Non invariant zeta - function regularization in quantum Liouville theory 1

We consider two possible zeta-function regularization schemes of quantum Liouville theory. One refers to the Laplace-Beltrami operator covariant under conformal transformations, the other to the naive non invariant operator. The first produces an invariant regularization which however does not give rise to a theory invariant under the full conformal group. The other is equivalent to the regularization proposed by Zamolodchikov and Zamolodchikov and gives rise to a theory invariant under the full conformal group. Work supported in part by M.I.U.R. Quantum Liouville theory has been the subject of intense study following different lines of attack. While the bootstrap [1, 2, 3, 4, 5] starts from the requirement of obtaining a theory invariant under the full infinite dimensional conformal group, the more conventional field theory techniques like the hamiltonian and the functional approaches depend in a critical way on the regularization scheme adopted. In the hamiltonian treatment [6] for the theory compactified on a circle the normal ordering regularization gives rise to a theory invariant under the full infinite dimensional conformal group. It came somewhat of a surprise that in the functional approach the regularization which realizes the full conformal invariance is the non invariant regularization introduced by Zamolodchikov and Zamolodchikov (ZZ) [4]. In [7] it was shown that such a regularization provides the correct quantum dimensions to the vertex functions on the sphere at least to two loops while in [8] it was shown that such a result holds true to all order perturbation theory on the pseudosphere. Here we consider the approach in which the determinant of a non covariant operator is computed in the framework of the zeta function regularization and show that this procedure is equivalent to the non invariant regularization of the Green function at coincident points proposed by ZZ [4], and extensively used in [7, 8, 9, 10]. For definiteness we shall refer to the case of sphere topology. The complete action is given by SL[φB, χ] = Scl[φB] + Sq[φB, χ] where [7] Scl[φB] = lim εn→0 R→∞ 1 b2 [ 1 8π ∫ Γ ( 1 2 (∂aφB) 2 + 8πμbeB )