Determinant Formula for Parabolic Verma Modules of Lie Superalgebras

TL;DR

This paper establishes a determinant formula for parabolic Verma modules of Lie superalgebras, extending classical results and enabling new insights into supersymmetric conformal field theories.

Contribution

It proves a conjectured determinant formula for parabolic Verma modules of Lie superalgebras, generalizing prior results for non-super and non-parabolic cases.

Findings

Derived a general determinant formula for parabolic Verma modules

Extended Jantzen's and Kac's formulas to superalgebras

Provided irreducibility criteria for these modules

Abstract

We prove a determinant formula for a parabolic Verma module of a Lie superalgebra, previously conjectured by the second author. Our determinant formula generalizes the previous results of Jantzen for a parabolic Verma module of a (non-super) Lie algebra, and of Kac concerning a (non-parabolic) Verma module for a Lie superalgebra. The resulting formula is expected to have a variety of applications in the study of higher-dimensional supersymmetric conformal field theories. We also discuss irreducibility criteria for the Verma module.