  Recently, boundary conformal field theories (BCFTs) have attracted much
attention in the context of quantum gravity. This is because a BCFT can be dual
to gravity coupled to a heat bath CFT, known as the island model. On this
background, it would be interesting to explore the duality between the boundary
and the braneworld. However, this seems to be a challenging problem. The reason
is because although there has been much study of rational BCFTs, there has been
comparatively little study of irrational BCFTs, and irrational BCFTs are
expected to be the boundary duals of the braneworlds. For this reason, we
explore properties of boundary ingredients: the boundary primary spectrum, the
boundary-boundary-boundary OPE coefficients and the bulk-boundary OPE
coefficients. For this purpose, the conformal bootstrap is extremely useful.
This is the first step in providing an understanding of BCFTs in the context of
braneworld holography by using the conformal bootstrap. The techniques
developed in this paper may be useful for further investigation of irrational
BCFTs.
