Lattice regularisation of a non-compact boundary conformal field theory

Abstract

Non-compact Conformal Field Theories (CFTs) are central to several aspects of string theory and condensed matter physics. They are characterised, in particular, by the appearance of a continuum of conformal dimensions. Surprisingly, such CFTs have been identified as the continuum limits of lattice models with a finite number of degrees of freedom per site. However, results have so far been restricted to the case of periodic boundary conditions, precluding the exploration via lattice models of aspects of non-compact boundary CFTs and the corresponding D-brane constructions.The present paper follows a series of previous works on a ℤ2-staggered XXZ spin chain, whose continuum limit is known to be a non-compact CFT related with the Euclidian black hole sigma model. By using the relationship of this spin chain with an integrable $$ {D}_2^2 $$ D 2 2 vertex model, we here identify integrable boundary conditions that lead to a continuous spectrum of boundary exponents, and thus correspond to non-compact branes. In the context of the Potts model on a square lattice, they correspond to wired boundary conditions at the physical antiferromagnetic critical point. The relations with the boundary parafermion theories are discussed as well. We are also able to identify a boundary renormalisation group flow from the non-compact boundary conditions to the previously studied compact ones.

Factorization identities and algebraic Bethe ansatz for $$ {D}_2^{(2)} $$ models

“…In particular, we construct the models' Bethe states, which had not been known, that would be needed to compute scalar products and correlation functions. Moreover, we prove previouslyproposed expressions for the models' eigenvalues and Bethe equations [16,[25][26][27][28]. The interesting degeneracies exhibited by these models are also explained.…”

“…We interpret this parameter as the rapidity of the boundary. We conjecture that this model, like the one in [16], has a non-compact continuum limit.…”

“…Here we express the D (2) 2 transfer matrices for the open spin chains considered in [12] and [16] as products of A (1) 1 transfer matrices. We then use these relations, which we call factorization identities, to solve the models by algebraic Bethe ansatz.…”

“…chains with two different sets of integrable boundary conditions, corresponding to the two possible values (namely, 0 and 1) of a certain parameter ε. We consider first the case ε = 1, which was studied in [16]. The factorization identity (4.10)-(4.11), whose derivation is presented in appendix A, involves a novel A (1) 1 transfer matrix (4.12).…”

“…The antiferromagnetic Potts model and the staggered six-vertex model [1][2][3][4][5][6][7][8][9][10][11] have recently been shown [12] to be related to the D (2) 2 R-matrix [13][14][15]. Even more recently, an open D (2) 2 spin chain with a particular integrable boundary condition has been shown [16] to have as its continuum limit a non-compact boundary conformal field theory, which possesses a continuous spectrum of conformal dimensions; it is closely related to the SL(2, R)/U(1) Euclidean black hole [17][18][19][20][21], see also [22][23][24].…”