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This review connects Fay identities and Hirota equations in integrable systems using geometric language. It reformulates these concepts within Topological Recursion, enabling new solution constructions from Riemann surface geometry.

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Area of Science:

  • Integrable Systems
  • Mathematical Physics
  • Geometric Methods

Background:

  • Fay identities and Hirota equations are fundamental in the study of integrable systems.
  • Topological Recursion is a recent formalism offering new perspectives on discrete and continuous systems.
  • A geometric reformulation is needed to bridge these areas.

Purpose of the Study:

  • To review and reformulate the relationship between Fay identities and Hirota equations.
  • To establish a geometric language compatible with Topological Recursion.
  • To explore constructions of solutions using Riemann surface geometry.

Main Methods:

  • Reformulating Hirota equations as trans-series.
  • Expressing Fay identities as spinor functional relations.
  • Utilizing geometric constructions based on Riemann surfaces.

Main Results:

  • A unified geometric framework for Fay identities and Hirota equations within Topological Recursion.
  • Demonstration of Hirota equations as trans-series and Fay identities as spinor functional relations.
  • Recollection of methods for constructing solutions from Riemann surface geometry.

Conclusions:

  • The geometric reformulation provides a powerful lens for understanding integrable systems.
  • This approach facilitates the construction of new solutions to Fay/Hirota equations.
  • The connection to Topological Recursion opens avenues for further research in mathematical physics.