Lectures on supersymmetry

Joseph Bernstein

September 17, 1996

Lecture 1. Why sypersymmetry

0. The aim of these lectures is to describe what physicists have in mind when they talk about supersymmetry (SUSY).

I should say that I am a mathematician, not a physicist, and these lectures are for mathematicians.

First I thought that these will be more or less straightforward course of lectures on supermanifolds. But then I (and David Kazhdan) realized that the situation is more interesting.

Physicists have constructed many models with supersymmetry; for some of them they found nice models formulated in terms of supermanifolds (super space in physical terminology). But for many interesting models with supersymmetry their “supermanifold” realization is unknown — and this may be an interesting problem for mathematicians.

1. What are sypersymmetries.

First, what are symmetries?

Physicists have three types of pictures for their theories.

(1) Classical picture – fields on a space-time + Lagrangian;

(2) Quantum field theory (QFT)

(3) Scattering theory (theory of S-matrix).

Usually there are many symmetries of the classical picture, but many of them give trivial symmetries of the scattering theory (of the S-matrix). It turns out that if we want to describe the symmetries of the S-matrix then the answer is very restrictive. In my lectures I will mostly deal only with classical picture. But in the first lecture I will describe an important result from the scattering theory which explains why physicists are so interested in supersymmetry.

2. The group P and its representations. 2.1.

Let V be a 4-dimensional real vector space (V ≃ ℝ^{4}) with a quadratic form of signature

( \begin{matrix}+ \\ & − \\ & & − \\ & & & − \end{matrix} ).

Let G denote the universal cover of the group SO(V): we have

1 → ℤ_{2} → G → SO(V) → 1
≀≀
Spin(V)

As a real group, G is isomoprhic to SL(2, ℂ). We will list now all finite-dimensional irreducible complex representations of G.

Let L denote a standard 2-dimensional complex representation of G (coming from the isomorphism G ≃ SL(2, ℂ)). The action of G on L preserves a volume form ω and we have L ≃ L^{∗}, as G-representations (^{∗} denotes the linear dual). Claim. Any irreducible representation of G has a form:

M_{i} ⊗ \overline{M}_{j}, where M_{k} denotes the k + 1-dimensional representation Sym^{k}(L) and where is the complex conjugation. ■

Example. The space V as a real representation of G identifies with the space of Hermitian forms on L.

We have

τ: V_{ℂ} ≃ M_{1} ⊗ \overline{M}_{1} = L ⊗ \overline{L}

2.2. Let P denote the Poincaré group, i.e. the semidirect product

P = G ⋉ V.

We will be interested in unitary representations of P. Note that the action of the central element z ∈ P (z = ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ) ∈ SL(2, ℂ) ≃ G) defines a ℤ_{2}-grading on the space of each such representation:

H = H_{+} ⊕ H_{−}

Let H be an irreducible unitary representation of P. Then H _{V} has a form ∫_{p∈V^{∗}} H_{p}dμ, where dμ is a G-invariant measure on V^{∗} concentrated on one G-orbit 𝒪 ⊆ V^{∗} ( in our case each orbit admits a G-invariant measure). The structure of a P -representation on ∫_{p∈V^{∗}} H_{p}dμ is determined by the action of Stab(p_{0}) ⊆ G (p_{0} is an arbitrary point of 𝒪) on H_{p_{0}}, which must be an irreducible unitary representaion of this group.

We say that an irreducible P -representation H is positive if the corresponding G-orbit 𝒪 ⊆ V^{∗} is one of the following three types:

a) 𝒪_{m}^{+}, m ∈ ℝ^{+∗}: {p ∈ V, < p, p >= m^{2}, p_{0} > 0};

b) 𝒪_{0}^{+}: {p ∈ V, < p, p >= 0, p_{0} > 0};

c) {0}.

Here we have used the identification V ≃ V^{∗} and an orientation of the time-coordinate x_{0} in V ≃ ℝ^{4}.

We say that a unitary representation H of P is “good” if it satisfies the following three properties:

1) It is a sum of finitely many irreducibles H = ⊕_{i} H;

2) Each of the H_{i}’s is positive in the above sense;

3) For each of the H_{i}’s the representations of stabilizers H_{p_{i}} are finite dimensional.

Example. Let H be an irreducible P -representation satisfying 2) and 3).

a) If 𝒪 = 𝒪_{m}^{+}, choose p = (m, 0, 0, 0); Stab_{p} = SU(2). Irreducible unitary representations of SU(2) are enumerated by positive half-integers and the corresponding P -representations will be denoted as H_{m,s} s ∈ \frac{1}{2} ℤ^{+}.

b) If 𝒪 = 𝒪_{0}^{+}, choose p = (1, 1, 0, 0); Stab_{p} = U(1) ⋉ N, where N is a nilpotent group. Irreducible Stab_{p}-representations are enumerated by half-integers and the corresponding P -representations will be denoted as H_{0,s} s ∈ \frac{1}{2} ℤ.

c) If 𝒪 = {0}, Stab_{0} = G and H_{p} must be trivial. Therefore, H is trivial too.

3. Scattering theory and Coleman-Mandula theorem.

3.1. The scattering picture consists of the following data:

1) H a “good” representation of P;

2) A unitary operator (called S-matrix) S: ℋ → ℋ, where ℋ: = Sym(H) such that S commutes with the action of P.

Here Sym(H) is understood in the super-sense, i.e.

Sym(H) ≃ Sym(H_{+}) ⊗ Λ(H_{−}).

Definition. An even symmetry of the S-matrix is an (unbounded) operator B: ℋ → ℋ such that

a) B is even and preserves the ℤ-grading on ℋ;

b) B satisfies the Leibnitz rule;

c) [B, S] = 0.

Example. Consider 𝔭 = Lie(P). Then 𝔭 acts n ℋ by symmetry operators.

A theorem of Coleman and Mandula classifies all possible symmetries of the S-matrix in a “non-degenerate” situation.

3.2. Assume that the pair (H, S) satisfies the following additional conditions:

a) H has irreducible summands with strictly positive masses only, i.e. only the orbits of the type 𝒪_{m}^{+} are allowed.

b) S is non-degenerate.

The last condition is rather obscure; let us, nevertheless, give a definition.

Let Σ = ∪_{i} 𝒪_{m}^{+} be the support of H in V^{∗} and consider the operator S^{2}: Sym^{2}H → Sym^{2}H

S^{2}: Sym^{2}H ↪^{i} ℋ ⟶^{S} ℋ ⟶^{pr} Sym^{2}H.

Sym^{2}H identifies with the space of L_{2}-sections of a vector bundle with fibers H_{p} ⊗H_{q} over the manifold Sym^{2} (Σ) ∖ Δ (Δ is the diagonal).

The operator S^{2} is given by a kernel

T_{p′,q′}^{p,q} (p ≠ q; p′ ≠ q′): H_{p} ⊗ H_{q} → H_{p′} ⊗ H_{q′}.

From the V ⊆ P -invariance of S it is easy to deduce that T_{p′,q′}^{p,q} is concentrated on the submanifold of (Sym^{2} (Σ) ∖ Δ) × (Sym^{2} (Σ) ∖ Δ) given by p + q − p′ − q′ = 0, i.e.

T_{p′,q′}^{p,q} = δ(p + q − p′ − q′)T̃{p,q \atop p′,q′}.

The non-degeneracy condition says that T̃{p,q \atop p′,q′} is a boundary value of an analytic function and that the matrix T̃{p,q \atop p′,q′} is non-degenerate if p is close to p′ and q is close to q′. Theorem (Coleman-Mandula). Let the pair (H, S) satisfy (a) and (b) as above. Then

a) Any symmetry B of S has a form B = B_{1} + B_{2}, where B_{1} ∈ 𝔭, [B_{2}, 𝔭] = 0.

b) The Lie-algebra of symmetries of S has a form 𝒮 = 𝔭 ⊕ I, where I is a finitedimensional, reductive Lie algebra. ■

3.3.

Let us now describe the extension of the Coleman-Mandula result to the case of nonnecessarily even symmetries. Definition. An odd symmetry of S is an operator B: ℋ → ℋ such that

a) B is odd and preserves the ℤ-grading on ℋ.

b) B satisfies the super-Leibnitz rule:

B(X ⊗ Y) = BX ⊗ Y + (−1)^{p(X)}X ⊗ BY.

c) [B, S] = 0.

Consider the Lie super-algebra 𝒮 = 𝒮¯0 ⊕ 𝒮¯1 of symmetries of S. We have the following result: Theorem (Haag, Lopuszansky, Sohnius). Let (H, S) satisfy a) and b) of 3.2. Then 𝒮 has the following structure:

1) 𝒮¯0 = 𝔭 ⊕ I (as in Coleman-Mandula theorem).

2) 𝒮¯1 is finite dimensional, [𝒮¯1, V] = 0 and as a representation of G it has a form

⊕_{i}(L ⊕ \overline{L}).

3) If Q ∈ L_{i}, Q′ ∈ L_{j}, we have

[Q, \overline{Q}′] = δ_{ij} · τ(Q, \overline{Q}) ∈ V_{ℂ} (τ of 2.1)

Here Q → \overline{Q} denotes the operation of passing to the adjoint operator acting on H.

4) [Q, Q′] = δ_{ij} · ι, where ι belongs to I and is central. ■