Krylov Complexity and $c$-function along RG Flows

Abstract

We investigate Krylov spread complexity along holographic renormalisation-group flows using the proposal that its growth rate is captured by the proper radial momentum of an in-falling massive probe. We focus on the second time derivative of the complexity $U$, which is determined locally by the redshift and radial metric functions of the dual geometry. For Lorentz-invariant domain-wall flows preserving the spacetime dimension, we derive a new relation between $U$ and the covariant central charge $c_{\text{cov}}$. The decrease of the covariant central function towards the infrared is accompanied by a monotonic increase of the complexity acceleration. For the top-down Dp-brane family, we obtain a universal relation. We then examine flows across dimensions, including a twisted compactification from four to two dimensions and from six to four dimensions. In these examples $c_{\mathrm{cov}}$ and $U$ are co-monotonic, in sharp contrast with the inverse correlation characteristic of fixed-dimensional flows. We argue that this reversal reflects the reorganisation, rather than simple depletion, of degrees of freedom into lower-dimensional sectors under compactification. Our results identify complexity acceleration as a sensitive geometric diagnostic connecting information spreading, holographic central functions and RG evolution.

20 pages, various figures