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Topological field theories associated with Calabi-Yau categories

Tristan Bozec, Damien Calaque, Sarah Scherotzke

Abstract

We construct symmetric monoidal higher categories of iterated Calabi-Yau cospans, that are noncommutative analogs of iterated lagrangian correspondences. We actually give a general (and functorial) procedure that applies to iterated nondegenerate cospans on certain comma categories. This allows us to factor the AKSZ fully extended TFT associated with the moduli of objects of a Calabi-Yau category (taking values in iterated lagrangian correspondences) through a fully extended TFT taking values in iterated Calabi-Yau cospans.

Topological field theories associated with Calabi-Yau categories

Abstract

We construct symmetric monoidal higher categories of iterated Calabi-Yau cospans, that are noncommutative analogs of iterated lagrangian correspondences. We actually give a general (and functorial) procedure that applies to iterated nondegenerate cospans on certain comma categories. This allows us to factor the AKSZ fully extended TFT associated with the moduli of objects of a Calabi-Yau category (taking values in iterated lagrangian correspondences) through a fully extended TFT taking values in iterated Calabi-Yau cospans.

Paper Structure

This paper contains 1 section, 3 theorems, 4 equations.

Table of Contents

  1. Proof of an $(\infty,2)$-categorical lemma

Key Result

Lemma A.1

Consider two cospans $\phi:\Upsilon\to\mathcal{A}$ and $\psi:\Upsilon\to\mathcal{B}$ in $\mathbf C_1^n(\Upsilon,\varnothing)$. Denote by $\mathcal{P}=\mathcal{A}\amalg_{\Upsilon}\mathcal{B}$ their composition in $\mathbf C_1^n(\varnothing,\varnothing)=\mathbf C_0^{n+1}$. Denote by $c:k[n]\to\mathcal

Theorems & Definitions (5)

  • Lemma A.1
  • proof
  • Corollary A.3
  • Proposition A.4
  • proof