Causal Poisson Brackets of the SL(2,R) WZNW Model and its Coset Theories

From the basic chiral and anti-chiral Poisson bracket algebra of the SL(2, R) WZNW model, non-equal time Poisson brackets are derived. Through Hamiltonian reduction we deduce the corresponding brackets for its coset theories. The analysis of WZNWmodels is usually reduced to a separate treatment of its chiral or anti-chiral components [1, 2, 3]. But much simpler results can arise when the contributions from the different chiralities are pieced together. This was observed recently for non-equal time Poisson brackets of a gauged WZNW theory, namely the Liouville theory [4]. Since WZNW models and their coset theories turn up in many areas of mathematics and physics, it is worthwhile to add the corresponding results for other WZNW theories. Here we consider the SL(2,R) WZNW model together with its three cosets, the Liouville theory and both the euclidean and Minkowskian black hole models. The general solution of the WZNW equations of motion gives g(τ, σ) as a product of chiral and anti-chiral fields g(z) and ḡ(z̄), where z = τ + σ, z̄ = τ − σ are light cone coordinates. For periodic boundary conditions, the chiral and anti-chiral fields have the monodromies g(z + 2π) = g(z)M and ḡ(z̄ − 2π) = Mḡ(z̄) with M ∈ SL(2,R). We use the following basis of the sl(2,R) algebra