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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 𝕄 = V = (ℝd, η = < ⋅,⋅ > ) be a pseudo-Euclidian vector space, e.g. 𝕄 = Minkowski space, the space-time of special relativity. A scalar field on 𝕄 is a function φ : 𝕄 → ℝ. The simplest Lagrangian for a scalar field is

 -1 2 Lbos(φ ) = ⟨grad φ,gradφ ⟩ = η (dφ,d φ) =: |dφ |.

It is invariant under any isometry φ ∈ Isom (𝕄 ) , since d (φ*φ) = φ *d φ . The corresponding Euler-Lagrange equations are linear:

0 = div gradφ =: Δ φ.

Δ is the pseudo-Euclidian version of the Laplacian.

The supersymmetry algebra. Suppose now that we have a non-degenerate bilinear form β on the spinor module S of V such that there exist σ,τ ∈ { ±1} such that:

  •  ′ ′ β (s,s) = σ β(s,s ) and
  • β (γvs,s′) = τ β(s,γvs′) ,

for all  ′ s,s ∈ S,v ∈ V , where γv : S → S is the Clifford multiplication by v ∈ V . All such forms have been determined in [AC].

If στ = +1 , which will be assumed from now on, we can define a symmetric vector-valued bilinear form

Γ = Γ β : S × S → V

by the equation

⟨Γ (s,s′),v⟩ = β (γ s,s′) ∀s, s′ ∈ S, v ∈ V. v

Γ is equivariant with respect to the connected spin group and defines an extension of the Poincaré algebra

𝔤0 = Lie Isom (𝕄 ) = so(V) + V

to a Lie superalgebra

𝔤 = 𝔤0 + 𝔤1 with 𝔤1 = S.

Such Lie superalgebras are called super-Poincaré algebras. (More generally, 𝔤1 could be a sum of spinor and semi-spinor modules.)

The supersymmetric scalar field. It turns out that the Lagrangian Lbos(φ) for a scalar field φ can be extended to a Lagrangian L (φ, ψ) depending on the additional spinor field ψ : 𝕄 → S in such a way that the action of Isom0 (𝕄 ) = SO0 (V ) ⋉ V on scalar fields φ is extended to an action of its double covering Spin0 (V ) ⋉ V on fields (φ,ψ ) preserving the Lagrangian L (φ,ψ ). Moreover, the infinitesimal action of 𝔤0 extends, roughly speaking, to an infinitesimal action of 𝔤 preserving L (φ, ψ) up to a divergence.

The formula for the Lagrangian is the following:

L (φ,ψ ) = η -1(dφ,dφ ) + β (ψ,D ψ ),

where D is the Dirac operator

 ∑ ∑ D ψ = γμ∂ μψ, γ μ = η μνγν with γν = γ∂ . ν

In this formula ψ has to be understood as an odd element of

Γ A(Σ ) := Γ (Σ ) ⊗ A,

where Σ = 𝕄 × S → 𝕄 is the trivial spinor bundle and A = ΛE is the exterior algebra of some auxiliary finite dimensional vector space E.

The bilinear form β : S × S → ℝ extends as follows to an even C ∞A (𝕄 ) -bilinear form

β : Γ A(Σ) × Γ A (Σ) → C∞A (𝕄 ) = C ∞(𝕄 ) ⊗ A.

Let (ε ) a be a basis of S , β := β(ε ,ε ) ab a b and

 ∑ ∑ ψ = εaψa , ψ′ = εaψ ′a ∈ Γ A(Σ ) = Γ (Σ) ⊗ A = S ⊗ C ∞A (𝕄 ).

Then on defines

 ∑ β(ψ,ψ ′) := β ψ ψ′. ab a b

For homogeneous elements ψ, ψ′ of degree ψ˜, ˜ψ ′ ∈ {0,1} we obtain

β (ψ, ψ ′) = (- 1)˜ψ˜ψ′σ β(ψ ′,ψ ).


This implies

 ′ ∑ μ ′ ∑ μ ′ ′ β(ψ, D ψ ) = β(ψ, γ ∂μψ ) = τ β (γ ψ, ∂μψ ) ≡ - τβ(D ψ, ψ ) (mod div ) ^D ψ˜ψ′ = - τσ (- 1)(=˜ψ˜ψ′)β(ψ ′,D ψ ) = - (- 1)˜ψ˜ψ′β(ψ′,D ψ). ◟◝◜◞ =+1

In particular,

 ˜ β (ψ,D ψ ) ≡ - (- 1)ψβ(ψ, D ψ) (mod div ).

Hence β(ψ, D ψ) is a divergence if ψ is even. The Euler-Lagrange equations are again linear:

{ Δ φ = 0, D ψ = 0.

This is why the classical field theory defined by the Lagrangian L (φ,ψ ) for the scalar field φ and its fermionic superpartner ψ is called free. It is easy to check the Spin0 (V ) ⋉ V -invariance of L (φ,ψ ).

Verification of supersymmetry. We shall now define the supersymmetry transformations and check the invariance of the Lagrangian L (φ, ψ) up to a divergence.

For any odd constant spinor

 ∑ λ = εaλa ∈ S ⊗ ΛoddE ( ~= 𝔤1 ⊗ ΛoddE ⊂ (𝔤 ⊗ ΛE )0)

we define a vector field X on the the infinite-dimensional vector space of fields. The value X (φ,ψ) = (δφ,δψ ) of X at (φ, ψ) is

{ δφ := - β(ψ, λ) ∈ C ∞A (𝕄 )0 δψ := γgrad φλ ∈ Γ A (Σ )1

Let us check that this infinitesimal transformation preserves the Lagrangian up to a divergence:

δL (φ,ψ ) ≡ 2η-1(dδφ, dφ) + 2β(δψ, D ψ) (mod div).


Here we have used that, by (1):

β (ψ,D δψ ) ≡ - (- 1)˜ψ^δψ β(δψ,D ψ ) (mod div) = β(δψ, D ψ). ◟--◝◜ --◞ =+1

The calculation of the two terms in (2) yields:

 -1 - 1 ∑ μν η (dδφ,d φ) = - η (β (dψ,λ ),d φ) = - η β(∂μψ, λ)∂νφ ∑ μ ∑ μ β(δψ,D ψ ) = β(γgrad φλ,γ ∂μψ) = τ β (γ γgrad φλ,∂μψ ) ∑ μν τ-∑ μ μ = - τ η (∂ νφ)β(λ,∂ μψ) + 2 β((γ γgrad φ - γgrad φγ )λ,∂ μψ) (by the Clifford relation ) ∑ τ ∑ = + τσ ημν(∂νφ)β (∂ μψ,λ) + -- (∂ νφ)β([γμ,γν]λ,∂μψ ) ∑ ∑ 2 ≡ ημν(∂νφ )β (∂μψ,λ ) - τ- (∂μ∂νφ) β( [γμ,γ ν] λ,ψ ) (mod div) 2◟ ---◝◜---◞ ◟--◝◜-◞ symm. skew -symm. ∑ μν -1 = η (∂νφ )β (∂μψ,λ ) = - η (dδφ,dφ ).

This shows that δL(φ,ψ ) ≡ 0 (mod div) .

2. Sigma-models

The linear supersymmetric sigma-model. Instead of considering one scalar field φ and its superpartner ψ we may consider n scalar fields φi and n spinor fields ψi on 𝕄 (i = 1,...,n ). The following Lagrangian is supersymmetric:

 n 1 n 1 n ∑ -1 i j i j L (φ ,...,φ ,ψ ,...,ψ ) = gij(η (dφ ,dφ ) + β(ψ ,D ψ )), i,j=1

where g ij 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

 1 n n φ = (φ ,...,φ ) : 𝕄 → ℝ

is harmonic, where the target carries the flat metric g = (gij).

Non-linear supersymmetric sigma-models. Next we consider maps

φ : 𝕄 → (M, g)

into a curved pseudo-Riemannian manifold (M, g ) . The Lagrangian

Lbos(φ) = |dφ|2 := (gφ ⊗ η- 1)(dφ, dφ)

is called the non-linear bosonic sigma-model. The Euler-Lagrange equation of Lbos is the harmonic map equation for φ .

It is natural to ask:
Does there exist a supersymmetric non-linear sigma-model, i.e. a supersymmetric extension L (φ,ψ ) of the bosonic sigma-model Lbos(φ) ?

It turns out that one cannot expect a positive answer for arbitrary target (M, g ).

Restrictions on the target geometry. Supersymmetry imposes restrictions on the target geometry, which depend on the dimension d of space-time and on the signature of the space-time metric η. In the case of 4-dimensional Minkowski space the restriction is that (M, g) is a (possibly indefinite) Kähler manifold [Z]. The corresponding supersymmetric sigma-model is of the form

 - 1 φ L (φ,ψ) = (gφ ⊗ η )(dφ, dφ) + (gφ ⊗ β)(ψ,D ψ ) + Q (φ,ψ ),

where ψ ∈ Γ (φ*TM ⊗ Σ ), ψ = ψ1,0 ⊕ ψ1,0, A ℝ with ψ1,0 ∈ Γ (φ*T M 1,0 ⊗ Σ ) A ℂ and  φ ∑ μ φ D = γ ∇ ∂μ, where  φ ∇ is the natural connection in  * φ TM ⊗ Σ and Q is a term quartic in the fermions constructed out of the curvature-tensor Rg of g , using that (M, g ) is Kähler and S = ℂ2.

Extended supersymmetry, special geometry. The super-Poincaré algebra 𝔤 = 𝔤N=1 = 𝔤0 + 𝔤1 underlying the above non-linear supersymmetric sigma-model on four-dimensional Minkowski space is minimal, in the sense that 𝔤1 = S is an irreducible Spin(1,3) -module. The real dimension of S is four.

There exists another super-Poincaré algebra 𝔤 = 𝔤 = 𝔤 + 𝔤 N=2 0 1 for which 𝔤 = S ⊗ ℝ2 1 is a sum of two irreducible submodules. Note that 𝔤N=1 is not a subalgebra of 𝔤N=2. In fact, the Spin (V) -submodules  2 S ⊗ v ⊂ S ⊗ ℝ ,  2 v ∈ ℝ , are commutative subalgebras, i.e. [S ⊗ v,S ⊗ v] = 0.

Field theories admitting the extended super-Poincaré algebra 𝔤N=2 as supersymmetry algebra are called N = 2 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:

  1. Theories with vector multiplets: the target geometry is (affine) special Kähler [DV], see [C] for a survey on special Kähler manifolds.
  2. 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 𝔤′N=2 = 𝔤′0 + 𝔤′1 of the Minkowskian N = 2 super-Poincaré algebra 𝔤 N=2 , for which 𝔤′= so(4) + ℝ4 0 is the Lie algebra of Killing vector fields of the four-dimensional Euclidian space and  ′ ~ 𝔤N=2 ⊗ ℂ = 𝔤N=2 ⊗ ℂ . The special geometry associated with field theories admitting the Lie superalgebra  ′ 𝔤 N=2 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 σ -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
Zentrum für Mathematische Physik
Universität Hamburg
Bundesstrasse 55
D- 20146 Hamburg

Recibido: 31 de agosto de 2005
Aceptado: 2 de octubre de 2006

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