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Title: Tensor networks, p-adic fields, and algebraic curves: arithmetic and the AdS 3/CFT 2 correspondence

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. Herein, 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 anticipate 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 themore » 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« less

Authors:
 [1];  [2];  [3];  [4]
  1. California Inst. of Technology (CalTech), Pasadena, CA (United States)
  2. California Inst. of Technology (CalTech), Pasadena, CA (United States); Univ. of Toronto, ON (Canada); Perimeter Inst. for Theoretical Physics, Waterloo, ON (Canada)
  3. California Inst. of Technology (CalTech), Pasadena, CA (United States); Heidelberg Univ. (Germany)
  4. California Inst. of Technology (CalTech), Pasadena, CA (United States); Brandeis Univ., Waltham, MA (United States); Brown Univ., Providence, RI (United States)
Publication Date:
Research Org.:
California Institute of Technology (CalTech), Pasadena, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), High Energy Physics (HEP); National Science Foundation (NSF)
OSTI Identifier:
1598352
Grant/Contract Number:  
SC0011632; DMS-1201512; PHY-1205440
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Advances in Theoretical and Mathematical Physics
Additional Journal Information:
Journal Volume: 22; Journal Issue: 1; Journal ID: ISSN 1095-0761
Publisher:
International Press
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING

Citation Formats

Heydeman, Matthew, Marcolli, Matilde, Saberi, Ingmar A., and Stoica, Bogdan. Tensor networks, p-adic fields, and algebraic curves: arithmetic and the AdS3/CFT2 correspondence. United States: N. p., 2018. Web. doi:10.4310/ATMP.2018.v22.n1.a4.
Heydeman, Matthew, Marcolli, Matilde, Saberi, Ingmar A., & Stoica, Bogdan. Tensor networks, p-adic fields, and algebraic curves: arithmetic and the AdS3/CFT2 correspondence. United States. https://doi.org/10.4310/ATMP.2018.v22.n1.a4
Heydeman, Matthew, Marcolli, Matilde, Saberi, Ingmar A., and Stoica, Bogdan. Fri . "Tensor networks, p-adic fields, and algebraic curves: arithmetic and the AdS3/CFT2 correspondence". United States. https://doi.org/10.4310/ATMP.2018.v22.n1.a4. https://www.osti.gov/servlets/purl/1598352.
@article{osti_1598352,
title = {Tensor networks, p-adic fields, and algebraic curves: arithmetic and the AdS3/CFT2 correspondence},
author = {Heydeman, Matthew and Marcolli, Matilde and Saberi, Ingmar A. and Stoica, Bogdan},
abstractNote = {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. Herein, 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 anticipate 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},
doi = {10.4310/ATMP.2018.v22.n1.a4},
url = {https://www.osti.gov/biblio/1598352}, journal = {Advances in Theoretical and Mathematical Physics},
issn = {1095-0761},
number = 1,
volume = 22,
place = {United States},
year = {2018},
month = {9}
}

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Works referencing / citing this record:

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Recursion relations in p  -adic Mellin Space
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Anderson localization on the Bethe lattice using cages and the Wegner flow
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