Unoriented 2-dimensional TQFTs and the category $\operatorname{Rep}(S_t\wr \mathbb Z_2)$

TL;DR

This paper constructs a family of unoriented 2D cobordism theories and shows how certain specializations relate to Deligne categories, including the category of representations of the wreath product of symmetric groups with Z2.

Contribution

It introduces a new family of unoriented 2D cobordism theories and establishes their connection to Deligne categories, including the representation category of $S_t \,\wr\, \mathbb Z_2$.

Findings

Specializations yield equivalences with Deligne categories.

One specialization recovers the category $ ext{Rep}(S_t\wr \mathbb Z_2)$.

The construction links cobordism theories with algebraic categories.

Abstract

We construct a family of unoriented 2-dimensional cobordism theories parametrized by certain triples of sequences. We also prove that some specializations of these sequences yield equivalences with an exterior product of Deligne categories. In particular, one of these specializations recovers the generalized Deligne category $\operatorname{Rep}(S_t\wr \mathbb Z_2)$.