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# Advances in Theoretical and Mathematical Physics

## Volume 21 (2017)

### Number 7

### Special Issue: Proceedings of the Strings 2016 Conference in Beijing

Guest Editors: J. Maldacena (Institute for Advanced Study), H. Ooguri (California Institute of Technology), H. Babak (Harvard University), S. Li (Tsinghua University), W. Song (Tsinghua University), and H. Lin (Tsinghua University)

### A $p$-adic version of AdS/CFT

Pages: 1655 – 1678

DOI: http://dx.doi.org/10.4310/ATMP.2017.v21.n7.a3

#### Author

#### Abstract

In this summary of my talk at Strings 2016, I explain how classical dynamics on an infinite tree graph can be dual to a conformal field theory defined over the $p$-adic numbers. An informal introduction to $p$-adic numbers is followed by a presentation of results on holographic three- and four-point functions. The simplicity of $p$-adic field theories and their similarity to ordinary field theories are illustrated through comparisons of holographic correlators and computations of simple loop diagrams on the field theory side. I close with a discussion of challenges and directions for future work.

I am grateful to the organizers of Strings 2016 for having put together a stimulating conference and for hosting me as a speaker. I particularly thank J. Knaute, S. Parikh, A. Samberg, and P. Witaszczyk for a stimulating collaboration. This work was supported in part by the Department of Energy under Grant No. DE-FG02-91ER40671.

Published 19 March 2018