Geodesic
The duality of quantum Liouville field theory
Teoretičeskaâ i matematičeskaâ fizika, Tome 123 (2000) no. 2, pp. 299-307
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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. 
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L. O'Raifeartaigh; J. M. Pawlowski; V. V. Sreedhar. The duality of quantum Liouville field theory. Teoretičeskaâ i matematičeskaâ fizika, Tome 123 (2000) no. 2, pp. 299-307. http://geodesic.mathdoc.fr/item/TMF_2000_123_2_a10/

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