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Toward holography on biregular trees

Arkapal Mondal*, Sarthak Parikh†, Pulak Pradhan‡, and Ritu Sengar§

  • *Contact author: phz228027@physics.iitd.ac.in
  • †Contact author: sarthak@physics.iitd.ac.in
  • ‡Contact author: pulakp133@gmail.com
  • §Contact author: ritusengar2021@gmail.com
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Phys. Rev. D 112, 086011 – Published 17 October, 2025

DOI: https://doi.org/10.1103/ms53-88pp

Abstract

We study scalar field theory on biregular trees, as a new model for discrete holography. Biregular trees are discrete symmetric spaces associated with the bulk isometry group SU(3) over the unramified quadratic extension of a nonarchimedean field. The bulk-to-bulk and bulk-to-boundary propagators exhibit distinct features absent on the regular tree or continuum AdS spaces, arising from the semihomogeneous nature of the bulk space. We compute the two- and three-point correlators of the putative boundary dual. The three-point correlator exhibits a nontrivial “tensor structure” via dependence on the homogeneity degree of a unique bulk point specified in terms of boundary insertion points. The computed operator product expansion coefficients show dependence on zeta functions associated with the unramified quadratic extension of a nonarchimedean field. This work initiates the formulation of holography on a family of discrete holographic spaces that exhibit features of both flat space and negatively curved space.

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References (65)

  1. F. Pastawski, B. Yoshida, D. Harlow, and J. Preskill, Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence, J. High Energy Phys. 06 (2015) 149.
  2. S. S. Gubser, J. Knaute, S. Parikh, A. Samberg, and P. Witaszczyk, p-adic AdS/CFT, Commun. Math. Phys. 352, 1019 (2017).
  3. M. Heydeman, M. Marcolli, I. Saberi, and B. Stoica, Tensor networks, p-adic fields, and algebraic curves: Arithmetic and the AdS3/CFT2 correspondence, Adv. Theor. Math. Phys. 22, 93 (2018).
  4. L.-Y. Hung, W. Li, and C. M. Melby-Thompson, p-adic CFT is a holographic tensor network, J. High Energy Phys. 04 (2019) 170.
  5. M. Axenides, E. Floratos, and S. Nicolis, The arithmetic geometry of AdS2 and its continuum limit, SIGMA 17, 004 (2021).
  6. P. Basteiro, G. Di Giulio, J. Erdmenger, J. Karl, R. Meyer, and Z.-Y. Xian, Towards explicit discrete holography: Aperiodic spin chains from hyperbolic tilings, SciPost Phys. 13, 103 (2022).
  7. F. Bruhat and J. Tits, Groupes réductifs sur un corps local: I. Données radicielles valuées, Publ. Math. l’IHÉS 41, 5 (1972).
  8. J.-P. Serre, Trees (Springer, Berlin, Heidelberg, 1980), Original French edition published by Société Mathématique France, 1977.
  9. P. Abramenko and K. S. Brown, Buildings: Theory and Applications. Graduate Texts in Mathematics (Springer, New York, NY, 2008).
  10. Y. I. Manin and M. Marcolli, Holography principle and arithmetic of algebraic curves, Adv. Theor. Math. Phys. 5, 617 (2002).
  11. P. Sarnak, Letter to Aaronson and Pollington on the Solvay-Kitaev theorem and golden gates (2015), https://publications.ias.edu/sites/default/files/Letter%20-%20golden%20gates%20march_0.pdf.
  12. S. Evra and O. Parzanchevski, Ramanujan complexes and golden gates in PU(3), Geom. Funct. Anal. 32, 193 (2022).
  13. D. I. Cartwright and T. Steger, Application of the Bruhat-Tits tree of SU3(h) to some Ã2 groups, J. Aust. Math. Soc. 64, 329 (1998).
  14. F. Bouaziz-Kellil, Représentations sphériques des groupes agissant transitivement sur un arbre semi-homogène, Bull. Soc. Math. Fr. 116, 255 (1988).
  15. A. Figá-Talamanca and C. Nebbia, Harmonic Analysis and Representation Theory for Groups Acting on Homogenous Trees, London Mathematical Society Lecture Note Series (Cambridge University Press, Cambridge, England, 1991).
  16. P. Gérardin and K. Lai, Asymptotic behaviour of eigenfunctions on semi-homogeneous tree, Pac. J. Math. 196, 415 (2000).
  17. E. C. Tarabusi and M. A. Picardello, Spherical functions and spectrum of the Laplacian on semi-homogeneous trees, arXiv:2208.00910.
  18. S. S. Gubser, M. Heydeman, C. Jepsen, M. Marcolli, S. Parikh, I. Saberi, B. Stoica, and B. Trundy, Edge length dynamics on graphs with applications to p-adic AdS/CFT, J. High Energy Phys. 06 (2017) 157.
  19. F. Q. Gouvêa, p-adic Numbers: An Introduction, edition 3rd, Universitext (Springer, Cham, Switzerland, 2020).
  20. E. Gesteau, M. Marcolli, and S. Parikh, Holographic tensor networks from hyperbolic buildings, J. High Energy Phys. 10 (2022) 169.
  21. M. Heydeman, M. Marcolli, S. Parikh, and I. Saberi, Nonarchimedean holographic entropy from networks of perfect tensors, Adv. Theor. Math. Phys. 25, 591 (2021).
  22. S. Stadler, CAT(0) spaces of higher rank I, Geom. Funct. Anal. 34, 512 (2024).
  23. S. Stadler, CAT(0) spaces of higher rank II, Inventiones Mathematicae 235, 709 (2024).
  24. L. Ji, Buildings and their applications in geometry and topology, Asian J. Math. 10, 11 (2006).
  25. Y. I. Manin, p-Adic automorphic functions, J. Sov. Math. 5, 279 (1976).
  26. D. Ghoshal, p-adic string theories provide lattice discretization to the ordinary string worldsheet, Phys. Rev. Lett. 97, 151601 (2006).
  27. L. Brekke and P. G. O. Freund, p-adic numbers in physics, Phys. Rep. 233, 1 (1993).
  28. O. King and G. Robertson, On the K-theory of boundary C*-algebras of Ã2 groups, J. K-Theor. 9, 521 (2012).
  29. A. Borel and L. Ji, Compactifications of Symmetric and Locally Symmetric Spaces, Mathematics: Theory & Applications (Birkhäuser, Boston, MA, 2006).
  30. L. Ji, From symmetric spaces to buildings curve complexes and outer spaces, Innovations Incidence Geom. 10, 33 (2009).
  31. P.-E. Caprace and N. Monod, An indiscrete Bieberbach theorem: From amenable CAT(0) groups to tits buildings, J. l’École Polytech.- Math. 2, 333 (2015).
  32. R. Dalal, S. Evra, and O. Parzanchevski, Multi-qubit golden gates, arXiv:2509.09047.
  33. S. Cappellini, Fibrations and congruence towers of arithmetic hyperbolic manifolds, Master’s thesis, Universita di Pisa, 2020.
  34. S. S. Gubser and S. Parikh, Geodesic bulk diagrams on the Bruhat–Tits tree, Phys. Rev. D 96, 066024 (2017).
  35. S. S. Gubser, C. Jepsen, S. Parikh, and B. Trundy, O(N) and O(N) and O(N), J. High Energy Phys. 11 (2017) 107.
  36. C. B. Jepsen and S. Parikh, p-adic Mellin amplitudes, J. High Energy Phys. 04 (2019) 101.
  37. C. B. Jepsen and S. Parikh, Recursion relations in p-adic Mellin space, J. Phys. A 52, 285401 (2019).
  38. C. B. Jepsen and S. Parikh, Propagator identities, holographic conformal blocks, and higher-point AdS diagrams, J. High Energy Phys. 10 (2019) 268.
  39. S. S. Gubser, M. Heydeman, C. Jepsen, S. Parikh, I. Saberi, B. Stoica, and B. Trundy, Melonic theories over diverse number systems, Phys. Rev. D 98, 126007 (2018).
  40. P. Cartier, Fonctions Harmoniques sur un arbre. Symposia Mathematica, Vol. IX (Academic Press, London, 1972), pp. 203–270.
  41. A. V. Zabrodin, Non-archimedean strings and Bruhat-Tits trees, Commun. Math. Phys. 123, 463 (1989).
  42. S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428, 105 (1998).
  43. V. Balasubramanian, P. Kraus, A. E. Lawrence, and S. P. Trivedi, Holographic probes of anti-de Sitter space-times, Phys. Rev. D 59, 104021 (1999).
  44. T. Banks, M. R. Douglas, G. T. Horowitz, and E. J. Martinec, AdS dynamics from conformal field theory, arXiv:hep-th/9808016.
  45. D. Harlow and D. Stanford, Operator dictionaries and wave functions in AdS/CFT and dS/CFT, arXiv:1104.2621.
  46. P. E. Bradley, Vladimirov-Pearson operators on ζ-regular ultrametric Cantor sets, arXiv:2504.20753.
  47. E. Melzer, Nonarchimedean conformal field theories, Int. J. Mod. Phys. A 04, 4877 (1989).
  48. D. Z. Freedman, S. D. Mathur, A. Matusis, and L. Rastelli, Correlation functions in the CFT(d) / AdS(d+1) correspondence, Nucl. Phys. B546, 96 (1999).
  49. E. Hijano, P. Kraus, E. Perlmutter, and R. Snively, Witten diagrams revisited: The AdS geometry of conformal blocks, J. High Energy Phys. 01 (2016) 146.
  50. M. Ronan, Lectures on Buildings: Updated and Revised. University of Chicago Press, Chicago, 2009.
  51. J. Figueroa-O’Farrill, E. Have, S. Prohazka, and J. Salzer, Carrollian and celestial spaces at infinity, J. High Energy Phys. 09 (2022) 007.
  52. L. Donnay, Celestial holography: An asymptotic symmetry perspective, Phys. Rep. 1073, 1 (2024).
  53. S. Pasterski, A chapter on celestial holography, arXiv:2310.04932.
  54. A. Bagchi, A. Banerjee, P. Dhivakar, S. Mondal, and A. Shukla, The Carrollian Kaleidoscope, arXiv:2506.16164.
  55. A. Huang and C. B. Jepsen, Finite temperature at finite places, J. High Energy Phys. 03 (2025) 097.
  56. I. M. Gel’fand, M. I. Graev, and I. Piatetski-Shapiro, Representation Theory and Automorphic Functions (Saunders, London, 1968).
  57. S. S. Gubser, C. Jepsen, and B. Trundy, Spin in p-adic AdS/CFT, J. Phys. A 52, 144004 (2019).
  58. P. G. O. Freund and M. Olson, Nonarchimedean strings, Phys. Lett. B 199, 186 (1987).
  59. P. G. O. Freund and E. Witten, Adelic string amplitudes, Phys. Lett. B 199, 191 (1987).
  60. M. Baker, An introduction to Berkovich analytic spaces and non-archimedean potential theory on curves, in p-adic Geometry: Lectures from the 2007 Arizona Winter School, edited by D. Savitt and D. S. Thakur, University Lecture Series (American Mathematical Society, Providence, RI, 2008), Vol. 45.
  61. A. Goldfarb, On PBerk1 and coarse-grainings of non-archimedean product spaces, arXiv:2306.10227.
  62. J. J. Heckman, Speculations on physical discretization and arithmetic geometry (2023), Working Paper https://www.jjheckman.com/uploads/2/0/6/7/20676186/samizdatv14.pdf.
  63. H. Yan, Hyperbolic fracton model, subsystem symmetry, and holography, Phys. Rev. B 99, 155126 (2019).
  64. H. Yan, Hyperbolic fracton model, subsystem symmetry, and holography II: The dual eight-vertex model, Phys. Rev. B 100, 245138 (2019).
  65. H. Yan, C. B. Jepsen, and Y. Oz, p-adic holography from the hyperbolic fracton model, J. High Energy Phys. 08 (2025) 096.

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Phys. Rev. D 112, 086011– Published 17 October, 2025
  • Received 12 August 2025
  • Accepted 30 September 2025
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