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Half-wormholes in nearly AdS$_2$ holography

by Antonio M. García-García, Victor Godet

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Authors (as registered SciPost users): Victor Godet
Submission information
Preprint Link: scipost_202109_00031v2  (pdf)
Date submitted: 2022-01-15 07:51
Submitted by: Godet, Victor
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
Approaches: Theoretical, Computational


We find half-wormhole solutions in Jackiw-Teitelboim gravity by allowing the geometry to end on a spacetime D-brane with specific boundary conditions. This theory also contains a Euclidean wormhole which leads to a factorization problem. We propose that half-wormholes provide a gravitational picture for how factorization is restored and show that the Euclidean wormhole emerges from averaging over the boundary conditions. The wormhole is known to be dual to a Sachdev-Ye-Kitaev (SYK) model with random complex couplings. We find that the free energy of the half-wormhole is strikingly similar to that of a single realization of this SYK model. These results suggest that the gravitational path integral computes an average over spacetime D-brane boundary conditions.

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minor changes detailed in the replies to the referees

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Anonymous Report 1 on 2022-1-22 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202109_00031v2, delivered 2022-01-22, doi: 10.21468/SciPost.Report.4220


What I wanted to say is that $z$ is an overall shift in the SYK definition of energy: for a generic tensor $J_{ijkl}$, the sum $\sum_{ijkl} J_{ijkl} \psi_i \psi_k \psi_k \psi_l$ can be separated into two parts. The antisymmetric part of $J_{ijkl}$ gives the standard SYK hamiltonian.
The symmetric part of $J_{ijkl}$ yields a constant, since fermions anticommute: $\{\psi_i, \psi_j\} = \delta_{ij}$. Since all fermion operators disappear, the residual sum over $ijkl$ yields $z$. This is a purely analytic argument. Interestingly, eq. (4.8) is indeed the shift in the ground state energy, after cancelling $\arctan$ and $\tan$ and factors of $T$. So one does not need to perform a numerical fit to obtain a precise relation between $j_0$ and $z$(modulo subtleties of taking a $\log $ of a complex $Z$, presumably this is why you have $\arctan(\
tan)$ in eq. (4.8) ). I strongly believe that Section 6 should emphasize this point.

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Author:  Victor Godet  on 2022-02-02  [id 2143]

(in reply to Report 1 on 2022-01-22)

Thanks for your comment. I agree that $z$ can be interpreted as a coupling-dependent ground state energy. I have modified the end of paragraph below 6.2 to reflect this, reproduced here:

This quantity has the interpretation of a coupling-dependent ground state energy. For example, a simple way to change the value of $z$ is to shift the couplings by a fixed complex constant. Overall, this just adds a constant to the Hamiltonian after using the anticommutation relations of the fermions. Our purpose in isolating this simple parameter is to make a comparison with the gravity side. We will see that $z$ appears to be closely related with the zero mode $j_0$ of $j(\tau)$.

The pattern in the imaginary part of F is indeed a consequence of the branch cut in the argument. This effect results from the fact that the partition is a sum of two exponentials, so we have schematically

$$ \mathrm{Im}(F) = -T \,\mathrm{arg} \, Z,\qquad Z = e^{S_0} +e^{S_1} e^{ i j_0/T} $$

For $S_1> S_0$, i.e. when the half-wormhole dominates over the black hole, $Z$ circles around the origin as $T$ is changed and this is what gives the saw pattern. The dependence of $j_0$ is indeed that of a ground state energy. In gravity, this is the (complex) ground state energy of the half-wormhole. I believe that this distinctive behaviour is a hint that the SYK model at fixed couplings also contains the half-wormhole saddle-point, since such behaviour is most easily accounted by added the second oscillatory exponential in $Z$ (the contribution of the half-wormhole). Also let me point out, as we discuss in section 6, that the identification of $j_0$ and $z$ is mostly heuristic and based on our numerical results. To obtain a precise relationship, one would need to go beyond numerics in SYK.

Anonymous on 2022-02-06  [id 2161]

(in reply to Victor Godet on 2022-02-02 [id 2143])

I thank Authors for the quick response. I recommend the paper for publication.

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