The Two-exponential Liouville Theory and the Uniqueness of the Three-point Function

It is shown that in the two-exponential version of Liouville theory the coefficients of the three-point functions of vertex operators can be determined uniquely using the translational invariance of the path integral measure and the self-consistency of the two-point functions. The result agrees with that obtained using conformal bootstrap methods. Reflection symmetry and a previously conjectured relationship between the dimensional parameters of the theory and the overall scale are derived.Comment: Plain TeX File; 15 Page

Duality in Liouville Theory as a Reduced Symmetry

The origin of the rather mysterious duality symmetry found in quantum Liouville theory is investigated by considering the Liouville theory as the reduction of a WZW-like theory in which the form of the potential for the Cartan field is not fixed a priori. It is shown that in the classical theory conformal invariance places no condition on the form of the potential, but the conformal invariance of the classical reduction requires that it be an exponential. In contrast, the quantum theory requires that, even before reduction, the potential be a sum of two exponentials. The duality of these two exponentials is the fore-runner of the Liouville duality. An interpretation for the reflection symmetry found in quantum Liouville theory is also obtained along similar lines.Comment: Plain TeX file; 9 page

Brief Resume of Seiberg-Witten Theory

Talk presented by the second author at the Inaugural Coference of the Asia Pacific Center for Theoretical Physics, Seoul, June 1996. The purpose of this note is to give a resume of the Seiberg-Witten theory in the simplest possible mathematical terms.Comment: 10 pages, LaTe

Path Integral Formulation of the Conformal Wess-Zumino-Witten to Liouville Reduction

The quantum Wess-Zumino-Witten $\to$ Liouville reduction is formulated using the phase space path integral method of Batalin, Fradkin, and Vilkovisky, adapted to theories on compact two dimensional manifolds. The importance of the zero modes of the Lagrange multipliers in producing the Liouville potential and the WZW anomaly, and in proving gauge invariance, is emphasised. A previous problem concerning the gauge dependence of the Virasoro centre is solved.Comment: Plain TeX file, 15 page

Conformally Invariant Path Integral Formulation of the Wess-Zumino-Witten →\to Liouville Reduction

The path integral description of the Wess-Zumino-Witten $\to$ Liouville reduction is formulated in a manner that exhibits the conformal invariance explicitly at each stage of the reduction process. The description requires a conformally invariant generalization of the phase space path integral methods of Batalin, Fradkin, and Vilkovisky for systems with first class constraints. The conformal anomaly is incorporated in a natural way and a generalization of the Fradkin-Vilkovisky theorem regarding gauge independence is proved. This generalised formalism should apply to all conformally invariant reductions in all dimensions. A previous problem concerning the gauge dependence of the centre of the Virasoro algebra of the reduced theory is solved.Comment: Plain TeX file; 28 Page

W-Algebras

W-algebras are defined as polynomial extensions of the Virasoro algebra by primary fields, and they occur in a natural manner in the context of two-dimensional integrable systems, notably in the KdV and Toda systems. Their occurrence in those theories can be traced to their being the residual symmetry algebras when certain first-class constraints are placed on Kac-Moody (KM) algebras. In particular their occurrence in 2-dimensional Toda theories is explained by the fact that the Toda theories can be regarded as constrained Wess.. Zumino-Novikov-Witten (WZNW) theories. The general form of such first-class constraint for WZNW theories is investigated, and is shown to lead to a wider class of two-dimensional integrable systems, all of which have W-algebras as symmetry algebras

Duality in Quantum Liouville Theory

The quantisation of the two-dimensional Liouville field theory is investigated using the path integral, on the sphere, in the large radius limit. The general form of the $N$-point functions of vertex operators is found and the three-point function is derived explicitly. In previous work it was inferred that the three-point function should possess a two-dimensional lattice of poles in the parameter space (as opposed to a one-dimensional lattice one would expect from the standard Liouville potential). Here we argue that the two-dimensionality of the lattice has its origin in the duality of the quantum mechanical Liouville states and we incorporate this duality into the path integral by using a two-exponential potential. Contrary to what one might expect, this does not violate conformal invariance; and has the great advantage of producing the two-dimensional lattice in a natural way.Comment: Plain TeX File; 36 page