This thesis is a description or most of the work I have done on Supersymmetry and Supergravity over the past two years. The motivation to deal with supersymmetry is actually found in the mathematical structure of the theories. Much more important, in the opinion of the author, is the possibility of building supersymmetric theories of gravity [2], where the metric tensor is unified through supersymmetry with matter fields.
Moreover, when interpreted in four dimensions through the compactification of six of the nine spatial dimensions, they appear as It concerns the spontaneous breaking of the local symmetry of measured theories of extended supergravity. The recipe for closing the supersymmetric algebra outside the shell can be found in the best-known case of the Lorentz algebra.
Indeed, assuming that a formulation of N=B supergravity can be found in terms of N=B superfields, the first possible occurrence of ultraviolet divergences in the S matrix of the theory increases from three to seven loops. The author's contributions are partly in the discussion of algebras, in a generalization to.
Supersymmetry algebras
As is well known, the Poincare algebra (1.2.1) can be recovered by taking a singular limit (contraction) of one of the two simple algebras. In D=10, as we have seen, the super-Poincare algebra does not have a simple extension just because this is incompatible with the Weyl property of the spinorial charge Q. On the other hand, in eleven dimensions it is a sim- There exists a extension of the super Poincare algebra.
This is the Osp(ll32) algebra and is a minimal simple extension of the super-Poincare algebra in eleven dimensions. This is what makes it possible to study the representations of the super-Poincare algebra by fixing the canonical moments and looking at the representations of the corresponding small groups. To extend the discussion to the example of super-Poincare algebras, it is also necessary to distinguish between massive and massless states.
The massless states in D= 11 are therefore ordered by the representations of this transversal group S0(9), which now becomes the corresponding part of the super-Poincare algebra. As in the four-dimensional case, the next step consists of choosing an explicit representation of the Dirac algebra and subjecting the charge Q to the Majoran condition.
Field theory models with simple supersymmetry
The discussion of particle representations of the super-Poincare algebra given in Section 3 also tells us why this happened. This result, however, should not be very surprising, since it also occurs in the more well-known case of Lorentz. This model describes the interactions of the gravitational field (described here with the help of vierbein vm JJ.).
However, the set of local symmetries of the Einstein action is correspondingly increased by the addition of the local Lorentz symmetry in the tangent space, which effectively measures the six additional components. Using this expression and the conventional (i.e. no rotation) expression of Christoffel symbols in terms of metrics. The natural candidate for the gravitino transformation is .. the gauge invariance generalization of Eq. 1.4.37) to include covariance with respect to general coordinate transformations, i.e. Along with this transformation, we need the vm JJ.• transformation and the CJp transformation.
An important point to note, however, is that the addition of the coupling 'f/;w'f/1 in Eq. The request to cancel these conditions then determines the vielbein transformation, which disappears after Fierz's rearrangement. This, in turn, corrects the sign of the cosmological term, which must be negative.
Then, which requires that the variation of all terms cancel quadratically in 1/1 leads to the change of the supersymmetry transformation from 1/1 into . and it fixes the action apart from quartic spinor terms arising from the variation of the quadrilateral in the mass and kinetic terms. Then, to make the supersymmetry local, one has to add the Noether term. to cancel the remaining AC terms from the variation of the kinetic term. But the effect of the new transformation for the gravitino on the remaining terms in the action then leads to the addition of a mass term for 1/IJ.I.
There are several index provisions which are equivalent to that shown in Eq. l.A.2), corresponding to various possible choices of indices ((3 1,. In this section we summarize our conventions for the eleven-dimensional Dirac algebra, which we use in the discussion of algebras and in the discussion of supergravity in Eleven dimensions We now discuss some properties of the matrices f M valid in the representation (l.B.6), (1.B.7) and which are used in the text.
AIJ 0 KIJ
In the ten-dimensional theory. however, the e = 0 parts of rpi are components of the ten-dimensional vector and are thus gauge fields. It should be noted that the form (2.5.5) of the covariant derivatives follows from the constraints. At e = e = 0, Fii, p>ii and F/ contain the part of the non-abelian field strength F ~ that corresponds to the.
To go further, it is useful to see in Eq. 2.5.10) the explicit form of the field strengths in terms of the fields. Indeed, most of the terms in the action can be guessed, using eq. 2.5.11) and compare it with the four-dimensional action given in Eq. To complete the construction of the action, it is convenient to use the last term in eq. expand.
It is invariant under the direct product of the four-dimensional Lorentz group and an S0(6) R:j SU(4) group corresponding to spatial rotations in the extra dimensions. The modified transformations in eq. 2.5.25) are not a symmetry of ten-dimensional action as they appear today. The other propagators differ from those of the four-dimensional theory in the corresponding Feynman-type meter only by the obvious substitution of 0 4.
The component fields are defined from the covariant derivatives and field strengths as. However, due to the extended nature of the strings, the diagrams are two-dimensional surfaces. 3.2.1) then requires that the amplitudes of the tree are completely symmetric during the exchange of the labels of the external states.
These conditions follow from the structure of the remains of the poles in the various intermediate channels. The corresponding symmetry of the tree amplitudes then follows by rotating all their external legs. They correspond to the anti-Hermitian matrices A, which are the matrices of the determinative representations of the classical algebras U(N), SO(N) and USp(2N).
That is, we need to find all subsets of the n 2 GL(N,C) matrices that are closed under multiplication and, when multiplied by arbitrary complex factors, reproduce the entire set of GL(N,C) matrices. This can be written using the twist condition (3.2.3). where m1 and mJ denote the mass levels of the intermediate states.
