Conformal boundary state for the rectangular geometry

TL;DR

This paper develops an explicit formalism for conformal boundary states in rectangular geometries within 1+1d conformal field theories, validated through analytical and numerical comparisons.

Contribution

It provides a general explicit expression for conformal boundary states in rectangular geometries applicable to any CFT, enhancing understanding of boundary conditions in these systems.

Findings

Derived an explicit boundary state expression for homogeneous boundary conditions.

Validated the formalism against known partition functions and numerical simulations.

Confirmed the approach's consistency with free theory coherent states.

Abstract

We discuss conformal field theories (CFTs) in rectangular geometries, and develop a formalism that involves a conformal boundary state for the 1+1d open system. We focus on the case of homogeneous boundary conditions (no insertion of a boundary condition changing operator), for which we derive an explicit expression of the associated boundary state, valid for any arbitrary CFT. We check the validity of our solution, comparing it with known results for partition functions, numerical simulations of lattice discretizations, and coherent state expressions for free theories.