Integrated vertex operators

It is interesting to follow what happens when we deform .

In string theory, such deformations correspond to integrated vertex operators.

or equivalently: 


← local operator of conformal dimension

In order for this new to satisfy the Master Equation to the first order in we require:


Such deformations correspond to deforming the target space. A natural question is:

as we deform , what happens to our subspace ?

It turns out that we also have to deform . For every we have to find such that:



deformation of

deformation of

symmetry undeformed

Moreover, the quadratic function defined by the equation also deforms:

Remember that the construction of equivariant form requires solving the equation:

The solution deforms: 

linear in

quadratic in

And the string measure deforms:

In the base form we substitute for the curvature of the connection — the 2-form. Therefore:
when we turn on the vertex operator, we should also deform the 2-form part of the measure
If the theory has ghost number, then has ghost number zero and has ghost number .