p -adic CFT is a holographic tensor network
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Springer
Received: March 1, 2019 Accepted: April 16, 2019 Published: April 30, 2019
p-adic CFT is a holographic tensor network
a
State Key Laboratory of Surface Physics and Department of Physics, Fudan University, 220 Handan Road, 200433 Shanghai, P.R. China b Department of Physics and Center for Field Theory and Particle Physics, Fudan University, Handan Road, 200433 Shanghai, P.R. China c Institute for Nanoelectronic devices and Quantum computing, Fudan University, 200433 Shanghai, China d Institute of Theoretical Physics, Chinese Academy of Sciences, 100190 Beijing, P.R. China e Institut f¨ ur Theoretische Physik und Astrophysik, Julius-Maximilians-Universit¨ at W¨ urzburg, Am Hubland, 97074 W¨ urzburg, Germany
E-mail: [email protected], [email protected], [email protected] Abstract: The p-adic AdS/CFT correspondence relates a CFT living on the p-adic numbers to a system living on the Bruhat-Tits tree. Modifying our earlier proposal [1] for a tensor network realization of p-adic AdS/CFT, we prove that the path integral of a p-adic CFT is equivalent to a tensor network on the Bruhat-Tits tree, in the sense that the tensor network reproduces all correlation functions of the p-adic CFT. Our rules give an explicit tensor network for any p-adic CFT (as axiomatized by Melzer), and can be applied not only to the p-adic plane, but also to compute any correlation functions on higher genus p-adic curves. Finally, we apply them to define and study RG flows in p-adic CFTs, establishing in particular that any IR fixed point is itself a p-adic CFT. Keywords: AdS-CFT Correspondence, Conformal Field Theory ArXiv ePrint: 1902.01411
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP04(2019)170
JHEP04(2019)170
Ling-Yan Hung,a,b,c Wei Lid and Charles M. Melby-Thompsone
Contents 1
2 A review of p-adic CFT and the p-adic AdS/CFT correspondence 2.1 p-adic numbers and projective space 2.2 p-adic CFT 2.3 p-adic AdS/CFT 2.3.1 Bruhat-Tits tree 2.3.2 Bulk action principle 2.4 The wavefunction approach
3 3 4 6 6 7 8
3 p-adic CFT as a tensor network 3.1 p-adic CFT tensor networks on the Bruhat-Tits tree 3.2 The regularized generating functional 3.3 Sources and the AdS/CFT dictionary 3.4 Boundary correlation functions 3.4.1 2-point function 3.4.2 3-point function 3.4.3 4-point function and onwards
9 9 13 16 18 18 19 20
4 p-adic CFT at higher genus 4.1 Genus one 4.1.1 Genus-one partition function 4.1.2 Correlation functions at genus one 4.2 Cutting and sewing 4.3 Higher genus
22 23 23 24 25 27
5 Renormalization group flow 5.1 RG flows and fixed points in p-adic CFT 5.2 Example: the Ising fusion model
28 28 31
6 Discussion
34
1
Introduction
The field of rational numbers Q can be completed with respect to two different types of norms [2]: (1) the Euclidean or Archimedean norm, giving the real field R; and (2) the p-adic or non-Archimedean norm for any prime p, yielding the p-adic number field Qp . Much of the importance of p-adic numbers in mathematics is derived from the loc
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