The Mini-Superspace Limit of the SL(2,C)/SU(2)-WZNW Model

TL;DR

The paper studies the mini-superspace (zero-mode) limit of the noncompact SL(2,C)/SU(2) WZNW model, recasting the problem as quantum mechanics on the hyperbolic space H3^+ and analyzing it via harmonic analysis. It constructs a complete spectral framework: a delta-normalized plane-wave basis, explicit SL(2,C) generators as differential operators, and a transparent operator-state correspondence that separates normalizable from non-normalizable states. Correlation functions are explored through a baby-CFT bootstrap and overlaps, yielding explicit two- and three-point data, OPEs, and a four-point factorization into s-channel blocks with hypergeometric structure; crossing symmetry ties these to t-channel blocks and informs fusion rules. Appendices provide a rigorous spectral decomposition, self-adjointness analysis, and explicit fusion kernels, showing how non-normalizable intermediate states contribute in a well-defined manner and how mini-superspace results connect to the full WZNW theory.

Abstract

Many qualitatively new features of WZNW models associated to noncompact cosets are due to zero modes with continuous spectrum. Insight may be gained by reducing the theory to its zero-mode sector, the mini-superspace limit. This will be discussed in some detail for the example of SL(2,C)/SU(2)-WZNW model. The mini-superspace limit of this model can be formulated as baby-CFT. Spectrum, structure constants and fusion rules as well as factorization of four point functions are obtained from the harmonic analysis on SL(2,C)/SU(2). The issues of operator-state correspondence or the appearance of non-normalizable intermediate states in correlation functions can be discussed transparently in this context.