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
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
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