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Euclidean Wormholes and Gravitational States

Alexandre Belin

TL;DR

This work reframes Euclidean wormholes as a gravitational mechanism to prepare semi-classical states, by slicing a genus-$2$ wormhole on a time-reflection symmetric surface and analytically continuing to obtain Lorentzian initial data. It proposes a concrete microscopic CFT description for the resulting wormhole state as a diagonal projection of half-genus-$2$ OPE blocks, showing its norm reproduces the genus-$2$ wormhole partition function and that it has an order-one overlap with the thermofield double when the initial data agree. The paper also analyzes the overlap from bulk gluing arguments and explores symmetry and Lorentzian-factorization aspects, including a Lorentzian version of the factorization puzzle where connected L–R correlators arise despite a factorized reduced state. Collectively, these results highlight non-uniqueness in semi-classical state preparations corresponding to the same initial data and motivate further study of wormhole contributions, moduli, and one-loop effects in holographic quantum gravity. The findings illuminate the nuanced relationship between Euclidean preparations, bulk initial data, and boundary CFT descriptions, with implications for nonperturbative gravity and black hole information questions.

Abstract

Euclidean wormholes are known to encode important non-perturbative effects in the physics of quantum black holes. In this paper, we discuss the slicing of Euclidean wormholes along a time-reflection symmetric slice which treats half of the Euclidean geometry as a gravitational machinery to produce a semi-classical state. This type of state preparation is different from Hartle-Hawking states prepared with the CFT path integral, such as the thermofield-double state. Nevertheless, the two different types of states have order one overlaps provided the gravitational data agrees on the initial data slice. This raises an interesting puzzle: one can easily construct an infinite family of semi-classical states that have order one overlap with the thermofield double state, while having a very different Euclidean preparation. We provide a microscopic description of wormhole states in the dual CFT and reformulate the factorization puzzle in the language of entanglement and the Hilbert space.

Euclidean Wormholes and Gravitational States

TL;DR

This work reframes Euclidean wormholes as a gravitational mechanism to prepare semi-classical states, by slicing a genus- wormhole on a time-reflection symmetric surface and analytically continuing to obtain Lorentzian initial data. It proposes a concrete microscopic CFT description for the resulting wormhole state as a diagonal projection of half-genus- OPE blocks, showing its norm reproduces the genus- wormhole partition function and that it has an order-one overlap with the thermofield double when the initial data agree. The paper also analyzes the overlap from bulk gluing arguments and explores symmetry and Lorentzian-factorization aspects, including a Lorentzian version of the factorization puzzle where connected L–R correlators arise despite a factorized reduced state. Collectively, these results highlight non-uniqueness in semi-classical state preparations corresponding to the same initial data and motivate further study of wormhole contributions, moduli, and one-loop effects in holographic quantum gravity. The findings illuminate the nuanced relationship between Euclidean preparations, bulk initial data, and boundary CFT descriptions, with implications for nonperturbative gravity and black hole information questions.

Abstract

Euclidean wormholes are known to encode important non-perturbative effects in the physics of quantum black holes. In this paper, we discuss the slicing of Euclidean wormholes along a time-reflection symmetric slice which treats half of the Euclidean geometry as a gravitational machinery to produce a semi-classical state. This type of state preparation is different from Hartle-Hawking states prepared with the CFT path integral, such as the thermofield-double state. Nevertheless, the two different types of states have order one overlaps provided the gravitational data agrees on the initial data slice. This raises an interesting puzzle: one can easily construct an infinite family of semi-classical states that have order one overlap with the thermofield double state, while having a very different Euclidean preparation. We provide a microscopic description of wormhole states in the dual CFT and reformulate the factorization puzzle in the language of entanglement and the Hilbert space.

Paper Structure

This paper contains 8 sections, 38 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Semi-classical states in AdS/CFT
  3. Slicing the Euclidean wormhole and the overlap with BTZ
  4. A Microscopic description of the wormhole state
  5. The overlap with the TFD state
  6. Comments on the $U(1)$ symmetry.
  7. A Lorentzian version of the factorization puzzle
  8. Discussion

Figures (3)

  • Figure 1: On the left, a slicing of the genus-two wormhole on a time-reflection symmetric slice. On the right, the geometry of that slice, with topology circle times an interval.
  • Figure 2: The gluing of the two geometries. On the left, we have half the Euclidean wormhole geometry. The two asymptotically AdS boundaries are the solid black line and the dashed line. The three-geometry is the filling between these two surfaces. We see that the dashed line links the other handle in a non-trivial way. On the right, half the BTZ geometry (the bagel slicing). We see that we can smoothly glue it at the top of half the wormhole. The single boundary of half-BTZ (an annulus) then glues together the two half genus-2 surface, into a single boundary of genus 2.
  • Figure 3: The state $\ket{\psi_{\frac{1}{2}g=2}}$ is prepared by doing the Euclidean path integral on half of a genus-2 surface. This state has two parameters, related to the two moduli of the surface. One is related to the length of the handle, while the other is related to the length of the cylinder extending from it.