Introduction

In the low energy limit of superstring theory, spacetime fields satisfy supergravity (SUGRA) equations of motion, which are super-analogues of the Einstein equations. It is one of the main principles of string theory, that these target space equations of motion are equivalent to the BRST invariance of the string worldsheet theory. When they are satisfied, the space of fields is an infinite-dimensional -manifold (a manifold with an odd nilpotent vector field Alexandrov:1995kv). But in the case of pure spinor string, the sigma-model also defines a finite-dimensional -manifold. Indeed, the action of the BRST operator on matter fields and pure spinor ghosts does not contain worldsheet derivatives. (The worldsheet derivatives will appear when we consider the action on the conjugate momenta to matter fields and pure spinor ghosts, but they can be considered separately.) This means that, if we think of the pure spinor ghosts as part of target space, the BRST operator defines on the target space an odd nilpotent vector field, which we denote . In other words, the target space of the pure spinor sigma-model (a finite-dimenisional supermanifold) is a -manifold. Moreover, in generic space-time (for example in , but not in flat space-time) the energy-momentum tensor and the -ghost can also be interpreted as symmetric tensors on the target space (see Mikhailov:2017mdo).

How to classify a generic odd nilpotent vector field ? A vector field can usually be “simplified” by a clever choice of coordinates. This is called “normal form”. If a vector field is non-vanishing, one can choose coordinates so that the it is where is one of fermionic coordinates. If vanishes at some point, then the normal form would be (in the notations of Alexandrov:1995kv) . But in out case, the target space is not a smooth supermanifold, because pure spinor ghosts live on a cone. The vector vanishes precisely at the singular locus, and the problem of classification of normal forms is a nontrivial cohomological computation. This is what we will do in this paper. We will find that the space of equivalence classes of odd nilpotent vector fields in a vicinity of the singular locus is equivalent to the space of the classical SUGRA solutions. This is true modulo some “zero modes” — a finite-dimensional subspaces of soultions (see Mikhailov:2014qka) which we ignore in this paper.

Some details of our computations can be found in the HTML version of this paper.

    Definition of
    Our results
    Relation to partial -structures
    Divergence of a nilpotent vector field
        Requiring
        Vanishing of obstacles to nonlinear solution

Definition of

The particular singularity which we are interested in can be described as follows. Consider the space

with bosonic coordinates

(

running from 1 to 10) and

(

and

both running from 1 to 16), and fermionic

and

, with the constraint:

(1)

where

are ten-dimensional gamma-matrices. These constraints are called “pure spinor constraints”. We understand Eqs. (1) as specifying the singular locus in

, from the point of view of differential geometry. All we need from these equations is to know how

deviates from being smooth. The singular locus is the tip of the cone (1):

(2)

Pure spinor constraints (1) are invariant under the action of the group

(3)

The diagonal

(4)

is called “ghost number symmetry”. Infinitesimal ghost number symmetry is generated by

Consider an odd vector field

satisfying the following properties:

has ghost number 1, i.e.:
(5)
is “smooth” in the sense that it can be obtained as a restriction to the cone (1) of a smooth (but not nilpotent) vector field in the space parametrized by unconstrained
is zero at

We want to classify such vector fields modulo coordinate transformations. Coordinate transformations are supermaps

such that

satisfy the same constraints (1).

Such a vector field is one of the geometrical structures associated to the pure spinor superstring worldsheet theory Berkovits:2001ue,Guttenberg:2008ic. In particular, flat background (empty ten-dimensional spacetime) corresponds to

(6)

String worldsheet theory also has, besides

, some other structures which are less geometrically transparent (various couplings in the string worldsheet sigma-model). All these structures should satisfy certain consistency conditions.

Question: is it true, that just a nilpotent vector field already includes, as various coefficients in its normal form, all the supergravity fields, and the supergravity equations of motion are automatically satisfied (i.e. follow from )?

This may be false in two ways. First, it could be that some supergravity fields do not enter as coefficients in the normal form of

(i.e. they would only appear as some couplings in the sigma-model, but would not enter in

). Second, it could be that just

would not be enough to impose SUGRA equations of motion (i.e. one would have to also require the

-invariance of the worldsheet sigma-model action).

Our results

In this paper we will derive the normal form of

as a deformation of

(7)

Our analysis will be restricted to the terms linear in

(i.e.

). It turns out that

is parameterized by some tensor fields satisfying certain hyperbolic partial differential equations. These fields are in one-to-one correspondence with the fields of the Type II SUGRA, and our hyperbolic equations are the equations of motion of the linearized Type II SUGRA.

It is useful to compare to the pure spinor description of the super-Yang-Mills equations. The super-Yang-Mills equations are equivalent to having an odd nilpotent operator:

(8)

where

are generators of the gauge group, and

is vector potential. Zero solution corresponds to

. In this sense, the SYM solutions can be considered as deformations of the differential operator:

where the leading symbol (i.e. the derivatives) remains undeformed. Here we consider, instead, the deformations of the leading symbol.

Relation to partial -structures

The variables

and

parametrize the normal direction to the singularity locus

The first infinitesimal neighborhood is a bundle over

with the fiber

— the product of two cones. Filling the cones, we obtain a vector bundle over

with the fiber

. The vector field

is power series in

, with zero at the tip of

. The derivative of

at the zero locus defines a linear map:

This map is not an isomorphism, since the image of

only covers a

-dimensional subbundle of

. We can interpret

where

is a partial frame bundle of

and

is given by Eq. (3). It was shown in Mikhailov:2015sva that

defines a connection in a partial

-structure on

with some constraints on torsion, modulo some equivalence relation.

The relation to previous work on SUGRA constraints Nilsson:1981bn, Howe:1983sra, Witten:1985nt, Shapiro:1986xp, Chau:1988sm, Bergshoeff:1990mr, Bergshoeff:1991ei, Howe:1991bx, Howe:1991mf can be established along these lines.

Divergence of a nilpotent vector field

Let us fix some volume form

, an integral form on

. Then we can consider, for any vector field

, its divergence

. This is by definition the Lie derivative of

along

In particular, for a nilpotent odd vector field

, we can consider the cohomology class:

This cohomology class does not depend on the choice of the volume form

The cohomology class plays two importan roles in our approach. First, they allow to reduce the study of to the first infinitesimal neighborhood of the singularity locus given by Eq. (2). Second, it allows to prove that there is no obstacle to extending the linearized deformations (the term in Eq. (7)) to higher orders in . We will now explain this.

Requiring

We required that

has ghost number one, see Eq. (5). But the “ghost number”

is coordinate-dependent. It is not invariant under a change of coordinates:

In fact, we can relax Eq. (5) by replacing it with the following two requirements:

(9)

(10)

Indeed Eq. (9) implies that

is a vector field of ghost number one plus vector fields of ghost numbers two and higher. Although we have not computed

, the results of Mikhailov:2014qka suggest that

is an isomorphism, and

is zero modulo finite-dimensional spaces. Then, Eq. (10) implies that the terms of the ghost number higher than one in

can be removed by the coordinate redefinition. In other words, the normal form of

can be always choosen to be of the ghost nuber one. It is enough to study the first infinitesimal neighborhood of the singularity locus.

Vanishing of obstacles to nonlinear solution

It is necessary to extend this analysis to full nonlinear SUGRA equations, i.e. higher order terms in Eq. (7). The potential obstacle to extending linearized solutions to the solution of the nonlinear equation lies in . We will not compute in this paper, but the results of Mikhailov:2014qka suggest that is actually nonzero. (We would expect it to be roughly isomorphic to which we compute here.) But we also know that the actual obstacle is zero, because of the consistency of the nonlinear supergravity equations of Berkovits:2001ue. This means that must be a coboundary. In our language, this can be proven in the following way. Let us choose so that the divergence of is zero. The divergence of , and therefore of , is -exact (this statement does not depend on the choice of ). This is because has ghost number 1. The cohomology at ghost number 1 is finite-dimensional, and in fact those with nonzero are non-physical (see Mikhailov:2014qka and references there). At the same time, the divergence of the elements of is nonzero. Therefore the obstacle actually vanishes.

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