String amplitudes in AdS and their limits

Thesis

Scattering amplitudes capture the dynamics and interactions of particles and strings, serving as powerful tools to test predictions and explore conjectures, while revealing deep symmetries and mathematical structures. Recent decades have seen dramatic advances in understanding and computing flat space amplitudes, revealing unexpected patterns. This thesis explores how similar structures emerge in curved backgrounds. In particular, we use the AdS/CFT correspondence to define on-shell amplitudes in AdS from conformal correlators in the boundary CFT, referred to as holographic correlators.

First, we bootstrap higher-point holographic correlators, neglecting stringy corrections. This allows us to probe deeper into the dual CFT. We present our results for the six-point supergluon amplitude in AdS₅, where the answer is fixed completely by the flat space limit and factorisation, and outline the strategy for the graviton case.

We then present the AdS Virasoro–Shapiro program, which addresses the challenge of computing string amplitudes in curved spacetimes, where no direct worldsheet formulation exists, even at tree-level. We study the tree-level amplitude of four gravitons in Type IIB superstring theory on AdS₅ × S⁵, dual to 4d N = 4 SYM. The AdS Virasoro-Shapiro amplitude is defined by a curvature expansion around flat space, organised as a series in 1/R, where R is the radius of AdS. Motivated by the appearance of odd zeta values in the low-energy expansion in flat space, a conjectural single-valuedness in AdS has guided recent advances. We contribute by analysing the high-energy and Regge limits, where exact computations confirm a worldsheet structure involving single-valued logarithms.

To conclude, we conduct a detailed analysis of the mathematical structure governing strings in AdS backgrounds, making the deep interplay with number theory explicit. A key result is an AdS generalisation of the celebrated stringy KLT relations, expressing closed-string amplitudes as bilinears of open-string ones via an exactly computable Kernel. We furthermore show that the building blocks for open-string amplitudes are given by Aomoto-Gelfand hypergeometric functions.

Authors: Nocchi, M

• DOI: 10.5287/ora-9oxrknemz
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: holographic correlators; number theory; scattering amplitudes; AdS/CFT
• Subjects: Mathematical physics