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Integrable and Conformal Boundary Conditions for \(\widehat{s\ell}\)(2) A–D–E Lattice Models and Unitary Minimal Conformal Field Theories

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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 \(\widehat{s\ell }\)(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.

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Authors and Affiliations

  1. C. N. Yang Institute for Theoretical Physics, State University of New York, Stony Brook, New York, 11794-3840

    Roger E. Behrend

  2. Department of Mathematics and Statistics, University of Melbourne, Parkville, Victoria, 3052, Australia

    Paul A. Pearce

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  1. Roger E. Behrend
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  2. Paul A. Pearce
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Behrend, R.E., Pearce, P.A. Integrable and Conformal Boundary Conditions for \(\widehat{s\ell}\)(2) A–D–E Lattice Models and Unitary Minimal Conformal Field Theories. Journal of Statistical Physics 102, 577–640 (2001). https://doi.org/10.1023/A:1004890600991

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