p-adic AdS/CFT correspondence



Duality in string theory

Arithmetic geometry



A version of the AdS/CFT correspondence (specifically: AdS3/CFT2 duality) for p-adic physics, obtained by passing on the conformal field theory-side from the complex geometry, hence (via GAGA)algebraic geometry of 2d CFT to arithmetic geometry over p-adic numbers, and on the string theory-side to p-adic string theory (?)

Here the arithmetic geometry-version of the bulk AdS spacetime is identified with the Bruhat-Tits tree for the projective general linear group PGL(2, p)PGL(2,\mathbb{Q}_p):

graphics from Casselman 14

This may be regarded (at some finite depth truncation) as a tensor network state:

graphics from Sati-Schreiber 19c

(As in HMSS 16, HLM 19. But maybe one wants the Poincaré-dual networks, instead, as in HMPS 18?)

There is the suggestion that BTZ black holes are encoded by networks that look like Bruhat-Tits trees towards the boundaries and like matrix product states towards the interior:

(As in ESZ 19.)


Suggestion to identify the Bruhat-Tits tree T pT_p with anti de Sitter spacetime in the p-adic AdS/CFT correspondence:

  • Steven Gubser, Johannes Knaute, Sarthak Parikh, Andreas Samberg, Przemek Witaszczyk, pp-adic AdS/CFT, Communications in Mathematical Physics volume 352, pages 1019–1059 (2017) (arXiv:1605.01061)

  • Steven Gubser, Sarthak Parikh, Geodesic bulk diagrams on the Bruhat-Tits tree (arXiv:1704.01149)

Relation to tensor networks:

  • Matthew Heydeman, Matilde Marcolli, Ingmar Saberi, Bogdan Stoica, Tensor networks, pp-adic fields, and algebraic curves: arithmetic and the AdS 3/CFT 2AdS_3/CFT_2 correspondence (arXiv:1605.07639)

  • Arpan Bhattacharyya, Ling-Yan Hung, Yang Lei, Wei Li, Tensor network and (pp-adic) AdS/CFT, JHEP 1801 (2018) 139 (arXiv:1703.05445)

  • Ling-Yan Hung, Wei Li, Charles M. Melby-Thompson, pp-adic CFT is a holographic tensor network (arXiv:1902.01411)

  • Lin Chen, Xirong Liu, Ling-Yan Hung, Emergent Einstein Equation in p-adic CFT Tensor Networks (arXiv:2102.12022)

  • Lin Chen, Xirong Liu, Ling-Yan Hung, Bending the Bruhat-Tits Tree I:Tensor Network and Emergent Einstein Equations (arXiv:2102.12023)

  • Lin Chen, Xirong Liu, Ling-Yan Hung, Bending the Bruhat-Tits Tree II: the p-adic BTZ Black hole and Local Diffeomorephism on the Bruhat-Tits Tree (arXiv:2102.12024)

  • Feng Qu, The effective field theory on a finite boundary of the Bruhat-Tits tree (arXiv:2103.02882)

Including spinors:

  • Steven Gubser, Christian Jepsen, Brian Trundy, Spin in pp-adic AdS/CFT, Journal of Physics A: Mathematical and Theoretical, Volume 52, Number 14 (arXiv:1811.02538)

Including BTZ black holes:

An expository account is given in

Proposed realization of aspects of p-adic AdS/CFT correspondence in solid-state physics:

  • Gregory Bentsen, Tomohiro Hashizume, Anton S. Buyskikh, Emily J. Davis, Andrew J. Daley, Steven Gubser, Monika Schleier-Smith, Treelike interactions and fast scrambling with cold atoms, Phys. Rev. Lett. 123, 130601 (2019) (arXiv:1905.11430)

Last revised on March 5, 2021 at 00:07:47. See the history of this page for a list of all contributions to it.