Virasoro constraints

We introduce an “inner commutator” map :

With this notation .

Notice that is a nondegenerate symmetric scalar product, but this scalar product is not positive definite.

Let denote the conjugate to with respect to this scalar product. It turns out that .

On the other hand, is non-zero. In fact can be identified with the action of the quadratic Casimir of on : .

Note that on is nilpotent: . Therefore:
  • The space of linearized -deformations is .

In string theory (the Virasoro constraint).