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Additivity of constructible factorization algebras over manifolds with corners

Victor Carmona, Anja Švraka

TL;DR

The paper addresses the additivity of constructible factorization algebras on products of manifolds with corners, solving Ginot’s conjecture in this setting. It develops a derived Dunn additivity for generalized Swiss-cheese operads and provides an alternative, streamlined hyperdescent proof for constructible factorization algebras on corners, culminating in a global additivity result: $\text{Fact}^{\mathsf{cbl}}_{X\times Y}(\mathcal{V})\simeq \text{Fact}^{\mathsf{cbl}}_{X}(\text{Fact}^{\mathsf{cbl}}_{Y}(\mathcal{V}))$ via pushforward along $\pi_X$. The approach blends operadic additivity for $\mathbb{E}_{p,q}$ with descent theory, using localization theorems for $\mathsf{Bsc}_X$, hyperdescent, and a directed-integral/disintegration framework. These results establish a robust foundation for constructing and comparing higher Morita theories and observables in perturbative QFT with defects on corners, and they set the stage for generalizing additivity to broader conically smooth stratified manifolds and flag-type singularities.

Abstract

We prove the statement in the title, solving in this way a conjecture stated by Ginot for manifolds with corners. Along the way, we establish a derived Swiss-cheese additivity theorem and an alternative proof for the hyperdescent of factorization algebras over those manifolds.

Additivity of constructible factorization algebras over manifolds with corners

TL;DR

The paper addresses the additivity of constructible factorization algebras on products of manifolds with corners, solving Ginot’s conjecture in this setting. It develops a derived Dunn additivity for generalized Swiss-cheese operads and provides an alternative, streamlined hyperdescent proof for constructible factorization algebras on corners, culminating in a global additivity result: via pushforward along . The approach blends operadic additivity for with descent theory, using localization theorems for , hyperdescent, and a directed-integral/disintegration framework. These results establish a robust foundation for constructing and comparing higher Morita theories and observables in perturbative QFT with defects on corners, and they set the stage for generalizing additivity to broader conically smooth stratified manifolds and flag-type singularities.

Abstract

We prove the statement in the title, solving in this way a conjecture stated by Ginot for manifolds with corners. Along the way, we establish a derived Swiss-cheese additivity theorem and an alternative proof for the hyperdescent of factorization algebras over those manifolds.

Paper Structure

This paper contains 21 sections, 49 theorems, 150 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Applications.
  3. Related work.
  4. Outline.
  5. Conventions.
  6. Acknowledgements.
  7. Background on Swiss-cheese and related operads
  8. Additivity for generalized Swiss-cheese operads
  9. Additivity for constructible factorization algebras
  10. Yet another localization theorem.
  11. Hyperdescent of constructible factorization algebras.
  12. From local to global additivity.
  13. Digression: a word on additivity for flag/linear singularities.
  14. Manifolds with corners
  15. Tangent and frame bundles.
  16. ...and 6 more sections

Key Result

Theorem A

(Global additivity; see Theorem thm: Global additivity) Let $\EuScript{V}$ be a presentable symmetric monoidal $\infty$-category and let $X$, $Y$ be two manifolds with corners. Then, there is an equivalence of symmetric monoidal $\infty$-categories induced by pushforward of factorization algebras along $\uppi_X\colon X\times Y\to X$.

Figures (6)

  • Figure 1: An instance of a horizontal (left) and a vertical product (right) of $\mathdutchcal{F}$ and the corresponding configuration of disjoint open subsets in $X\times Y$.
  • Figure 2: Image of a multimorphism $c\in\mathbbst{E}_{1,1}\genfrac{[}{]}{0pt}{}{\{\hbox{$\diameter$} , \{1\}\}}{\{1\}}$. The resulting collection of open subsets of $\square^{\,1,1}$ can also be viewed as an object in ${\mathsf{Cube}^{\otimes}_{{1},{1}}}$, denoted $(U,V)$.
  • Figure 3: Image of a map in $\mathop{\mathrm{\mathsf{Rect}}}\nolimits \left( \bigsqcup_{i\in \langle 3\rangle^{\circ}} \square^{\,p, R_i}, X \right)$ with $(R_1 , R_2, R_3) = (\{1\}, \diameter, \diameter)$ and $(Q_1, Q_2, Q_3) = (\{1\}, \diameter, \{1\})$, both seen as objects of $\mathbbst{E}^{\otimes}_{1,1}$.
  • Figure 4: Open subsets associated with the image of the open rectilinear embedding $c$ from Figure \ref{['fig: figure 1']} along $\upmu_{\hbox{$\diameter$}},\upmu_{\{1\}}\colon \mathbbst{E}_{1,1}\longrightarrow \mathbbst{E}_{1,2}$; both maps use $\square^{\,1,2}\cong \square^{\, 1,1}\times \square^{0,1}$.
  • Figure 5: Illustration of the $(j,k)$-component of a morphism in the image of the bifunctor $\upmu\colon \mathbbst{E}^{\otimes}_{1,1}\times \mathbbst{E}^{\otimes}_{0,1}\longrightarrow \mathbbst{E}^{\otimes}_{1,2}$. The accent in $\{1'\}$ denotes that the object belongs to the second factor $\mathbbst{E}^{\otimes}_{0,1}$.
  • ...and 1 more figures

Theorems & Definitions (128)

  • Theorem A
  • Theorem B
  • Theorem C
  • Corollary A
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.5
  • Definition 2.6
  • Remark 2.7
  • ...and 118 more