Contact 4d Chern-Simons theory: Generalities

We refine and generalize the results of e-Print: 2307.10428 [hep-th], where evidence in favor of applying the non-Abelian localization method to handle the 4d Chern-Simons theory path integral formulation was presented. We show, via duality manipulations and invoking some symplectic geometry results, both inspired by the Beasley-Witten work e-Print: 0503126 [hep-th], that the path integral of a regularized version of the 4d Chern-Simons theory, formally takes the canonical symplectic form required by the method of non-Abelian localization. The new theory is defined on a deformed quotient space and interpolates between the conventional 3d Chern-Simons theory on a Seifert manifold M e-Print: 0503126 [hep-th], trivially embedded into $\mathbb{R}\times \text{M}$, and the Costello-Yamazaki e-Print: 1908.02289 [hep-th] 4d Chern-Simons theory defined on the same 4d manifold. It is also shown that the regularized theory is consistent, following an idea of Beasley e-Print: 0911.2687 [hep-th], with the insertion of coadjoint orbit defects of the 1d Chern-Simons theory type. This approach opens the possibility for using exact path integral methods to explore the quantum integrable structure of certain 2d integrable sigma models of the non-ultralocal type, which are widely known to be somehow immune to the use of more traditional quantization methods, like the algebraic Bethe ansatz.

💡 Research Summary

The paper presents a systematic refinement and generalization of the 4‑dimensional Chern‑Simons (4d CS) theory originally introduced by Costello and Yamazaki, with the explicit aim of making the theory amenable to the powerful technique of non‑Abelian localization. The authors begin by reviewing the standard 4d CS action
(S_{\text{4d‑CS}} = i c \int_{M} \omega_{C} \wedge \text{CS}(A)),
where (M = \Sigma \times C) (with (\Sigma = \mathbb{R}\times S^{1}) a cylinder) and (\omega_{C}) is a meromorphic (1,0)‑form on the Riemann surface (C). This formulation enjoys a (1,0)‑shift symmetry that removes the (A_{z}) component, and the bulk and boundary equations of motion are well known. However, two obstacles prevent a direct application of non‑Abelian localization: (i) the theory is built on a complex, non‑compact gauge group, so the natural symplectic structure is not Kähler but pseudo‑Kähler; (ii) the boundary conditions derived from the original action do not match those required to reproduce the Lax connections of non‑ultralocal integrable field theories (IFTs).

To overcome these issues the authors introduce a regularized 4d CS theory. The key step is to replace the original background (\Sigma \times C) by a deformed quotient space (M = \mathbb{R} \times \mathcal{M}), where (\mathcal{M}) is a non‑trivial (S^{1})‑bundle over (C) with first Chern class (n\neq 0). A one‑form (\omega_{\zeta}) depending on a deformation parameter (\zeta) is introduced; in the limit (\zeta\to 0) it reduces to the original twist form (\omega_{C}). This replacement guarantees a non‑vanishing top‑form (\gamma_{\text{top}} \sim n\zeta, d\tau\wedge\alpha\wedge\pi^{*}\sigma_{C}), which is essential for defining a non‑degenerate inner product on the space of connections and for constructing a Hamiltonian group action.

The regularized action reads
(S_{\text{reg}} = i c \int_{M} \omega_{\zeta} \wedge \text{CS}(A)).
When (\zeta\to 0) the action reproduces the original Costello‑Yamazaki theory, ensuring that the physical content (e.g. the pole‑defect surface and associated 2d IFTs) is preserved.

Next, the authors perform a duality transformation that rewrites the regularized action in a purely quadratic, symplectic form. Introducing a symplectic two‑form (\hat\Omega) on the infinite‑dimensional space (\mathcal{A}) of gauge connections and a moment map (\mu) for the Hamiltonian action of the gauge group, they obtain the identity

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