Modules Over Affine Lie Superalgebras

TL;DR

This paper extends key concepts from Kac-Moody algebra theory to affine Lie superalgebras, providing explicit formulas for weights and singular vectors, and explores their representations and modular properties relevant to superconformal algebras.

Contribution

It generalizes the Kac-Kazhdan formula and Felder BRST complex construction to affine Lie superalgebras, advancing understanding of their modules and representations.

Findings

Generalized Kac-Kazhdan formula for superalgebras

Explicit singular vectors in Verma modules

Decomposition of admissible representations and modular transformations

Abstract

Modules over affine Lie superalgebras ${\cal G}$ are studied, in particular, for ${\cal G}=\hat{OSP(1,2)}$. It is shown that on studying Verma modules, much of the results in Kac-Moody algebra can be generalized to the super case. Of most importance are the generalized Kac-Kazhdan formula and the Malikov-Feigin-Fuchs construction, which give the weights and the explicit form of the singular vectors in the Verma module over affine Kac-Moody superalgebras. We have also considered the decomposition of the admissible representation of $\hat{OSP(1,2)}$ into that of $\hat{SL(2)}\otimes$Virasoro algebra, through which we get the modular transformations on the torus and the fusion rules. Different boundary conditions on the torus correspond to the different modings of the current superalgebra and characters or super-characters, which might be relevant to the Hamiltonian reduction resulting in Neveu-Schwarz or Ramond superconformal algebras. Finally, the Felder BRST complex, which consists of Wakimoto modules by the free field realization, is constructed.