R\'enyi entropy of single-character CFTs on the torus

We introduce a nonperturbative approach to calculate the R\'enyi entropy of a single interval on the torus for single-character (meromorphic) conformal field theories. Our prescription uses the Wro\'nskian method of Mathur, Mukhi, and Sen [Nucl. Phys. B312, 15 (1989)], in which we construct differential equations for torus conformal blocks of the twist two-point function. As an illustrative example, we provide a detailed calculation of the second R\'enyi entropy for the $\rm E_{8,1}$ Wess-Zumino-Witten (WZW) model. We find that the $\mathbb Z_2$ cyclic orbifold of a meromorphic conformal field theory (CFT) results in a four-character CFT which realizes the toric code modular tensor category. The $\mathbb Z_2$ cyclic orbifold of the $\rm E_{8,1}$ WZW model, however, yields a three-character CFT since two of the characters coincide. We then compute the torus conformal blocks and find that the twist two-point function, and therefore the R\'enyi entropy, is two-periodic along each cycle of the torus. The second R\'enyi entropy for a single interval of the $\rm E_{8,1}$ WZW model has the universal logarithmic divergent behavior in the decompactification limit of the torus, as expected as well as the interval approaches the size of the cycle of the torus. Furthermore, we see that the $q$-expansion is UV finite, apart from the leading universal logarithmic divergence. We also find that there is a divergence as the size of the entangling interval approaches the cycle of the torus, suggesting that gluing two tori along an interval the size of a cycle is a singular limit.