Abstract
We revisit the line of non-unitary theories that interpolate between the Virasoro minimal models. Numerical bootstrap applications have brought about interest in the four-point function involving the scalar primary of lowest dimension. Using recent progress in harmonic analysis on the conformal group, we prove the conjecture that global conformal blocks in this correlator appear with positive coefficients. We also compute many such coefficients in the simplest mixed correlator system. Finally, we comment on the status of using global conformal blocks to isolate the truly unitary points on this line.
Highlights
Conformal field theories (CFTs) in two dimensions enjoy invariance under two copies of the Virasoro algebra—an algebra defined by 1⁄2Lm; Ln 1⁄4 ðm −nÞLmþn þ c 12 mðm ð1:1Þ where c is the central charge
In addition to providing an exact solution, representation theory of the Virasoro algebra enabled the authors of Refs. [2,3,4,5] to show that these models are the only unitary CFTs in two dimensions with c < 1
It has become known more recently that one can see hints of the special role played by minimal models without exploiting Virasoro symmetry at all [6,7,8,9,10,11,12,13]
Summary
Conformal field theories (CFTs) in two dimensions enjoy invariance under two copies of the Virasoro algebra—an algebra defined by 1⁄2Lm; Ln. [2,3,4,5] to show that these models are the only unitary CFTs in two dimensions with c < 1. In order to have a unitary two-dimensional (2D) CFT with c < 1, it is necessary that all primary operators have conformal weights equal to Kac’s formula hr;sðcÞ for some ðr; sÞ. The Kac table of degenerate weights is given by c. Each of these Verma modules has a null state at level rs. In the operator product expansion (OPE) of primary operators φr;s and φr0;s0, the new conformal families that appear are captured in the fusion rule
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