Revista de la Unión Matemática Argentina
versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.47 n.1 Bahía Blanca ene./jun. 2006
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 pseudo-Euclidian space: the free supersymmetric scalar field. A straightforward generalisation is the linear supersymmetric sigma-model, the target manifold of which is flat. The generalisation to curved targets is non-trivial 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.
The bosonic scalar field. Let be a pseudo-Euclidian vector space, e.g. Minkowski space, the space-time of special relativity. A scalar field on is a function The simplest Lagrangian for a scalar field is
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 vector-valued bilinear form
Such Lie superalgebras are called super-Poincaré algebras. (More generally, could be a sum of spinor and semi-spinor 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
For any odd constant spinor
Let us check that this infinitesimal transformation preserves the Lagrangian up to a divergence:
Here we have used that, by (1):
The calculation of the two terms in (2) yields:
This shows that .
The linear supersymmetric sigma-model. 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 non-degenerate. The above Lagrangian is called the linear supersymmetric sigma-model. The Euler Lagrange equations for the scalar fields imply that the map
It is natural to ask:
Does there exist a supersymmetric non-linear sigma-model, i.e. a supersymmetric extension of the bosonic sigma-model ?
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 space-time and on the signature of the space-time metric In the case of 4-dimensional Minkowski space the restriction is that is a (possibly indefinite) Kähler manifold [Z]. The corresponding supersymmetric sigma-model 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 curvature-tensor of , using that is Kähler and
Extended supersymmetry, special geometry. The super-Poincaré algebra underlying the above non-linear supersymmetric sigma-model on four-dimensional Minkowski space is minimal, in the sense that is an irreducible -module. The real dimension of is four.
There exists another super-Poincaré 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 super-Poincaré 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 hyper-Kähler, as follows from results about two-dimensional sigma-models [AF].
There exists also a Euclidian version of the Minkowskian super-Poincaré algebra , for which is the Lie algebra of Killing vector fields of the four-dimensional 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, 477-510. [ Links ]
[AF] L. Alvarez-Gaumé and D. Freedman, Geometrical structure and ultraviolet finiteness in the supersymmetric -model, Commun. Math. Phys. 80 (1981), 443-451. [ Links ]
[C] V. Cortés, Special Kähler manifolds: a survey, Rend. Circ. Mat. Palermo (2) 66 (2001), 11-18. [ Links ]
[DF] P. Deligne and D. S. Freed, Supersolutions, Quantum fields and strings: a course for mathematicians, Vol. 1, 227-355, Amer. Math. Soc., Providence, RI, 1999. [ Links ]
[DV] B. de Wit and A. Van Proeyen, Potentials and symmetries of general gauged N = 2 supergravity-Yang-Mills models, Nuclear Phys. B 245 (1984), no. 1, 89-117. [ Links ]
[Z] B. Zumino, Supersymmetry and Kähler manifolds, Phys. Lett. 87B, no. 3, 203-206. [ Links ]
Schwerpunkt Analysis und Differentialgeometrie
Zentrum für Mathematische Physik
D- 20146 Hamburg
Recibido: 31 de agosto de 2005
Aceptado: 2 de octubre de 2006