Advances in Theoretical and Mathematical Physics

Volume 22 (2018)

Number 1

Tensor networks, $p$-adic fields, and algebraic curves: arithmetic and the $\mathrm{AdS}_3 / \mathrm{CFT}_2$ correspondence

Pages: 93 – 176

DOI: http://dx.doi.org/10.4310/ATMP.2018.v22.n1.a4

Authors

Matthew Heydeman (Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, Ca., U.S.A.)

Matilde Marcolli (Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, Ca., U.S.A.; Department of Mathematics, University of Toronto, Ontario, Canada; and Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada)

Ingmar A. Saberi (Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, Ca., U.S.A.; and Mathematisches Institut, Ruprecht-Karls-Universität, Heidelberg, Germany)

Bogdan Stoica (Institute for Theoretical Physics, California Institute of Technology, Pasadena, Ca., U.S.A.; School of Physics, Brandeis University, Waltham, Massachusetts, U.S.A.; and Dept. of Physics, Brown University, Providence, Rhode Island, U.S.A.)

Abstract

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 $\mathrm{AdS} / \mathrm{CFT}$ correspondence according to this principle; the result is a formulation of holography in which the bulk geometry is discrete—the Bruhat–Tits tree for $\mathrm{PGL} (2, \mathbb{Q}_p)$—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 $\mathrm{CFT}$. Higher-genus bulk geometries (the $\mathrm{BTZ}$ 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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