On the Duality of Quantum Liouville Field Theory

Authors: L. O'RAIFEARTAIGH, J. M. PAWLOWSKI, V. V. SREEDHAR

Abstract: It has been found empirically that the Virasoro centre and 3-point functions of quantum Liouville field theory with potential $e^{2b\phi(x)}$ and external primary fields exp($\alpha\phi(x)$) 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 3-point functions) are reviewed in the path-integral formalism. The duality stems from the fact that the quantum relationship between the $\alpha$ and the conformal weights $\Delta_\alpha$ is two-to-one. As a result the quantum Liouville potential may actually contain two exponentials (with related parameters). It is shown that in the two-exponential theory the duality appears in a natural way and that an important extrapolation which was previously conjectured can be proved.

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