Continuous Tambara-Yamagami tensor categories
Adrià Marín-Salvador
TL;DR
The paper develops a framework for continuous tensor categories as algebra objects in the Morita bicategory of C*-algebras, enabling a natural generalization of Tambara–Yamagami categories from finite abelian groups to locally compact abelian groups G. It constructs continuous TY categories via A=C_0(G)⊕C with a TY_G correspondence and associates associators to a continuous symmetric nondegenerate bicharacter χ and a sign ξ, ultimately proving a full classification by (χ,ξ). It also proves an automatic continuity result: any TY W*-tensor category for G is equivalent to the image of a continuous TY category under the forgetful functor, so the TY fusion rules force continuity of associators. The results connect operator-algebraic models with topological group duality and yield a precise, structurally rich classification that parallels the finite TY theory while accommodating direct integrals and topological objects.
Abstract
We present a new model for continuous tensor categories as algebra objects in the Morita bicategory of $\mathrm{C}^*$-algebras. In this setting, we generalize the construction of Tambara-Yamagami tensor categories from finite abelian groups to locally compact abelian groups, and provide a classification of continuous Tambara-Yamagami tensor categories for a locally compact group $G$. A continuous Tambara-Yamagami tensor category associated to a locally compact group $G$ is a continuous tensor category that has a single non-invertible simple object $τ$ such that $τ\otimes τ$ decomposes as a direct integral indexed over $G$, meaning $τ\otimesτ\cong L^2(G)$. We show that continuous Tambara-Yamagami tensor categories for $G$ are classified by a continuous symmetric nondegenerate bicharacter $χ: G\times G\to U(1)$ and a sign $ξ\in\{\pm 1\}$. We also prove that, if a $\mathrm{W}^*$-tensor category $\mathcal{C}$ obeys the Tambara-Yamagami fusion rules, then its associators are automatically continuous in the sense that $\mathcal{C}$ is obtained from a continuous tensor category by forgetting its topology.