We follow the notations in Bedoya:2010qz. The superconformal algebra has -grading:

Bars over subindices are to remind that they are mod 4. Geometrically, can be identified with the tangent space to the bosonic , which is the direct sum of the tangent space to and the tangent space to :

Therefore elements of are vectors from this tangent space. We can also consider the tangent space to the full superspace :

— this is a direct sum of three vector bundles. We parametrize a point in by modulo the equivalence relation:

We are identifying representations of , such as , , , with the corresponding vector bundles over the coset space (22). In fact, the worldsheet field takes values in the fibers of and takes values in the fibers of . The pure spinor conditions define the cones and :

Here denotes the anticommutator (the Lie superalgebra operation) of elements of . It should not be confused with neither the odd Poisson bracket, nor the even Poisson bracket corresponding to of Section Master Actions quadratic-linear in antifields. Again, we identify and as bundles over super-AdS. (They are not vector bundles, because their fibers are cones and not linear spaces.) We will denote:

where the prefix PS on the LHS stands for “Pure spinors” (and on the RHS for “Projective” and “Special”).

In Appendix The projector we construct -invariant surjective maps of bundles (“projectors”):

They are rational functions of and .

The action of the AdS sigma-model has the following form Berkovits:2000fe:

where are the -components of . We write instead of and instead of , just to highlight the -grading. (And also because neither is strictly speaking left-moving, nor is right-moving.) The covariant derivative is defined as follows:

Since and both satisfy the pure spinor constraints, the corresponding conjugate momenta are defined up to “gauge transformations”:

where and are arbitrary sections of the pullback to the worldsheet of . The BRST transformations are defined up to gauge transformations corresponding to the equivalence relation (23). It is possible to fix this ambiguity so that:

The first line in Eq. (25) is by itself not BRST invariant. Modulo total derivatives, its BRST variation is:

This cancels with the BRST variation of the second line in Eq. (25).

On the other hand, we observe that:

Notice that the projector drops out on the RHS because is automatically tangent to the cone. Comparing this to (31) we see that the following expression:

is BRST invariant. It does not contain neither derivatives of pure spinors, nor their conjugate momenta.

We define:

(See Appendix The projector for notations. We use the fact that .) These expressions satisfy (Appendix BRST variation of the -tensor):

Notice that:

and is diffeomorphism-invariant (and therefore degenerate!). The BRST invariance of can be verified explicitly as follows:

Consider the action of the BRST operator given by Eq (29) on . It is nilpotent only up to the -gauge transformation by . We have so far worked on the factorspace by gauge transformations. This means that we think of the group element and pure spinors as defined only modulo the gauge transformation:

It turns out that the action of these gauge transformations on the BV phase space is somewhat nontrivial, see Section Gluing charts. We will now just fix the gauge, postponing the discussion of gauge transformations to Section Gluing charts. Let us parametrize the group element by :

where , and , and impose the following gauge fixing condition:

Since Eq. (36) does not contain derivatives, this gauge is “ghostless”, the Faddeev-Popov procedure is not needed

. In this gauge fixed formalism, the BRST operator includes the gauge fixing term ( cp. Eqs. (28), (29), (30)):

where is some function of , and , defined by Eqs. (37) and (36); schematically This is usually called “the compensating gauge transformation”. It automatically satisfies:

Gauge fixing is only possible locally in . In order for our constructions to work globally, we will cover with patches and gauge-fix over each patch. Then we have to glue overlapping patches. We will explain how to do this in Section Gluing charts.

The BRST symmetry of the pure spinor superstring in is nilpotent only on-shell. More precisely, the only deviation from the nilpotence arises when we act on the conjugate momenta of the pure spinors:

(while the action of on the matter fields is zero even off-shell). This means that the BV Master Action contains a term quadratic in the antifields:

In this formula and stand for matter fields ( and ) and their antifields, and is given by Eq. (25). The matter fields are essentially and where with , , :

Their BRST transformation is read from Eq. (37). We observe that the action is of the same type as described in Section Master Actions quadratic-linear in antifields. The Poisson bivector is:

The 2-form discussed in Section Master Actions quadratic-linear in antifields can be choosen as follows:

The projector is needed to make invariant with respect to the gauge transformations (26) and (27). We take the following generating function satisfying Eq. (14):

The new “classical action” is given by Eq (32). (We will provide more details for a slightly more general calculation in Section Regularization.) It is, indeed, constant along the symplectic leaves of , as the fields are not present in this new Lagrangian at all. The new BV action is:

where runs over and the action of on is the same as it was in the original -model. The new BV phase space is smaller, it only contains . The BRST operator is now nilpotent off-shell; the dependence of the BV action on the antifields is linear. The fields enter only through their combination invariant under local rescalings (they enter through ). This in particular implies that the BRST symmetry is now a local symmetry.