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Three-dimensional topological field theory and symplectic algebraic geometry II

Anton Kapustin, Lev Rozansky

TL;DR

This work develops a comprehensive, multi-layered framework for 3-dimensional Rozansky–Witten (RW) boundary theories by building a hierarchy of 2- and 3-categories associated to holomorphic symplectic manifolds. It connects algebraic models (matrix factorizations, curved DGAs, and Dolbeault complexes) with geometric data (lagrangian submanifolds, fibrations, and lagrangian correspondences), and it shows how deformations of holomorphic symplectic structures induce deep, controlled deformations of the morphisms and their compositions via Maurer–Cartan-type formalisms. A central theme is locality: morphisms are determined by tubular neighborhoods and intersections of supports, enabling precise localization and a coherent passage between the algebraic and geometric viewpoints, including Knorrer periodicity, Legendre transforms, and derived categorical sheaves. The paper also outlines a presheaf-based, micro-local approach to define the 2-category of boundary conditions on general holomorphic symplectic manifolds, and it connects to categorified algebraic geometry and RW models, including graded enhancements and skyscraper-type boundary data. Collectively, these results pave the way for a robust, deformation-aware, higher-categorical understanding of boundary conditions in holomorphic symplectic settings and their physical realizations in RW-type theories.

Abstract

Motivated by the path integral analysis of boundary conditions in a 3-dimensional topological sigma-model, we suggest a definition of the 2-category associated with a holomorphic symplectic manifold X and study its properties. The simplest objects of this 2-category are holomorphic lagrangian submanifolds of X. We pay special attention to the case when X is the total space of the cotangent bundle of a complex manifold U or a deformation thereof. In the latter case the endomorphism category of the zero section is a monoidal category which is an A-infinity deformation of the 2-periodic derived category of U.

Three-dimensional topological field theory and symplectic algebraic geometry II

TL;DR

This work develops a comprehensive, multi-layered framework for 3-dimensional Rozansky–Witten (RW) boundary theories by building a hierarchy of 2- and 3-categories associated to holomorphic symplectic manifolds. It connects algebraic models (matrix factorizations, curved DGAs, and Dolbeault complexes) with geometric data (lagrangian submanifolds, fibrations, and lagrangian correspondences), and it shows how deformations of holomorphic symplectic structures induce deep, controlled deformations of the morphisms and their compositions via Maurer–Cartan-type formalisms. A central theme is locality: morphisms are determined by tubular neighborhoods and intersections of supports, enabling precise localization and a coherent passage between the algebraic and geometric viewpoints, including Knorrer periodicity, Legendre transforms, and derived categorical sheaves. The paper also outlines a presheaf-based, micro-local approach to define the 2-category of boundary conditions on general holomorphic symplectic manifolds, and it connects to categorified algebraic geometry and RW models, including graded enhancements and skyscraper-type boundary data. Collectively, these results pave the way for a robust, deformation-aware, higher-categorical understanding of boundary conditions in holomorphic symplectic settings and their physical realizations in RW-type theories.

Abstract

Motivated by the path integral analysis of boundary conditions in a 3-dimensional topological sigma-model, we suggest a definition of the 2-category associated with a holomorphic symplectic manifold X and study its properties. The simplest objects of this 2-category are holomorphic lagrangian submanifolds of X. We pay special attention to the case when X is the total space of the cotangent bundle of a complex manifold U or a deformation thereof. In the latter case the endomorphism category of the zero section is a monoidal category which is an A-infinity deformation of the 2-periodic derived category of U.

Paper Structure

This paper contains 68 sections, 3 theorems, 341 equations.

Table of Contents

  1. Introduction
  2. Sigma-models and categories
  3. The 3-category of holomorphic symplectic manifolds
  4. Algebraic approach
  5. Geometric approach: the case of a cotangent bundle
  6. Relation between algebraic and geometric approaches
  7. The 2-category of a deformed cotangent bundle
  8. Geometric approach: the general case
  9. A presheaf definition of the 2-category $\ddot{ \mathsf{L} } ( X , \omega )$
  10. Derived categorical sheaves
  11. The 3-category of affine spaces
  12. 2-periodic perfect derived category of a curved differential graded algebra
  13. Categories of matrix factorizations
  14. Definition of the category
  15. Koszul matrix factorization s
  16. ...and 53 more sections

Key Result

Corollary 5.2

If $\beta =0$, then $\mathop{{\rm deg}}\nolimits _{ \partial W } \lambda _{i} \geq 2$ for $i\geq 1$.

Theorems & Definitions (4)

  • Conjecture 5.1
  • Corollary 5.2
  • Corollary 5.3
  • Corollary 5.4