The linear property of genus-$g$, $n$-point, $b$-boundary, $c$-crosscap correlation functions in two-dimensional conformal field theory

TL;DR

This paper introduces a method to express complex genus-$g$, $n$-point, boundary, and crosscap correlation functions in 2D conformal field theory as linear combinations of simpler bulk functions, revealing a pole structure and utilizing free field realizations.

Contribution

It demonstrates that these correlation functions are infinite linear combinations of bulk functions and provides a practical method to compute the linear coefficients using free field realizations.

Findings

Linear coefficients have a single pole structure at degenerate limits.

Correlation functions can be expressed as infinite linear combinations of bulk functions.

The method is applied to Virasoro and $N=1$ minimal models.

Abstract

We propose a method to challenge the calculation of genus-$g$, bulk $n$-point, $b$-boundary, $c$-crosscap correlation functions with $x$ boundary operators $\mathcal{F}_{g,n,b,c}^{x}$ in two-dimensional conformal field theories (CFT$_2$). We show that $\mathcal{F}_{g,n,b,c}^{x}$ are infinite linear combinations of genus-$g$, bulk $(n+b+c)$-point functions $\mathcal{F}_{g,(n+b+c)}$, and try to obtain the linear coefficients in this work. We show the existence of a single pole structure in the linear coefficients at degenerate limits. A practical method to obtain the infinite linear coefficients is the free field realizations of Ishibashi states. We review the results in Virasoro minimal models $\mathcal{M}(p,p')$ and extend it to the $N=1$ minimal models $\mathcal{SM}(p,p')$.