Consistency of operator product expansions of Boundary 2d CFT and Swiss-cheese operad

TL;DR

This paper demonstrates that in boundary 2D conformal field theories with certain symmetries, operator product expansions are well-behaved and converge to real analytic functions, linking algebraic structures to analytic properties.

Contribution

It constructs an action of the Swiss-cheese operad's fundamental groupoid on module categories and proves convergence and analyticity of OPEs in boundary CFTs with $C_1$-cofinite symmetry.

Findings

Operator product expansions converge absolutely in boundary CFTs.

Correlation functions are independent of OPE order and parentheses.

Established a connection between operad actions and analytic properties of CFTs.

Abstract

In this paper, we construct an action of the fundamental groupoid of the Swiss-cheese operad (the parenthesized permutation and braid operad) on $C_1$-cofinite module categories of a vertex operator algebra. Based on this result, we show that when a boundary conformal field theory has locally $C_1$-cofinite chiral symmetry, all operator product expansions converge absolutely on open regions in the configuration spaces of the upper half-plane and define real analytic functions (the correlation functions), independent of orders and parentheses of OPEs.