A braided monoidal category for free super-bosons

Ingo Runkel

The chiral conformal field theory of free super-bosons is generated by weight
one currents whose mode algebra is the affinisation of an abelian Lie
super-algebra h with non-degenerate super-symmetric pairing. The mode algebras
of a single free boson and of a single pair of symplectic fermions arise for
even|odd dimension 1|0 and 0|2 of h, respectively.
In this paper, the representations of the untwisted mode algebra of free
super-bosons are equipped with a tensor product, a braiding, and an associator.
In the symplectic fermion case, i.e. if h is purely odd, the braided monoidal
structure is extended to representations of the Z/2Z-twisted mode algebra. The
tensor product is obtained by computing spaces of vertex operators. The
braiding and associator are determined by explicit calculations from three- and
four-point conformal blocks.