Holographic correlators and scattering amplitudes in N=4 and beyond

Abstract

In the first part of this thesis we study string corrections to one-loop amplitudes of single-particle half-BPS operators O_p in AdS₅ × S⁵ . The tree-level correlators (dual to AdS scattering amplitudes via the AdS/CFT correspondence) in supergravity enjoy an accidental 10d conformal symmetry. Consequently, one observes a partial degeneracy in the spectrum of anomalous dimensions of double-trace operators and at the same time equality of many different correlators for different external charges p_i=1,2,3,4. The one-loop contribution is expected to lift such bonus properties, and its precise form can be predicted from tree-level data and consistency with the operator product expansion. Here we present a closed-form Mellin space formula for ⟨O_p1 O_p2 O_p3 O_p4 ⟩ at order λ^−3/2 in the expansion around large λ valid for arbitrary external charges p_i. Our formula makes explicit the lifting of the bonus degeneracy among different correlators through a feature we refer to as ‘sphere splitting’. While tree-level Mellin amplitudes come with a single crossing symmetric kernel, which defines the pole structure of the AdS₅ × S⁵ amplitude, our one-loop amplitude naturally splits the S⁵ part into two separate contributions. The amplitude also exhibits a remarkable consistency with the corresponding flat space IIB amplitude through the large p limit. In the second part of this thesis we study the relation between the branch cut structure of scattering amplitudes in planar N = 4 SYM and Grassmannian cluster algebras using the novel language of Gröbner theory. We detail how to extract the familiar A-coordinates and their respective adjacency conditions from the Gröbner fan of the Plücker ideal. Having established this connection we apply similar techniques to the case of non dual conformal invariant five-point kinematics where we extract the full non-planar symbol alphabet relevant for the construction of five-point integrals/amplitudes. Finally, we continue to study the connection between cluster algebras and scattering amplitudes by considering the family of partial flag cluster algebras F (2, 4, n) in order to extract information on the symbol alphabet for amplitudes with five-point and six-point non dual conformal invariant kinematics.

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