Realizations of Simple Affine Vertex Algebras and Their Modules: The Cases \({\widehat{sl(2)}}\) and \({\widehat{osp(1,2)}}\)

Abstract

We study the embeddings of the simple admissible affine vertex algebras \({V_k(sl(2))}\) and \({V_k({osp}(1,2))}\), \({k \notin {\mathbb Z}_{\ge 0}}\), into the tensor product of rational Virasoro and N = 1 Neveu–Schwarz vertex algebra with lattice vertex algebras. By using these realizations we construct a family of weight, logarithmic, and Whittaker \({\widehat{sl(2)}}\) and \({\widehat{osp(1,2)}}\)-modules. As an application, we construct all irreducible degenerate Whittaker modules for \({V_k(sl(2))}\).

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Acknowledgements

The author is partially supported by the Croatian Science Foundation under the Project 2634 and by the QuantiXLie Centre of Excellence, a Project cofinanced by the Croatian Government and European Union through the European Regional Development Fund—the Competitiveness and Cohesion Operational Programme (Grant KK.01.1.1.01.0004).We would like to thank T. Creutzig, A.Milas, G. Radobolja, D. Ridout, and S. Wood for valuable discussions. Finally, we thank the referee for a careful reading of the paper and helpful comments.

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Adamović, D. Realizations of Simple Affine Vertex Algebras and Their Modules: The Cases \({\widehat{sl(2)}}\) and \({\widehat{osp(1,2)}}\). Commun. Math. Phys. 366, 1025–1067 (2019). https://doi.org/10.1007/s00220-019-03328-4

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