Unintegrated vertices

Definition 1: “Operators” are reasonably nice expressions built on
the sigma-model fields and their derivatives, modulo the -model classical equations of motion.

The sigma-model fields are typically matter fields and ghosts .
The ghosts satisfy the pure spinor constraints:

Definition 2: The conformal dimension of an operator is defined as the total number of derivatives, plus the number of .

The BRST operator , being a symmetry of the theory, acts on such expressions.

Definition 3: The ghost number of an operator is the total number of ’s minus the total number of .

Definition 4:
Unintegrated vertex is an operator of the conformal dimension zero
and ghost number two, annihilated by .
The gauge transformation of the vertex operator is:

Unintegrated operators form the cohomology of the BRST operator at the ghost number two.
This is, essentially, the Hilbert space of states in the worldsheet sigma-model

It is a fundamental principle of the string theory, that every state corresponds to some deformation of the worldsheet theory. This leads to the concept of integrated vertex, which we will now describe.