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

TL;DR

To address the distribution of OPE data in 2D CFTs, the paper introduces spectral functions defined by truncating Virasoro blocks and uses semidefinite programming to bound them. It provides strong numerical evidence that Liouville theory governs scalar OPE data for $c>1$ and argues for Liouville being the unique unitary $c>1$ CFT with bounded spin primaries, via modular constraints. The modular bootstrap is similarly used to bound modular spectral functions, with results at large $c$ showing convergence to pure gravity predictions from thermal AdS3 and BTZ saddles. The work highlights a potential universality of the BTZ spectral density in large-$c$ and large-gap regimes and connects Liouville uniqueness to holographic expectations.

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.

Figures (15)

• Figure 1: Upper and lower bounds on the spectral function from linear functionals of increasing derivative order (from green to red), assuming only scalar primaries for $c=8$ with $\Delta_\phi / \Delta_0 = {3 \over 4}, {7 \over 8}, 1, {24 \over 7}$. In all cases, the shaded regions are excluded and the black curve denotes the corresponding spectral function of (analytically continued) Liouville theory.
• Figure 2: Upper and lower bounds on the mixed correlator spectral function for $c=8$ and $({\Delta_1\over\Delta_0},{\Delta_2\over\Delta_0})=({5\over 9},{8\over 9}),(1,{7\over 8}),(1,{12\over 7})$. The black curve denotes the (analytically continued) DOZZ spectral function. In (c), a small gap of $\Delta_{\rm gap}=0.01$ has been imposed to explicitly exclude the vacuum channel which would correspond to a singular conformal block for the mixed correlator.
• Figure 3: (a) Plot of $f_N(\Delta_*)$ for $c=8$, $\Delta_\phi = {7\over 12}$, as $N$ ranges from $N=1$ (blue) to $N=25$ (red) with step of 2. (b) Comparison of $f_N(\Delta_*)$ ($N=27$ in solid blue and $N=33$ in solid red) with the exact DOZZ spectral function (dashed, black) for $c=2$, $\Delta_\phi = {55\over 12}$.
• Figure 4: Comparison of $f_{N}(\Delta_*)$ (solid, red) with the exact DOZZ spectral function (dashed, blue) for external operator dimension $\Delta_\phi$, and in the mixed correlator case, external operator dimensions $\Delta_1$ and $\Delta_2$ ($\Delta_0 \equiv {c-1\over 12}$ is the Liouville threshold as before).
• Figure 5: Plot of $E_N$ for as a function of $1/N$, $c=8$, $\Delta_\phi={7\over12}$, $\Delta_*=0.8$, $N\leq 25$. The dashed curve is a linear fit for $N\geq 11$.