Servicios Personalizados
Revista
Articulo
Indicadores
-
Citado por SciELO
Links relacionados
-
Similares en SciELO
Compartir
Revista de la Unión Matemática Argentina
versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637
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 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.
1. The free supersymmetric scalar field
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
![φ ∈ Isom (𝕄 )](/img/revistas/ruma/v47n1/1a035x.png)
![d (φ*φ) = φ *d φ](/img/revistas/ruma/v47n1/1a036x.png)
![Δ](/img/revistas/ruma/v47n1/1a038x.png)
The supersymmetry algebra. Suppose now that we have a non-degenerate 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 vector-valued bilinear form
![⟨Γ (s,s′),v⟩ = β (γ s,s′) ∀s, s′ ∈ S, v ∈ V. v](/img/revistas/ruma/v47n1/1a0320x.png)
![Γ](/img/revistas/ruma/v47n1/1a0321x.png)
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:
![D](/img/revistas/ruma/v47n1/1a0338x.png)
In this formula has to be understood as an odd element of
![Σ = 𝕄 × S → 𝕄](/img/revistas/ruma/v47n1/1a0342x.png)
![A = ΛE](/img/revistas/ruma/v47n1/1a0343x.png)
![E.](/img/revistas/ruma/v47n1/1a0344x.png)
The bilinear form extends as follows to an even
-bilinear form
![(ε ) a](/img/revistas/ruma/v47n1/1a0348x.png)
![S](/img/revistas/ruma/v47n1/1a0349x.png)
![β := β(ε ,ε ) ab a b](/img/revistas/ruma/v47n1/1a0350x.png)
For homogeneous elements of degree
we obtain
| (1) |
This implies
In particular,
![β(ψ, D ψ)](/img/revistas/ruma/v47n1/1a0358x.png)
![ψ](/img/revistas/ruma/v47n1/1a0359x.png)
![L (φ,ψ )](/img/revistas/ruma/v47n1/1a0361x.png)
![φ](/img/revistas/ruma/v47n1/1a0362x.png)
![ψ](/img/revistas/ruma/v47n1/1a0363x.png)
![Spin0 (V ) ⋉ V](/img/revistas/ruma/v47n1/1a0364x.png)
![L (φ,ψ ).](/img/revistas/ruma/v47n1/1a0365x.png)
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
![X](/img/revistas/ruma/v47n1/1a0368x.png)
![X (φ,ψ) = (δφ,δψ )](/img/revistas/ruma/v47n1/1a0369x.png)
![X](/img/revistas/ruma/v47n1/1a0370x.png)
![(φ, ψ)](/img/revistas/ruma/v47n1/1a0371x.png)
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 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
![g = (gij).](/img/revistas/ruma/v47n1/1a0386x.png)
Non-linear supersymmetric sigma-models. Next we consider maps
![(M, g )](/img/revistas/ruma/v47n1/1a0388x.png)
![Lbos](/img/revistas/ruma/v47n1/1a0390x.png)
![φ](/img/revistas/ruma/v47n1/1a0391x.png)
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 ]
Vicente Cortés
Department Mathematik
Schwerpunkt Analysis und Differentialgeometrie
und
Zentrum für Mathematische Physik
Universität Hamburg
Bundesstrasse 55
D- 20146 Hamburg
cortes@math.uni-hamburg.de
Recibido: 31 de agosto de 2005
Aceptado: 2 de octubre de 2006