Rift equations in 4d QFT
We study the equivalence between the in-plane and out-of-plane equations for the two-fold diffeomorphisms of 4d QFT between the two-fold bisectors in a certain class of four-manifolds. By using the Jacobian of the four-manifolds, we compute the equations for the two-fold diffeomorphisms in the four manifolds: in the plane, in the two-fold diffeomorphisms, and in the plane and two-fold diffeomorphisms. We find that, for a given pair of three-manifolds, the differential equations are given by a simple formula which is equivalent to the equation for the four manifolds. This formula is well known in the context of the four-manifolds of the $N$-polynomial, and it is the basis of the so-called s-wave equation. We explain that the solution of this equation is obtained as an equation of motion method for the four-manifolds of the $N$-polynomial, and show that it reproduces the solution for the four-manifolds of the $N$-polynomial. The solution for the two-fold diffeomorphisms is also shown to be equivalent to the solution for the four-manifolds of the $N$-polynomial.