Superalgebras, constraints and partition functions

TL;DR

This paper explores Borcherds superalgebras derived from finite-dimensional Lie algebras, focusing on their structure, constraints, and partition functions, with applications to pure spinors, exceptional geometry, and tensor hierarchies.

Contribution

It introduces a covariant formulation of Serre relations and links the algebra's positive level generators to partition functions of constrained bosonic variables, simplifying their construction.

Findings

Expressed Serre relations covariantly

Connected generator spectra to partition functions

Simplified algebra construction at arbitrary levels

Abstract

We consider Borcherds superalgebras obtained from semisimple finite-dimensional Lie algebras by adding an odd null root to the simple roots. The additional Serre relations can be expressed in a covariant way. The spectrum of generators at positive levels are associated to partition functions for a certain set of constrained bosonic variables, the constraints on which are complementary to the Serre relations in the symmetric product. We give some examples, focusing on superalgebras related to pure spinors, exceptional geometry and tensor hierarchies, of how construction of the content of the algebra at arbitrary levels is simplified.