Free Energy from Replica Wormholes

TL;DR

The paper investigates how Euclidean replica wormholes affect the gravitational path integral’s observable of free energy, arguing that in certain regimes the connected topologies dominate and signal ensemble averaging in gravity. By studying two-dimensional models—a CGHS variant and JT gravity—it demonstrates that naive annealed calculations can display pathologies at low temperature, which are ameliorated but not fully resolved by including replica wormholes; the correct analytic continuation to m → 0 appears to require replica-symmetry breaking, drawing an explicit analogy with spin glass physics. The authors outline a spin-glass-like framework (RSB) as a potential mechanism to obtain the proper quenched free energy, and discuss broader implications for ensemble averaging, nonperturbative completions, and the emergence of semiclassical gravity in higher dimensions. This work connects gravity, random matrix theory, and spin-glass theory to explore how wormholes might reflect deeper disorder-averaged structures in quantum gravity. The study suggests concrete directions for future work, including implementing a gravity-appropriate Parisi ansatz and examining higher-dimensional analogs where ensemble averaging may or may not be essential depending on UV completions.

Abstract

Euclidean wormholes -- geometries which connect disconnected boundaries -- present a challenge to a standard quantum mechanical interpretation of the theory. One potential resolution is that the gravitational path integral computes the ensemble average of many theories. The connected topologies contribute to the simplest possible observable: the free energy, which is computed using a replica trick. This is distinct from the replica trick used to compute entanglement entropies, and appears in the computation of any extensive quantity. We argue that both JT gravity and a simplified version of CGHS admit a regime where the contribution of connected replica wormholes to the free energy is larger than that of disconnected topologies. In both theories we find evidence of replica symmetry breaking, which is reminiscent of the behavior of certain spin glasses. We discuss possible insights about ensemble averaging in gravity from this perspective.

Figures (7)

• Figure 1: A computation of the Renyi entropy \ref{['eq:Renyiaverage']} from the GPI requires an additional replica trick, involving computing the GPI with the boundary $B_n^m$ shown here. Each of the columns is an $n$-sheeted geometry $B_n$ constructed by slicing $n$ copies of $B$ along the region $R$ and then identifying these copies cyclically along the cut. $B_n^m$ consists of $m$ copies of this multi-sheeted geometry. The disorder-averaged von Neumann entropy is computed in the double limit $m \to 0$, $n \to 1$.
• Figure 2: The only topologies that can appear in the $\widehat{\mathrm{CGHS}}$ path integral are the disk and the cylinder.
• Figure 3: The annealed free energy $F_\mathrm{ann}$ for $S_0 = 7$. \ref{['subfig:JTgenusconverge']}: From top to bottom, the solid blue curves show the result after including up to genus $g = 0,1,2,3,4$, and 5 in the genus expansion \ref{['eq:JTgenusexpansion']}; the dashed red curve shows the result obtained from the low-temperature expansion \ref{['eq:JTlowtemp']} truncated to $\ell \leq 2$. \ref{['subfig:JTellconverge']}: From top to bottom, the dashed red curves show the result after including up to $\ell = 0,1$, and 2 in the low-temperature expansion \ref{['eq:JTlowtemp']}; the solid blue curve shows the result obtained from keeping up to $g \leq 5$ in the genus expansion \ref{['eq:JTgenusexpansion']}. The local maximum at $e^{2S_0/3} T \approx 0.7$ is robust against the inclusion of higher order perturbative as well as doubly non-perturbative effects.
• Figure 4: The low-temperature behavior of the JT gravity free energy $\overline{F}_M$ for various $M$; here we take $S_0 = 7$, and the contour $C$ in \ref{['eq:factorialintegrals']} is the unit circle. From top left to bottom, the energy is computed using topologies with genus up to zero, one, or two. The blue, orange, green, red, and purple curves correspond to $M = 1,2,3,4,5$, respectively.
• Figure 5: The low-temperature behavior of the JT gravity free energy $\overline{F}_M$ obtained using two different analytic continuations to non-integer $m$: on the left we used the continuation \ref{['eq:factorialintegrals']} with the contour $C$ taken to be the unit circle (this is the same as Figure \ref{['subfig:JTfreeenergyS0g0second']}), while on the right we used \ref{['eq:gammamultiplication']}. The qualitative features agree, but quantitative details do not. The blue, orange, green, red, and purple curves correspond to $M = 1,2,3,4,5$, respectively, and we take $S_0 = 7$.