Eleven Dimensional Supergravity

The dimension D = 11 is the maximal dimension for which one can realize supersymmetry in terms of an ordinary supergravity theory. That a supergravity theory can exist in at most eleven dimensions can be explained by considering the more familiar irreducible representations of supersymmetry. A still higher and more ambitious stage of unification deals with the possibility of combining grand unified and gravity theories into a superunified theory. The conjectured 11-dimensional M-theory is required to have 11-dimensional supergravity as a low energy limit. Among the various supergravity theories, 11-dimensional supergravity plays a special and significant role.

D=11 Supergravity

The low-energy approximation of M-theory is given by the eleven-dimensional supergravity which describes the dynamics of the N= 1 supermultiplet in eleven dimensions. This contains the metric and an antisymmetric tensor field as bosonic components and the gravitino, which is a Majorana spinor in eleven dimensions, as their fermionic superpartner. We use D for the spinor covariant derivative. The indices denote curved eleven-dimensional indices. The supergravity action can be written as 

 

The action is invariant under the supersymmetry transformations. For the gravitino this takes the form

The relevant term can be written as

The symmetry of this supergravity theory is given by the supergroup OSp(1|32) which gives the subgroups O(1) for the bosonic symmetry and Sp(32) for the fermion symmetry. This is because spinors need 32 components in 11 dimensions.
 
The total action followed by an account of the boundary conditions is

The version of equation which describes the supersymmetric variation of the action and includes four fermi terms is

This boundary condition represents a significant difference between the present model and the construction of Horava and Witten. One significant difference is that torsion can be generated by the matter fields. This affects the connection which is governs the motion of gauginos. The chirality condition on the gravitino and the boundary condition on the three form play a special role in the supersymmetry of the theory. The supersymmetry transformation rules are almost conventional

In the new theory, the boundary conditions are

If we write the gravitino variation in the interior, then integration by parts adds a term to other boundary terms, where

A slight rearrangement gives

Now we are ready to examine the variation under supersymmetry transformations. Firstly, using the transformation and gamma-matrix identities

The variation of the action under the supersymmetry transformations can be obtained by combining the boundary terms with terms from the variation of the matter multiplet and with terms which arise from the interior. The invariance of 11-dimensional supergravity ensures that the volume terms in the variation cancel. Boundary terms arise from the interior when partial integration has to be used.


Horava-Witten Theory

Supersymmetric theories for systems with boundaries have been of great interest. The most notable example of this is the 11D Horava-Witten construction, also known as Heterotic M-Theory. Horava-Witten theory can be formulated as an expansion in the 11-dimensional gravitational coupling. To lowest order in this expansion, Horava-Witten theory is 11-dimensional supergravity with the fields restricted under the action. In the upstairs picture, the action is

The terms which are quartic in the gravitino can be absorbed into the definition of supercovariant objects. The condition

means that the gravitino is chiral from a 10-dimensional perspective, and so the theory has a gravitational anomaly localized on the fixed planes. The action is invariant under the local supersymmetry transformations

Cancellation of the gravitational anomaly requires the introduction of one Yang-Mills supermultiplet on each orbifold fixed plane. The minimal Yang-Mills action is

This action is invariant under the global supersymmetry transformations

 The challenge is then to add interactions and modify the supersymmetry transformation laws so that

is locally supersymmetric. This involves coupling the gravitino to the Yang-Mills supercurrent. However, since the gravitino lives in the 11-dimensional bulk, while the Yang-Mills supermultiplets live on the 10-dimensional fixed planes, a locally supersymmetric theory cannot be achieved simply by adding interactions on the fixed planes. To achieve local supersymmetry, the Bianchi identity must be modified to read

With the modified Bianchi identity, the total action can be made locally supersymmetric. However, having gained supersymmetry, Yang Mills gauge invariance has been lost. The modified Bianchi identity implies that the 4-form field strength is invariant under the infinitesimal gauge transformations

if the field strenght transforms as

The quantum theory is anomaly free. Gauge, gravitational and mixed anomalies are cancelled with a refinement of the standard Green-Schwarz mechanism. The Horava-Witten action is

This action is invariant under the local supersymmetry transformations

The terms appearing in the above boundary conditions and Bianchi identity are not required by the low energy theory, but must be present in the full quantum M-theory. The 11 dimensional Horava-Witten M-theory offers a fundamental framework for the construction of phenomenologically viable models. The theory has a number of interesting and unusual features.


Duality-symmetric D=11 Supergravity
 
The duality–symmetric (doubled field) action for the complete D = 11 supergravity has been constructed. The construction of these actions is based on the covariant techniques. To study interactions of branes with supergravity backgrounds, and to derive effective brane actions from corresponding supergravities, it is therefore desirable to have a formulation of supergravities in which the standard and the dual fields enter the action in a duality–symmetric way.
 
The duality-symmetric action for D = 11 supergravity is

or in a more symmetric form

Modulo the last term in the first action is the conventional D = 11 supergravity action written in the same notation as in the original paper except for the coefficient in front of the Einstein-Hilbert term and the coefficient in the definition of the spin connection.We should also point out that the duality-symmetric version of D = 11 supergravity has the following structure

                          
where we have the Einstein-Hilbert term, the fermion kinetic term and the specific term of the form which contains the information on the duality relations. To conclude this section let us recall that in the conventional Cremmer-Julia-Scherk formulation of D = 11 supergravity

The solution of the total action is

The general variation of the last term in the previous equation is

   
A universal, duality–symmetric, formulation of maximal D = 10 and D = 11 supergravities has proved to be useful for the understanding of many aspects of superstring and M–theory including their symmetry structure and the dynamics of various branes constituting an intrinsic part of these theories.
 

The 11-dimensional theory generated considerable exaltation as the first potential candidate for the theory of everything. The action and boundary conditions provide a supersymmetric theory which is a natural candidate for the a low energy limit of M-theory. In search of signatures of purely M-theoretic effects one may try to go beyond the limiting approximation of ordinary 11D supergravity, by including higher-order derivative curvature corrections. Supersymmetry, provided it will prove restrictive enough, is at present our best hope for addressing such corrections directly in eleven dimensions. The known supersymmetry transformations of eleven dimensional supergravity lead to symmetries of the theory indicating the consistency of supergravity. Our understanding of M-theory is still very limited, mainly due to the lack of powerful methods to probe it at the quantum level. One approach to encoding information about M-theory is through its low energy effective field theory.
                                     The maximal supergravity is the classical limit of M-Theory 

Bulk Branes in Supergravity

We construct a consistent supersymmetric action for brane chiral and vector multiplets in a six-dimensional chiral gauged supergravity. A nonzero brane tension can be accommodated by allowing for a brane-localized Fayet-Iliopoulos term proportional to the brane tension. When the brane chiral multiplet is charged under the bulk, we obtain a nontrivial coupling to the extra component of the gauge field strength and a singular scalar self-interaction term. In this article, we consider the supersymmetrisation of the brane tension action in a way compatible with the bulk SUSY in 6D Salam-Sezgin supergravity. We find that a branelocalised Fayet-Ilioupolos term proportional to each brane tension must be introduced to cancel the SUSY variation of the brane tension term. In particular, brane world models in six dimensions have drawn much attention because the brane tension generates a nonzero deficit angle in extra dimensions without curving the 4D spacetime. This feature has been first pursued in the framework of spontaneous compactification due to gauge fluxes in 6D Einstein-Maxwell theory, but ended up with a fine-tuning condition for the brane tension due to flux quantization or conservation. Furthermore, we still need some symmetry to ensure that a bulk tuning condition is stable against the quantum corrections. In the context of D = 5, N = 2, Yang-Mills Supergavity compactified on orbifold we consider the supersymmetric coupling of matter fields propagating on the brane. To solve the discrepancy between the bulk actions with limited field content and the wide range of brane actions that involve all the possible forms, we have constructed a new formulation of IIA supergravity up to quartic order in fermions. In the process of pursuing this goal we have realized that all descriptions of the effective field theory of Type IIA string theory available in the literature are inefficient for our purpose. The realization of the total system in supergravity is rather obscure.

Brane Multiplets in D = 5 , N = 2 Supergravity

In the on-shell formulation of D = 5 , N = 2 supergravity, compactified on orbifold, we extend the results describing the interaction of the bulk fields with matter which is assumed to be confined on the brane. We consider a five-dimensional Yang-Mills supergravity model. The field content of the model is

The supergravity multiplet consists of the fünfbein, two gravitini and the graviphoton. Moreover, there exist vector multiplets, counting the Yang-Mills fields. The spinor and the scalar fields included in the vector multiplets are presented. The bulk Lagrangian is

Recalling the linearized supersymmetry transformations of the bulk fields

 

The requirement of N = 1 local supersymmetry invariance on the branes determines the on-shell couplings of these fields to the gravity and gauge multiplets. These can be found following Nöther’s procedure. This procedure is used in the on-shell formulation of local supersymmetry, where the role of the gauge field is played by the gravitino, while the gauge current is the supercurrent. However in the case of supersymmetry besides the modification of the Lagrangian the transformation laws should be also modified accordingly.

The original Lagrangian is

with the brane part including the interactions of the matter fields, localized on the brane, with the projections of the bulk fields, gravity and gauge fields, on the brane.

We must modify the original theory by adding new terms, so that the total Lagrangian is

Using the gravitino transformation law one gets

 

The above results are easily extended in the case that the original brane action has the structure of a general model

Summarizing, the interactions of a set of chiral multiplets localized on the brane designated by the index, with the bulk gravity and gauge fields are found to be

It then turns that with this interpretation the terms in the action are exactly those encountered in the ordinary D=4 supergravity involving the interactions of the chiral fields among themselves and their interactions with the gravity and gauge multiplets. Exception to it are additional terms where bulk fields, involving the radion multiplet, interact with the multiplets on the brane.


Bulk Branes in Salam-Sezgin Supergravity
 
We present a new anomaly-free gauged N = 1 supergravity model in six dimensions. We construct a consistent supersymmetric action for brane chiral and vector multiplets in a six-dimensional chiral gauged supergravity. When the brane chiral multiplet is charged under the bulk, we obtain a nontrivial coupling to the extra component of the gauge field strength and a singular scalar self-interaction term. The six-dimensional Salam-Sezgin supergravity consists of gravity coupled to a dilaton field, a Kalb-Ramond (KR) field, along with the SUSY fermionic partners, the gravitino, the dilatino. Moreover, it also contains a bulk vector multiplet that gauges the R-symmetry of six-dimensional supergravity. All the bulk fermions are 6D Weyl. In order to do this analysis, we need the spinor part of the action and in particular the part that is quadratic in fermionic terms. This is given by

  
The complete bulk Langrangian up to four fermion terms is

The local N = 2 SUSY transformations are

We can show that the action for the Lagrangian is invariant under the above SUSY transformations up to the trilinear fermion terms and the Bianchi identities as follows

Thus, as will be seen later, the SUSY variation of the brane action can be cancelled with the bulk variation by modifying the Bianchi identities.

We consider a nonzero brane tension as well as brane matter multiplets: a brane chiral multiplet, the superfield of which has an charge, and a brane vector multiplet. Then, by employing the Noether method for the local SUSY, we find that the supersymmetric action for the bulk-brane system is composed of the original bulk action  with the field strength tensors being replaced by the modified field strenght tensors, respectively, and the brane action as follows

with

We introduce a gravitino mass term on the brane. Then, the brane action is supplemented by the supersymmetric gravitino mass terms as

Let us add to the bulk Lagrangian a term for a brane located at the position, where includes the internal space 2D coordinate. This brane Lagrangian will be given by

 

The SUSY transformation of the brane action is non-vanishing as follows

The gravitino contains a piece of the gauge field strength as

 

As a consequence of introducing the localised FI terms, we have seen that the brane tension action is made compatible with the bulk SUSY transformations. The supersymmetric action of the bulk and brane system up to four fermion terms is

                                                                                                               

All the fermionic SUSY transformations are modified as

The important ingredient of the above modifications is that we have a brane term linear in the brane-localised FI term. In other words, there is a brane coupling to the magnetic flux, which is proportional to the brane tension.

There are some known anomaly-free models including the non-abelian gauge fields in 6D gauged supergravity. In these cases, an abelian flux can be also turned on in the direction of the non-abelian gauge fields. For instance, in the model with hyperino, the U(1) contained in E₆ can also develop a nonzero flux, still maintaining the warped solution that was obtained for the Salam-Sezgin supergravity.  

 Bulk and Brane System in Type IIA Supergravity 

We will present here the new dual formulation with action, available for the IIA case only. A proper action will be constructed in this formulation. It is this formulation that we will apply in our construction of the bulk and brane system. We will call this the dual formulation. Note that the standard action for IIA supergravity can be obtained by integrating out the dual potentials in the action.
 
There are also terms quartic in fermions in the action. It turns out that it is important to take them into account to check the T-duality invariance of partition sum. We collect 4-fermionic terms in D=10 IIA supergravity action which are obtained from circle reduction of the D=11 action

where

We are in the position to read of the values of the sources and constants introduced in the dualization procedure and sum the different contributions to obtain the full action of the bulk and brane system in the Einstein frame

The abbreviations used in the above are

The source current arising from the dualization of the two-form is given by

In the total action the kinetic terms of the axionic scalars contain the gauge covariant derivatives

The bulk action has a set of shift symmetries

                            
which are gauged upon inclusion of a D6-brane, resulting in the covariant derivatives given above.

In the sequel we will assume that this is the case so that the part of the Lagrangian including the kinetic terms of the scalar fields encoded in the total action may be written as

Let us now take a closer look at the scalar potential evaluated for the F-fl at con figuration in which we performed our dimensional reduction. From what we said above it reduces to

and depends on the Kähler structure moduli as well as the dualization constants. First we stress again that the Kähler moduli should be viewed as fixed at the F-flat value consistent with the calibration condition and are thus not to be treated dynamically in this potential.


In the context of D = 5, N = 2, Yang-Mills Supergavity compactified on orbifold we consider the supersymmetric coupling of matter fields propagating on the brane. Working in the onshell scheme we have derived the terms of the brane action which are relevant for studying the mechanisms of supersymmetry and gauge symmetry breaking. This rather complicated task, along with the derivation of additional terms coupling the brane fields to the radion multiplet, and other even combination of odd fields, which complete the Lagrangian. The complete brane action including these terms and the mechanisms of supersymmetry and the gauge symmetry breaking in particular unified models, in which both gravity and gauge forces propagate in the bulk. We have constructed a consistent SUSY action for brane matter multiplets in a 6D chiral gauged supergravity. Introducing brane chiral multiplets charged under the U(1)R, we derived the supersymmetric coupling to the brane by modifying both the gauge field strength and the field strength for the KR field together with the necessary modifications of the fermionic SUSY transformations. We studied in detail the spectrum in the general warped background where codimension-two branes were supporting the necessary conical singularities. The above property for the massless gravitino and its mass suppression with extra operators, should also hold for the other fermionic states of the spectrum. In particular the gauginos which correspond to the directions of isometry of the internal space should have the same feature. This procedure offers an alternative way to obtain light fermions in models with extra dimensions. We have constructed new formulations of Type IIA supergravity. We constructed bulk theories with a unified treatment of all potentials. Due to the doubling of degrees of freedom one had to impose extra duality constraints and thus a proper action was not possible. Using these actions all bulk and brane systems can be described. Apart from being a tool to understand the supersymmetric domain walls we were interested, it can be expected that the new effective theories of D = 10 supergravity will have more general applications in the future.