Amplitudes at Strong Coupling as Hyper-Kähler Scalars.

Area of Science:

• High Energy Physics
• String Theory
• Quantum Field Theory

Background:

• The Alday-Maldacena conjecture proposes an equivalence between string amplitudes in AdS5×S5 and null polygonal Wilson loops in planar N=4 super-Yang-Mills (SYM) theory.
• At strong coupling, this conjecture equates SYM amplitudes with minimal surface areas in anti-de Sitter space.

Purpose of the Study:

• To investigate the remainder function for minimal surfaces in AdS3.
• To identify the mathematical structures governing these amplitudes at strong coupling.
• To connect these structures to pseudo-hyper-Kähler geometry and twistor theory.

Main Methods:

• Analysis of minimal surfaces in AdS3.
• Derivation of the Lax form for the integrable system satisfied by the remainder function.
• Development of a new perspective on "Y systems" to define a pseudo-hyper-Kähler structure on kinematic data.

Main Results:

• The nontrivial part of the amplitudes (remainder function) for minimal surfaces in AdS3 satisfies an integrable system of nonlinear differential equations.
• A new pseudo-hyper-Kähler structure is defined on the space of kinematic data via a twistor space derived from Y-system equations.
• The remainder function is identified as the (pseudo-)Kähler scalar for this geometry.

Conclusions:

• The study establishes a connection between string amplitudes, minimal surfaces, and integrable systems.
• A novel pseudo-hyper-Kähler geometry and its twistor theory are introduced, offering new insights into the AdS/CFT correspondence.
• This work provides a framework for extending conjectures on nonperturbative amplitudes using strong coupling structures.