The half-signature extension of the 3D bordism bicategory and its representations

Abstract

We construct the half-signature bordism bicategory $\mathbf{Bord}_{1,2,3}^\mathrm{sig/2}$ as a central extension of the 3-dimensional oriented bordism bicategory by $\mathbb{Z}$. This is an index $2$ subextension of the signature bordism bicategory. Following methods of Bartlett--Douglas--Schommer-Pries--Vicary, we construct a presentation of this bicategory and hence show that its linear representations are classified exactly by modular tensor categories.