Abstract
We consider the Liouville theory in fixed Euclidean AdS2 background. Expanded near the minimum of the potential the elementary field has mass squared 2 and (assuming the standard Dirichlet b.c.) corresponds to a dimension 2 operator at the boundary. We provide strong evidence for the conjecture that the boundary correlators of the Liouville field are the same as the correlators of the holomorphic stress tensor (or the Virasoro generator with the same central charge) on a half-plane or a disc restricted to the boundary. This relation was first observed at the leading semiclassical order (tree-level Witten diagrams in AdS2) in [19] and here we demonstrate its validity also at the one-loop level. We also discuss arguments that may lead to its general proof.
Highlights
Defined on a fixed curved 2d background it is a Weyl-covariant quantum theory with the central charge c = 1 + 6 Q2
We provide strong evidence for the conjecture that the boundary correlators of the Liouville field are the same as the correlators of the holomorphic stress tensor on a half-plane or a disc restricted to the boundary
This relation was first observed at the leading semiclassical order in [19] and here we demonstrate its validity at the one-loop level
Summary
The correlators in the Liouville theory (1.1) in AdS2 background may be computed using two alternative approaches. In the first (the “ZZ formulation” [5]) one starts with the Liouville action on a flat upper half plane (or flat disc) and expands it near a non-trivial non-constant solution [1, 2] preserving the SL(2, R) symmetry. In the second (the “AdS formulation”) one starts directly with the Liouville action (1.1) in the AdS2 background and expands near the constant minimum of the curved-space potential.. The two approaches are classically equivalent by a field redefinition but imply the use of different regularizations at the quantum level.
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