Title: Exact Transition Amplitudes in Liouville CFT and the Cigar

Abstract: We study exact transition amplitudes in certain 2D CFTs — Liouville theory and the SL(2,R)/U(1) coset model describing Witten's 2D black hole. Using bootstrapped conserved currents and a generating functional approach based on canonical transformations, we compute these transitions both recursively and via functional integral representations. The tree-level evaluation of the generating functional reproduces exact results, while providing a diagrammatic framework for loop corrections. We also find that the statistics of Liouville transition elements at higher oscillator levels exhibit level repulsion.