Symmetries of a rigid braided category

Abstract

We identify natural symmetries of each rigid higher braided category. Specifically, we construct a functorial action by the continuous group $\Omega \mathsf{O}(n)$ on each ${\mathcal{E}}_{n-1}$-monoidal $(g,d)$-category $\mathcal{R}$ in which each object is dualizable (for $n\ge 2$, $d\ge 0$, $d\le g\le \infty$). This action determines a canonical action by the continuous group $\Omega \mathbb{R}{\mathbb{P}}^{n-1}$ on the moduli space of objects of each such $\mathcal{R}$. In cases where the parameters $n$, $d$, and $g$ are small, we compare these continuous symmetries to known symmetries, which manifest as categorical identities.

20 pages, preliminary version, comments and suggested citations are welcome