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

.