# Cost of holographic path integrals

### Submission summary

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

### Abstract

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.

###### Current status:
Accepted in target Journal

Editorial decision: For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)

### 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.

### Submission & Refereeing History

Resubmission 2203.08842v3 on 11 October 2022
Submission 2203.08842v2 on 29 June 2022

## Reports on this Submission

### 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: -

### Report

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)
Category:
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 $\tilde{M}$.