Abstract
We relate two formalisms recently proposed for describing classical integrable field theories. The first (Costello and Yamazaki in Gauge Theory and Integrability, III, 2019) is based on the action of four-dimensional Chern–Simons theory introduced and studied by Costello, Witten and Yamazaki. The second (Costello and Yamazaki, in Gauge Theory and Integrability, III, 2017) makes use of classical generalised Gaudin models associated with untwisted affine Kac–Moody algebras.
Highlights
Introduction and summaryIt was shown by Costello [11,12], and further developed recently in [13,14,48] by Costello, Witten and Yamazaki that various integrable lattice models can be understood as originating from a four-dimensional variant of Chern–Simons theory on the product M := ×C of a real two-dimensional manifold and a Riemann surface C equipped with a non-vanishing meromorphic 1-form ω
The purpose of this note is to show that the framework of [15] is intimately related to the description of classical integrable field theories that we proposed in [47], which is based on Gaudin models associated with untwisted affine Kac–Moody algebras
Recalling that we identified L = 4π i zin Sect. 3.1, it follows that the Hamiltonian (2.19) of four-dimensional Chern–Simons theory on the reduced phase space is given by a linear combination of the quadratic Gaudin Hamiltonians, explicitly
Summary
It was shown by Costello [11,12], and further developed recently in [13,14,48] by Costello, Witten and Yamazaki that various integrable lattice models can be understood as originating from a four-dimensional variant of Chern–Simons theory on the product M := ×C of a real two-dimensional manifold and a Riemann surface C equipped with a non-vanishing meromorphic 1-form ω. Very recently in [15], Costello and Yamazaki extended this approach to describe integrable field theories on , with spectral plane C, by starting from the same variant of Chern–Simons theory on × C as in [11,12,13,14,48].
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