Derived loop stacks and categorification of orbifold products

TL;DR

This paper constructs a categorification of the virtual orbifold product using derived loop stacks, connecting it to Drinfeld centers and advancing the understanding of orbifold cohomology theories.

Contribution

It introduces a new categorification approach for the virtual orbifold product based on derived geometry, linking it to monoidal category centers.

Findings

Establishes a geometric framework for orbifold product categorification.

Connects virtual orbifold products to Drinfeld centers of monoidal categories.

Answers a question posed by Hinich regarding these connections.

Abstract

The existence of interesting multiplicative cohomology theories for orbifolds was first suggested by string theorists. Orbifold products have been intensely studied by mathematicians for the last fifteen years. In this paper we focus on the virtual orbifold product that was first introduced in Lupercio et al. (2007). We construct a categorification of the virtual orbifold product that leverages the geometry of derived loop stacks. By work of Ben-Zvi, Francis and Nadler, this reveals connections between virtual orbifold products and Drinfeld centers of monoidal categories, thus answering a question of Hinich.