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.
Highlights
In this paper we introduce the spectral function method, which allows for constraining not just the gap or the first few OPE coefficients but the distribution of OPE coefficients over a wide range of scaling dimensions
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
Summary
We truncate the conformal block to this order as an approximation of the exact block It follows from the associativity of OPE that the four-point function is crossing symmetric, which amounts to the crossing equation. The upper and lower bounds on the spectral function derived from the above minimization procedure using linear functionals up to derivative order N will be denoted fN+(∆∗) and fN−(∆∗), respectively. For the application to theories with only scalar primaries in the few subsections, we do not need to worry about the spin truncation being sufficiently large In this case, we must be especially careful in taking the truncation on the q-series of the conformal blocks to be sufficiently large, as the corrections to the approximate blocks would introduce nonzero spin primary contributions.
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