Evolution With(out) Time: Relational Holography & BPS Complexity Growth in $\mathcal{N}=2$ Double-Scaled SYK

TL;DR

This work develops a relational holographic framework for the $ abla$N=2$ double-scaled SYK model, incorporating quantum reference frames to dress bulk observables relative to clocks defined by boundary time and R-charge. It introduces a novel BPS-specific spread complexity, built from a doubled Krylov construction, which exactly matches the bulk BPS wormhole length in the semiclassical limit and captures supersymmetric corrections beyond that limit. The authors connect relational bulk observables to boundary chord numbers, present a detailed analysis of both BPS and non-BPS sectors, and contrast the zero-time-evolution BPS case with time-evolving Hartle–Hawking states. The work also maps out non-BPS scenarios in bosonic subspaces, showing semiclassical agreement with known gravity results while highlighting quantum corrections from SUSY. Finally, it discusses extensions to matter insertions, deformations of relational holography, and broader implications for SUSY holography and potential bulk duals beyond JT supergravity.

Abstract

How do we describe non-trivial bulk measurements relative to an observer (i.e. relationally) when both the observer and the system it probes may/may not evolve in time? How can we interpret this holographically; particularly for zero-energy BPS states in supersymmetric theories? We address these questions, in the $\mathcal{N}=2$ double-scaled SYK model and its putative bulk dual by: (i) formulating a holographic procedure in the language of quantum reference frames to gravitationally dress bulk observables to ``clocks'' parametrized by both boundary time and R-charge; and (ii) proposing a \emph{new measure of Krylov complexity} with R-charge in the boundary theory that probes zero-energy BPS states. Holographically, this proposal reproduces a relational bulk observable, a BPS wormhole length. We contrast this to the Krylov complexity for Hartle-Hawking states with non-trivial time flow. The latter reproduces the same observable as for the bosonic DSSYK in the semiclassical limit, while its quantum fluctuations can capture supersymmetric corrections depending on the specific initial state.