Equivariant Master Equation
Let us reformulate. The equivariant Master Equation is:
is the odd moment map of
We should also require
-covariant dependence on
Deformations of solutions (integrated vertex operators)
Such solutions admit deformations, corresponding to the deformations of string background.
Infinitesimal deformations can be represented in the form:
satisfies the equation:
This is Cartan model. If we relax the
, and allow it to also depend on
then it is:
is the differential of the Weyl algebra of
Another point of view on the deformation complex
Instead of -invariant , let us consider arbitrary satisfying .
This is “unintegrated vertex operator”.
We want to construct from a closed PDF on .
The insertion procedure would require integration of this PDF over an appropriate cycle.
we would write something like:
But there is no
... We only have
. In bosonic
, and it works:
But will it work in general case, when
is nonlinear in
It turns out that the answer is “yes”, although the naive
(whatever it would mean) will not work. But actually exists
a generalization of the Kalkman formula which only requires a representation of
We know that is Maurer-Cartan (because the differential given
by Eq. (6)
is the differential in the deformation complex of the equivariant half-density!).
Then we consider the Kalkman embedding:
Let us put and .
Let us put
But, consider some vertex operator
which is not
will not contain any
, and will satisfy the Master Equation for
This is the same as to say:
In other words, we have constructed a closed differential form on
When we are inserting an unintegrated vertex
, we have to integrate this closed
form over an appropriate cycle in