The spectrum of a quantum Lifshitz black hole in two dimensions

We examine the low-energy spectrum of a four-dimensional near-extremal black hole that arises as a solution to a low energy effective theory of heterotic string theory. The effective two-dimensional gravitational description exhibits features of Lifshitz symmetry, which break the usual $\text{SL}(2,\mathbb{R})$ invariance down to $U(1)$. For this effective two-dimensional gravitational description, we derive a one-dimensional Schwarzian-like action that inherits the $U(1)$ symmetry. The Schwarzian-like description allows us to compute a logarithmic correction to the entropy through a saddle-point approximation of the two-dimensional partition function. This logarithmic correction modifies the density of states, lifting the delta-function divergence at extremality, and removes the exponential ground-state degeneracy seen in the semiclassical analysis. Furthermore, the prefactor of the logarithmic term is $\frac{1}{2}$, rather than $\frac{3}{2}$ found for the $\text{SL}(2,\mathbb{R})$ invariant description, indeed reflecting having fewer symmetries.