| Summary: | 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.
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