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

• URL: http://arxiv.org/abs/2510.11777v1
• Date: Mon, 13 Oct 2025 18:00:00 GMT
• Title: Evolution With(out) Time: Relational Holography & BPS Complexity Growth in $\mathcal{N}=2$ Double-Scaled SYK
• Authors: Sergio E. Aguilar-Gutierrez,
• Abstract summary: Holographically, this proposal reproduces a relational bulk observable, a BPS wormhole length.<n>We contrast this to the Krylov complexity for Hartle-Hawking states with non-trivial time flow.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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