochschild chains of A-infinity categories with Calabi-Yau structures, connecting string topology to categorical formal neighborhoods.

Findings

Constructed an algebraic model for the Goresky-Hingston coproduct.

Established a categorical formal punctured neighborhood of infinity.

Explicitly computed the coproduct for spheres.

Abstract

We construct and study an algebraic analogue of the loop coproduct in string topology, also known as the Goresky-Hingston coproduct. Our algebraic setup, which under this analogy takes the place of the complex of chains on the free loop space of a possibly non-simply connected manifold, is the Hochschild chain complex of a smooth $A_{\infty}$-category equipped with a pre-Calabi-Yau structure and a trivialization of a version of the Chern character of its diagonal bimodule. The algebraic analogue of the loop coproduct is part of a more general mapping cone construction, which we describe in terms of the categorical formal punctured neighborhood of infinity associated to the underlying smooth $A_\infty$-category. We use a graphical formalism for $A_\infty$-categories and bimodules to describe explicit models for the operations and homotopies involved. We also compute explicitly the algebraic coproduct in the context of the string topology of spheres.