Spheres as Frobenius objects

TL;DR

This paper explores the Frobenius structures of spheres across all dimensions within cobordism categories, revealing a uniform pattern where spheres are Frobenius objects, with the 0-sphere forming a symmetric Frobenius object linked to Brauer algebras.

Contribution

It generalizes the Frobenius structure of spheres to all dimensions and connects the 0-sphere to matrix Frobenius algebras via topological quantum field theory.

Findings

All spheres in dimensions d≥1 are commutative Frobenius objects in cobordism categories.

The 0-sphere is a symmetric Frobenius object, not commutative.

The 1-dimensional TQFT maps the 0-sphere to a matrix Frobenius algebra related to Brauer algebras.

Abstract

Following the pattern of the Frobenius structure usually assigned to the 1-dimensional sphere, we investigate the Frobenius structures of spheres in all other dimensions. Starting from dimension $d=1$, all the spheres are commutative Frobenius objects in categories whose arrows are ${(d+1)}$-dimensional cobordisms. With respect to the language of Frobenius objects, there is no distinction between these spheres---they are all free of additional equations formulated in this language. The corresponding structure makes out of the 0-dimensional sphere not a commutative but a symmetric Frobenius object. This sphere is mapped to a matrix Frobenius algebra by a 1-dimensional topological quantum field theory, which corresponds to the representation of a class of diagrammatic algebras given by Richard Brauer.