Lattice regularisation of a non-compact boundary conformal field theory

Non-compact Conformal Field Theories (CFTs) are central to several aspects of string theory and condensed matter physics. They characterised, in particular, by the appearance a continuum conformal dimensions. Surprisingly, such CFTs have been identified as limits lattice models with finite number degrees freedom per site. However, results so far restricted case periodic boundary conditions, precluding exploration via non-compact corresponding D-brane constructions. The present paper follows series previous works on $\mathbb{Z}_2$-staggered XXZ spin chain, whose limit is known be CFT related Euclidian black hole sigma model. By using relationship this chain an integrable $D_2^2$ vertex model, we here identify conditions that lead continuous spectrum exponents, thus correspond branes. In context Potts model square lattice, they wired at physical antiferromagnetic critical point. relations parafermion theories discussed well. We also able renormalisation group flow from previously studied compact ones.