Bootstrapping the Spectral Function: On the Uniqueness of Liouville and the Universality of BTZ

Abstract

We introduce spectral functions that capture the distribution of OPE coefficients and density of states in two-dimensional conformal field theories, and show that nontrivial upper and lower bounds on the spectral function can be obtained from semidefinite programming. We find substantial numerical evidence indicating that OPEs involving only scalar Virasoro primaries in a c>1 CFT are necessarily governed by the structure constants of Liouville theory. Combining this with analytic results in modular bootstrap, we conjecture that Liouville theory is the unique unitary c>1 CFT whose primaries have bounded spins. We also use the spectral function method to study modular constraints on CFT spectra, and discuss some implications of our results on CFTs of large c and large gap, in particular, to what extent the BTZ spectral density is universal.

arXiv comment44 pages, 15 figures

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• The Virasoro Minimal String
• Bootstrapping Non-Invertible Symmetries
• Fast Conformal Bootstrap and Constraints on 3d Gravity
• Entanglement Entropy, OTOC and Bootstrap in 2D CFTs from Regge and Light Cone Limits of Multi-point Conformal Block
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• Analytic Bootstrap in 2D Boundary Conformal Field Theory: Towards Braneworld Holography
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