Abstract
We consider the generalization of a matrix integral with arbitrary spectral curve ρ0(E) to a 0+1D theory of matrix quantum mechanics (MQM). Using recent techniques for 1D quantum systems at large-N, we formulate a hydrodynamical effective theory for the eigenvalues. The result is a simple 2D free boson BCFT on a curved background, describing the quantum fluctuations of the eigenvalues around ρ0(E), which is now the large-N limit of the quantum expectation value of the eigenvalue density operator \( \hat{\rho}(E) \).
The average over the ensemble of random matrices becomes a quantum expectation value. Equal-time density correlations reproduce the results (including non-perturbative corrections) of random matrix theory. This suggests an interpretation of JT gravity as dual to a one-time-point reduction of MQM.
As an application, we compute the Rényi entropy associated to a bipartition of the eigenvalues. We match a previous result by Hartnoll and Mazenc for the c = 1 matrix model dual to two-dimensional string theory and extend it to arbitrary ρ0(E). The hydrodynamical theory provides a clear picture of the emergence of spacetime in two dimensional string theory. The entropy is naturally finite and displays a large amount of short range entanglement, proportional to the microcanonical entropy. We also compute the reduced density matrix for a subset of n < N eigenvalues.
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Acknowledgments
We would like to thank Bruno Balthazar, Shaun Hampton, Vladimir Kazakov, Pierfrancesco Urbani and Takato Yoshimura for useful discussions and comments on the manuscript. GDU’s research is supported by ERC Starting Grant 853507. GDU would like to thank École Normale Supérieure, where this project started, for previous support in the form of a LABEX ENS-ICFP scholarship.
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Di Ubaldo, G., Policastro, G. Ensemble averaging in JT gravity from entanglement in Matrix Quantum Mechanics. J. High Energ. Phys. 2023, 122 (2023). https://doi.org/10.1007/JHEP07(2023)122
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DOI: https://doi.org/10.1007/JHEP07(2023)122