Abstract
There are various definitions of the concept of complexity in Quantum Field Theory as well as for finite quantum systems. For several of them there are conjectured holographic bulk duals. In this work we establish an entry in the AdS/CFT dictionary for one such class of complexity, namely Krylov or K-complexity. For this purpose we work in the double-scaled SYK model which is dual in a certain limit to JT gravity, a theory of gravity in AdS2. In particular, states on the boundary have a clear geometrical definition in the bulk. We use this result to show that Krylov complexity of the infinite-temperature thermofield double state on the boundary of AdS2 has a precise bulk description in JT gravity, namely the length of the two-sided wormhole. We do this by showing that the Krylov basis elements, which are eigenstates of the Krylov complexity operator, are mapped to length eigenstates in the bulk theory by subjecting K-complexity to the bulk-boundary map identifying the bulk/boundary Hilbert spaces. Our result makes extensive use of chord diagram techniques and identifies the Krylov basis of the boundary quantum system with fixed chord number states building the bulk gravitational Hilbert space.
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Acknowledgments
It is a pleasure to thank Vijay Balasubramanian, José Barbón, Micha Berkooz, Damián Galante, Vladimir Navrolansky and Edward Witten for insightful discussions. This work has been supported in part by the Fonds National Suisse de la Recherche Scientifique (Schweizerischer Nationalfonds zur Förderung der wissenschaftlichen Forschung) through Project Grant 200020_182513, and the NCCR 51NF40-141869, The Mathematics of Physics (SwissMAP). ER would like to thank the special fund for high energy physics of the PBC for partial support of this work. RS would like to thank Vladimir Narovlansky for a discussion about chord diagrams some years ago. We would like to thank José Barbón for some insightful correspondence on the continuum limit of Krylov space.
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Rabinovici, E., Sánchez-Garrido, A., Shir, R. et al. A bulk manifestation of Krylov complexity. J. High Energ. Phys. 2023, 213 (2023). https://doi.org/10.1007/JHEP08(2023)213
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DOI: https://doi.org/10.1007/JHEP08(2023)213