Contents

### Context

#### Duality in string theory

duality in string theory

general mechanisms

string-fivebrane duality

string-string dualities

M-theory

F-theory

string-QFT duality

QFT-QFT duality:

# Contents

## Idea

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 $PGL(2,\mathbb{Q}_p)$:

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.)

## References

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

• Steven Gubser, Johannes Knaute, Sarthak Parikh, Andreas Samberg, Przemek Witaszczyk, $p$-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, $p$-adic fields, and algebraic curves: arithmetic and the $AdS_3/CFT_2$ correspondence (arXiv:1605.07639)

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

• Ling-Yan Hung, Wei Li, Charles M. Melby-Thompson, $p$-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 $p$-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