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:

For each element there is a corresponding vector field on . Let us
consider . It is a vector field on :

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”:

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)):

It satisfies the Maurer-Cartan equation:

Ambiguity in the choice of translates to gauge transformations of :

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