Topological Field Theories and the Algebraic Structures of the Two-Sphere

TL;DR

This paper explores algebraic structures arising from bordisms of the 2-sphere in 3D topological field theories, introducing P-monoids and L-monoids, and analyzing their relations and constraints.

Contribution

It introduces two new types of monoids, P-monoids and L-monoids, with prime structures, and develops an $ abla$-operad framework capturing these structures.

Findings

P-monoids and L-monoids are equivalent and are commutative Frobenius monoids.

Prime structures satisfy countable 'legs relations' with Frobenius structures.

Prime endomorphisms act by multiplication by prime units, simplifying the structures.

Abstract

We give two presentations for bordisms of $S^2$ in the 3-dimensional oriented bordism category $\operatorname{Cob}(3) $, encoding the algebraic structures on $S^2$. After passing through topological field theories, we define two kinds of monoids which we call P-monoids and L-monoids. In addition to both being commutative Frobenius monoids, P-monoids are equipped with a class of endomorphisms while L-monoids are equipped with a class of unit morphisms, all of which are labelled by closed oriented irreducible prime 3-manifolds. They turn out to be equivalent. The new prime structures satisfy some countable relations with the commutative Frobenius structure, the most notable of which we call "legs relations." We then restrict to the setting of algebras and show that the legs relations place strong constraints on the new prime endomorphisms which forces them to act by multiplications by prime units, rendering the additional prime structures remarkably simple. We also propose an $\infty$-operad which encodes these prime structures and contains the $\infty$-little 3-cube operad as a sub-operad.% We briefly discuss the relations between P/L-algebras and J-algebras which classify 3-dimensional TFTs.