Is Action Complexity better for de Sitter space in Jackiw-Teitelboim gravity?

TL;DR

The paper demonstrates that complexity=action in two-dimensional de Sitter space within Jackiw-Teitelboim gravity captures the dilaton’s growth and yields a finite-time divergence, consistent with its higher-dimensional origin from dS$_3$. By deriving the correct dS$_2$ action through dimensional reduction and evaluating the Wheeler-DeWitt patch, the authors show CA diverges at a finite time with late-time growth scaling as $\\phi/G_2$, mirroring the dilaton’s role as an effective gravitational strength. They expose a Weyl-frame sensitivity in complexity=volume (CV), with two explicit frames giving different time evolutions, including a divergent volume in the $w=2$ case. To resolve this, they propose a refined volume using a Weyl factor $\\Omega(\\phi)=\\phi^2$, which reproduces the CA behavior and thereby links CV more closely to the dilaton-enhanced dynamics observed in higher dimensions. Overall, the work clarifies the necessity of the dilaton in JT gravity for holographic complexity and opens paths to connect with higher-dimensional dS physics and SYK-like duals.

Abstract

Volume complexity in dS$_2$ remains $O(1)$ up to a critical time, after which it suddenly diverges. On the other hand, for the dS$_2$ solution in JT gravity there is a linear dilaton which smoothly grows towards the future infinity. From the dimensional reduction viewpoint, the growth of the dilaton is due to the expansion of the orthogonal sphere in higher-dimensional dS$_d$ ($d \ge 3$). Since in higher dimensions complexity becomes very large even before the critical time, by properly taking into account the dilaton, the same behavior is expected for complexity in dS$_2$ JT gravity. We show that this expectation is met by complexity = action (CA) conjecture. For this purpose, we obtain an appropriate action for dS$_2$ in JT gravity, by dimensional reduction from dS$_3$. In addition, we discuss complexity = "refined volume" where we choose an appropriate Weyl field-redefinition such that refined volume avoids the discontinuous jump in time evolution.

Figures (3)

• Figure 1: Penrose diagram for dS spacetime. The time coordinate $t$ runs upwards on the right and downwards on the left. Coordinate axes for the EF coordinates on the right and the left are shown in red.
• Figure 2: Penrose diagram for dS spacetime showing the WDW patch (red region). The stretched horizons at $r = \rho L$ are represented by red curves.
• Figure 3: Plot of $V(\tau)$ for fixed values of $w$. As $\tau \to \tau_{\infty}$, the refined volume behaves differently depending on $w$. The $w=0$ result is bounded by $\pi$. Instead, the refined volume for $w=2$ diverges, due to the additional Weyl (dilaton) contribution from the point near the future infinity. We set $\rho=0.7$ and $\tau_{\infty}={\rm arctanh} (\rho) = 0.88$.