Sum rules and three point functions

TL;DR

This paper derives holographic sum rules for R-current spectral densities in D3, M2, and M5-brane theories at finite chemical potential, linking spectral integrals to long and short distance physics, and extracting structure constants from OPE data.

Contribution

It introduces holographic sum rules relating spectral densities to OPE data, enabling the calculation of structure constants in theories with finite chemical potential.

Findings

Sum rules relate spectral integrals to hydrodynamics and short-distance physics.

Sum rules incorporate OPE data, revealing structure constants.

3-point functions from sum rules match Witten diagram calculations.

Abstract

Sum rules constraining the R-current spectral densities are derived holographically for the case of D3-branes, M2-branes and M5-branes all at finite chemical potentials. In each of the cases the sum rule relates a certain integral of the spectral density over the frequency to terms which depend both on long distance physics, hydrodynamics and short distance physics of the theory. The terms which which depend on the short distance physics result from the presence of certain chiral primaries in the OPE of two R-currents which are turned on at finite chemical potential. Since these sum rules contain information of the OPE they provide an alternate method to obtain the structure constants of the two R-currents and the chiral primary. As a consistency check we show that the 3 point function derived from the sum rule precisely matches with that obtained using Witten diagrams.