Integrable and Conformal Boundary Conditions for sl(2) A-D-E Lattice Models and Unitary Minimal Conformal Field Theories

TL;DR

This paper develops integrable boundary conditions for A-D-E lattice models, linking them to conformal boundary conditions in sl(2) unitary minimal conformal field theories, and demonstrates their completeness in the continuum limit.

Contribution

It constructs boundary Boltzmann weights satisfying the boundary Yang-Baxter equation for A-D-E models and establishes their correspondence with conformal boundary conditions.

Findings

Boundary weights satisfy the boundary Yang-Baxter equation.

Specialized boundary weights decompose into two-spin edge weights.

Boundary conditions correspond to all conformal boundary conditions in the continuum limit.

Abstract

Integrable boundary conditions are studied for critical A-D-E and general graph-based lattice models of statistical mechanics. In particular, using techniques associated with the Temperley-Lieb algebra and fusion, a set of boundary Boltzmann weights which satisfies the boundary Yang-Baxter equation is obtained for each boundary condition. When appropriately specialized, these boundary weights, each of which depends on three spins, decompose into more natural two-spin edge weights. The specialized boundary conditions for the A-D-E cases are naturally in one-to-one correspondence with the conformal boundary conditions of sl(2) unitary minimal conformal field theories. Supported by this and further evidence, we conclude that, in the continuum scaling limit, the integrable boundary conditions provide realizations of the complete set of conformal boundary conditions in the corresponding field theories.