Probabilistic construction of non compactified imaginary Liouville field theory

Abstract

We propose a probabilistic construction of imaginary Liouville Field Theory based on a real (non-compactified) Gaussian Free Field. We argue that our theory is the first explicit Lagrangian field theory that reproduces the imaginary DOZZ structure constants without requiring a neutrality constraint. Our proposal is supported by exact results for the imaginary Gaussian Multiplicative Chaos on the circle, and by numerical simulations on the sphere. In particular, we show that the three-point functions of the theory agree remarkably well with the imaginary DOZZ structure constants.

Figures (9)

• Figure 1: In the $c$ complex plane, the regions where $\text{Re}\left[\mathcal{A}(c)\right]$ takes smaller and smaller negative values, where $\mathcal{A}(c)= \mathbf{i} \beta s c / 2 + \log{\mathcal{G}^{\mathbb{S}}_{\beta}(\mu e^{\mathbf{i} \beta/2 c})}$, are shaded in different blue tones. One can see that on the $\mathcal{U}$ contour (shown in black) the one-point function (\ref{['1-pt_path_integral']}) is well defined. The parameters used in the above picture are $\beta=0.6$, $s=0.37$ and $\mu=1$.
• Figure 2: Contour lines of the real part of $\mathbf{i} (\bar{\alpha}-Q) c+\log(\mathcal{G}^{\hat{\mathbb{C}}}_{\beta,\boldsymbol{\alpha}}(e^{\mathbf{i} \beta c}))$, as from numerical simulations with $N=12000$ independent samples of the chaos, $\beta=0.4$, $\alpha_1=\alpha_2=-4.0$, $\alpha_3=-2.4$.
• Figure 3: Overview of the match between the imaginary DOZZ formula and the numerical simulations of LFT$_{\mathbf{i}\beta}$ three-point function. The relative deviation is shown in the insert. The area $\alpha_3\leq Q$ is shaded in gray, and the poles and zeros of $C^\text{ImDOZZ}$ are marked by vertical lines. See the supplementary material for further details.
• Figure 4: We show the distribution of values of $M^{\mathbb{S}}_{\beta,\epsilon}$ for $\beta=0.43$
• Figure 5: We show the behavior of $\mathcal{G}^{\mathbb{S}}_{\beta}(e^{\mathbf{i} \frac{\beta}{2}c})$ along the $\mathcal{U}$ contour, obtained by numerical simulations. On the left $c\in [0,\frac{4\pi}{\beta}]$, and on the right on the vertical line $c\in [0,-\mathbf{i} \infty]$.