Deformation complex
Equivariant Master Equation
Let us reformulate. The equivariant Master Equation is:
where
is the odd moment map of
.
We should also require
-covariant dependence on
.
We write:
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:
where
satisfies the equation:
This is Cartan model. If we relax the
-covarance of
, and allow it to also depend on
then it is:
where
is the differential of the Weyl algebra of
formed by
letters
and
(sometimes
is denoted
and
is denoted
).
The differential:
is nilpotent.
Another point of view on the deformation complex
Unintegrated vertices
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.
Following the
〚
Kalkman element〛
we would write something like:
But there is no
... We only have
. In bosonic
string
, and it works:
But will it work in general case, when
is nonlinear in
and
?
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
:
Derivation
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:
We have:
This implies:
Let us put 
and 
.
Let us put
and
.
But, consider some vertex operator
which is
not -invariant.
Our
will not contain any
and
, and will satisfy the Master Equation for
vertex operators:
Then:
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
.
,
is not minimal, because linearized diffrential is nonzero:
. We can turn it into a minimal algebra by adding a central element
and saying
,
. This is Koszul dual to the Lie superalgebra
.
; any representation of
, but not vice-versa.
.
