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Tensor networks, p-adic fields, and algebraic curves: arithmetic and the AdS_3/CFT_2 correspondence

Heydeman, Matthew and Marcolli, Matilde and Saberi, Ingmar A. and Stoica, Bogdan (2018) Tensor networks, p-adic fields, and algebraic curves: arithmetic and the AdS_3/CFT_2 correspondence. Advances in Theoretical and Mathematical Physics, 22 (1). pp. 93-176. ISSN 1095-0761. doi:10.4310/ATMP.2018.v22.n1.a4.

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One of the many remarkable properties of conformal field theory in two dimensions is its connection to algebraic geometry. Since every compact Riemann surface is a projective algebraic curve, many constructions of interest in physics (which a priori depend on the analytic structure of the spacetime) can be formulated in purely algebraic language. This opens the door to interesting generalizations, obtained by taking another choice of field: for instance, the p-adics. We generalize the AdS/CFT correspondence according to this principle; the result is a formulation of holography in which the bulk geometry is discrete—the Bruhat–Tits tree for PGL(2,Qp)—but the group of bulk isometries nonetheless agrees with that of boundary conformal transformations and is not broken by discretization. We suggest that this forms the natural geometric setting for tensor networks that have been proposed as models of bulk reconstruction via quantum error correcting codes; in certain cases, geodesics in the Bruhat–Tits tree reproduce those constructed using quantum error correction. Other aspects of holography also hold: Standard holographic results for massive free scalar fields in a fixed background carry over to the tree, whose vertical direction can be interpreted as a renormalization-group scale for modes in the boundary CFT. Higher-genus bulk geometries (the BTZ black hole and its generalizations) can be understood straightforwardly in our setting, and the Ryu–Takayanagi formula for the entanglement entropy appears naturally.

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Heydeman, Matthew0000-0001-7033-9075
Additional Information:© 2018 International Press. The authors wish to thank N. Bao, H. Kim, M. Koloğlu, T. McKinney, N. Hunter-Jones, B. Michel, M. M. Năstăsescu, H. Ooguri, and A. Turzillo for helpful conversations as this work was being prepared. I.A.S. is also grateful to the Partnership Mathematics and Physics of Universität Heidelberg, the University of Bristol, and the Gone Fishing meeting at the University of Colorado, Boulder for hospitality. We are especially grateful to the anonymous referee, for careful reading and thorough and useful comments. The work of M.H., I.A.S., and B.S. is supported by the United States Department of Energy under the grant DE-SC0011632, as well as by the Walter Burke Institute for Theoretical Physics at Caltech. M.M. is partially supported by NSF grant DMS-1707882, by NSERC Discovery Grant RGPIN-2018-04937 and Accelerator Supplement grant RGPAS-2018-522593, and by the Perimeter Institute for Theoretical Physics.
Group:Walter Burke Institute for Theoretical Physics
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Department of Energy (DOE)DE-SC0011632
Walter Burke Institute for Theoretical Physics, CaltechUNSPECIFIED
Natural Sciences and Engineering Research Council of Canada (NSERC)RGPIN-2018-04937
Natural Sciences and Engineering Research Council of Canada (NSERC)RGPAS-2018-522593
Perimeter Institute for Theoretical PhysicsUNSPECIFIED
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Record Number:CaltechAUTHORS:20160615-160129721
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:67954
Deposited By: Joy Painter
Deposited On:15 Jun 2016 23:10
Last Modified:11 Nov 2021 03:57

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