Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Higher categories of push-pull spans, II: Matrix factorizations

Lorenzo Riva

TL;DR

The paper develops a bridge between matrix factorizations and the derived symplectic 2-category CRW by constructing a symmetric monoidal 2-functor $ rak e$ from the 2-category $ extbf{MF}$ of matrix factorizations to the homotopy 2-category $ extbf{h}_2 extbf{CRW}$. It shows that the image lands in affine cotangent stacks $ ext{T}^ullet b K^n$ and proves that all objects in the target are 2-dualizable, enabling explicit computations of 2d topological field theories associated to affine symplectic derived stacks, including a canonical Serre-trivialization. The work reveals that while a full 3-dualizability lift is obstructed for affine cases like $ ext{T}^ullet b K^n$, the constructed TFTs recover Hochschild-type invariants such as $ ext{Spec HC}(A)$ on $S^1$ and intricate genus-$g$ expressions, thus connecting matrix-factorization data to categorical field theory in a precise algebraic framework. An open algebraic question about reconstructing matrix factorizations from endomorphism dg-modules over $A_V$ is posed for future study, signaling deeper insights into the derived Morita aspects of the correspondence.

Abstract

This is the second part of a project aimed at formalizing Rozansky-Witten models in the functorial field theory framework. In the first part we constructed a symmetric monoidal $(\infty, 3)$-category $\mathscr{CRW}$ of commutative Rozansky-Witten models with the goal of approximating the $3$-category of Kapustin and Rozansky. In this paper we extend work of Brunner, Carqueville, Fragkos, and Roggenkamp on the affine Rozansky-Witten models: we exhibit a functor connecting their $2$-category of matrix factorizations with the homotopy $2$-category of $\mathscr{CRW}$, and calculate the associated TFTs.

Higher categories of push-pull spans, II: Matrix factorizations

TL;DR

The paper develops a bridge between matrix factorizations and the derived symplectic 2-category CRW by constructing a symmetric monoidal 2-functor from the 2-category of matrix factorizations to the homotopy 2-category . It shows that the image lands in affine cotangent stacks and proves that all objects in the target are 2-dualizable, enabling explicit computations of 2d topological field theories associated to affine symplectic derived stacks, including a canonical Serre-trivialization. The work reveals that while a full 3-dualizability lift is obstructed for affine cases like , the constructed TFTs recover Hochschild-type invariants such as on and intricate genus- expressions, thus connecting matrix-factorization data to categorical field theory in a precise algebraic framework. An open algebraic question about reconstructing matrix factorizations from endomorphism dg-modules over is posed for future study, signaling deeper insights into the derived Morita aspects of the correspondence.

Abstract

This is the second part of a project aimed at formalizing Rozansky-Witten models in the functorial field theory framework. In the first part we constructed a symmetric monoidal -category of commutative Rozansky-Witten models with the goal of approximating the -category of Kapustin and Rozansky. In this paper we extend work of Brunner, Carqueville, Fragkos, and Roggenkamp on the affine Rozansky-Witten models: we exhibit a functor connecting their -category of matrix factorizations with the homotopy -category of , and calculate the associated TFTs.
Paper Structure (19 sections, 15 theorems, 84 equations, 3 figures)

This paper contains 19 sections, 15 theorems, 84 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Brief review of $\mathscr{CRW}$ and dualizability results
  3. The basics
  4. $2$-dualizability in $\mathscr{CRW}$
  5. The affine part
  6. $2$-dimensional TFTs associated to affine symplectic derived stacks
  7. The Serre automorphism
  8. The $0$- and $1$-dimensional manifolds
  9. The $2$-dimensional manifolds.
  10. Failure of $3$-dualizability
  11. Aside: cotangent stacks and affine planes
  12. Matrix factorizations
  13. The $2$-category $\mathbf{MF}$
  14. A useful quasi-isomorphism
  15. Constructing the functor
  16. ...and 4 more sections

Key Result

Theorem 1

Fix an algebraically closed field $\mathbb{K}$ of characteristic $0$. Then there exists a symmetric monoidal $2$-functor with target the homotopy $2$-category of $\mathscr{CRW}$. The image lands in the subcategory of $\mathbf{h}_2 \mathscr{CRW}$ spanned by the cotangent stacks of the form $\mathrm{T}^\ast \mathbb{K}^{n}$ for $n \geq 0$, and in fact $\mathfrak{e}$ is surjective on those objects.

Figures (3)

  • Figure 1: The Serre automorphism, on the left, and a way to "tighten the twist" to a straight line, on the right.
  • Figure 2: $\mathsf{coev}$ on the left and $\mathsf{ev}$ on the right.
  • Figure 3: From left to right: $\mathsf{sad}$, $\mathsf{cosad}$, $\mathsf{cap}$, and $\mathsf{cocap}$. We omitted the $+$ and $-$ distinguishing the framing on the points since applying $\mathcal{Z}_A$ nullifies the distinction due to the ambidexterity of adjoints in this $2$-category.

Theorems & Definitions (25)

  • Theorem 1: \ref{['thm:main']}
  • Theorem 2: \ref{['thm:2-dual-crw']}
  • Theorem 3
  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 15 more