Constructing new open-closed TQFTs from the interpolation of symmetric monoidal categories

Abstract

For any symmetric monoidal category $\mathcal{D}$, Lauda and Pfeiffer showed the equivalence between the $\mathcal{D}$-valued open-closed 2-dimensional TQFTs and the so-called knowledgeable Frobenius algebras (KFAs) in $\mathcal{D}$. Each KFA in $\mathcal{D}=\mathbf{Vec}_{\mathbb{K}}$ provides a sequence of scalars indexed by the set $\mathbb{N}^2$ of diffeomorphism classes of connected endocobordisms of the empty set, given by evaluation by the associated TQFT on each such cobordism class. From an arbitrary sequence $\chi=(\chi_{g,w})_{g,w\in\mathbb{N}}$, we build a symmetric monoidal category $\mathcal{C}_{\chi}$ -- with unit object $\textbf{1}$ satisfying $\text{End}_{\mathcal{C}_{\chi}}(\textbf{1})\cong \mathbb{K}$ -- generated by a KFA object affording this sequence. We then determine which sequences $\chi$ produce semisimple abelian categories $\mathcal{C}_{\chi}$ with finite-dimensional hom-spaces. These form a family of categories interpolating the categories of representations of automorphism groups of certain KFAs in $\mathbf{Vec}_{\mathbb{K}}$.