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arXiv:2502.02633 · Journal of High Energy Physics

Geometric modular flows in 2d CFT and beyond

Jacqueline Caminiti, Federico Capeccia, Luca Ciambelli, Robert Cobb Myers

Published Aug 20, 2025• 9 citations
arXivHTMLPDF

Abstract

A bstract We study geometric modular flows in two-dimensional conformal field theories. We explore which states exhibit a geometric modular flow with respect to a causally complete subregion and, conversely, how to construct a state from a given geometric modular flow. Given suitable boundary conditions, we find that generic geometric modular flows in the Rindler wedge are conformally equivalent. Based on this insight, we show how conformal unitaries can be used to explicitly construct a state for each flow. We analyze these states, deriving general formulas for the energy density and entanglement entropy. We also consider geometric flows beyond the Rindler wedge setting, and in higher dimensions.

DOI
10.1007/jhep08(2025)166

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We study the reconstruction of bulk operators in the entanglement wedge in terms of low energy operators localized in the respective boundary region. To leading order in N, the dual boundary operators are constructed from the modular flow of single trace operators in the boundary subregion. The appearance of modular evolved boundary operators can be understood due to the equality between bulk and boundary modular flows and explicit formulas for bulk operators can be found with a complete understanding of the action of bulk modular flow, a difficult but in principle solvable task. We also obtain an expression when the bulk operator is located on the Ryu-Takayanagi surface which only depends on the bulk to boundary correlator and does not require the explicit use of bulk modular flow. This expression generalizes the geodesic operator/OPE block dictionary to general states and boundary regions.
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Abstract We enumerate the cases in 2d conformal field theory where the logarithm of the reduced density matrix (the entanglement or modular Hamiltonian) may be written as an integral over the energy-momentum tensor times a local weight. These include known examples and new ones corresponding to the time-dependent scenarios of a global and local quench. In these latter cases the entanglement Hamiltonian depends on the momentum density as well as the energy density. In all cases the entanglement spectrum is that of the appropriate boundary CFT. We emphasize the role of boundary conditions at the entangling surface and the appearance of boundary entropies as universal O (1) terms in the entanglement entropy.