Genus-$g$, $n$-point, $b$-boundary, $c$-crosscap correlation functions of two-dimensional conformal field theory: Definition and general properties

Abstract

We propose a systematic definition of genus-$g$, $n$-point, $b$-boundary, $c$-crosscap correlation functions $\mathcal{F}_{g,n,b,c}^{x}$ with $x$ boundary operators, for general two-dimensional conformal field theory. The $\mathcal{F}_{g,n,b,c}^{x}$ are defined as the inner products of the surface states with the tensor product of $n$ asymptotic states, $b$ boundary states and $c$ crosscap states, where the $b$ defining boundary states carry a total of $x$ boundary operators. The definition implies that $\mathcal{F}_{g,n,b,c}^{x}$ are infinite linear combinations of genus-$g$, $(n+b+c)$-point functions $\mathcal{F}_{g,(n+b+c)}$. A single pole structure is identified in the expansion coefficients $\mathcal{F}_{g,n,b,c}^{0}$, which is analogous to the single pole structure in the conformal block decomposition. This linear property unifies boundary and bulk CFTs. The consistency conditions of $\mathcal{F}_{g,n,b,c}^{x}$ are discussed: conventional upper-half plane (UHP) and crosscap constraints can be reformulated as extra constraints on a collection of bulk correlation functions.Comment: 41 pages + references (Minor corrections + references added comparing to v1