Exact Operator Solution for Liouville Theory with q A Root of Unity

The exact operator solution for quantum Liouville theory constructed for the generic quantum deformation parameter q is extended to the case with q being a root of unity. The screening charge operator becomes nilpotent in such cases and arbitrary Liouville exponentials can be obtained in finite polynomials of the screening charge. There are mainly two reasons that the investigation of the quantum Liouville theory is particularly interesting and important. One is related to the fact that it describes the dynamics of the conformal mode of the world-sheet metrics of string theory at off-critical dimensions [1]. Though we do not yet fully understand the subcritical string theory, exact results for 2D quantum gravity coupled to conformal matters have been obtained so far [2, 3]. ∗ The other is that it is an exactly solvable conformally invariant 2D field theory that admits exact operator analyses [5, 6, 7]. One can apply and examine the techniques of 2D conformal field theory [8] for the Liouville theory and obtain, in principle, the exact results for such as the correlation functions including the Liuoville vertices. This program has been vigorousely pursued by Gervais and his collaborators [9, 10, 11] with the guide of quantum group structure of the theory [12]. In our recent paper [13] we have investigated the Liouville theory based on the canonical free field method [5, 6, 7] and given a proof that the exact operator solution conjectured by Otto and Weigt [7] is correct to all orders in the cosmological constant. The Liouville coupling there is assumed to take generic values such that the quantum deformation parameter q is not a root of unity. The Liouville exponentials become infinite series in the screening charge. This makes further analysis rather complicated. One could develop a systematic expansion by the cosmological constant. This corresponds to treating the cosmological term as perturbation. It is not clear whether this is justified or not. The situation, however, changes drastically when q is a root of unity. In such cases the screening charge becomes nilpotent and arbitrary Liouville exponentials can be obtained in closed forms as finite polynomials. In this note we will investigate the exact operator solution in such cases by extending the previous results [13]. Let us begin with a brief review of [13]. It was argued that the canonical transformation from the Liouville field to a free field described by the classical solution can be extended to quantum theory and the exact operator expression for the interacting Liouville field, which satisfies the field equations and the equal-time commutation relations, can be obtained in terms of the free See recent review articles [4] for subsequent developments. In the path integral approach most results are obtained by treating the cosmological term as perturbation.