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

TL;DR

This paper analytically solves the critical Ashkin-Teller model by computing structure constants and crossing symmetry for both affine and Virasoro primary fields, clarifying conformal blocks at c=1.

Contribution

It provides the first explicit analytic computation of three-point constants and crossing symmetry for Virasoro primaries in the Ashkin-Teller model at c=1, including degenerate fields.

Findings

Computed three-point structure constants for the model.

Proved crossing symmetry for four-point functions involving Virasoro primaries.

Clarified the analytic structure of conformal blocks at c=1, especially for degenerate fields.

Abstract

We revisit the critical two-dimensional Ashkin-Teller model, i.e. the $\mathbb{Z}_2$ orbifold of the compactified free boson CFT at $c=1$. 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 $c=1$. 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.