Segal's semigroup, Virasoro structure, and Ward identities in Compactified Imaginary Liouville Theory

Abstract

Compactified Imaginary Liouville Theory (CILT for short) is a probabilistically constructed Conformal Field Theory (CFT), expected to have connections to minimal models, Coulomb gas integrals, and loop models. In this paper, we study the quantization of Segal's semigroup given by CILT. Each element $\mathbf T_f$ in the quantized semigroup corresponds to a conformal annulus: the unit disk $\mathbb D$ with $f(\mathbb D^\circ)$ removed, where $f:\mathbb D\to f(\mathbb D)\subset \mathbb D^\circ$ is a contracting biholomorphism. This semigroup encodes the local conformal structure of the CFT, in particular the Virasoro symmetry with central charge $<1$ for CILT. We first relate the quantized Segal's semigroup to Segal's amplitudes of annuli and recover the Virasoro generators $\mathbf L_n,\widetilde{\mathbf L}_n$ ($n\geq0$) of CILT as generators of this semigroup. Compared to the real Liouville CFT, the main difficulty here is the absence of positivity, which we address through estimates for the imaginary Gaussian Multiplicative Chaos and non-self-adjoint operators. Using Segal's gluing axiom as in the real case, we differentiate the amplitude and use conformal covariance to derive the Ward identities in terms of all Virasoro operators $\mathbf L_n,\widetilde{\mathbf L}_n$ ($n\in\mathbb Z$). We also outline the parallel theory for Boundary CILT (BCILT), concerning Segal's semigroup of semiannuli. These results provide essential tools for the conformal bootstrap program of (B)CILT, boundary BPZ equations, boundary structure constants, and the global construction of conformal blocks.