Cost of holographic path integrals

TL;DR

This paper explores various proposals for quantifying the cost of holographic path integrals, connecting gravitational path integrals with $Tar T$ deformed CFTs, and analyzing their properties and implications.

Contribution

It introduces a broad class of cost proposals for holographic path integrals, including UV-finite options, and relates them to existing complexity conjectures and gravitational action.

Findings

Identifies cost proposals that reduce to CV and CV2.0 complexity conjectures.

Shows that bounded cost proposals based on gravitational action are not found.

Provides a Lorentzian Gauss-Bonnet theorem applicable with conical singularities.

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.