Two-loop supergravity on AdS$_5\times$S$^5$ from CFT

TL;DR

This paper constructs the two-loop four-graviton supermultiplet amplitude in AdS$_5\times$S$^5$, revealing new integral structures, symmetries, and explicit anomalous dimension corrections, advancing the understanding of AdS/CFT correspondence at higher loops.

Contribution

It introduces a novel ansatz-based method for calculating two-loop amplitudes in AdS$_5\times$S$^5$, identifying zigzag integrals and crossing symmetries, and provides explicit anomalous dimensions including harmonic sums.

Findings

Confirmed Huang and Yuan's ansatz for the amplitude.

Identified zigzag integrals for leading logarithmic discontinuities.

Derived explicit two-loop anomalous dimensions with harmonic sums.

Abstract

We describe a construction of the two-loop amplitude of four graviton supermultiplets in AdS$_5\times$S$^5$. We start from an ansatz for a preamplitude from which we generate the full amplitude under the action of a specific Casimir operator. The ansatz captures a recent ansatz of Huang and Yuan and we confirm their result through similar constraints. The form of the result suggests that all ambiguities are captured by the preamplitude which determines the result up to tree-level ambiguities only. We identify a class of four-dimensional `zigzag' integrals which are perfectly adapted to describing the leading logarithmic discontinuity to all orders. We also observe that a bonus crossing symmetry of the preamplitude follows from the transformation properties of the Casimir operator. Combined with the zigzag integrals this allows us to construct a crossing symmetric function with the correct leading logarithmic discontinuities in all channels. From the two-loop result we extract an explicit expression for the two-loop correction to the anomalous dimensions of twist-four operators of generic spin which includes dependence on (alternating) nested harmonic sums up to weight three. We also revisit the prescription of the bulk-point limit of AdS amplitudes and show how it recovers the full flat-space amplitude, not just its discontinuity. With this extended notion of the bulk-point limit we reproduce the scale-dependent logarithmic threshold terms of type IIB string theory in flat-space.