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.