Learn About Supersymmetry and Deligne’s Theorem

 

In 2002, Pierre Deligne proved a remarkable theorem on what mathematically is called Tannakian reconstruction of tensor categories. Here I give an informal explanation what this theorem says and why it has profound relevance for theoretical particle physics:

Deligne’s theorem on tensor categories combined with Wigner’s classification of fundamental particles implies a strong motivation for expecting that fundamental high energy physics exhibits supersymmetry.

I explain this in a moment. But that said, before continuing I should make the following

Side remark.

Recall that what these days are being constrained more and more by experiment are models of “low energy supersymmetry”: scenarios where a fundamental high energy supergravity theory sits in a vacuum with the exceptional property that a global supersymmetry transformation survives. Results such as Deligne’s theorem have nothing to say about the complicated process of stagewise spontaneous symmetry breaking of a high energy theory down to the low energy effective theory of its vacua. Instead, they say something (via the reasoning explained in a moment) about the mathematical principles which underlie fundamental physics fundamentally, i.e. at high energy. Present experiments, for better or worse, say nothing about high energy supersymmetry.

Incidentally, it is also high energy supersymmetry, namely supergravity, which is actually predicted by string theory (this is a theorem: the spectrum of the fermionic “spinning string” miraculously exhibits local spacetime supersymmetry), while low energy supersymmetry needs to be imposed by hand in string theory (namely by assuming Calabi-Yau compactifications, there is no mechanism in the theory that would single them out).

End of side remark.

Now first recall the idea of Wigner’s classification of fundamental particles. In order to bring out the fundamental force of Wigner classification, I begin by recalling some basics of the fundamental relevance of local spacetime symmetry groups:

Given a symmetry group $G$ and a subgroup $H \hookrightarrow G$, we may regard this as implicitly defining a local model of spacetime: We think of $G$ as the group of symmetries of the would-be spacetime and of $H$ as the subgroup of symmetries that fix a given point. Assuming that $G$ acts transitively, this means that space itself is the coset

$$
X = G/H
$$

For instance if $X = \mathbb{R}^{d-1,1}$ is Minkowski spacetime, then its isometry group $G = \mathrm{Iso}(\mathbb{R}^{d-1,1})$ is the Poincaré group and $H = O(d-1,1)$ is the Lorentz group. But it also makes sense to consider alternative local spacetime symmetry groups, such as $G = O(d-1,2)$ the anti-de Sitter group. etc. The idea of characterizing local spacetimes as the coset of its local symmetry group by the stabilizer of any one of its points is called Klein geometry.

To globalize this, consider a manifold $X$ whose tangent spaces look like $G/H$, and such that the structure group of its tangent bundle is reduced to the action of $H$. This is called Cartan geometry.

For the previous example where $G/H$ is Poincaré/Lorentz, then Cartan geometry is equivalently pseudo-Riemannian geometry: the reduction of the structure group to the Lorentz group is equivalently a “vielbein field” that defines a metric, hence a field configuration of gravity. For other choices of $G$ and $H$ the same construction unifies essentially all concepts of geometry ever considered, see the table of examples here.

This is a powerful formulation of a spacetime geometry that regards spacetime symmetry groups as more fundamental than spacetime itself. In physics literature, it is essentially known as the first-order formulation of gravity.

Incidentally, this is also the way to obtain super-spacetimes: simply replace the Poincaré group with its super-group extension: the super-Poincaré group (super-Cartan geometry) But why would one consider that? We get to this in a moment.

 

Now as we consider quantum fields covariantly on such a spacetime, then locally all fields transform linearly under the symmetry group $G$, hence they form linear representations of the group $G$.

Given two $G$-representations, we may form their tensor product to obtain a new representation. Physically this corresponds to combining two fields to the joint field of the composite system.

Based on this, Wigner suggested that the elementary particle species are to be identified with the irreducible representations of $G$, those which are not the direct sum of two non-zero representations.

Indeed, if one computes, in the above example, the irreducible unitary representations of the Poincaré group, then one finds that these are labeled by the quantum numbers of elementary particles seen in the experiment, mass and spin, and helicity for massless particles.

One may do the same for other model spacetimes, such as (anti-)de Sitter spacetimes. Then the particle content is given by the irreducible representations of the corresponding symmetry groups, the (anti-)de Sitter groups, etc.

The point of this digression via Klein geometry and Cartan geometry is to make the following important point: the spacetime symmetry group is more fundamental than the spacetime itself.

Therefore we should not be asking:

What are all possible types of spacetimes over which we could consider Wigner classification of particles?

but we should ask:

What are all possible symmetry groups such that their irreducible representations behave like elementary particle species?

This is the question that Deligne’s theorem on tensor categories is the answer to.

To give a precise answer, one first needs to make the question precise. But it is well known how to do this:

1. A collection of things (for us: particle species)
2. which may be tensored together (for us: compound systems may be formed)
3. where two things may be exchanged in the tensor product (two particles may be exchanged)
4. such that exchanging twice is the identity operation,
5. and such that everything has a dual under tensoring (for every particle there is an anti-particle);
6. such that the homomorphisms between things (for us: the possible interaction vertices between particle species) form vector spaces;

is said to be a linear tensor category.

We also add the following condition, which physically is completely obvious, but necessary to make explicit to prove the theorem below:

• Everything consists of a finite number of particle species, and the compound of $n$ copies of  $N$ particle species contains at most $N^n$ copies of fundamental particle species. Mathematically this is the condition of “subexponential growth”, see here for the mathematical detail.

A key example of tensor categories are categories of finite-dimensional representations of groups, but not all tensor categories are necessary of this form. The question for those which are is called Tannaka duality: the problem of starting with a given tensor category and reconstructing the group that it is the category of representations of.

The case of interest to us here is that of tensor categories which are $\mathbb{C}$-linear, hence where the spaces of particle interaction vertices are complex vector spaces. More generally we could consider $k$-linear tensor categories, for $k$ any field of characteristic 0.

Deligne studied the question:

Under which conditions is such a tensor category the representation category of some group, and if so, of which kind of group?

Phrased in terms of our setup this question is:

Given any collection of things that behave like particle species and spaces of interaction vertices between these, under which condition is there a local spacetime symmetry group such that these are the particles in the corresponding Wigner classification of quanta on that spacetime, and what kinds of spacetime symmetry groups arise this way?

Now the answer to Deligne’s theorem on tensor categories is this:

1. Every  $k$-linear tensor category is of this form;
2. the class of groups arising this way is precisely the (algebraic) super-groups.

This is due to

Pierre Deligne, Catégorie Tensorielle, Moscow Math. Journal 2 (2002) no. 2, 227-248. (pdf)

based on

Pierre Deligne,  Catégories Tannakiennes , Grothendieck Festschrift, vol. II, Birkhäuser Progress in Math. 87 (1990) pp.111-195.

reviewed in

Victor Ostrik, Tensor categories (after P. Deligne) (arXiv:math/0401347)

and in

Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, section 9.11 in Tensor categories, Mathematical Surveys and Monographs, Volume 205, American Mathematical Society, 2015 (pdf)

Phrased in terms of our setup this means

1. Every sensible collection of particle species and spaces of interaction vertices between them is the collection of elementary particles in the Wigner classification for some local spacetime symmetry group;
2. the local spacetime symmetry groups appearing this way are precisely super-symmetry groups.

Notice here that a super-group is understood to be a group that may contain odd-graded components. So also an ordinary group is a super-group in this sense. The statement does not say that spacetime symmetry groups need to have odd super graded components (that would evidently be false). But it says that the largest possible class of those groups that are sensible as local spacetime symmetry groups is precisely the class of possible supergroups. Not more. Not less.

Hence Deligne’s theorem — when regarded as a statement about local spacetime symmetry via Wigner classification as above — is a much stronger statement than for instance the Coleman-Mandula theorem + Haag-Lopuszanski-Sohnius theorem which is traditionally invoked as a motivation for supersymmetry:

For Coleman-Mandula+Haag-Lopuszanski-Sohnius to be a motivation for supersymmetry, you first of all already need to believe that spacetime symmetries and internal symmetries ought to be unified. Even if you already believe this, then the theorem only tells you that supersymmetry is one possibility to achieve this unification, there might still be an infinitude of other possibilities that you haven’t considered yet.

For Deligne’s theorem the conclusion is much stronger:

First of all, the only thing we need to believe about physics, for it to give us information, is an utmost minimum: that particle species transform linearly under spacetime symmetry groups. For this to be wrong at some fundamental scale we would have to suppose non-linear modifications of quantum physics or other dramatic breakdowns of everything that is known about the foundations of fundamental physics.

Second, it says not just that local spacetime supersymmetry is one possibility to have sensible particle content under Wigner classification, but that the class of (algebraic) super-groups precisely exhausts the moduli space of possible consistent local spacetime symmetry groups.

This does not prove that fundamentally local spacetime symmetry is non-trivial supersymmetry. But it means that it is well motivated to expect that it might be one.