From Diamond Gaugings to Dualisations

TL;DR

This work develops a Cartan-geometry formulation of gauged holomorphic Chern–Simons theory on twistor space to systematize gaugings across the twistor-space diamond. By decomposing the 6d connection into a Cartan piece and an unbroken gauge piece, it makes gauge covariance manifest and eliminates the need for ad hoc edge modes, with boundary terms arising naturally. Applying the framework to meromorphic pole data tied to λ-deformations, the authors show that gauging induces a flat-connection constraint in 2d, yielding non-Abelian dualisations of λ-models rather than ordinary gauged versions, and they work out explicit 4d and 2d reductions including a detailed SU(2)/U(1) example. The results clarify the higher-dimensional origin of these integrable models, reveal a Buscher-type duality structure, and point to rich future directions in dualities, boundary conditions, and quantum aspects within the twistor-space holography of integrable systems.

Abstract

We revisit the proposal that coupling two six-dimensional holomorphic Chern-Simons theories generates gaugings throughout the twistor-space diamond relating 6d hCS, 4d self-dual Yang-Mills, 4d Chern-Simons, and 2d integrable models. In previous work this mechanism was demonstrated only in a special case, leaving its general status unclear. By reformulating the construction in the language of Cartan geometry, we expose the underlying gauge structure and show that the argument extends to generic choices of meromorphic data. We then apply this to the pole structure that yields the well-studied $λ$-deformations of the WZW model. The coupled 6d system indeed induces gaugings of the associated $λ$-models, but necessarily introduces Lagrange multipliers enforcing flatness of the gauged connection. The resulting two-dimensional theories are therefore non-Abelian dualisations rather than ordinary gauged $λ$-models.

Figures (1)

• Figure 1: The diamond of theories spawned by \ref{['hCS6_action_A']}. The black straight arrows denote localization on $\mathbb{CP}^1$ while the red wavy ones dimensional reduction along certain directions in $\mathbb{R}^4$.