Secrets of the Metric In N=4 and N=2* Geometries

TL;DR

This paper explores the detailed relationship between supergravity metrics and field theory operators in N=4 and N=2* theories, revealing how brane probing and coordinate choices encode gauge invariants and uncovering unexpected logarithmic renormalization effects.

Contribution

It provides a detailed analysis of the supergravity duals of N=4 and N=2* theories, emphasizing the role of brane probing in identifying gauge-invariant coordinates and operators, and extends previous work to include the entire moduli space.

Findings

Brane probing reveals the moduli space and gauge coupling functional form.

Coordinates aligned with field theory simplify the metric and operator extraction.

Logarithmic renormalization occurs even in the UV where N=4 SYM is recovered.

Abstract

The metric of the gravity dual of a field theory should contain precisely the same information as the field theory. We discuss this connection in the N=4 theory where a scalar vev may be introducedat the level of 5d supergravity and the solutions lifted to 10d. We stress the role of brane probing in finding the coordinates appropriate to the field theory. In these coordinates the metric parametrizes the gauge invariant operators of the field theory and either side of the duality is uniquely determined by the other. We follow this same chain of computations for the 10d lift of the N=2* geometry of Pilch and Warner. The brane probe of the metric reveals the 2d moduli space and the functional form of the gauge coupling. In the coordinates appropriate to the field theory the metric on moduli space takes a very simple form and one can read off the gravity predictions for operators in the field theory. Surprisingly there is logarithmic renormalization even in the far UV where the field theory reverts to N=4 super Yang-Mills. We extend the analysis of Buchel et al to find the D3 brane source distribution that generates the supergravity prediction for the gauge coupling for the whole class of solutions corresponding to different points on moduli space. This distribution does not account for the logarithmic behaviour in the rest of the metric though. We discuss possible resolutions of the discrepancy.