Complexity and the Hilbert space dimension of 3D gravity

by Jonathan Karl (University of Würzburg)

Thursday 4 Jun 2026, 14:15 → 15:45 Europe/Berlin

Abbeanum/1-HS2 - Straubel-HS (TPI, FSU Jena)

In this talk we determine the size and structure of the Hilbert space for near-extremal black holes in three spacetime dimensions, using a quantum dynamical Krylov complexity approach. In particular, we compute the spread of an evolving thermofield double state over the accessible part of the Hilbert space. For early times, the dynamics of the system coincide with those of a quantum particle moving on the SL(2,R) group manifold. The late-time behavior of the black hole receives large quantum corrections, induced by wormhole effects in the semiclassical path integral, leading to the saturation of spread complexity. The saturation value determines the Hilbert space dimension, which we find to be exponential in the Bekenstein–Hawking entropy. Our results introduce a new way to compute the Hilbert space dimension of complex interacting systems from the saturating value of spread complexity.