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.
