Curved String Topology and Tangential Fukaya Categories

TL;DR

This paper explores curved deformations of algebraic structures related to simply connected manifolds, establishing criteria for when associated categories are smooth, proper, and Calabi-Yau, and connecting these to topological quantum field theories and Floer theory.

Contribution

It introduces explicit criteria for the smoothness, properness, and Calabi-Yau property of categories derived from curved deformations of cochain algebras, linking algebraic and Floer theoretic perspectives.

Findings

Criteria for when curved module categories are smooth, proper, and CY.

Explicit Floer theoretic interpretations for projective spaces.

Construction of a Fukaya category with tangency conditions.

Abstract

Given a simply connected manifold M such that its cochain algebra, C^\star(M), is a pure Sullivan dga, this paper considers curved deformations of the algebra C_\star({\Omega}M) and consider when the category of curved modules over these algebras becomes fully dualizable. For simple manifolds, like products of spheres, we are able to give an explicit criterion for when the resulting category of curved modules is smooth, proper and CY and thus gives rise to a TQFT. We give Floer theoretic interpretations of these theories for projective spaces and their products, which involve defining a Fukaya category which counts holomorphic disks with prescribed tangencies to a divisor.