Ammanamanchi Ramesh Chandra, Jan de Boer, Mario Flory, Michal P. Heller, Sergio HÃ¶rtner, Andrew Rolph
SciPost Phys. 14, 061 (2023) ·
published 5 April 2023

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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 notnecessarilygeodesic 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 codimension which satisfy the physical properties of cost, and (3) the proposals are by construction UVfinite. 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 actionbased proposals, we study bulk hypersurfaces with a constant intrinsic curvature of a specific value and give a Lorentzian version of the GaussBonnet theorem valid in the presence of conical singularities.
Johanna Erdmenger, Mario Flory, Marius Gerbershagen, Michal P. Heller, AnnaLena Weigel
SciPost Phys. 13, 061 (2022) ·
published 22 September 2022

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Holographic complexity proposals have sparked interest in quantifying the cost of state preparation in quantum field theories and its possible dual gravitational manifestations. The most basic ingredient in defining complexity is the notion of a class of circuits that, when acting on a given reference state, all produce a desired target state. In the present work we build on studies of circuits performing local conformal transformations in general twodimensional conformal field theories and construct the exact gravity dual to such circuits. In our approach to holographic complexity, the gravity dual to the optimal circuit is the one that minimizes an externally chosen cost assigned to each circuit. Our results provide a basis for studying exact gravity duals to circuit costs from first principles.