Logarithmic conformal field theories of type $B_n,\ell=4$ and symplectic fermions

Abstract: There are important conjectures about logarithmic conformal field theories (LCFT), which are constructed as 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 $B_n/\sqrt{2}$ as a first working example beyond $A_1/\sqrt{p}$. 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 to rescaled $A_1^n$, which are $n$ copies of the even part of one pair. The new long screenings give the global $C_n$-symmetry. The extension is due to a degeneracy in this particular case: Rescaled long roots still have even integer norm. The associated quantum group of divided powers has similar degeneracies [Lent16]: It contains the small quantum group of type $A_1^n$ and the Lie algebra $C_n$. Recent results [FGR17b] 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 $D_n,D_4,A_2$ with larger global symmetry $B_n,F_4,G_2$.