Renormalization Group Flows from Holography—Supersymmetry and a c-Theorem

Renormalization Group Flows from Holography—Supersymmetry and a c-Theorem

April, 1999 | D. Z. Freedman, S. S. Gubser, K. Pilch, N. P. Warner
This paper explores the renormalization group (RG) flows in N = 4 super-Yang-Mills theory through the lens of holography, focusing on supersymmetry and a c-theorem. The authors derive first-order equations for a supersymmetric kink solution in five-dimensional N = 8 gauged supergravity, which interpolates between an exterior anti-de Sitter (AdS) region with maximal supersymmetry and an interior AdS region with one quarter of the maximal supersymmetry. The kink preserves one eighth of supersymmetry and is interpreted as describing the RG flow of N = 4 super-Yang-Mills theory broken to N = 1 by adding a mass term to one of the three adjoint chiral superfields. The kink solution is shown to correspond to a c-function that is positive and monotonic under the weak energy condition, and it coincides with the trace anomaly coefficients of the holographically dual field theory in limits where conformal invariance is recovered. The authors also demonstrate that the kink solution preserves N = 1 supersymmetry throughout the flow and matches the trace anomaly coefficients computed in both the field theory and the supergravity dual. The paper further discusses the general properties of supergravity kinks and their relation to the c-theorem, showing that the monotonicity of the c-function follows from the weak energy condition and Einstein's equations. The analysis is extended to a general class of supersymmetric flows, and the authors argue that the supergravity dual of the mass-deformed N = 4 theory is the kink solution that interpolates between the maximally supersymmetric and one-quarter supersymmetric critical points of N = 8 gauged supergravity. The paper concludes with a discussion of the holographic correspondence between the supergravity kink and the field theory RG flow, emphasizing the role of supersymmetry and the c-theorem in establishing the duality.This paper explores the renormalization group (RG) flows in N = 4 super-Yang-Mills theory through the lens of holography, focusing on supersymmetry and a c-theorem. The authors derive first-order equations for a supersymmetric kink solution in five-dimensional N = 8 gauged supergravity, which interpolates between an exterior anti-de Sitter (AdS) region with maximal supersymmetry and an interior AdS region with one quarter of the maximal supersymmetry. The kink preserves one eighth of supersymmetry and is interpreted as describing the RG flow of N = 4 super-Yang-Mills theory broken to N = 1 by adding a mass term to one of the three adjoint chiral superfields. The kink solution is shown to correspond to a c-function that is positive and monotonic under the weak energy condition, and it coincides with the trace anomaly coefficients of the holographically dual field theory in limits where conformal invariance is recovered. The authors also demonstrate that the kink solution preserves N = 1 supersymmetry throughout the flow and matches the trace anomaly coefficients computed in both the field theory and the supergravity dual. The paper further discusses the general properties of supergravity kinks and their relation to the c-theorem, showing that the monotonicity of the c-function follows from the weak energy condition and Einstein's equations. The analysis is extended to a general class of supersymmetric flows, and the authors argue that the supergravity dual of the mass-deformed N = 4 theory is the kink solution that interpolates between the maximally supersymmetric and one-quarter supersymmetric critical points of N = 8 gauged supergravity. The paper concludes with a discussion of the holographic correspondence between the supergravity kink and the field theory RG flow, emphasizing the role of supersymmetry and the c-theorem in establishing the duality.
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