## Cost of holographic path integrals

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

SciPost Phys. 14, 061 (2023) · published 5 April 2023

- doi: 10.21468/SciPostPhys.14.4.061
- Submissions/Reports

### 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 $T\overline{T}$ 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.

### Authors / Affiliations: mappings to Contributors and Organizations

See all Organizations.-
^{1}A. Ramesh Chandra, -
^{1}Jan de Boer, -
^{2}Mario Flory, -
^{3}Michal P. Heller, -
^{4}Sergio Hörtner, -
^{1}Andrew Rolph

^{1}Universiteit van Amsterdam / University of Amsterdam [UvA]^{2}Instituto de Física Teórica / Institute for Theoretical Physics [IFT]^{3}Universiteit Gent / Ghent University^{4}Max-Planck-Institut für Gravitationsphysik / Max Planck Institute for Gravitational Physics [AEI]

- European Regional Development Fund [ERDF]
- European Research Council [ERC]
- FP7 Seventh Framework Programme (FP7) (through Organization: European Commission [EC])
- Fundación Ramón Areces
- Ministerio de Ciencia e Innovación
- Nederlandse Organisatie voor Wetenschappelijk Onderzoek / Netherlands Organisation for Scientific Research [NWO]
- Secretaría de Estado de Investigacion, Desarrollo e Innovacion (through Organization: Ministerio de Economía, Industria y Competitividad / Ministry of Economy, Industry and Competitiveness [MINECO])