Stringy Effects and the Role of the Singularity in Holographic Complexity

There has been considerable recent interest in holographic complexity. The two leading conjectures on this subject hold that the quantum complexity of the boundary thermofield double state should be dual to either the volume of the Einstein-Rosen bridge connecting the two sides (CV conjecture) or to the action of the Wheeler-de-Witt patch of the bulk spacetime (CA conjecture). Although these conjectures are frequently studied in the context of pure Einstein gravity, from the perspective of string theory it is also natural to consider models of gravity in which general relativity is perturbed by higher powers of the Riemann tensor, suppressed by powers of the string length; in a holographic context, these corrections are dual to corrections in inverse powers of the 't Hooft coupling. In this paper, we investigate the CV and CA conjectures in two stringy models of higher-curvature gravity. We find that the CV complexification rate remains well-behaved, but conversely that these corrections induce new divergences in the CA complexification rate that are absent in pure Einstein gravity. These divergences are intrinsically linked to the singularity, and appear to be generic in higher curvature theories. To the best of our knowledge, infinities originating at the singularity have not yet been observed elsewhere in the literature. We argue that these divergences imply that, in the CA picture, the complexification rate of the boundary theory is a nonanalytic function of the 't Hooft coupling.