Geometric Properties of W-Algebras and the Toda model

The W-algebra minimal models on hyperelliptic Riemann surfaces are constructed. Using a proposal by Polyakov, we reduce the partition function of the Toda field theory on the hyperelliptic surface to a product of partition functions: one of a free field theory on the sphere with inserted Toda vertex operators and one of a free scalar field theory with antiperiodic boundary conditions with inserted twist fields.     

Liouville theory on a Z N Riemann surface

Liouville field theory on a Z _N curve is studied. It is shown that the partition function on such surfaces reduces to Liouville correlation functions on a sphere and to a free scalar theory with Ramond boundary conditions. Pis’ma Zh. éksp. Teor. Fiz. 66, No. 2, 85–90 (25 July 1997)     

Remarks on A 2 Toda theory

We study the Toda field theory with finite Lie algebras using an extension of the Goulian-Li technique. In this way, we show that, after integrating over the zero mode in the correlation functions of the exponential fields, the resulting correlation function resembles that of a free theory. Furthermore, it is shown that for some ratios of the charges of the exponential fields the four-point correlation functions which contain a degenerate field satisfy the Riemann ordinary differential equation. Using this fact and the crossing symmetry, we derive a set of functional equations for the structure constants of the A ₂ Toda field theory.