Open-Closed String Field Theory from Calabi-Yau Categories and its Applications to Enumerative Geometry

TL;DR

This thesis develops categorical methods linking enumerative geometry and gauge theories, constructing algebraic morphisms that generalize closed string field theory to open-closed settings, with applications to quantization and holography.

Contribution

It introduces a formality L_infinity-morphism relating Calabi-Yau categories and objects, extending categorical enumerative invariants to open-closed string field theory.

Findings

Established a relation between graph complexes and Calabi-Yau categories.

Constructed a formality L_infinity-morphism depending on a Hodge filtration splitting.

Proposed a categorical framework for 'Twisted Holography' using Calabi-Yau categories.

Abstract

The overarching goal of this thesis was to develop categorical methods that connect enumerative geometry, as studied in mirror symmetry, with large $N$ gauge theories. In the first part, we established a relation between graph complexes, Calabi-Yau $A_\infty$-categories, and Kontsevich's cocycle construction. The next main result is the construction of a formality $L_\infty$-morphism relating algebraic structures built from a Calabi-Yau category and one of its objects; this morphism depends on a splitting of the non-commutative Hodge filtration. This generalizes the approach of categorical enumerative invariants from the closed to the open-closed setting. From a physics perspective, closed categorical enumerative invariants are encoded by the partition function of the associated closed string field theory (SFT). We explain how our open-closed morphism is an ingredient in quantizing the large N open SFT associated to an object of a Calabi-Yau category. In the final part of this thesis, based on an algebraic approach to open and closed backreacted SFT, we propose ideas towards a categorical formulation of 'Twisted Holography' at the level of partition functions, given as input a Calabi-Yau category and one of its objects.