Obstacle in terms of the -ghost

Remember our obstacle is a bilinear map:

#### 1` `Conjecture about the obstacle

where is defined in (4)

This means that when we deform the obstacle at the second
order in will be controlled by

#### 2` `Conjecture about the relation between and

is an isomorphism

#### 3` `Consistency of the action

From these conjectures follows that the actual obstacle vanishes. Indeed, is a derivation of . (But not of the product!) Therefore, when is physical (i.e. ) we automatically have . But the kernel of on the ghost number three cohomology is trivial. Therefore the obstacle actually vanishes.

#### 4` `Physical and nonphysical are mutually obstructed

where is the symmetry corresponding to the nonphysicality of .

This explains how it could be that the equation for the dilaton is consistent for almost all deformations, but becomes inconsistent on nonphysical. (“How come the obstacles to solving for form a finite-dimensional space?”) We now see that nonphysical deformations form a separate branch. |