The imaginary Toda field theory

TL;DR

This paper studies the imaginary Toda field theory with higher spin symmetry, computing structure constants for scalar and non-scalar fields using bootstrap methods, revealing new solutions not obtained by analytic continuation.

Contribution

It introduces the first bootstrap solution for the imaginary Toda field theory, including explicit structure constants for scalar and non-scalar primaries, and classifies non-scalar fields by permutation group conjugacy classes.

Findings

Computed structure constants involving scalar fields.

Classified non-scalar primary fields by permutation conjugacy classes.

Discovered solutions not obtainable through analytic continuation.

Abstract

We consider the two-dimensional $\mathfrak{sl}_n$ quantum Toda field theory with an imaginary background charge. This conformal field theory has a higher spin symmetry ($W_n$ algebra), a central charge $c \leq n-1$ and a continuous spectrum. Using the conformal bootstrap, we compute structure constants involving two arbitrary scalar fields and a semi-degenerate field of Wyllard type. The solution obtained is not the analytic continuation of the usual Toda three-point function. Non-scalar primary fields and their three-point functions are also discussed. Non-scalar primary fields are classified by conjugacy classes of the permutation group $\mathfrak{S}_n$, and their structure constants are computed explicitly, up to an overall factor.