p-adic Mellin amplitudes

Jepsen, Christian Baadsgaard; Parikh, Sarthak

doi:10.1007/jhep04(2019)101

Cited by 20 publications

(56 citation statements)

References 68 publications

(193 reference statements)

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“…We note that consistent with Ref. [4], we are using non-standard normalizations for the bulk-to-boundary and bulk-to-bulk propagators, so that the p-adic and real bulk-to-bulk 7 propagators satisfy the equations 4 Q p n □z 0 ,z +m 2 ∆ G ∆ (z 0 , z; w 0 , w) = 2ν ∆ ζ p (2∆ − 2h)δ(z 0 , z; w 0 , w) ,…”

Section: Mellin Amplitudes and Pre-amplitudessupporting

confidence: 93%

“…leading also to N (N − 3)/2 independent components. The integration measure in (2.12) is over the independent Mellin variables, 14) and the precise contour prescription is similar in both cases, with the main distinction being that the Mellin variables in p-adic Mellin space live on a "complex cylinder", R × S 1 where the imaginary direction is periodically identified: γ ij ∼ γ ij + iπ log p [4]. This distinction arises for the following reason.…”

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confidence: 99%

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Recursion relations in p -adic Mellin Space

Jepsen¹,

Parikh²

2019

J. Phys. A: Math. Theor.

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In this work, we formulate a set of rules for writing down p-adic Mellin amplitudes at tree-level. The rules lead to closed-form expressions for Mellin amplitudes for arbitrary scalar bulk diagrams. The prescription is recursive in nature, with two different physical interpretations: one as a recursion on the number of internal lines in the diagram, and the other as reminiscent of on-shell BCFW recursion for flat-space amplitudes, especially when viewed in auxiliary momentum space. The prescriptions are proven in full generality, and their close connection with Feynman rules for real Mellin amplitudes is explained. We also show that the integrands in the Mellin-Barnes representation of both real and p-adic Mellin amplitudes, the so-called pre-amplitudes, can be constructed according to virtually identical rules, and that these pre-amplitudes themselves may be re-expressed as products of particular Mellin amplitudes with complexified conformal dimensions.

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Section: Mellin Amplitudes and Pre-amplitudessupporting

confidence: 93%

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confidence: 99%

Recursion relations in p -adic Mellin Space

Jepsen¹,

Parikh²

2019

J. Phys. A: Math. Theor.

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“…(∆a−n/2+1) k , and summing over all k, we recover (65). Moreover, we note that (66) is more general than (63); thus proving (66) indirectly furnishes a proof for the inverse relation (30) for arbitrary n.…”

Section: (∆A) 2kmentioning

confidence: 82%

Holographic dual of the five-point conformal block

Parikh

2019

J. High Energ. Phys.

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109

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We present the holographic object which computes the five-point global conformal block in arbitrary dimensions for external and exchanged scalar operators. This object is interpreted as a weighted sum over infinitely many five-point geodesic bulk diagrams. These five-point geodesic bulk diagrams provide a generalization of their previously studied fourpoint counterparts. We prove our claim by showing that the aforementioned sum over geodesic bulk diagrams is the appropriate eigenfunction of the conformal Casimir operator with the right boundary conditions. This result rests on crucial inspiration from a much simpler p-adic version of the problem set up on the Bruhat-Tits tree.

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“…The three p-adic propagator identities (2.8), (2.9), and (2.10), originally given in Ref [72]. and found by direct computation on the Bruhat-Tits tree, can also be derived in a manner parallel to the computations over the reals shown in this appendix using the p-adic Schwinger-parametrization and Mellin representation developed in Ref [108]. (though various infinite series encountered in the following calculations get collapsed to just the leading term of the series).…”

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confidence: 98%

“…Also, owing to the periodicity of ζ p in the imaginary direction, in the p-adic case the complex variables c j are not integrated over a line in the complex plane but along a contour that wraps around a cylindrical manifold with circumference π/ log p; see Ref [108]. where the necessary p-adic split representation was first worked out.…”

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confidence: 99%

Propagator identities, holographic conformal blocks, and higher-point AdS diagrams

Jepsen

Parikh

2019

J. High Energ. Phys.

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Conformal blocks are the fundamental, theory-independent building blocks in any CFT, so it is important to understand their holographic representation in the context of AdS/CFT. We describe how to systematically extract the holographic objects which compute higherpoint global (scalar) conformal blocks in arbitrary spacetime dimensions, extending the result for the four-point block, known in the literature as a geodesic Witten diagram, to five-and six-point blocks. The main new tools which allow us to obtain such representations are various higher-point propagator identities, which can be interpreted as generalizations of the well-known flat space star-triangle identity, and which compute integrals over products of three bulk-to-bulk and/or bulk-to-boundary propagators in negatively curved spacetime. Using the holographic representation of the higher-point conformal blocks and higher-point propagator identities, we develop geodesic diagram techniques to obtain the explicit directchannel conformal block decomposition of a broad class of higher-point AdS diagrams in a scalar effective bulk theory, with closed-form expressions for the decomposition coefficients. These methods require only certain elementary manipulations and no bulk integration, and furthermore provide quite trivially a simple algebraic origin of the logarithmic singularities of higher-point tree-level AdS diagrams. We also provide a more compact repackaging in terms of the spectral decomposition of the same diagrams, as well as an independent discussion on the closely related but computationally simpler framework over p-adics which admits comparable statements for all previously mentioned results.

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p-adic Mellin amplitudes

Cited by 20 publications

References 68 publications

Recursion relations in p -adic Mellin Space

Recursion relations in p -adic Mellin Space

Holographic dual of the five-point conformal block

Propagator identities, holographic conformal blocks, and higher-point AdS diagrams

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