Sum rules and three point functions

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.