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

Jakob Ulmer

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.

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

Abstract

The overarching goal of this thesis was to develop categorical methods that connect enumerative geometry, as studied in mirror symmetry, with large gauge theories. In the first part, we established a relation between graph complexes, Calabi-Yau -categories, and Kontsevich's cocycle construction. The next main result is the construction of a formality -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.
Paper Structure (63 sections, 111 theorems, 557 equations, 1 figure)

This paper contains 63 sections, 111 theorems, 557 equations, 1 figure.

Table of Contents

  1. Introduction and Summary
  2. Enumerative and Homological Mirror Symmetry
  3. Closed Categorical Enumerative Invariants
  4. Results Obtained in this Thesis
  5. Kontsevich's Cocycle Construction and the Quantized Loday-Quillen-Tsygan Theorem
  6. Open-Closed Circle-Action Formality Morphism
  7. Towards Twisted Holography from Calabi-Yau Categories
  8. Overview over the Chapters
  9. Some Related Works
  10. Topological Conformal Field Theories
  11. Cyclic and Modular Operads
  12. Open Sector
  13. Closed Sector: $S^1$-framed modular operads
  14. Barannikov's work
  15. Algebraic Preliminaries
  16. ...and 48 more sections

Key Result

Theorem 1.1.3

Considering the category with one object with endomorphisms given by the algebra of rational numbers, then we have $HH^*(\mathbb{Q})\cong \mathbb{Q}$, there is a unique splitting $s$ and the coefficient of is

Figures (1)

  • Figure 1: An example of composition in $\mathcal{O}_\Lambda$, from Cos07a

Theorems & Definitions (374)

  • Conjecture 1.1.2: AmTu22, conjecture 6.12
  • Theorem 1.1.3: Tu21
  • Remark 1.2.3
  • Theorem 1.2.7: Cos05CaTu24
  • Theorem 1.2.10: CaTu24
  • Definition 1.2.12
  • Remark 1.2.14
  • Remark 1.2.15
  • Remark 1.3.6
  • Remark 1.3.9
  • ...and 364 more