Towards Open-Closed Categorical Enumerative Invariants: Circle-Action Formality Morphism

TL;DR

This paper constructs an open-closed formality morphism in Calabi-Yau categories, aiming to connect categorical invariants with open-closed Gromov-Witten invariants and advance quantization of open string field theory.

Contribution

It introduces an open-closed formality morphism that extends existing formality structures, providing a new algebraic framework for open-closed Gromov-Witten invariants.

Findings

Constructed an open-closed formality morphism.

Proposed a conjectural link to open-closed Gromov-Witten invariants.

Outlined a step towards quantizing large N open string field theory.

Abstract

Categorical enumerative invariants of a Calabi-Yau category, encoded as the partition function of the associated closed string field theory (SFT), conjecturally equal Gromov-Witten invariants when applied to Fukaya categories. Part of this theory is a formality $L_\infty$ morphism which depends on a splitting of the non-commutative Hodge filtration. Our main result is providing an open-closed formality morphism; the algebraic structures involved conjecturallygive a home to open-closed GW invariants. We explain how the open-closed morphism is an ingredient towards quantizing the large $N$ open SFT of an object of a Calabi-Yau category.