  Certain vertex algebras and Lie algebras arising in superstring theory are
investigated.
  We show that the Fock space of a compactified Neveu-Schwarz superstring, i.e.
a Neveu-Schwarz superstring moving on a torus, carries the structure of a
vertex superalgebra with a Neveu-Schwarz element. This implies that the
physical states of such a string form a Lie algebra. The same is true for the
GSO-projected states. The structure of these Lie algebras is investigated in
detail. In particular there is a natural invariant form on them. In case that
the torus has Lorentzian signature the quotient of these Lie algebras by the
kernel of this form is a generalized Kac-Moody algebra. The roots can be easily
described. If the dimension of space-time is smaller than or equal to 10 we can
even determine their multiplicities.
