On the gl(1|1) Wess-Zumino-Witten model

Abstract

We continue the study of the gl(1|1) Wess-Zumino-Witten model. The Knizhnik-Zamolodchikov equations for the one, two, three and four point functions are analyzed, for vertex operators corresponding to typical and projective representations. We demonstrate their interplay with the logarithmic global conformal Ward identities. We compute the four point function for one projective and three typical representations. Three coupled first order Knizhnik-Zamolodchikov equations are integrated consecutively in terms of generalized hypergeometric functions, and we assemble the solutions into a local correlator. Moreover, we prove crossing symmetry of the four point function of four typical representations at generic momenta. Throughout, the map between the gl(1|1) Wess-Zumino-Witten model and symplectic fermions is exploited and extended.

The integer quantum Hall plateau transition is a current algebra after all

“…Please be warned that other definitions (cf. [29]) of the GL(r|r) WZW model give Z = 0 , which is unacceptable and unphysical for our purposes. Let us finish this subsection with two more remarks on (56).…”

“…First of all, in view of dissenting proposals in the literature (see [29] for a recent reference), we must offer some basic clarification as to what is meant by a GL(r|s) WZW model. Notwithstanding its misleading name, such a field theory does not have any Lie supergroup GL(r|s) or U(r|s) ⊂ GL(r|s) for its real target space (before complexification).…”

“…Different statements exist in the literature, cf. [29,30], where the questionable choice of a Lie supergroup is made for the target space.…”

The free field representation for the GL(1∣1) WZW model revisited

“…If was realized [4,5] that the model can be reformulated as a system of symplectic fermions [6]. This description provides a more direct way to obtain correlation functions and structure constants [7] than the traditional Dotsenko-Fateev type free field realization [8,9]. Despite this fact we are going to show that the original free field representation is consistent and fully describes the theory on a sphere.…”