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.
Highlights
Recent work [9, 10] extended the use of p-adics in physics to a new arena: the AdS/CFT correspondence
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
The role of bulk geometry is played by the Bruhat-Tits tree — a (p + 1)-valent tree with symmetry group PGL(2, Qp) whose asymptotic boundary is Qp [11]
Summary
Recent reviews of the BT tree can be found in [1, 9, 10], so here we limit ourselves to those features most relevant to the current discussion. As with Archimedean CFT, the points of the p-adic bulk geometry are taken to be in one-to-one correspondence with the maximal compact subgroups of the conformal group PGL2(K). These subgroups are conjugates of PGL2(ZK), where ZK = {x ∈ K : |x| ≤ 1} is the ring of p-adic integers. In this description, z tells us the accuracy with which we must specify x, meaning the parametrization is not unique: (z, x) = (z, x ) whenever.
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