Supergravity in AdS space

Holographic renormgroup

Consider Type IIB SUGRA in and supersymmetric Yang-Mills on the boundary. We can proceed in two ways, which are equivalent because of the AdS/CFT duality:

- Renormgroup flow on the boundary We choose some map from the space of linearized deformations of the SYM theory to the space of finite deformations. There is no way to fix such a map preserving , so we want to study the deviation from -invariance, in the context of Section Obstacles to linearization of symmetry.

- Classical solutions of SUGRA in the bulk We fix some map from the space of solutions of the linearized SUGRA equations to the space of nonlinear solutions. Then we study the deviation of this map from being -invariant as in Section Obstacles to linearization of symmetry.

Here we will discuss this second approach. In fact, is a subalgebra of a larger superalgebra, the superalgebra of gauge transformations of supergravity. This requires a generalization of the formalism of Section Obstacles to linearization of symmetry which we will describe in Section Geometrical abstraction.

Gauge transformations of supergravity

The precise description of the gauge transformations of supergravity depends, generally speaking, on the formalism. Any theory of supergravity necessarily includes the group of space-time super-diffeomorphisms (= coordinate transformations), as gauge transformations. Those theories which have B-field should also include gauge transformation of the B-field. In the case of bosonic string they have been recently discussed in Schulgin:2014gra. In bosonic string, gauge transformations correspond to BRST exact vertices Schulgin:2014gra.

In the pure spinor formalism, we are not aware of any reference discussing specifically gauge transformations. Apriori the coordinate transformations should also include transformations of pure spinor ghosts; they are in fact gauge-fixed Berkovits:2001ue.

Our approach to holographic renormgroup is based on the study of the normal form of the action of the group of SUGRA gauge transformations in the vicinity of the AdS solution. We will now develop a geometrical abstraction for that. The construction parallels Section Geometrical abstraction, the main difference being that instead of a point we have to consider a degenerate orbit . We will now explain the details.

Geometrical abstraction

Consider a Lie supergroup acting on a supermanifold with a subgroup preserving a point . Moreover, we will assume that:

Let denote the orbit of :

Introduce the coordinates on , and coordinates in the transverse direction to . Consider the normal bundle of :

The action of on induces the action of on . Let denotes the formal neighborhood of in . It also comes with an action of .

Consider the following question: can we find a family of maps, parameterized by :

commuting with the action of ?

It turns out that the obstacle exists already at the linearized level. Normal bundle is not the same as the first infinitesimal neighborhood. The space of functions on is not the same, as a representation of , as the space of functions on constant-linear on fibers — see Eq. (53) below.

We will now proceed to the study of the obstacles to the existence of .

Normal form of the action of

Locally near the point , the normal bundle to can be trivialized. Let denote the coordinates on , and coordinates in the fiber. Then, the equation of is:

General vector field tangent to

Let us fix some satisfying Eqs. (47), (48), (49).

Since acts on , every element of the Lie algebra defines a vector field on .

Then, is a vector field on :

The power of correlates with the degree in -expansion. As an abbreviation, we will omit in the following formulas.

Big Maurer-Cartan equation

Let us introduce the Faddeev-Popov ghosts for . Let us denote:

Reduction to

We want to reduce from to , for the following reasons:

Therefore, we will now investigate the reduction to . We will start by concentrating on the vicinity of the point .

Let us concentrate on the vicinity of the point . Suppose that at the point : . Again, we will separate into linear and non-linear part. For (i.e. the stabilizer of ), , therefore:

The tensor defines a cocycle — an element of . Its class in is the obstacle to finding a -invariant map . If this obstacle is zero, then by a linear coordinate redefinition we can put , i.e. . In fact, we want to remove all , not only . The first non-vanishing coefficient defines a class in:

If nonzero, this is a cohomological invariant. It is not clear to us how to interpret such invariants on the field theory side. It was proven in Mikhailov:2011si that the sequence (53) splits for of large enough spin. In Section Description of beta-deformation in pure spinor formalism we will prove that it splits for beta-deformation, which is the representation of the smallest nonzero spin. Motivated by these observations, we will assume in the following discussion that (53) always splits. Moreover, we assume that all the obstacles in (54) vanish. (Not that the whole cohomology group vanishes, but the actual invariant is zero.) If true, this is a nonlinear analogue of the covariance property of the vertex discussed in Mikhailov:2011si.

Assuming that there are no obstacles of the type Eq. (54)

and

If the obstacles of the type Eq. (54) are all zero up to some value of , then we can remove all with , so we are left with:

Geometrically this means that we can find a -invariant submanifold which intersects transversally at the point . It is given by the equation . In this case, for defines a vector field on , and a solution of the Maurer-Cartan equation. Let be the Faddeev-Popov ghost for . Restriction of to is:

We then observe that satisfies the Maurer-Cartan equation:

Is an invariant? Deformations of .

Therefore, an action of on the vicinity of defines some solution of the Maurer-Cartan Eq. (55). A gauge transformation corresponds a change of coordinates on .

But is it true that modulo such gauge transformations is an invariant? Generally speaking the answer is negative, for the following reason: may be deformable, different choices of leading to different , apriori not related by a gauge transformations of the form .

Therefore, we have to study the deformations of .

Geometrical picture

Deformations of a -invariant submanifold are given by -invariant sections of its normal bundle . An exact sequence

The space can be naturally identified with the tangent space to the space of solutions of the Maurer-Cartan Eq. (55) modulo gauge transformations:

Therefore, when , solution of MC equation depends on the choice of . This means that we have, generally speaking, a family of invariants.

In coordinates

Remember that is given by the equations:

Infinitesimal deformations of can be obtained as fluxes by vector fields of the form

(57) where commute with . In other words, the tensor field defines an element of . The flux of such vector fields preserves the condition that for . To keep for , we need to add to ((57)) some terms of higher order in (the in Eq. ((57))); the existence of such terms depends on the vanishing of .

For example, consider a vector field :

where is a -invariant tensor in . The commutator contains terms:

and under the assumption of vanishing exists :

Therefore we can construct, order by order in -expansion, a vector field:

defining a -invariant section of the normal bundle of , and therefore a deformation of as a -invariant submanifold. The corresponding deformation of is:

We do not see any apriori reason why this could be absorbed by a gauge transformation, i.e. why would exist such that .

Conclusion

We have studied the problem of classifying the normal forms of the action of a Lie supergroup in the vicinity of an orbit with nontrivial stabilizer. As invariants of the action, we found families of equivalence classes of solutions of MC equations modulo gauge transformations. These families are parameterized by -invariant submanifolds .

Remember that is a finite-dimensional Lie superalgebra, while is infinite-dimensional, and therefore is infinite-dimensional. One would think that the deformations of “along ” will be “as complicated as ”. That would be ugly. But in fact, the space of deformations of is “no more complicated than ”. For examle, when is finite-dimensional, the space of deformations is also finite-dimensional at each order in . Roughly speaking, at the order , it is . Even though may be non-zero, it is certainly nicer than .

In other words, the deformations of do not involve the dependence on all , but only on finite-dimensional subspaces, the images of intertwining operators .

Pure spinor formalism

Here we will briefly outline the pure spinor implementation of supergravity in the vicinity of AdS. We use the notations of Mikhailov:2011si. Let be the subalgebra preserving a point in . In the pure spinor formalism, the space of vertex operators (cochains) of ghost number transforms in an induced representation of :

where is the space of homogeneous polynomials of the order on the pure spinor variable. The space of linearized SUGRA solutions corresponds to the cohomology at ghost number :

This defines an extension of with :

The extension is nontrivial iff defines a nontrivial class in . This class is an obstacle to the existence of a covariant vertex.

We have seen (Section Reduction to ) that, more generally, are obstacles to finding a -invariant gauge for nonlinear solutions. We will now show that , although nonzero, is in some sense small.

Notice that fits in a short exact sequence or representations, with the corresponding long exact sequence of cohomologies:

Shapiro’s lemma implies that . For example, for we have:

because is semisimple as a representation of , and . Therefore:

Notice that is in the following exact sequences (the pure spinor cohomology at ghost number one, , corresponds to the global symmetries .):

BRST exact vertices of ghost number one, i.e. , fit in the following exact sequences:

These observations together imply that fits into a short exact sequence of linear spaces:

This means that is a direct sum:

Notice that is a factorspace of a subspace of . In this sense, is “lesser than”:

This is the “upper limit” on the cohomology group containing the obstacle to the existence of at the order in -expansion.

When In particular corresponds to the obstacle to the existence of a covariant vertex, i.e. to the splitting of the short exact sequence of Eq. (58). In Section Description of beta-deformation in pure spinor formalism we will show that the actual obstacle is zero in the case of linearized beta-deformation (which is the representation with smallest spin). At the same time, results of Mikhailov:2011si imply that the obstacle is zero for linearized solutions with large enough spin. The conjectured existence of covariant vertices is the interpolation between these two cases.

If does not exist

We do not have a proof that the obstacle in defined in Section Reduction to is actually zero. If it is not zero, then we cannot restrict to (because does not exist). Then we must study the full expansion of of Eq. (52), in powers of both and .

However, we do not need to take into account all . We only need those which represent the obstacle in . Therefore, the complication is not actually as bad as it could have been. The main point is that, even though seems an ugly infinite-dimensional space, its cohomologies are reduced to expressions like (59) by the “magic” of the Shapiro’s lemma.

In the rest of this Section we will accept as a working hypothesis that exists. Also, we will leave open the question of non-uniqueness of .

Normalizable SUGRA solutions “Normalizable” means decreasing sufficiently rapidly near the boundary.

All linearized normalizable SUGRA solutions are periodic in the global time . They approximate some complete (nonlinear) solutions. The nonlinear solutions are not periodic. But, since linearized solutions are periodic, we can define the monodromy transformation as in Section Monodromy transformation.

The space of normalizable ( i.e. rapidly decreasing at the boundary) solutions has a symplectic form. This is true at the linearized level as well as for the non-linear solutions. Then we can choose a map so that it preserves the symplectic structure

. Therefore, we can now identify with the corresponding Hamiltonian, which we denote , or just . The MC equation (13) becomes:

Remember that is of cubic and higher order in the coordinates on .

Given the monodromy matrix of Eq. (17) we can (in perturbation theory) define a vector field such whose flux generates it:

It has some Hamiltonian which is of cubic and higher order in the coordinates on . In some sense, the quantization of should give the spectrum of anomalous dimensions. This program is complicated, though, by a non-straigthforward action of the symmetry, see Section Symmetries.

Non-normalizable SUGRA solutions The non-normalizable SUGRA solutions correspond to the deformations of the boundary theory. Consider the following element of — the symmetry group of :

Suppose that our deformation is invariant under :

Transformation , as it acts on the boundary of AdS

Indeed, let us consider the retarded wave generated by an insertion of some operator at the point on the boudary. It is the same as retarded boundary-to-bulk propagator. It is a generalized function with support on the future light cone of . Let be a light-like vector encoding the point on the boundary. Then the boundary-to-bulk propagator, in sufficiently small neighborhood of is, schematically:

where with corresponds to a point inside AdS; is the conformal dimension of . The future light cone gets re-focused at . The free solution (61) gets then reflected from the boundary at the point , and when reflected changes sign. Therefore, in order to cancel the reflection, we have to put the same operator at the point .

However, the corresponding nonlinear solution may or may not be periodic. If it is not periodic, then the deviation from periodicity is characterized by the monodromy of Eq. (17). In any case, the solution of the Maurer-Cartan equation is more fundamental than the monodromy transformation.

Simplest non-periodic linearized solution

(This subsection is a side remark.)

As we mentioned in Section Normalizable SUGRA solutions, all normalizable linearized solutions are periodic in global time .

But of course, this is not true for non-normalizable solutions. (Indeed, nothing prevents us from considering non-periodic boundry conditions at the boundary of AdS. There exist corresponding solutions, which are not periodic.) As a simplest example, consider the dilaton linearly dependent on :

This is a solution of SUGRA only at the linearized level. Indeed, the energy is nonzero, and it will deform the metric. It would be interesting to see if it approximates some solution of nonlinear equations with the following property: the action of on it is the shift of dilaton.

If we act on of Eq. (62) by generators of , we get an infinite-dimensional representation. This infinite-dimensional representation contains a 1-dimensional invariant subspace, because the action of gives constant. (But the action of and of Eqs. (33), (34) results in expressions like , etc, an infinite-dimensional space.)