nLab Bruhat-Tits tree

Idea

(…)

Examples

The Bruhat-Tits tree of $SL(2)$

The Bruhat-Tits tree of the special linear group $SL(2)$:

graphics from Casselman 14

In the 2-adic AdS/CFT correspondence this tree plays the role of anti de Sitter spacetime. This may be regarded (at some finite depth truncation) as a tree-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?)

Related concepts

• building

• p-adic AdS/CFT correspondence

• tensor network state, holographic entanglement entropy

References

General

• Bill Casselman, The Bruhat-Tits tree of $\mathrm{SL}(2)$, 2014 (pdf, pdf)

In $p$-adic string theory

Suggestion that the disk worldsheet of the open p-adic string is to be identified with the Bruhat-Tits tree $T_p$:

• Anton Zabrodin, Non-Archimedean strings and Bruhat-Tits trees, Comm. Math. Phys.

  Volume 123, Number 3 (1989), 463-483 (euclid.cmp/1104178891)

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)

Discussion of Bruhat-Tits-tree tensor networks in this context of the p-adic AdS/CFT correspondence:

• 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)