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#### Cost of holographic path integrals

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##### Citation

Chandra, A. R., de Boer, J., Flory, M., Heller, M. P., Hörtner, S., & Rolph, A. (2023).
Cost of holographic path integrals.* SciPost Physics,* *14*(4):
061. doi:10.21468/SciPostPhys.14.4.061.

Cite as: https://hdl.handle.net/21.11116/0000-000A-240E-F

##### 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.

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.