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. 
L. O'Raifeartaigh, J. M. Pawlowski, and V. V. Sreedhar