Logarithmic conformal field theories of type B n, ℓ = 4 and symplectic fermions

Abstract

There are important conjectures about logarithmic conformal field theories (LCFT’s), which are constructed as a kernel of screening operators acting on the vertex algebra of the rescaled root lattice of a finite-dimensional semisimple complex Lie algebra. In particular, their representation theory should be equivalent to the representation theory of an associated small quantum group. This article solves the case of the rescaled root lattice Bn/2 as a first working example beyond A1/p. We discuss the kernel of short screening operators, its representations, and graded characters. Our main result is that this vertex algebra is isomorphic to a well-known example: The even part of n pairs of symplectic fermions. In the screening operator approach, this vertex algebra appears as an extension of the vertex algebra associated with A1n/2, which are n copies of the even part of one pair of symplectic fermions. The new long screenings give the new global Cn-symmetry. The extension is due to a degeneracy in this particular case: Rescaled long roots still have an even integer norm. For the associated quantum group of divided powers, the first author has previously encountered matching degeneracies: It contains the small quantum group of type A1n and the Lie algebra Cn. Recent results by Farsad, Gainutdinov, and Runkel on symplectic fermions suggest finally the conjectured category equivalence to this quantum group. We also study the other degenerate cases of a quantum group, giving extensions of LCFT’s of type Dn, D4, A2 with larger global symmetry Bn, F4, G2.