Super Liouville Theory with Boundary

TL;DR

This work extends the bosonic Liouville framework to N=1 super Liouville theory on surfaces with and without boundaries, using degenerate representations to compute sphere and disc correlators and to classify boundary states. It reveals two inequivalent ways to impose boundary conditions on the supercurrent, leading to distinct NS and RR boundary sectors and showing that modular methods alone do not fix all RR Cardy states. The authors derive explicit bulk and boundary one-point and two-point functions, including detailed recursion relations for degenerate boundary states and reflection coefficients, expressed through Upsilon and related special functions, and connect these to the open-string density via annulus amplitudes. The results illuminate how supersymmetry alters reflection, fusion, and boundary dynamics compared to the bosonic case, and set the stage for further exploration of boundary phenomena in non-compact, supersymmetric CFTs.

Abstract

We study N=1 super Liouville theory on worldsheets with and without boundary. Some basic correlation functions on a sphere or a disc are obtained using the properties of degenerate representations of superconformal algebra. Boundary states are classified by using the modular transformation property of annulus partition functions, but there are some of those whose wave functions cannot be obtained from the analysis of modular property. There are two ways of putting boundary condition on supercurrent, and it turns out that the two choices lead to different boundary states in quality. Some properties of boundary vertex operators are also presented. The boundary degenerate operators are shown to connect two boundary states in a way slightly complicated than the bosonic case.