The non-rational limit of D-series minimal models

Abstract

We study the limit of D-series minimal models when the central charge tends to a generic irrational value c\in (-\infty, 1)c∈(−∞,1). We find that the limit theory’s diagonal three-point structure constant differs from that of Liouville theory by a distribution factor, which is given by a divergent Verlinde formula. Nevertheless, correlation functions that involve both non-diagonal and diagonal fields are smooth functions of the diagonal fields’ conformal dimensions. The limit theory is a non-trivial example of a non-diagonal, non-rational, solved two-dimensional conformal field theory.

Geometrical four-point functions in the two-dimensional critical Q-state Potts model: connections with the RSOS models

“…It will turn out that the difference between the two types of correlation functions is numerically small, and probably indiscernible by Monte-Carlo methods [5], although they are certainly detectable by the transfer matrix techniques developed in [2] and used in the present paper. The quantities defined and studied in [1,[3][4][5][6] will prove to be skewed versions of the true correlation functions (2.6), as we shall explain in detail in section 7.…”

“…where x is a finite number. In such a limit, it was argued in [4,6] that the levels of the null vectors, which are removed in irreducible modules of minimal models, go to infinity, and one obtains Verma modules with the same conformal dimensions: the non-diagonal sector contains fields with conformal dimensions (h r,s , h r,−s ) where r ∈ Z + 1 2 , s ∈ 2Z, and the spectrum in the diagonal sector becomes continuous. The limit spectrum was then used in a conformal block expansion for the numerical bootstrap of the four-point function (18), and the results obtained were found to be in reasonable agreement with Monte-Carlo simulations [5].…”

“…2 Interestingly, it was found afterwards in [4] that the spectrum of [1] could be obtained from a certain limit of minimal models when the central charge is taken to be an irrational JHEP05(2020)156 number (see section 2.4 below). The corresponding CFT was further elucidated analytically in [3,6]. The main question however remains: what statistical physics model does the spectrum in [1] actually describe, if it is not the Potts model, and why does it give results apparently so close numerically [5] to those of the Potts model?…”

“…20) which is in fact the four-point function (2.18), (2.22) studied in details in[1,[3][4][5][6]. In this case, since the only three-point coupling involving φ p/2 and φp /2 is C p/2,p/2,p/2 , and since λp /2 = 0, considerable simplification occurs in the cluster expansion of this correlator: diagrams where φ p/2 and φp /2 are in the same cluster with no other insertions are given a vanishing weight and thus disappear.…”

“…A rather than the minimal model, as there might be several modular invariants 6. With the identification of the parameters as explained in footnote 4, the p, q in[4] is also switched with respect to ours and the limit (2.19) correspond to the limit p q → β 2 in[4].…”