Lax integrability and holomorphic-topological gauge theory (Lecture 1)

Abstract: The Lax formalism provides a powerful and unifying framework for describing classical integrable field theories in various spacetime dimensions. Its central object, the Lax matrix, depends on the spacetime coordinates and meromorphically on an auxiliary complex variable known as the spectral parameter.

In a series of recent seminal works, Costello, Witten and Yamazaki have shown that the Lax formalism admits a natural and elegant geometric origin in higher-dimensional holomorphic-topological gauge theory. In this setting, the spectral parameter is incorporated into the spacetime geometry and the Lax matrix arises as a specific component of the gauge field.

In these lectures I will give an introduction to this connection between the Lax formalism and holomorphic-topological gauge theories.

Lecture 1 - (1d IFTs) Lax pairs encode the integrable structure of finite-dimensional integrable systems, i.e. 1-dimensional integrable field theories, such as the closed Toda chain or the Gaudin model on a product of coadjoint orbits. After reviewing this formalism, I will explain how the framework of spectral parameter dependent Lax pairs naturally emerges from 3-dimensional holomorphic-topological BF theory.

Lecture 2 - (2d IFTs) Lax connections are an affine generalisation of Lax pairs which encode the integrable structure of 2-dimensional integrable field theories. I will review their deep connection to affine Gaudin models in the Hamiltonian formalism and explain how 4-dimensional holomorphic-topological Chern-Simons theory captures the same structure from a Lagrangian perspective.

Lecture 3 - (≥ 3d IFTs) In 3 dimensions and above there is no general, universally accepted definition of integrability. I will explain how the framework of holomorphic-topological gauge theories in 5-dimensions and above can be used as a guiding principle for formulating appropriate higher-dimensional analogues of Lax integrability. In particular, I will introduce 5-dimensional holomorphic-topological 2-Chern-Simons theory as a potential higher gauge-theoretic framework for describing 3-dimensional integrable field theories.

high energy physicsmathematical physicsnonlinear sciences

Audience: researchers in the topic

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Nagoya IAR workshop on Unification of Integrable Systems

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