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

## TL;DR

This paper develops spectral functions to analyze 2D conformal field theories, providing bounds and evidence that Liouville theory uniquely governs scalar OPEs in certain regimes, and explores the universality of BTZ spectral density.

## Contribution

It introduces spectral functions constrained by semidefinite programming and provides evidence for Liouville theory's uniqueness in scalar OPEs for c>1 CFTs.

## Key findings

- Spectral functions yield bounds on OPE coefficients and density of states.
- Numerical evidence suggests Liouville theory governs scalar OPEs in c>1 CFTs.
- Discussion on the universality of BTZ spectral density in large c regimes.

## 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.