Structure of the obstacle

Some map always exists, the question is, can we pick a -invariant one?

Let us choose some . Any map can be thought of as a family of maps: parametrized by satisfying:  (1)  (2)  (3)

For each element there is a corresponding vector field on . Let us consider . It is a vector field on : (4)
Because of Eq. (2), the scaling degree of correlates with the power of . In particular, is a linear vector field on the linear space . This is the action of on . (The is a quadratic vector field: , and is qubic, etc..)

Can we choose so that ?

We would like to reformulate this question as a cohomological problem.

For two elements and of , we have: Let us introduce the “ghosts” and the “BRST operator”:    (5) This defines the differential in the Lia algebra cohomology complex of with values in the space of vector fields on having zero of at least second order at the point . (The action of the second term, , is by the commutator of vector fields.)

Let us define as the sum of nonlinear terms in ( cf. Eq. (4)): (6)
It satisfies the Maurer-Cartan equation: (7)

Ambiguity in the choice of translates to gauge transformations of : (8)

Obstacle to the existence of -invariant is a solution of the MC Eq. (7) modulo gauge transformations defined in Eq. (8)

In particular, if the first non-removable term is , then the corresponding obstacle is: 