Virasoro constraintsWe 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).