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Abstract

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.