Free fermion resolution of supergroup WZNW models

Abstract

Extending our earlier work on PSL(2|2), we explain how to reduce the solution of WZNW models on general type I supergroups to those defined on the bosonic subgroup. The new analysis covers in particular the supergroups GL(M |N ) along with several close relatives such as PSL(N |N ), certain Poincaré supergroups and the series OSP (2|2N ). The technical foundation for this remarkable progress is a special Feigin-Fuchs type representation which allows to keep the bosonic symmetry manifest instead of reducing it to free fields. In preparation for the field theory analysis, we shall exploit a minisuperspace analogue of the resulting free fermion construction to deduce the spectrum of the Laplacian on type I supergroups. The latter is shown to be non-diagonalizable. After lifting these results to the full WZNW model, we address various issues of the field theory, including its modular invariance and the computation of correlation functions. In agreement with previous findings, supergroup WZNW models allow to study chiral and non-chiral aspects of logarithmic conformal field theory within a geometric framework. We shall briefly indicate how insights from WZNW models carry over to non-geometric examples, such as e.g. the W(p) triplet models.

Structure constants of the OSP(1|2) WZNW model

“…While bulk 2-point functions of the latter model have been studied [33] and a conjecture for bulk 3-point functions was formulated in [34], higher correlators are not yet available. In this context, it may also be worthwhile investigating the precise relation between the OSP(2|2) WZNW model discussed above and the SL(1|2) theory that has been solved in [35,18]. The OSP(2|2) WZNW model was also investigated in the condensed matter literature, see e.g.…”

“…For WZNW models on type I supergroups a special parametrization could be found [18] in which the interaction terms are at most quadratic in the fermionic fields. It is a basic feature of the type II case that such a simplification cannot be achieved.…”

“…Because there exist no bosonic elements with degree δ = +2, anti-commutators between fermionic elements of degree δ = +1 vanish. This feature of type I superalgebras was exploited in [18] and leads to a solution of the corresponding WZNW models. For type II superalgebras the situation is different.…”

“…By definition, these contain bosonic generators of degree δ = ±2 with respect to the distinguished Z-gradation. Consequently, anti-commutators between fermionic elements of degree δ = +1 need no longer vanish, a property that spoils a successful solution along the lines of [18]. Our idea here is to remove the problematic generators of degree δ = ±2 through reduction to a local field theory with a lower dimensional target space.…”

Yangian superalgebras in conformal field theory

“…The state of the art of sigma models on target superspace is that WZW models on supergroups are fairly well understood due to their Kac-Moody symmetry, see [5,6,7,8,9,10,11,12,13] for bulk and [13,14,15,16,17,18,19] for boundary models. The situation changes when Kac-Moody symmetry ceases to be present.…”