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.
Highlights
Bulk CFT involves coupling left- and right-moving chiral structures, and this coupling can be nontrivial [2]. Is it possible to study the bulk theory without starting from the chiral theory? We propose a positive answer to this question, in the case of CFTs based on the Virasoro algebra at generic central charge
The degenerate fields that allowed us to predict the nontrivial conformal blocks, lead to linear relations between certain structure constants [20]. These relations may be viewed as emanating from an “interchiral” symmetry algebra that is larger than the product of the left and right Virasoro algebra [8]
In two dimensions, logarithmizing Verma modules with null vectors leads to so-called staggered Virasoro modules [11]
Summary
Following the lead of its non-logarithmic counterpart, the study of two-dimensional logarithmic CFT has led to a good understanding of chiral structures, in particular logarithmic representations and their fusion rules [1]. We propose a positive answer to this question, in the case of CFTs based on the Virasoro algebra at generic central charge This case is relatively simple algebraically, and it is motivated by the O(n) model and the Q-state Potts model. We can differentiate, and apply any linear operation: in particular, following [5], we will linearly combine primary fields with descendant fields This approach is effective for computing correlation functions and conformal blocks. In the cases of the O(n) model and of the Q-state Potts model, this idea was recently used in [7], it was only worked out for a subset of logarithmic representations We will follow this idea more systematically, and write a conjecture for the structures of all logarithmic representations in these models at generic central charge. According to forthcoming work by Grans-Samuelsson et al, predictions from the lattice discretization of the Q-state Potts model seem to converge towards the same logarithmic structures [10]
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