On construction of correlation numbers in super Minimal Liouville Gravity in the Ramond sector

TL;DR

We develop a BRST-cohomology framework for the Ramond sector of $\mathcal{N}=1$ SMLG and construct Ramond physical fields $\mathbb{R}_a$ together with NS fields, enabling analytic three-point correlators through a factorized form $I_{RRN}=\Omega_R(b)N_R(a_1)N_R(a_2)N_{NS}(a_3)$. Central to the approach are ground-ring operators $\mathbb{O}_{m,n}$ and their logarithmic partners $\mathbb{O}'_{m,n}$, whose Higher Equations of Motion representations yield BRST-closed structures that reduce moduli integrals to boundary terms. The Ramond leg factor $N_R(a)$ and the overall normalization $\Omega_R(b)$ are explicit, and a consistency check via a ground-ring computation reproduces the three-point result. The work lays a foundation for analytic Ramond-involving multi-point amplitudes and paves the way for four-point calculations and tests of the continuous/discrete (matrix-model) duality in the Ramond sector.

Abstract

We study the construction of correlation numbers in super minimal Liouville gravity. In particular, we construct the fundamental physical fields in the Ramond sector and compute the three-point correlation number involving two physical fields in the Ramond sector and one in the NS sector. Furthermore, we establish the relation between Ramond physical fields and the elements of the ground ring. Using the higher equations of motion of super Liouville theory, this relation leads to a new representation of the Ramond physical fields. This formulation enables a direct analytic computation of correlation numbers involving Ramond field insertions. As an application, we demonstrate the method in the simplest case of a three-point correlation function.