Rectangular amplitudes, conformal blocks, and applications to loop models

TL;DR

This paper develops a conformal field theory framework for analyzing partition functions on rectangles, applies it to loop models, and derives new probability formulas for self-avoiding walks, enhancing understanding of critical systems with boundary conditions.

Contribution

It introduces a general formalism of rectangle boundary states in conformal field theory and applies it to loop models, providing new analytical and numerical results.

Findings

Explicit rectangular amplitudes for free theories

Modular properties of these amplitudes analyzed

New probability formulas for self-avoiding walks derived

Abstract

In this paper we continue the investigation of partition functions of critical systems on a rectangle initiated in [R. Bondesan et al, Nucl.Phys.B862:553-575,2012]. Here we develop a general formalism of rectangle boundary states using conformal field theory, adapted to describe geometries supporting different boundary conditions. We discuss the computation of rectangular amplitudes and their modular properties, presenting explicit results for the case of free theories. In a second part of the paper we focus on applications to loop models, discussing in details lattice discretizations using both numerical and analytical calculations. These results allow to interpret geometrically conformal blocks, and as an application we derive new probability formulas for self-avoiding walks.