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Cost of holographic path integrals

by A. Ramesh Chandra, Jan de Boer, Mario Flory, Michal P. Heller, Sergio Hörtner, Andrew Rolph

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Submission summary

Authors (as registered SciPost users): A. Ramesh Chandra · Mario Flory · Andrew Rolph
Submission information
Preprint Link:  (pdf)
Date accepted: 2022-12-01
Date submitted: 2022-10-11 09:54
Submitted by: Chandra, A. Ramesh
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


We consider proposals for the cost of holographic path integrals. Gravitational path integrals within finite radial cutoff surfaces have a precise map to path integrals in $T\bar T$ deformed holographic CFTs. In Nielsen's geometric formulation cost is the length of a not-necessarily-geodesic path in a metric space of operators. Our cost proposals differ from holographic state complexity proposals in that (1) the boundary dual is cost, a quantity that can be `optimised' to state complexity, (2) the set of proposals is large: all functions on all bulk subregions of any co-dimension which satisfy the physical properties of cost, and (3) the proposals are by construction UV-finite. The optimal path integral that prepares a given state is that with minimal cost, and cost proposals which reduce to the CV and CV2.0 complexity conjectures when the path integral is optimised are found, while bounded cost proposals based on gravitational action are not found. Related to our analysis of gravitational action-based proposals, we study bulk hypersurfaces with a constant intrinsic curvature of a specific value and give a Lorentzian version of the Gauss-Bonnet theorem valid in the presence of conical singularities.

Published as SciPost Phys. 14, 061 (2023)

List of changes

1. Added a new figure (figure 7) in section 4.2, to explain the discussion of obtaining CV2.0 from cost=bulk volume.
2. Expanded the paragraph at the beginning of section 5 to connect it better with the rest of the paper.
3. The highly coarse-grained initial state $\Sigma_1$ is changed to a point, instead of being an empty set.
4. Minor corrections: fixed typos and rephrased slightly confusing statements.

Reports on this Submission

Anonymous Report 2 on 2022-10-17 (Invited Report)


The authors have addressed all of my concerns well. I wholeheartedly recommend the article be accepted for publication.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Anonymous Report 1 on 2022-10-17 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2203.08842v3, delivered 2022-10-17, doi: 10.21468/SciPost.Report.5913


I think the authors have addressed most of my comments and questions in their replies.

My last question is related to the previous point 5 and the new figure 17 in section 4.2.
I thought before that the timelike surface $\tilde{M}$ is defined by connecting $\partial \Sigma_1$ and $\partial \Sigma_2$.
From the new figure 17 and the description around eq.(4.12). I guess my previous
understanding is wrong. As shown in figure 17, the timelike surface $\tilde{M}$ (with respect to which the optimization is performed) could even be piecewise or discontinuous.
If this is correct, I guess it is better to stress this point at the beginning of the paper.

And I'm happy to recommend the paper for publication.

  • validity: good
  • significance: good
  • originality: high
  • clarity: good
  • formatting: good
  • grammar: excellent

Author:  A. Ramesh Chandra  on 2022-11-03  [id 2981]

(in reply to Report 1 on 2022-10-17)
answer to question

Thank you for the above question. It is not incorrect to think of $\tilde{M}$ as a surface connecting $\partial\Sigma_1$ and $\partial\Sigma_2$. More importantly, as we optimise a given cost, we keep $\partial\tilde{M} = \partial\Sigma_1\cup\partial\Sigma_2$ fixed. This condition is still satisfied in the case illustrated in new figure 7. Here, the minimising (timelike) $\tilde{M}$ could be a disjoint union depending on the location of $\Sigma_1$. In the limit as $\Sigma_1$ is taken to a point, the contribution from smaller component of $\tilde{M}$ shrinks to zero, and we remain with a connected $\tilde{M}$.

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