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Revista de la Unión Matemática Argentina
versión Online ISSN 16699637
Rev. Unión Mat. Argent. v.47 n.1 Bahía Blanca ene./jun. 2006
An introduction to supersymmetry
Vicente Cortés
Abstract: This is a short introduction to supersymmetry based on the first of two lectures given at the II Workshop in Differential Geometry, La Falda, Córdoba, 2005.
The aim of this note is to explain — in mathematical terms and based on simple examples — some of the basic ideas involved in classical supersymmetric field theories. It should provide some helpful background for the, more advanced, discussion of geometrical aspects of supersymmetric field theories on Euclidian space, which is the theme of a second paper in this volume. Supersymmetric field theories on Minkowski space are discussed in great detail in the paper [DF], written for mathematicians. We shall not attempt here to give a reasonably complete list of papers written for physicists.
Our exposition starts with the simplest supersymmetric field theory on a pseudoEuclidian space: the free supersymmetric scalar field. A straightforward generalisation is the linear supersymmetric sigmamodel, the target manifold of which is flat. The generalisation to curved targets is nontrivial and leads to geometrical constraints imposed by supersymmetry.
Acknowledgement I thank Lars Schäfer for his assistance in the preparation of the computer presentation, which I gave at the II Workshop in Differential Geometry, La Falda, Córdoba, 2005.
1. The free supersymmetric scalar field
The bosonic scalar field. Let be a pseudoEuclidian vector space, e.g. Minkowski space, the spacetime of special relativity. A scalar field on is a function The simplest Lagrangian for a scalar field is
The supersymmetry algebra. Suppose now that we have a nondegenerate bilinear form on the spinor module of such that there exist such that:
 and
 ,
for all , where is the Clifford multiplication by . All such forms have been determined in [AC].
If , which will be assumed from now on, we can define a symmetric vectorvalued bilinear form
Such Lie superalgebras are called superPoincaré algebras. (More generally, could be a sum of spinor and semispinor modules.)
The supersymmetric scalar field. It turns out that the Lagrangian for a scalar field can be extended to a Lagrangian depending on the additional spinor field in such a way that the action of on scalar fields is extended to an action of its double covering on fields preserving the Lagrangian Moreover, the infinitesimal action of extends, roughly speaking, to an infinitesimal action of preserving up to a divergence.
The formula for the Lagrangian is the following:
In this formula has to be understood as an odd element of
The bilinear form extends as follows to an even bilinear form
For homogeneous elements of degree we obtain
 (1) 
This implies
In particular,
Verification of supersymmetry. We shall now define the supersymmetry transformations and check the invariance of the Lagrangian up to a divergence.
For any odd constant spinor
Let us check that this infinitesimal transformation preserves the Lagrangian up to a divergence:
 (2) 
Here we have used that, by (1):

The calculation of the two terms in (2) yields:
This shows that .
The linear supersymmetric sigmamodel. Instead of considering one scalar field and its superpartner we may consider n scalar fields and n spinor fields on (). The following Lagrangian is supersymmetric:
where is a constant symmetric matrix, which we assume to be nondegenerate. The above Lagrangian is called the linear supersymmetric sigmamodel. The Euler Lagrange equations for the scalar fields imply that the map
Nonlinear supersymmetric sigmamodels. Next we consider maps
It is natural to ask:
Does there exist a supersymmetric nonlinear sigmamodel, i.e. a supersymmetric extension of the bosonic sigmamodel ?
It turns out that one cannot expect a positive answer for arbitrary target
Restrictions on the target geometry. Supersymmetry imposes restrictions on the target geometry, which depend on the dimension of spacetime and on the signature of the spacetime metric In the case of 4dimensional Minkowski space the restriction is that is a (possibly indefinite) Kähler manifold [Z]. The corresponding supersymmetric sigmamodel is of the form
where with and where is the natural connection in and is a term quartic in the fermions constructed out of the curvaturetensor of , using that is Kähler and
Extended supersymmetry, special geometry. The superPoincaré algebra underlying the above nonlinear supersymmetric sigmamodel on fourdimensional Minkowski space is minimal, in the sense that is an irreducible module. The real dimension of is four.
There exists another superPoincaré algebra for which is a sum of two irreducible submodules. Note that is not a subalgebra of In fact, the submodules , , are commutative subalgebras, i.e.
Field theories admitting the extended superPoincaré algebra as supersymmetry algebra are called supersymmetric theories. The target geometry of such theories is called special geometry. The geometry depends on the field content of the theory. There are two fundamental cases:
 Theories with vector multiplets: the target geometry is (affine) special Kähler [DV], see [C] for a survey on special Kähler manifolds.
 Theories with hypermultiplets: the target geometry is hyperKähler, as follows from results about twodimensional sigmamodels [AF].
There exists also a Euclidian version of the Minkowskian superPoincaré algebra , for which is the Lie algebra of Killing vector fields of the fourdimensional Euclidian space and . The special geometry associated with field theories admitting the Lie superalgebra as supersymmetry algebra is discussed in a second contribution to this volume.
[AC] D. V. Alekseevsky and V. Cortés, Classification of N(super)extended Poincaré algebras and bilinear invariants of the spinor representation of Spin(p,q), Comm. Math. Phys. 183 (1997), no. 3, 477510. [ Links ]
[AF] L. AlvarezGaumé and D. Freedman, Geometrical structure and ultraviolet finiteness in the supersymmetric model, Commun. Math. Phys. 80 (1981), 443451. [ Links ]
[C] V. Cortés, Special Kähler manifolds: a survey, Rend. Circ. Mat. Palermo (2) 66 (2001), 1118. [ Links ]
[DF] P. Deligne and D. S. Freed, Supersolutions, Quantum fields and strings: a course for mathematicians, Vol. 1, 227355, Amer. Math. Soc., Providence, RI, 1999. [ Links ]
[DV] B. de Wit and A. Van Proeyen, Potentials and symmetries of general gauged N = 2 supergravityYangMills models, Nuclear Phys. B 245 (1984), no. 1, 89117. [ Links ]
[Z] B. Zumino, Supersymmetry and Kähler manifolds, Phys. Lett. 87B, no. 3, 203206. [ Links ]
Vicente Cortés
Department Mathematik
Schwerpunkt Analysis und Differentialgeometrie
und
Zentrum für Mathematische Physik
Universität Hamburg
Bundesstrasse 55
D 20146 Hamburg
cortes@math.unihamburg.de
Recibido: 31 de agosto de 2005
Aceptado: 2 de octubre de 2006