Comments on the Quantum Field Theory of the Coulomb Gas Formalism

TL;DR

This work embeds the holomorphic Coulomb gas description of (2,p) minimal models into a standard quantum-field-theoretic framework by treating the free timelike linear dilaton as compact and then performing Felder’s BRST quotient. By combining Feigin–Fuchs free-field resolution, Dotsenko–Fateev screening, and Felder’s BRST construction within a Lagrangian setting, the authors show how minimal-model primaries and their fusion data arise from a BRST-reduced spectrum and how the torus partition function matches Rocha-Caridi characters. The dynamics, accessed via marginal deformations, reproduce Coulomb gas integrals through conformal perturbation theory, while the zero-mode quantum mechanics of the compact timelike dilaton explains finite resonance correlators and an infinite tower of plane-wave states that collapse to the minimal-model content under BRST projection. Subtleties with reflection identifications, truncation of the OPE, and higher-genus issues are discussed, with a detailed treatment of the (2,p) case and conjectures for general (p,p') models. The results illuminate connections to JT gravity and minimal strings, and provide a concrete QFT framework for the longstanding Coulomb gas formalism of minimal models.

Abstract

The holomorphic Coulomb gas formalism is a set of rules for computing minimal model observables using free field techniques. We attempt to derive and clarify these rules using standard techniques of QFT. We begin with a careful examination of the timelike linear dilaton. Although the background charge $Q$ breaks the scalar field's continuous shift symmetry, the exponential of the action is invariant under a discrete shift because $Q$ is imaginary. Gauging this symmetry makes the dilaton compact and introduces winding modes into the spectrum. One of these winding operators corresponds to the BRST current first introduced by Felder. The cohomology of this BRST charge isolates the irreducible representations of the Virasoro algebra within the linear dilaton Fock space, and the supertrace in the BRST complex reproduces the minimal model partition function. The model at the radius $R=\sqrt{pp'}$ has two marginal operators corresponding to the Dotsenko-Fateev screening charges. Deforming by them, we obtain a model that might be called a "BRST quotiented compact timelike Liouville theory." The Hamiltonian of the zero-mode quantum mechanics is not Hermitian, but it is $PT$-symmetric and exactly solvable. Its eigenfunctions have support on an infinite number of plane waves, suggesting an infinite reduction in the number of independent states in the full QFT. Applying conformal perturbation theory to the exponential interactions reproduces the Coulomb gas calculations of minimal model correlators. In contrast to spacelike Liouville, these resonance correlators are finite because the zero mode is compact. We comment on subtleties regarding the reflection operator identification, as well as naive violations of truncation in correlators with multiple reflection operators inserted. This work is part of an attempt to understand the relationship between JT gravity and the $(2,p)$ minimal string.

Figures (9)

• Figure 1: Structure of reducible Verma modules. An arrow from one point to the next indicates that the highest weight vector of the second module is a singular vector of the first module.
• Figure 2: Route to the Kac determinant.
• Figure 3: Feigin-Fuchs characterization of the Fock space structure.
• Figure 4: Successive quotients of the Fock space.
• Figure 5: Verma module structure versus Fock space structure.