D-Dimensional Log Gravity

TL;DR

The paper investigates D-dimensional Einstein gravity augmented with curvature-squared terms at a critical point where only massless gravitons propagate (e.g., in D=4 with $\beta= -3\alpha= -\ell^2/2$). By analyzing AdS-wave (logarithmic) perturbations, it shows that the linearized equations degenerate at the critical point, producing logarithmic modes $F\sim \log r$ and requiring modified Fefferman–Graham boundary data with a second source, indicating a holographic dual to a logarithmic CFT in the boundary. The construction is extended to general D, where a critical condition $\beta=-\frac{4(D-1)}{D}\alpha$ yields logarithmic sectors with $F=f_4+f_3 r^{D-1}+(f_2+f_1 r^{D-1})\log r$; additionally, for special parameter choices, Schrödinger-type non-relativistic backgrounds arise, broadening holographic applications to non-relativistic CFTs. These results provide a gravitational realization of logarithmic CFTs in higher dimensions and suggest further avenues for exploring non-Einstein and non-relativistic black-hole solutions within quadratic-curvature gravity.

Abstract

We study Einstein gravity in dimensions $D\geq 4$ modified by curvature squared at critical point where the theory contains only massless gravitons. We show that at the critical point a new mode appears leading to a logarithmic gravity in the theory. The corresponding logarithmic solution may provide a gravity description of logarithmic CFT in higher dimensions. We note also that for special values of the parameters of the theory, the model admits solutions with non-relativistic isometries.