We propose a systematic definition of genus-g, n-point, b-boundary,
c-crosscap correlation functions Fg,n,b,cxβ with x
boundary operators, for general two-dimensional conformal field theory. The
Fg,n,b,cxβ 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
Fg,n,b,cxβ are infinite linear combinations of genus-g,
(n+b+c)-point functions Fg,(n+b+c)β. A single pole structure is
identified in the expansion coefficients Fg,n,b,c0β, 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 Fg,n,b,cxβ 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