Ground state degeneracy and module category

Abstract

We develop a systematic method to classify connected étale algebras $A$'s in (possibly degenerate) pre-modular category $\mathcal{B}$. In particular, we find the category of $A$-modules, ${\mathcal{B}}_A$, have ranks bounded from above by $\lfloor \text{FPdim}(\mathcal{B})\rfloor$. For demonstration, we classify connected étale algebras in some $\mathcal{B}$'s, which appear in physics. Physically, the results constrain (or fix) ground state degeneracies of (certain) $\mathcal{B}$-symmetric gapped phases. We study massive deformations of rational conformal field theories such as minimal models and Wess-Zumino-Witten models. In most of our examples, the classification suggests the symmetries $\mathcal{B}$'s are spontaneously broken.

28 pages + Appendix, 4 figures, 6 tables; v2: minor changes