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The $N = 1$ $N = 1$ super Heisenberg–Virasoro vertex algebra at level zero

Dražen Adamović

Department of Mathematics, Faculty of Science, University of Zagreb, Bijenicka 30, 10 000 Zagreb, Croatia

E-mail Address: adamovic@math.hr

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Berislav Jandrić

Department of Mathematics, Faculty of Science, University of Zagreb, Bijenicka 30, 10 000 Zagreb, Croatia

E-mail Address: jandricberislav@gmail.com

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, and

Gordan Radobolja

Faculty of Science, University of Split, Ruđera Boškovića 33, 21 000 Split, Croatia

E-mail Address: gordan@pmfst.hr

corresponding author.

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https://doi.org/10.1142/S0219498823500032Cited by:1 (Source: Crossref)

Abstract

We study the representation theory of the $N = 1$ $N = 1$ super Heisenberg–Virasoro vertex algebra at level zero, which extends the previous work on the Heisenberg–Virasoro vertex algebra [D. Adamović and G. Radobolja, Free field realization of the twisted Heisenberg–Virasoro algebra at level zero and its applications, J. Pure Appl. Algebra 219(10) (2015) 4322–4342; D. Adamović and G. Radobolja, Self-dual and logarithmic representations of the twisted Heisenberg–Virasoro algebra at level zero, Commun. Contemp. Math. 21(2) (2019) 1850008; Y. Billig, Representations of the twisted Heisenberg–Virasoro algebra at level zero, Can. Math. Bull. 46(4) (2003) 529–537] to the super case. We calculated all characters of irreducible highest weight representations by investigating certain Fock space representations. Quite surprisingly, we found that the maximal submodules of certain Verma modules are generated by subsingular vectors. The formulas for singular and subsingular vectors are obtained using screening operators appearing in a study of certain logarithmic vertex algebras [D. Adamović and A. Milas, On W-algebras associated to (2, $p$ $p$ ) minimal models and their representations, Int. Math. Res. Notices 2010(20) (2010) 3896–3934].

Communicated by M. Gorelik

Keywords:

Super Heisenberg–Virasoro vertex algebra

AMSC: 17B69, 17B20, 17B65

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