Chirality Deconfinement Beyond the C = 1 Barrier of Two Dimensional Gravity

Abstract

In recent times we have witnessed important developments in our understanding of two dimensional gravity in the weak coupling regime, especially in what concerns the dressing by gravity of matter with central charge smaller than one. Nevertheless, the popular approaches seem to be unable to overcome the so called c = 1 barrier, powerful and elegant as they may be. The point of these two lectures is to show, following refs.[l, 2, 3, 4] how the operator approach to Liouville theory which I started to develop long ago with A. Neveu, remains applicable in the strong coupling regime. The principle is as follows. In the weak coupling regime, the Liouville exponentials are expressed in terms of chiral vertex operators, themselves constructed from a free field in two dimensions. This is the quantum version of the well known classical Bäcklund transformation. Locality and closure of their OPA (operator product algebra) uniquely determines this chiral decomposition. In the strong coupling regime, this latter construction looses meaning since the OPA of the Liouville exponentials involves operators and/or highest weight states with complex Virasoro weights. Nevertheless the general operator-family of chiral components may still be used, when truncation theorems[5, 1, 2] apply, that is with central charges C _L = 7, 13, 19. Indeed for these values there exist subfamilies of the above chiral operators which form closed operator algebras, only involving real Virasoro weights.