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We initiate a study of the boundary version of the square-lattice $Q$-statePotts antiferromagnet, with $Q \in [0,4]$ real, motivated by the fact that thecontinuum limit of the corresponding bulk model is a non-compact CFT, closelyrelated with the $SL(2,\mathbb{R})_k/U(1)$ Euclidian black-hole coset model.While various types of conformal boundary conditions (discrete and continuousbranes) 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 boundaryconditions (CBC) leading to a discrete boundary spectrum. The conformalboundary conditions (CBC) we find are of two types. The first is free boundaryPotts spins, for which we confirm an old conjecture for the generatingfunctions of conformal levels, and show them to be related to characters in anon-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 ofsize $Q_1$, or its complement of size $Q-Q_1$, at alternating sites along theboundary - is new, and turns out to be conformal in the antiferromagnetic caseonly. Using algebraic and numerical techniques, we show that the correspondingspectrum generating functions produce all the characters of discreterepresentations for the coset CFT. The normalizability bounds of the associateddiscrete states in the coset CFT are found to have a simple interpretation interms 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 boundaryconditions can be reformulated in terms of a RSOS height model. The spectrumgenerating 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 altconditions are needed to cover all the string functions.

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