Secrets of the metric in N = 4 and N = 2* geometries

The metric of the gravity dual of a field theory should contain precisely thesame information as the field theory. We discuss this connection in the N=4theory where a scalar vev may be introducedat the level of 5d supergravity andthe solutions lifted to 10d. We stress the role of brane probing in finding thecoordinates appropriate to the field theory. In these coordinates the metricparametrizes the gauge invariant operators of the field theory and either sideof the duality is uniquely determined by the other. We follow this same chainof computations for the 10d lift of the N=2* geometry of Pilch and Warner. Thebrane probe of the metric reveals the 2d moduli space and the functional formof the gauge coupling. In the coordinates appropriate to the field theory themetric on moduli space takes a very simple form and one can read off thegravity predictions for operators in the field theory. Surprisingly there islogarithmic renormalization even in the far UV where the field theory revertsto N=4 super Yang-Mills. We extend the analysis of Buchel et al to find the D3brane source distribution that generates the supergravity prediction for thegauge coupling for the whole class of solutions corresponding to differentpoints on moduli space. This distribution does not account for the logarithmicbehaviour in the rest of the metric though. We discuss possible resolutions ofthe discrepancy.Comment: 16 pages, references update