De Sitter holographic complexity from Krylov complexity in DSSYK

Abstract

We utilize the recent connection between the high energy limit of the double-scaled SYK model and two-dimensional de Sitter solutions of sine dilaton gravity to identify the length of a family of geodesics spanned between future and past infinities with Krylov spread complexity. This constitutes an explicit top-down microscopic realization of holographic complexity in a cosmological spacetime. Our identification is different from the existing holographic complexity proposals for de Sitter geometries which are anchored either on horizons as holographic screens or on timelike observers. This leads us to introduce and investigate a new cosmological holographic complexity proposal in any dimension. It is based on extremal timelike volumes anchored at the asymptotic past and future and at large values of the anchoring boundary coordinate grows linearly with growth rate proportional to the product of de Sitter entropy and temperature.

Figures (4)

• Figure 1: Parts of Penrose diagrams of maximally extended AdS$_2$ (Left) and dS$_2$ (Right) relevant to our setup. Weyl rescaling the metric relates the AdS BH interior to the dS Static patch (red), the AdS Rindler patch to the dS Milne patch (green), the AdS BH horizon to the dS cosmological horizon (dashed), and the AdS asymptotic spatial boundaries to dS future and past boundaries (blue). As depicted in red, AdS extremal codimension-1 spacelike slices evolving in boundary time $t$ are mapped to timelike extremal codimension-1 slices anchored at past and future infinity that evolve with the spacelike boundary coordinate $\chi$ in dS. Full Penrose diagrams of the maximal extensions are discussed in Harlow:2018tqv for AdS$_2$ and in Maldacena:2019cbz for dS$_2$.
• Figure 2:
• Figure 3: The volume of the timelike extremal slices in dS$_{d+1}$ exhibits quadratic to linear growth as a function of the space coordinate $w_0$ on the Euclidean CFTs at the boundaries. The dotted lines represent the large-$w_0$ growth rate \ref{['growthrate']}. The volume of the hypersurfaces has been regularized by subtracting the divergent contribution at $w_0=0$. Moreover, a vertical shift proportional to $d-2$ has been introduced for readability: $\mathcal{V}_\mathrm{reg}=\mathcal{V}(w_0)-\mathcal{V}(0)+i(d-2)$.
• Figure 4: Comparison between the proposals for complexity in dS$_{d+1}$ introduced in the present Letter and in Mohan:2025aiw. In these plots, $d=2$. Left panel: the imaginary part of the regularised volume (compare with Fig. \ref{['fig:higher_d_numerics']}) is compared by expressing both quantities as a function of $w_0$ as described in the body of this Supplemental Material. The asymptotic growth rate as $w_0\to \infty$ is the same, even if the small $w_0$ behavior differs. Right panel: the same comparison for the derivative of $\mathcal{V}_\mathrm{vreg}$, showing in both cases a transition from quadratic to linear growth.