On the duality of quantum liouville field theory - O'Raifeartaigh, Lochlainn; Pawlowski, Jan M.; Sreedhar, Vinnakota (International journal of modern physics. A, Particles and fields, gravitation, cosmology, 2000)
It has been found empirically that the Virasoro centre and 3-point functions of quantum Liouville field theory with potential e2bΦ(x) and external primary fields exp(αΦ(x)), are invariant with respect to the duality transformations ℏα→q-α 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 α and the conformal weights Δα 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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