Rectangular amplitudes, conformal blocks, and applications to loop models

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.

46 pages

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• Geometric mutual information at classical critical points
• Shape dependent finite-size effect of critical two-dimensional Ising model on a triangular lattice
• Logarithmic corrections to the free energy from sharp corners with angle $2\pi$
• Exact solution of the $2d$ dimer model: Corner free energy, correlation functions and combinatorics
• Extreme boundary conditions and random tilings
• Finite-size corrections for universal boundary entropy in bond percolation
• The critical phenomena of a single defect
• Exact finite-size corrections in the dimer model on a planar square lattice
• Geometrical mutual information at the tricritical point of the two-dimensional Blume-Capel model
• Universality and exact finite-size corrections for spanning trees on cobweb and fan networks