From super Weyl modules to the super Virasoro modules

Abstract

Based on the classification of irreducible modules over the Weyl superalgebra S, the restriction functor and the "twisting technique", we construct a class of new (generally non-weight) modules over the N = 1, 2 Ramond algebras R and T . The sufficient and necessary conditions for these modules to be irreducible are determined. In addition, lots of interesting examples for such irreducible modules over the N = 1, 2 Ramond algebras with different features are given, which include super intermediate series modules, U(CL 0 )-free modules of rank two, super degree two modules and so on. Finally, we obtain that some of these non-weight modules over the N = 1, 2 Ramond algebras are both new. Note that the Witt superalgebra, the N = 2 Neveu-Schwarz algebra and the N = 2 topological algebra are isomorphic to T by spectral flows. As a byproduct, we also give a characterization of the irreducibility of goal modules over these Lie superalgebras (namely, untwisted N = 2 superconformal algebras).

Representations of the twisted $N=2$ superconformal algebra

“…All simple Harish-Chandra modules over the untwisted N = 2 superconformal algebras were classified in [34]. Recently, some non-weight modules related to the U(h)-free modules over the N = 2 superconformal algebras were studied in [10,11,47]. But as far as we know, there are very few studies on non-weight modules for the twisted N = 2 superconformal algebra.…”

Simple restricted modules over the $N=1$ Ramond algebra as weak modules for vertex operator superalgebras

“…Recently, some non-weight modules as Whittaker modules and U(CL 0 ⊕ CG 0 )-free modules of rank 1 over the N = 1 Ramond algebra were studied in [18,28], respectively. Moreover, a class of non-weight modules over the N = 1 Ramond algebra were developed in [4], which include super intermediate series modules, U(CL 0 )-free modules of rank 2 and so on. As everyone knows, the N = 1 Ramond algebra can be seen as certain supersymmetric extensions of the Virasoro algebra.…”