Comments on the double cone wormhole

TL;DR

This work reframes the Saad–Shenker–Stanford double cone wormhole as a trace over the bulk two-sided black hole, clarifying how the normalization and spectral form factor ramp arise. It introduces the modified boost operator \tilde{K}, whose eigenvalues coincide with black hole quasinormal mode frequencies, and analyzes hydrodynamic contributions to the ramp via one-loop determinants. The JT gravity plus matter analysis reveals a backreaction that yields a wrong-sign deformation, motivating the proposal of imaginary couplings or alternative contours to realize a physically sensible geometry. A harmonic oscillator toy model then illuminates how a small iε-like deformation can drastically alter the spectrum and trace properties, illustrating the non-unitary nature of \tilde{K} and its implications for interpreting the double cone in large-N holography.

Abstract

In this paper we revisit the double cone wormhole introduced by Saad, Shenker and Stanford (SSS), which was shown to reproduce the ramp in the spectral form factor. As a first approximation we can say that this solution computes $\textrm{Tr}[e^{-iKT}]$, a trace of the "evolution" operator that generates Schwarzschild time translations on the two sided wormhole geometry. This point of view leads to a simple way to compute the normalization factor of the wormhole. When we have bulk matter fields, SSS suggested using a modified evolution $\tilde K$ which involves a slightly complex geometry, so that we are really computing $\textrm{Tr}[e^{-i\tilde{K}T}]$. We argue that, for general black holes, the spectrum of $\tilde K$ is given by quasinormal mode frequencies. We explain that this reproduces various features that were previously predicted from the spectral form factor on hydrodynamics grounds. We also give a general algebraic construction of the modified boost in terms of operators constructed from half sided modular inclusions. For the special case of JT gravity, we work out the backreaction of matter on the geometry of the double cone and find that it deforms the geometry in an undesirable direction. We finally give some comments on the possible physical interpretation of $\tilde K$.

Figures (7)

• Figure 1: (a) The geometry of a two-sided black hole. We have an symmetry $K=i\partial_t$ that acts like a boost near the horizon and like a time translation far away. (b) The double cone geometry involves quotienting the two-sided black hole such that $t$ has period $T$.
• Figure 2: SSS Saad:2018bqo proposed a prescription to regulate the double cone geometry, by deforming the slice of $\rho$ on which the path integral for quantum fields is defined from the real axis $\mathcal{C}$ to $\tilde{\mathcal{C}}$ that has a tiny excursion into the lower half plane.
• Figure 3: By changing $K$ into $\tilde{K}$, we are deforming the contour away from the real $\rho$ axis $\mathcal{C}$, into a contour $\tilde{\mathcal{C}}$ with constant and negative imaginary part. Furthermore, when the imaginary part becomes $-i\pi/2$, the metric on the contour becomes that of Euclidean global $AdS_2$.
• Figure 4: Definition of the subregions $R$, $R_+$, $R_-$.
• Figure 5: When we take into account the backreaction of the matter field, the physical boundaries of the spacetime are no longer on the real axis of $\rho$, but instead become in the upper half plane, at the two ends of the red line. The generalized SSS prescription would be to take a contour (shown in green) that takes a detour below $\rho=0$. Another possibility is to take a contour that is parametrized by real parameter $\tilde{\rho}$ (shown in blue).