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Amplitudes at Strong Coupling as Hyper-Kähler Scalars.

Hadleigh Frost1, Ömer Gürdoğan2, Lionel Mason1,3

  • 1The Mathematical Institute, University of Oxford, Woodstock Road, OX2 6GG, United Kingdom.

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|April 29, 2024
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Researchers explored the Alday-Maldacena conjecture, linking string theory and super-Yang-Mills (SYM) theory. They discovered the remainder function for minimal surfaces in AdS3 satisfies integrable equations, revealing a new pseudo-hyper-Kähler geometry.

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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.