Conformally invariant boundary conditions in the antiferromagnetic Potts model and the $SL(2,\mathbb{R})/U(1)$ sigma model

Abstract

We initiate a study of the boundary version of the square-lattice $Q$-state Potts antiferromagnet, with $Q\in [0,4]$ real, motivated by the fact that the continuum limit of the corresponding bulk model is a non-compact CFT, closely related with the $SL(2,\mathbb{R}{)}_k/U(1)$ Euclidian black-hole coset model. While various types of conformal boundary conditions (discrete and continuous branes) have been formally identified for the the $SL(2,\mathbb{R}{)}_k/U(1)$ coset CFT, we are only able in this work to identify conformal boundary conditions (CBC) leading to a discrete boundary spectrum. The conformal boundary conditions (CBC) we find are of two types. The first is free boundary Potts spins, for which we confirm an old conjecture for the generating functions of conformal levels, and show them to be related to characters in a non-linear deformation of the $W_{\infty }$ algebra. The second type of CBC - which corresponds to restricting the values of the Potts spins to a subset of size $Q_1$, or its complement of size $Q-Q_1$, at alternating sites along the boundary - is new, and turns out to be conformal in the antiferromagnetic case only. Using algebraic and numerical techniques, we show that the corresponding spectrum generating functions produce all the characters of discrete representations for the coset CFT. The normalizability bounds of the associated discrete states in the coset CFT are found to have a simple interpretation in terms of boundary phase transitions in the lattice model. For $\sqrt{Q}=2\cos \frac{\pi }{k}$, with $k\ge 3$ integer, we show also how our boundary conditions can be reformulated in terms of a RSOS height model. The spectrum generating functions are then identified with string functions of the compact $SU(2{)}_{k-2}/U(1)$ parafermion theory (with symmetry $Z_{k-2}$). The new alt conditions are needed to cover all the string functions.

added references for section 9

References in corpus (7)

• Conformal boundary loop models
• The two-boundary Temperley-Lieb algebra
• Non compact conformal field theory and the a_2^{(2)} (Izergin-Korepin) model in regime III
• Combinatorial aspects of boundary loop models
• On the scaling behaviour of the alternating spin chain
• Non compact continuum limit of two coupled Potts models
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• Poisson-Lie T-Duality of WZW Model via Current Algebra Deformation
• Anisotropic scaling of the two-dimensional Ising model II: surfaces and boundary fields
• The $D_3^{(2)}$ spin chain and its finite-size spectrum
• Boundary critical phenomena in the quantum Ashkin-Teller model
• $U_q[OSp(3|2)]$ quantum chains with quantum group invariant boundaries