On this page:
Equivariant Master Equation
Deformations of solutions (integrated vertex operators)
Another point of view on the deformation complex
Unintegrated vertices

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.

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 :




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:

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:


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 .