Abstract
We give an interpretation of holography in the form of the AdS/CFT correspondence in terms of homotopy algebras. A field theory such as a bulk gravity theory can be viewed as a homotopy Lie or L∞ algebra. We extend this dictionary to theories defined on manifolds with a boundary, including the conformal boundary of AdS, taking into account the cyclic structure needed to define an action with the correct boundary terms. Projecting fields to their boundary values then defines a homotopy retract, which in turn implies that the cyclic L∞ algebra of the bulk theory is equivalent, up to homotopy, to a cyclic L∞ algebra on the boundary. The resulting action is the ‘on-shell action’ conventionally computed via Witten diagrams that, according to AdS/CFT, yields the generating functional for the correlation functions of the dual CFT. These results are established with the help of new techniques regarding the homotopy transfer of cyclic L∞ algebras.
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Acknowledgments
We would like to thank Alex Arvanitakis, Roberto Bonezzi, Tomas Codina, Felipe Diaz-Jaramillo, Owen Gwiliiam, Allison Pinto, Ivo Sachs and Barton Zwiebach for useful discussions and collaborations on related topics.
This work is funded by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 771862).
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Chiaffrino, C., Ersoy, T. & Hohm, O. Holography as homotopy. J. High Energ. Phys. 2024, 161 (2024). https://doi.org/10.1007/JHEP09(2024)161
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DOI: https://doi.org/10.1007/JHEP09(2024)161