Three-Dimensional CFTs and RG Flow from Squashing M2-Brane Horizon

TL;DR

The paper provides a concrete holographic realization of RG flows between interacting $d=3$ CFTs via squashing deformations of the M2-brane horizon. By analyzing the $AdS_4\ times S^7$ vacua and their squashed counterpart, it identifies the squashing mode as an irrelevant operator at the $SO(8)$ fixed point (with $\Delta=4$) and as a relevant operator at the squashed fixed points (with $\Delta=4/3$ or $5/3$), corresponding to ${\cal N}=1$ and ${\cal N}=0$ theories, respectively. The authors derive a static domain-wall solution that captures the RG trajectory and discuss stability through BF bounds and the nonexistence of finite-action gravitational instantons, effectively establishing a holographic $c$-theorem. Overall, the work elucidates how geometric deformations of extra dimensions map to boundary operators, yielding a controlled RG flow between distinct 3D CFTs with different supersymmetry and global symmetry structures.

Abstract

Utilizing AdS/CFT correspondence in M-theory, an example of interacting d=3 conformal field theories and renormalization group flow between them is presented. Near-horizon geometry of N coincident M2-branes located on a conical singularity on eight-dimensional hyperkähler manifold or manifold with Spin(7) holonomy is, in large-N limit, AdS4*X7, where X7 is seven-sphere with squashing. Deformation from round $§_7$ to squashed one is known to lead to spontaneous breaking of N=8 local supersymmetry in gauged AdS4 supergravity to N=1, 0. Via AdS/CFT correspondence, it is interpreted as renormalization group flow from SO(5)*SO(3) symmetric UV fixed point to SO(8) symmetric IR fixed point. Evidences for the interpretation are found both from supergravity scalar potential and existence of interpolating static domain-wall thereof, and from conformal dimensions of relevant chiral primary operator at each fixed point.

Figures (3)

• Figure 1: Scalar potential $V(u, v)$ for $Q = 3$. The local minimum point and the saddle point correspond to the ${\bf S}_7$ and the $\widetilde{{\bf S}_7}$, respectively. Note that the steepest descent is nearly along the $v$ (squashing parameter) direction. Both ground-states are stable against $u$ (breathing) direction deformation.
• Figure 2: Schematic view of static domain wall interpolating the two vacua of $AdS_4$ supergravity. Note that interpolating scalar fields are critically damped between the two vacua.
• Figure 3: The RG flow along the $SO(5)\times SO(3)$ invariant direction for I: $\widetilde{{\bf S}^7} \rightarrow$${\bf S}^7$ and along the $SO(3) \times SO(3)$ invariant direction for II: $SO(8) \rightarrow SO(3) \times SO(3)$.