Hamiltonian analysis of an Effective action coupled to the Palatini action in 4 dimensions

We carry out the canonical analysis of the coupling of a $4$-dimensional effective action that arises from a dimensional reduction of a $7$-dimensional BF theory to the Palatini action in $4$-dimensions with cosmological constant, focusing on homogeneity and isotropy in all involved fields. We employ Ostrogradsky’s methodology because of the presence of second time derivatives. Through the analysis of consistency conditions, we identify three distinct and consistent possibilities. In all these scenarios, we explore the solution of second-class constraints in terms of pairs of canonical variables. Similarly, Hamiltonian analysis is performed only for the $4$-dimensional effective theory. The second-class constraints are solved in terms of canonical coordinates, revealing that this formulation aligns with one of the viable scenarios explored in the coupling to the $4$-dimensional Palatini action, entailing that the $4$-dimensional effective action nullifies gravity.