Abstract
We take the tensor network describing explicit p-adic conformal field theory partition functions proposed in [L.-Y. Hung et al., J. High Energy Phys. 04 (2019) 170JHEPFG1029-847910.1007/JHEP04(2019)170], and consider boundary conditions of the network describing a deformed Bruhat-Tits (BT) tree geometry. We demonstrate that this geometry satisfies an emergent graph Einstein equation in a unique way that is consistent with the bulk effective matter action encoding the same correlation function as the tensor network, at least in the perturbative limit order by order away from the pure BT tree. Moreover, the (perturbative) definition of the graph curvature in the mathematics [Y. Lin and S.-T. Yau, Tohoku Math. J. 63, 605 (2011)TOMJAM0040-873510.2748/tmj/1325886283; Y. Ollivier, J. Funct. Anal. 256, 810 (2009)JFUAAW0022-123610.1016/j.jfa.2008.11.001] and physics [S. S. Gubser et al., J. High Energy Phys. 06 (2017) 157JHEPFG1029-847910.1007/JHEP06(2017)157] literature naturally emerges from the consistency requirements of the emergent Einstein equation. This could provide new insights into the understanding of gravitational dynamics potentially encoded in more general tensor networks.
Highlights
High Energy Phys. 04 (2019) 170], and consider boundary conditions of the network describing a deformed Bruhat-Tits (BT) tree geometry. We demonstrate that this geometry satisfies an emergent graph Einstein equation in a unique way that is consistent with the bulk effective matter action encoding the same correlation function as the tensor network, at least in the perturbative limit order by order away from the pure BT tree
We present the emergence of a graph Einstein equation based on the proposed Tensor networks (TN) [20] of the p-adic AdS=conformal field theory (CFT) [21,22]
The bulk correlation functions hφaðxÞφbðyÞ Á Á Ái are defined as evaluation of the tensor network with the extra legs inserted at the appropriate vertices
Summary
Jxjp 1⁄4 p−v; ð2Þ which satisfies various axioms of norms [24]. Conformal symmetry is defined as the transformation x. Assuming aà 1⁄4 a and Cab1 1⁄4 δab, the edges need not be oriented, and the notations are less cluttered This assumption will be taken in the rest of the paper, but it is readily generalized to cases where a ≠ aà [20]. Two tensors at two vertices connected by an edge are contracted with the edge index weighted by p−Δa, where Δa is the conformal dimension of the corresponding primary labeled a. CFT and a dual bulk theory containing some bulk fields φa living on the BT tree in 1-1 correspondence with the primary operators. The bulk correlation functions hφaðxÞφbðyÞ Á Á Ái are defined as evaluation of the tensor network with the extra legs inserted at the appropriate vertices (see Fig. 1). One can readily show that these results are consistent with an emergent bulk matter field theory living on the BT tree [20,21], with action in the large mass limit [27]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
