A generalized 4d Chern-Simons theory

A generalization of the 4d Chern-Simons theory action introduced by Costello and Yamazaki is presented. We apply general arguments from symplectic geometry concerning the Hamiltonian action of a symmetry group on the space of gauge connections defined on a 4d manifold and construct an action functional that is quadratic in the moment map associated to the group action. The generalization relies on the use of contact 1-forms defined on non-trivial circle bundles over Riemann surfaces and mimics closely the approach used by Beasley and Witten to reformulate conventional 3d Chern-Simons theories on Seifert manifolds. We also show that the path integral of the generalized theory associated to integrable field theories of the PCM type, takes the canonical form of a symplectic integral over a subspace of the space of gauge connections, turning it a potential candidate for using the method of non-Abelian localization. Alternatively, this new quadratic completion of the 4d Chern-Simons theory can also be deduced in an intuitive way from manipulations similar to those used in T-duality. Further details on how to recover the original 4d Chern-Simons theory data, from the point of view of the Hamiltonian formalism applied to the generalized theory, are included as well.