The duality of quantum Liouville field theory

It has been found empirically that the Virasoro center and three-point func tions of quantum Liouville held theory with the potential exp(2b phi (x)) a nd the external primary fields exp(alpha phi (x)) are invariant with respec t to the duality transformations h alpha --> q - alpha where q = b(-1) + b. The steps leading to this result (via the Virasoro algebra and three-point functions) are reviewed in the path-integral formalism. The duality occurs because the quantum relationship between the a and the conformal weights D elta (alpha) is two-to-one. As a result, the quantum Liouville potential ca n actually contain two exponentials (with related parameters). In the two-e xponential theory, the duality appears naturally, and an important previous ly conjectured extrapolation can be proved.