Conformal Bootstrap to Renyi Entropy in 2D Liouville and Super-Liouville CFTs

We mainly study the R\'enyi entanglement entropy (REE) of the states excited by local operators in two dimensional irrational conformal field theories (CFTs), especially in Liouville field theory (LFT) and $\mathcal{N}=1$ super Liouville field theory (SLFT). In particular, we consider the excited states obtained by acting on the vacuum with primary operators. These states can be divided into three classes in LFT and SLFT. We show that the 2nd REE of such local excited states becomes divergent in early and late time limits. Choosing a target state and reference state in the same class, the variation of REE between target and reference states can be well defined. The difference of such variation of REE between in early and late time limit always coincides with the log of the ratio of fusion matrix element between target states and reference states. Furthermore, we also study the locally excited states by acting generic descendent operators on the vacuum and the difference of variation of REE will be sum of the log of the ratio of the fusion matrix element and some additional normalization factors. Because the identity operator does not live in Hilbert space of LFT and SLFT, we found that all these properties are quite different from those of excited states in 1+1 dimensional rational CFTs