This paper constructs topological field theories from representations of a certain product of the infinite symmetric group \(S_\infty\). A topological field theory is a functor from a topological category to an algebraic one.
The objects of the topological category are the nonnegative integers \(\mathbb{Z}_+\), and the morphisms are the \((\alpha, \beta)\)-boards, \(\alpha, \beta\in \mathbb{Z}_+\). An \((\alpha, \beta)\)-board is a compact, oriented, triangulated surface such that (a) the triangles are painted in the checker board fashion, and the edges of every triangle are painted in three different colors; (b) numbers \(1, \dots, \beta\) (resp. \(1, \dots, \alpha\)) are assigned to some black (resp. white) triangles; (c) every sphere that consists of two triangles has a number label.
The above topological category is canonically isomorphic to the category \(\mathbb{S}^{[3]}\) whose morphisms are defined as follows. Let \(S_\infty^{\text{fin}}\) be the finite permutation subgroup of \(S_\infty\) and \(S_\infty(\alpha)\) the subgroup that fixes \(1,\dots, \alpha\). Let \(\mathbb{G}^{[3]} = \{(gh_1, gh_2, gh_3) | g\in S_\infty, h_i\in S_\infty^{\text{fin}}\}\). Let \(K(\alpha)\) be the image of \(S_\infty(\alpha)\) under the diagonal embedding \(S_\infty\hookrightarrow \mathbb{G}^{[3]}\). The morphisms of \(\mathbb{S}^{[3]}\) are the double cosets \(K(\alpha)\backslash\mathbb{G}^{[3]}/K(\beta)\).
The author shows that every \(K(0)\)-spherical unitary representation of \(\mathbb{G}^{[3]}\) defines a functor from \(\mathbb{S}^{[3]}\) to the category of Hilbert spaces and bounded operators. The paper also contains a construction for such unitary representations.