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: (6)
is nilpotent.

Weyl algebra , 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 .

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:

Bosonic string:  Remember the 〚Kalkman element〛 required a representation of ; any representation of gives a representation of , but not vice-versa.

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 : (7)
 where 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: (8)
This implies: Remember that .

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 .