A braided monoidal category for free super-bosons

TL;DR

This paper constructs a braided monoidal category structure for the representations of free super-bosons and symplectic fermions, using explicit calculations of vertex operators and conformal blocks.

Contribution

It introduces a new braided monoidal category framework for free super-boson and symplectic fermion mode algebra representations, including twisted cases.

Findings

Established tensor product, braiding, and associator structures.

Extended monoidal structure to twisted mode algebra representations.

Explicit calculations of conformal blocks underpin the categorical structures.

Abstract

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.