Logarithmic CFT at generic central charge: from Liouville theory to the $Q$-state Potts model

Abstract

Using derivatives of primary fields (null or not) with respect to the conformal dimension, we build infinite families of non-trivial logarithmic representations of the conformal algebra at generic central charge, with Jordan blocks of dimension 2 or 3. Each representation comes with one free parameter, which takes fixed values under assumptions on the existence of degenerate fields. This parameter can be viewed as a simpler, normalization-independent redefinition of the logarithmic coupling. We compute the corresponding non-chiral conformal blocks, and show that they appear in limits of Liouville theory four-point functions.As an application, we describe the logarithmic structures of the critical two-dimensional O(n) and Q-state Potts models at generic central charge. The validity of our description is demonstrated by semi-analytically bootstrapping four-point connectivities in the Q-state Potts model to arbitrary precision. Moreover, we provide numerical evidence for the Delfino-Viti conjecture for the three-point connectivity. Our results hold for generic values of Q in the complex plane and beyond.

Particles, conformal invariance and criticality in pure and disordered systems

“…In contrast with the problem of boundary connectivity [7,8], the probability that k points in the bulk belong to a same percolation cluster [9] is not determined by the differential equations provided by the algebraic framework. Only in recent years the way to answer the question for k = 3 has been found [10], stimulating the study of the case k = 4 [11][12][13][14][15][16], which is sensitive to the whole spectrum of scaling dimensions of the theory 1 .…”

Correlation functions and quantum measures of descendant states

“…They cannot be part of the ADE classifiation of rational, unitary, modular invariant CFTs [11] but could e.g. be logarithmic [58]. In figure 5 we show the second SRD for states L −n |0 with n = 2, 3, 4, 5, 10 and c = 1/1000 to illustrate its non-convexity for all these states.…”

Analytic conformal bootstrap and Virasoro primary fields in the Ashkin-Teller model

“…In contrast, fields are not always completely determined by their left and right conformal dimensions. The fusion relation allows to eliminate conformal blocks from the crossing symmetry equation (18) in favor of the fusion kernel. Schematically ∀∆ t , ∆t ,…”

“…From Zamolodchikov's recursive representation, it is easy to deduce that the limit exists, because the fusion rules ensure Res ∆=∆ (r,s) F ∆ (c|(∆ i )|z) = 0. Conjecture 1 has long been used for numerically computing conformal blocks and testing crossing symmetry in models such as Generalized Minimal Models [7] or the Potts model [18], so there is little doubt that it is true.…”