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Three-dimensional topological field theory and symplectic algebraic geometry I

Anton Kapustin, Lev Rozansky, Natalia Saulina

TL;DR

The paper develops a comprehensive framework for boundary conditions and defects in the Rozansky–Witten theory, showing that boundary data correspond to complex Lagrangian submanifolds with Calabi–Yau fibrations and that the collection of boundary conditions forms a 2‑category whose morphisms are boundary line operators. Through circle and interval reductions, it reveals deep connections to curved and Landau–Ginzburg variants of the B‑model, and introduces a curved B‑model on boundary data to capture quantum corrections and deformations via curvings and the Atiyah class. The boundary–bulk map generalizes Chern characters to a categorified setting, while line and surface operators organize into a rich monoidal 2‑categorical structure that encodes fusion, braiding, and higher morphisms, with reductions to Landau–Ginzburg theories at intersections. The work ties these physical constructions to categorified algebraic geometry, proposing a framework in which 2‑categories of fibrations over a fixed Y model deformations of the monoidal category of coherent sheaves, thereby providing a concrete physical realization of deformation quantization for higher categorical structures. Overall, the RW boundary theory serves as a bridge between 3d topological field theory and modern categorified geometry, with potential applications to 3d mirror symmetry and quantum deformation theories in algebraic geometry.

Abstract

We study boundary conditions and defects in a three-dimensional topological sigma-model with a complex symplectic target space X (the Rozansky-Witten model). We show that boundary conditions correspond to complex Lagrangian submanifolds in X equipped with complex fibrations. The set of boundary conditions has the structure of a 2-category; morphisms in this 2-category are interpreted physically as one-dimensional defect lines separating parts of the boundary with different boundary conditions. This 2-category is a categorification of the Z/2-graded derived category of X; it is also related to categories of matrix factorizations and a categorification of deformation quantization (quantization of symmetric monoidal categories). In the appendix we describe a deformation of the B-model and the associated category of branes by forms of arbitrary even degree.

Three-dimensional topological field theory and symplectic algebraic geometry I

TL;DR

The paper develops a comprehensive framework for boundary conditions and defects in the Rozansky–Witten theory, showing that boundary data correspond to complex Lagrangian submanifolds with Calabi–Yau fibrations and that the collection of boundary conditions forms a 2‑category whose morphisms are boundary line operators. Through circle and interval reductions, it reveals deep connections to curved and Landau–Ginzburg variants of the B‑model, and introduces a curved B‑model on boundary data to capture quantum corrections and deformations via curvings and the Atiyah class. The boundary–bulk map generalizes Chern characters to a categorified setting, while line and surface operators organize into a rich monoidal 2‑categorical structure that encodes fusion, braiding, and higher morphisms, with reductions to Landau–Ginzburg theories at intersections. The work ties these physical constructions to categorified algebraic geometry, proposing a framework in which 2‑categories of fibrations over a fixed Y model deformations of the monoidal category of coherent sheaves, thereby providing a concrete physical realization of deformation quantization for higher categorical structures. Overall, the RW boundary theory serves as a bridge between 3d topological field theory and modern categorified geometry, with potential applications to 3d mirror symmetry and quantum deformation theories in algebraic geometry.

Abstract

We study boundary conditions and defects in a three-dimensional topological sigma-model with a complex symplectic target space X (the Rozansky-Witten model). We show that boundary conditions correspond to complex Lagrangian submanifolds in X equipped with complex fibrations. The set of boundary conditions has the structure of a 2-category; morphisms in this 2-category are interpreted physically as one-dimensional defect lines separating parts of the boundary with different boundary conditions. This 2-category is a categorification of the Z/2-graded derived category of X; it is also related to categories of matrix factorizations and a categorification of deformation quantization (quantization of symmetric monoidal categories). In the appendix we describe a deformation of the B-model and the associated category of branes by forms of arbitrary even degree.

Paper Structure

This paper contains 41 sections, 259 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Topological boundary conditions in the RW model
  3. Review of the RW model
  4. Reduction of the RW model on a circle
  5. Complex lagrangian submanifolds
  6. Deformations of boundary conditions
  7. Obstructions
  8. Calabi-Yau fibrations
  9. Boundary degrees of freedom
  10. The Ehresmann connection
  11. An affine connection on the fibration
  12. BRST transformations of boundary degrees of freedom
  13. The boundary action
  14. Boundary local observables
  15. Deformations of the boundary action
  16. ...and 26 more sections

Figures (4)

  • Figure 1: Fusion of surface operators.
  • Figure 2: Line operator $C$ is a morphism between surface operators $B_1$ and $B_2$.
  • Figure 3: Boundary line operators in 3d TFT correspond to branes in the effective 2d TFT obtained by reducing 3d TFT on an interval.
  • Figure 4: Boundary line operators in 3d TFT also correspond to bulk line operators in the effective 2d TFT obtained by reducing 3d TFT on an interval.