Abstract
In this paper, we propose a p-adic analog of Mellin amplitudes for scalar operators, and present the computation of the general contact amplitude as well as arbitrary-point tree-level amplitudes for bulk diagrams involving up to three internal lines, and along the way obtain the p-adic version of the split representation formula. These amplitudes share noteworthy similarities with the usual (real) Mellin amplitudes for scalars, but are also significantly simpler, admitting closed-form expressions where none are available over the reals. The dramatic simplicity can be attributed to the absence of descendant fields in the p-adic formulation.
Highlights
In this paper, we propose a p-adic analog of Mellin amplitudes for scalar operators, and present the computation of the general contact amplitude as well as arbitrarypoint tree-level amplitudes for bulk diagrams involving up to three internal lines, and along the way obtain the p-adic version of the split representation formula
Invariant cross-ratios constructed out of the boundary insertion points xi, the Mellin amplitude M depends on Mandelstam-like invariants defined in terms of the Mellin variables γij — the number of conformally invariant cross-ratios in position space matches the number of independent Mandelstam-like variables in Mellin space
Given the important role Mellin amplitudes have played in the usual AdS/CFT correspondence and the similarities between the position space correlators in the p-adic and real formulations of holography, it is natural to ask whether p-adic versions of Mellin space and Mellin amplitudes exist and whether they can prove as fruitful in the context of p-adic AdS/CFT
Summary
In the standard AdSn+1/CFTn formulation, to any N -point position space amplitude A({xi}) there corresponds a Mellin amplitude M, which is a function of complex Mellin variables γij, with indices i and j running from 1 to N. (Over the reals, the “fundamental domain” is the entire complex plane, and the contours run parallel to the imaginary axis from −i∞ to i∞, which curiously corresponds to taking the p → 1 limit in the p-adic formulation.7) Just like in eq (1.6), the contours are placed so that they separate out poles arising from different factors of the local zeta functions.
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