Extended TQFTs and Algebraic Geometry

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.

35 pages, 1 figure

References in corpus (5)

• Modular categories as representations of the 3-dimensional bordism 2-category
• Three-dimensional topological field theory and symplectic algebraic geometry I
• Constructing symmetric monoidal bicategories functorially
• BV quantization of the Rozansky-Witten model
• Obvious natural morphisms of sheaves are unique