Comments on the Quantum Field Theory of the Coulomb Gas Formalism

 I will revisit the quantum field theory of the Coulomb gas formalism, clarifying several important points along the way. The first key ingredient involves a peculiarity of the timelike linear dilaton: although the background charge Q breaks the scalar field’s continuous shift symmetry, the exponential of the action is still invariant under a discrete shift since Q is imaginary. Gauging this symmetry makes the linear dilaton compact and introduces winding modes into the spectrum. One of these winding operators corresponds to a BRST current first introduced by Felder, and the BRST cohomology singles out the minimal model operators within the linear dilaton theory. The model at radius R=pp′‾‾‾√R=pp′ has two marginal operators corresponding to the Dotsenko-Fateev screening charges. Deforming by them, one obtains a model that might be called a “BRST quotiented compact timelike Liouville theory” with many interesting properties which I will describe. Applying conformal perturbation theory to the exponential interactions reproduces the Coulomb gas calculations of minimal model correlators and allows for a kinematic derivation of the fusion rules. In contrast to spacelike Liouville, these resonance correlators are finite because the zero mode is compact.