Extended TQFTs and Algebraic Geometry

TL;DR

This paper explores the construction of Rozansky--Witten TQFTs within an extended categorical framework, revealing limitations when using reduced Noetherian schemes for full extension.

Contribution

It constructs a 2-category of schemes and sheaves, demonstrating how certain 2D TQFTs relate to Rozansky--Witten theory and identifying obstructions to full extension.

Findings

Constructs a 2-category of schemes and sheaves.

Shows existence of (1+1)-TQFTs matching Rozansky--Witten invariants.

Proves no extension to (1+1+1)-TQFTs for reduced Noetherian schemes.

Abstract

We study a potential method for constructing the Rozansky--Witten TQFT as an extended $(1+1+1)$-TQFT. We construct a $2$-category consisting of schemes, complexes of sheaves and sheaf morphisms and show that there are $(1+1)$-TQFTs valued in the truncation of this category which have state spaces that agree with the Rozansky--Witten TQFT. However, we also show that if such a TQFT is based on a reduced Noetherian scheme, it cannot be extended upwards to a $(1+1+1)$-TQFT.