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Domain walls of D=5 supergravity and fixed points of N=1 Super Yang Mills

Klaus Behrndt

TL;DR

This paper presents a holographic framework in which domain-wall solutions of five-dimensional gauged N=2 supergravity, with an abelian U(1) gauging of the R-symmetry and decoupled hypermultiplets, realize renormalization-group flows between AdS_5 fixed points of four-dimensional gauge theories with four supercharges. The flow is encoded by a superpotential W, yielding a monotonic C-function C(μ) ~ 1/(g|W|)^3 and beta-functions β^A = -(3 g^{AB} ∂_B W)/W, with fixed points at ∂_A W = 0 and AdS_5 geometries. The paper provides two explicit realizations: (i) a Z_k orbifold dual corresponding to N=2 SYM with a single fixed point, and (ii) an orientifold/CY construction yielding N=1 SYM with a phase transition (flop) that connects two fixed points, along with a quantitative comparison of central charges via c_-/c_+ = 24/25. These examples illustrate how geometry and field theory data map across the DW/QFT correspondence and demonstrate the non-perturbative beta-functions and c-theorem structure of the holographic RG flow.

Abstract

Employing the AdS/CFT correspondence, we give an explicit supergravity picture for the renormalization group flow of couplings 4-d super Yang Mills with four supercharges. The solution represents a domain wall of 5-d, N=2 supergravity, that interpolates between two (different) AdS_5 vacua and is obtained by gauging a U(1) subgroup of the SU(2) R-symmetry. On the supergravity side the domain wall couples only to scalar fields from vector mulitplets, but not to scalars from hyper multiplets. We discuss the c-theorem, the beta-functions and consider two examples: one is the sugra solution related to Z_k orbifolds (corresponding to N=2 SYM) and the other is an orientifold construction for an elliptically fibered CY with F_1 basis (corresponding to N=1 SYM).

Domain walls of D=5 supergravity and fixed points of N=1 Super Yang Mills

TL;DR

This paper presents a holographic framework in which domain-wall solutions of five-dimensional gauged N=2 supergravity, with an abelian U(1) gauging of the R-symmetry and decoupled hypermultiplets, realize renormalization-group flows between AdS_5 fixed points of four-dimensional gauge theories with four supercharges. The flow is encoded by a superpotential W, yielding a monotonic C-function C(μ) ~ 1/(g|W|)^3 and beta-functions β^A = -(3 g^{AB} ∂_B W)/W, with fixed points at ∂_A W = 0 and AdS_5 geometries. The paper provides two explicit realizations: (i) a Z_k orbifold dual corresponding to N=2 SYM with a single fixed point, and (ii) an orientifold/CY construction yielding N=1 SYM with a phase transition (flop) that connects two fixed points, along with a quantitative comparison of central charges via c_-/c_+ = 24/25. These examples illustrate how geometry and field theory data map across the DW/QFT correspondence and demonstrate the non-perturbative beta-functions and c-theorem structure of the holographic RG flow.

Abstract

Employing the AdS/CFT correspondence, we give an explicit supergravity picture for the renormalization group flow of couplings 4-d super Yang Mills with four supercharges. The solution represents a domain wall of 5-d, N=2 supergravity, that interpolates between two (different) AdS_5 vacua and is obtained by gauging a U(1) subgroup of the SU(2) R-symmetry. On the supergravity side the domain wall couples only to scalar fields from vector mulitplets, but not to scalars from hyper multiplets. We discuss the c-theorem, the beta-functions and consider two examples: one is the sugra solution related to Z_k orbifolds (corresponding to N=2 SYM) and the other is an orientifold construction for an elliptically fibered CY with F_1 basis (corresponding to N=1 SYM).

Paper Structure

This paper contains 9 sections, 84 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Abelian gauged supergravity in 5 dimensions
  3. General remarks
  4. A simple example
  5. Definitions and conventions of 5-d supergravity
  6. BPS domain walls in gauged $N$=2 supergravity
  7. Supersymmetric fixed points, RG flow and the $c$-theorem
  8. Examples
  9. Conclusion

Figures (2)

  • Figure 1: The scalar fields are not constant and therefore every point on the scalar manifold $M$ represents a different point in spacetime. A solution of the scalar field equation defines a trajectory $\Phi = \Phi(r)$ connecting two fixed point values. Supersymmetry is preserved, if along this trajectory the normal vector of $M$ remains parallel the harmonic function $H_I = h_I + {q_I \over r^2}$ which fixes the solution. Although $M$ is smooth and differentiable the trajectory may make turns related to source terms at the domain wall.
  • Figure 2: Case (a) shows a coupling that runs towards a non-vanishing IR of UV fixed point value $\Phi_0$ (the arrow indicates the IR flow). On the other side, case (b) shows a running coupling with two fixed points. This case is typical for N=1 super Yang-Mills model.