A Study of Holographic Renormalization Group Flows in d=6 and d=3

TL;DR

The paper addresses holographic RG flows in six dimensions using minimal gauged supergravity, mapping a UV operator of dimension $\Delta=4$ to a bulk scalar and analyzing whether RG trajectories correspond to deformations or vevs via the Hamilton-Jacobi framework. It identifies a true supersymmetric vacuum and a non-supersymmetric IR fixed point, and applies Gubser criteria to assess physical acceptability, including an explicit UV–IR flow constructed numerically. It then extends to squashed seven-manifold compactifications in M-theory, finds no generic interpolating flow in the chosen truncations, but exhibits a generating function $W$ with a squashed critical point indicating a vev for the squashing operator with possible operator dimensions $\Delta=\frac{5}{3}$ or $\Delta=\frac{4}{3}$ and a compliant runaway solution. Together, these results clarify how holographic RG flows can be implemented via Hamilton-Jacobi generating functions and illuminate the roles of deformation versus vev, with implications for $AdS_4/CFT_3$ holography from M-theory.

Abstract

We present an explicit study of the holographic renormalization group (RG) in six dimensions using minimal gauged supergravity. By perturbing the theory with the addition of a relevant operator of dimension four one flows to a non-supersymmetric conformal fixed point. There are also solutions describing non-conformal vacua of the same theory obtained by giving an expectation value to the operator. One such vacuum is supersymmetric and is obtained by using the true superpotential of the theory. We discuss the physical acceptability of these vacua by applying the criteria recently given by Gubser for the four dimensional case and find that those criteria give a clear physical picture in the six dimensional case as well. We use this example to comment on the role of the Hamilton-Jacobi equations in implementing the RG. We conclude with some remarks on AdS_4 and the status of three dimensional superconformal theories from squashed solutions of M-theory.

Figures (5)

• Figure 1: The potential $V(\phi)$ plotted against $\phi$. The point $\phi=0$ represents the UV theory and the point $\phi=-\log 2/\sqrt{5}\approx -0.309985$ the IR theory.
• Figure 2: Comparison of the two generating functions $W_{\mathrm{ir}}$ and $W_{\mathrm{susy}}$ between the IR and UV fixed points. Note that $W_{\mathrm{ir}}$ has a second extremum at the IR fixed point, whereas $W_{\mathrm{susy}}$ does not.
• Figure 3: The solution $\phi_{\mathrm{ir}}$ connecting the two fixed points plotted against the scale factor $y$.
• Figure 4: The asymptotic behavior of $\dot\phi/\phi$ shows that $\phi\approx e^{-y}$ when $W=W_{\mathrm{ir}}$ corresponding to a true deformation by ${\cal O}_\phi$.
• Figure 5: The supersymmetric runaway solution $\phi_{\mathrm{susy}}$ corresponding to a non-conformal vacuum where $\langle{\cal O}_\phi\rangle > 0$, plotted against the scale factor $y$ in the vicinity of the UV fixed point.