Infinite symmetric groups and combinatorial constructions of topological field theory type

TL;DR

This paper surveys combinatorial constructions of topological field theory related to infinite symmetric groups, involving categories of tiled surfaces and their representations as functors to Hilbert spaces, with various extensions.

Contribution

It introduces new categorical frameworks for infinite symmetric groups using combinatorial bordisms and extends these constructions to diverse geometric and algebraic objects.

Findings

Constructed categories with morphisms as tiled surfaces and bordisms

Established functorial assignments from group representations to Hilbert spaces

Presented various extensions including Brauer diagrams and pseudomanifolds

Abstract

The paper contains a survey of train constructions for infinite symmetric groups and related groups. For certain pairs (a group $G$, a subgroup $K$), we construct categories, whose morphisms are two-dimensional surfaces tiled by polygons and colored in a certain way. A product of morphisms is a gluing of combinatorial bordisms. For a unitary representation of $G$ we assign a functor from the category of bordisms to the category of Hilbert spaces and bounded operators. The construction has numerous variations, instead of surfaces there arise also one-dimensional objects of Brauer diagram type, multi-dimensional pseudomanifolds, bipartite graphs