Hamiltonian Analysis of Doubled 4d Chern-Simons

TL;DR

This work analyzes the Hamiltonian structure of doubled $4d$ Chern–Simons theory with boundary–defect couplings that identify two gauge fields, deriving its Poisson algebra and showing it takes the form of an affine Gaudin model subject to a first-class coset constraint. A key result is that the bulk Poisson bracket realizes a Gaudin/Maillet-type structure, while boundary/defect data generate Lax representations for coset integrable field theories, generalizing the Goddard–Kent–Olive construction to this higher-dimensional setting. The paper develops two complementary approaches to handle boundary identifications—edge modes and a Dirac-principle defect-constraint reformulation—leading to a reduced phase space described by defect currents and gauge-invariant observables. It further conjectures the existence of extended quantum groups and a doubled affine Harish-Chandra-type isomorphism, outlining rich directions for quantization, Takiff-extensions at higher pole orders, and coset-type integrable hierarchies arising from doubled $4d$CS.

Abstract

Motivated by a conjecture that doubled four-dimensional Chern-Simons produces new integrable models, we perform its Hamiltonian analysis and find the theory's Poisson algebra. This requires carefully accounting for a set of boundary conditions that identify two gauge fields. Two methods for doing so are given, one of which is based on edge-modes and the other on a recharacterisation of the boundary conditions as constraints. We find that the Poisson algebra is that of an affine Gaudin model subject to a constraint, generalising the Goddard-Kent-Olive construction (from conformal field theory) to the world of integrable models. We also conjecture the existence of extended quantum groups and a generalisation of the affine Harish-Chandra Isomorphism.