Abstract
We revisit the critical two-dimensional Ashkin–Teller model, i.e. the \mathbb{Z}_2ℤ2 orbifold of the compactified free boson CFT at c=1c=1. We solve the model on the plane by computing its three-point structure constants and proving crossing symmetry of four-point correlation functions. We do this not only for affine primary fields, but also for Virasoro primary fields, i.e. higher twist fields and degenerate fields. This leads us to clarify the analytic properties of Virasoro conformal blocks and fusion kernels at c=1c=1. We show that blocks with a degenerate channel field should be computed by taking limits in the central charge, rather than in the conformal dimension. In particular, Al. Zamolodchikov’s simple explicit expression for the blocks that appear in four-twist correlation functions is only valid in the non-degenerate case: degenerate blocks, starting with the identity block, are more complicated generalized theta functions.
Highlights
Crossing symmetry equations for four-point functions that involve degenerate fields are enough for determining the three-point structure constants of Liouville theory [7,11], even though degenerate fields are unphysical, i.e. they do not appear in operator product expansion (OPE)
Solving the compactified free boson and the Ashkin–Teller model from the point of view of their affine symmetry algebra was not very difficult: building on earlier work, we only had to determine the signs of structure constants, and to check crossing symmetry of four-point functions of the type 〈V V T T 〉
We have focused on finer observables in the same model, namely correlation functions of Virasoro primary fields
Summary
After Belavin, Polyakov and Zamolodchikov worked out the basics of the conformal bootstrap approach to two-dimensional CFT [1], the critical Ashkin–Teller model was one of the first theories to be solved. Per the conformal bootstrap’s basic doctrine, the model’s solution extensively relies on its affine symmetry algebra This means that only correlation functions of affine primary fields are explicitly known. There is an infinite series of Virasoro primary fields, called degenerate fields, which are affine descendants of the identity field Some correlation functions that involve the first few higher twist fields were computed by Apikyan and Al. Zamolodchikov [5], using the model’s affine symmetry. The relevant fusion kernels can be computed explicitly, which will allow us to determine the structure constants of arbitrary Virasoro primary fields. Degenerate fusion kernels can be computed using the same procedure This will allow us to write and solve enough crossing symmetry equations for determining all structure constants of the Ashkin–Teller model. The discrete fusion transformation (273) for conformal blocks that appear in four-point functions of the type 〈V V T T 〉
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