The duality of quantum Liouville field theory

It has been found empirically that the Virasoro center and three-point functions of quantum Liouville field theory with the potential $\exp\bigl(2b\phi(x)\bigr)$ and the external primary fields $\exp\bigl(\alpha\phi(x)\bigr)$ are invariant with respect to the duality transformations $\hbar\alpha\rightarrow q-\alpha$, 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 $\alpha$ and the conformal weights $\Delta_\alpha$ is two-to-one. As a result, the quantum Liouville potential can actually contain two exponentials (with related parameters). In the two-exponential theory, the duality appears naturally, and an important previously conjectured extrapolation can be proved.