Groupoids, Frobenius algebras and Poisson sigma models

TL;DR

This paper explores the mathematical relationships between groupoids and Frobenius algebras within the framework of Poisson sigma models with boundary, establishing new correspondences and structures in symplectic and Hilbert space categories.

Contribution

It introduces a correspondence between groupoids and relative Frobenius algebras, and defines weak monoids and relational symplectic groupoids in the context of Poisson sigma models.

Findings

Established a correspondence between groupoids in Set and relative Frobenius algebras in Rel.

Proved an adjunction between semigroupoids and H*-algebras.

Described structures in the extended symplectic category and Hilbert spaces.

Abstract

In this paper we discuss some connections between groupoids and Frobenius algebras specialized in the case of Poisson sigma models with boundary. We prove a correspondence between groupoids in the category Set and relative Frobenius algebras in the category Rel, as well as an adjunction between a special type of semigroupoids and relative H*-algebras. The connection between groupoids and Frobenius algebras is made explicit by introducing what we called weak monoids and relational symplectic groupoids, in the context of Poisson sigma models with boundary and in particular, describing such structures in the ex- tended symplectic category and the category of Hilbert spaces. This is part of a joint work with Alberto Cattaneo and Chris Heunen.