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Halfwormholes in nearly AdS$_2$ holography
by Antonio M. GarcíaGarcía, Victor Godet
This is not the current version.
Submission summary
As Contributors:  Victor Godet 
Preprint link:  scipost_202109_00031v1 
Date submitted:  20210928 06:20 
Submitted by:  Godet, Victor 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
We find halfwormhole solutions in JackiwTeitelboim gravity by allowing the geometry to end on a spacetime Dbrane with specific boundary conditions. This theory also contains a Euclidean wormhole which leads to a factorization problem. We propose that halfwormholes 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 SachdevYeKitaev (SYK) model with random complex couplings. We find that the free energy of the halfwormhole 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 Dbrane boundary conditions.
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Submission & Refereeing History
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Reports on this Submission
Anonymous Report 2 on 20211122 (Invited Report)
Report
This paper studies the wormhole and halfwormhole solutions in JT gravity coupled to a complex scalar field, and also proposes a possible interpretation of these solutions in the SYK model. The topic of the paper is interesting and the results are new and well written. Therefore, I recommend for publication in SciPost Physics once the authors address the following points. \\
$\bullet$ The complex couplings introduced in (2.13), (2.14) seem to give nonunitary evolution for each SYK system, even though the total Hamiltonian (2.12) is Hermitian. It might be explained in the earlier paper, but it's useful to explain again how the authors justify these type of couplings in that section. \\
$\bullet$ The SYK model has the zero mode sector which is described by the Schwarzian theory and dual to the pure JT gravity. The SYK model also has the nonzero mode sector, which seems to give an infinite number of matter fields in the dual gravity theory. In this paper, the authors consider a single scalar field (2.2) in the gravity theory. It's useful to explain how the authors think of this single scalar field (2.2) from the SYK point of view. (i.e. do they correspond to a particular nonzero mode in the SYK or any kinds of external degree of freedom?)\\
$\bullet$ Even though the pure JT gravity (without matter) is oneloop exact, I don't think the system of JT gravity coupled to a matter is oneloop exact. The interactions between the Schwarzian mode and the matter field (like (A.3)) give contributions for twoloop and higher.\\
Anonymous Report 1 on 20211025 (Invited Report)
Report
The paper studied halfwormhole configurations in a varian of SYK/JT with complex couplings. Despite the couplings being complex, the authors argued in their previous paper that the theory still retains certain hermitian properties and the partition function is well defined. The major interesting point of this model is that wormhole configurations can be easily seen. In the current paper they continue their investigation in order to find halfwormhole configurations.
The authors propose to introduce a brane in the AdS_2 which is dual to a particular choice of couplings on the SYK side. They proceed by studying the corresponding halfwormhole configurations in detail and demonstrate how they affect the factorization. Also they present some numerical calculations in SYK to support their proposal.
The paper is clearly written and is definitely interesting for the field, so I recommend it for publication. However, I believe a few questions should be clarified:
My main confusion is related to the quantity $z$ introduced in eq. (6.2). As far as I understand, SYK hamiltonian (6.1) depends only on the antisymmetric part of $J_{ijkl}, M_{ijkl}$. $z$ can only introduce an overall shift in the energy.
Also I have a few minor questions:
1. In Secton 2.1 it would be useful to emphasize whether there is a single complex field $\chi$ or a large number N of them. I expect that there should be a large number in order to compete with gravity fluctuations and
produce a large determinant in Section 3.2.2
2. As far as I can understand, the nice equation (5.16) does not rely on a particular choice of $j(\tau)$ action( (5.3) or (5.4) ). I think it would be very helpful for readers if it is emphasized around eq. (5.16).
3. Finally, it would be very interesting if the authors could comment whether something like (5.16) would hold for correlation functions. From conservative point of view, one could argue that the measure for $j(\tau)$ can always be finetuned to get (5.16), but matching the correlation function is nontrivial.
Author: Victor Godet on 20220115 [id 2097]
(in reply to Report 1 on 20211025)
We would like to thank the referee for his comments.
The quantity $z$ should be viewed as a coarsegrained version of a single realization $J_{ijkl}$ of the SYK couplings. We claim that the halfwormhole should be dual to a single realization of the SYK couplings, the choice of $J_{ijkl}$ corresponding to a choice of boundary condition $j(\tau)$. In trying to make this more quantitative, it is certainly be too ambitious to try to reproduce the exact value of $J_{ijkl}$ in gravity. For this reason, we consider a coarsegrained version: the "centerofmass" $z$ obtained by summing over $i,j,k,l$. As we argued in our paper, $z$ is dual to the zero mode $j_0$ of $j(\tau)$ and we give concrete numerical evidence for this. This is our motivation to introduce the quantity $z$.
 A single scalar field is enough in our case. This is because our wormhole is a classical solution, supported by a classical profile for the scalar field. This is different than the MaldacenaQi wormhole where the wormhole is supported by quantum effects so that a large number of scalar fields is required. In other words, the scalar contribution is not small but proportional to $k$, which we can take as large as we want.
 This is correct. The fact that the wormhole emerges from the average of two halfwormholes is a general fact of the "uniform limit" $J\to+\infty$ which doesn't depend on details of the ensemble for $j(\tau)$. We have added a sentence after (5.17) to emphasise this. Note that the following paragraph explains why this is the case from a path integral point of view.
 The fact that the wormhole emerges from the average of halfwormholes (equation (5.16)) would also work for correlation functions. As explained below Figure 6, our average has the simple interpretation of "finishing the path integral". That is, the path integral on the wormhole can be done in two steps: we can first integrate over all fields except the value $j(\tau)$ of the scalar at the geodesic boundary. Interpreting the result as the contributions of two halfwormholes, the final integral over $j(\tau)$ is then interpreted as an average over halfwormhole boundary contributions. This interpretation shows that this will also work for more general observables.
Author: Victor Godet on 20220115 [id 2098]
(in reply to Report 2 on 20211122)We would like to thank the referee for his comments.