Thermodynamics of the N=2* flow
Alex Buchel, James T. Liu
TL;DR
This paper investigates the finite-temperature thermodynamics of the N=2* SU(N) gauge theory at strong coupling using the non-extremal Pilch–Warner flow and its ten-dimensional uplift. It constructs the exact nonsingular non-extremal solution in five dimensions, uplifts it to ten dimensions, and analyzes the thermodynamics by computing the leading m/T correction to the free energy, finding it positive, and showing that the mass deformation induces a temperature-dependent gaugino condensate. A key finding is that extracting the N=2* thermodynamics from the dual gravity setup requires a holographic renormalization that preserves F = E − T S, yet the result TdS ≠ dE, i.e., a violation of the first law, suggesting a missing ingredient such as a UV-induced chemical potential. The authors discuss this paradox and its possible resolution, and outline the conditions for a BH vs PW phase transition, noting that the transition region remains to be settled by further analytical or numerical work.
Abstract
We discuss the thermodynamics of the N=2*, SU(N) gauge theory at large 't Hooft coupling. The tool we use is the non-extremal deformation of the supergravity solution of Pilch and Warner (PW) [hep-th/0004063], dual to N=4, SU(N) gauge theory softly broken to N=2. We construct the exact non-extremal solution in five-dimensional gauged supergravity and further uplift it to ten dimensions. Turning to the thermodynamics, we analytically compute the leading correction in m/T to the free energy of the non-extremal D3 branes due to the PW mass deformation, and find that it is positive. We also demonstrate that the mass deformation of the non-extremal D3 brane geometry induces a temperature dependent gaugino condensate. We find that the standard procedure of extracting the N=2* gauge theory thermodynamic quantities from the dual supergravity leads to a violation of the first law of thermodynamics. We speculate on a possible resolution of this paradox.