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Holographic Duals of a Family of N=1 Fixed Points

N. Halmagyi, K. Pilch, C. Romelsberger, N. P. Warner

TL;DR

<3-5 sentence high-level summary> The paper addresses constructing explicit IIB supergravity backgrounds that holographically realize the full family of ${\cal N}=1^*$ fixed points in a ${\mathbb Z}_2$ quiver gauge theory, interpolating between the KW and PW geometries. It employs Killing spinor methods with two projection conditions to derive a controlled, SU(2)-structured ansatz, reducing to a set of three first-order BPS equations and solving them numerically to demonstrate a continuous family of solutions. A key result is that the central charge remains proportional to the UV value by a factor of $c_{IR}=\frac{27}{32} c_{UV}$ across the whole family, validating the proposed RG-flow interpretation and offering insight into nontrivial dilaton profiles in the interior. The work thus provides a robust holographic framework for a broader class of ${\rm N}=1$ RG-flows and highlights the role of discrete symmetries in constraining ten-dimensional solutions.

Abstract

We construct a family of warped AdS_5 compactifications of IIB supergravity that are the holographic duals of the complete set of N=1 fixed points of a Z_2 quiver gauge theory. This family interpolates between the T^{1,1} compactification with no three-form flux and the Z_2 orbifold of the Pilch-Warner geometry which contains three-form flux. This family of solutions is constructed by making the most general Ansatz allowed by the symmetries of the field theory. We use Killing spinor methods because the symmetries impose two simple projection conditions on the Killing spinors, and these greatly reduce the problem. We see that generic interpolating solution has a nontrivial dilaton in the internal five-manifold. We calculate the central charge of the gauge theories from the supergravity backgrounds and find that it is 27/32 of the parent N=2, quiver gauge theory. We believe that the projection conditions that we derived here will be useful for a much larger class of N=1 holographic RG-flows.

Holographic Duals of a Family of N=1 Fixed Points

TL;DR

<3-5 sentence high-level summary> The paper addresses constructing explicit IIB supergravity backgrounds that holographically realize the full family of fixed points in a quiver gauge theory, interpolating between the KW and PW geometries. It employs Killing spinor methods with two projection conditions to derive a controlled, SU(2)-structured ansatz, reducing to a set of three first-order BPS equations and solving them numerically to demonstrate a continuous family of solutions. A key result is that the central charge remains proportional to the UV value by a factor of across the whole family, validating the proposed RG-flow interpretation and offering insight into nontrivial dilaton profiles in the interior. The work thus provides a robust holographic framework for a broader class of RG-flows and highlights the role of discrete symmetries in constraining ten-dimensional solutions.

Abstract

We construct a family of warped AdS_5 compactifications of IIB supergravity that are the holographic duals of the complete set of N=1 fixed points of a Z_2 quiver gauge theory. This family interpolates between the T^{1,1} compactification with no three-form flux and the Z_2 orbifold of the Pilch-Warner geometry which contains three-form flux. This family of solutions is constructed by making the most general Ansatz allowed by the symmetries of the field theory. We use Killing spinor methods because the symmetries impose two simple projection conditions on the Killing spinors, and these greatly reduce the problem. We see that generic interpolating solution has a nontrivial dilaton in the internal five-manifold. We calculate the central charge of the gauge theories from the supergravity backgrounds and find that it is 27/32 of the parent N=2, quiver gauge theory. We believe that the projection conditions that we derived here will be useful for a much larger class of N=1 holographic RG-flows.

Paper Structure

This paper contains 49 sections, 176 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Field theory considerations
  3. Realizing the global symmetries within supergravity
  4. Continous bosonic symmetries
  5. Supersymmetries
  6. Discrete symmetries
  7. Reduction to a five-dimensional problem
  8. Decomposing the metric and spinors
  9. Killing spinors on $AdS_5$
  10. The ten-dimensional Killing spinors
  11. The dilatino variation
  12. The gravitino variation
  13. Known solutions
  14. The $T^{1,1}$ solution
  15. The Pilch-Warner fixed point solution
  16. ...and 34 more sections

Figures (3)

  • Figure 1: Plots of the function $\beta(\alpha)$ against $\alpha$ for of six different values of the initial data. The curves merge into a dashed enveloping curve that shows $\beta(\alpha)$ for the Pilch-Warner solution. For the Klebanov-Witten solution one has $\beta \equiv 0$.
  • Figure 2: Plots of the function $g(\alpha)$ against $\alpha$ for of six different values of the initial data. The upper and lower dashed lines show the function, $g(\alpha)$, for the Pilch-Warner solution and for the Klebanov-Witten solution respectively. The horizontal dashed line shows the "target value", ${2 \over 3}$, for $g(\alpha)$ at $\alpha = \frac{\pi}{2}$.
  • Figure 3: Plots of the function $h(\alpha)$ against $\alpha$ for of six different values of the initial data. The lower dashed curve shows $h(\alpha)$ for the Klebanov-Witten solution, while $h(\alpha) \equiv 1$ for the Pilch-Warner solution.