The \(\text{GL}(1| 1)\) WZW-model: from supergeometry to logarithmic CFT. (English) Zbl 1192.81185
Summary: We present a complete solution of the WZW model on the supergroup \(\text{GL}(1|1)\). Our analysis begins with a careful study of its minisuperspace limit (“harmonic analysis on the supergroup”). Its spectrum is shown to contain indecomposable representations. This is interpreted as a geometric signal for the appearance of logarithms in the correlators of the full field theory. We then discuss the representation theory of the \(\text{gl}(1|1)\) current algebra and propose an Ansatz for the state space of the WZW model. The latter is established through an explicit computation of the correlation function. We show in particular, that the 4-point functions of the theory factorize on the proposed set of states and that the model possesses an interesting spectral flow symmetry. The note concludes with some remarks on generalizations to other supergroups.
MSC:
| 81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
| 22E70 | Applications of Lie groups to the sciences; explicit representations |
| 32C11 | Complex supergeometry |
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