Superalgebras, constraints and partition functions

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.

Exceptional geometry and Borcherds superalgebras

“…The corresponding infinite sequence of e n -representations agrees precisely with the level decomposition of B n for positive levels, which was later explained in [39]. The same representations also appear in the tensor hierarchies considered in [16][17][18], related to those appearing in gauged supergravity [40,41].…”

Generalized diffeomorphisms for E9

“…This is equivalent to the statement that |p is in a minimal R(λ)-orbit under g. This is discussed e.g. in [14], and a direct connection between minimal orbits and Borcherds superalgebras (the fermionic extensions of g) was made in [66].…”

E8 geometry

“…[42,52]), coinciding with the reducibility of the generalised diffeomorphisms [36], with parameter in R 1 , and also of tensor fields, with parameters in higher R k . The sequence of R k 's coincides with the generators of Borcherds superalgebras [53][54][55][56][57]. The sequence of ghosts, corresponding to reducibility, does not stop where the connection-free window closes, however.…”

“…By the method of ref. [57] (see also ref. [36], where the corresponding counting is performed for E 8 and for lower n), the relevant Borcherds algebra is related to a bosonic object λ in (10 .…”

“…(7.13) Z n (t) is also the inverse partition function for the subalgebra of the Borcherds superalgebra at positive levels [57], with generators in R k . The effective number of gauge parameters, modulo reducibility, equals the number of degrees of freedom in λ, and may be read off as the power of the pole of the partition function at t = 1.…”