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It seems that Deligne’s theorem on tensor categories could give a neat motivation for supersymmetry, in the concrete sense of passing attention from the unitary representations of the Poincaré group group to unitary representations of the super Poincaré group (“super multiplets”).
The theorem says that every sufficiently tame tensor category linear over a field of characteristic zero is – while not necessarily the representation category of an ordinary group – necessarily the representation category of a supergroup (where “group” means “algebraic group”).
Together with the observation of Wigner classification, which observes that elementary particles may be identified with irreducible representations of the spacetime isometry group, this leads to a suggestive syllogism:
A) elementary particles are irreducible objects of the representation categories of spacetime symmetry groups;
B) categories of this type are generally representations of super-groups;
C) hence the general kind of spacetime symmetry group with such a concept of elementary particles are super-symmetry groups.
But there are technical details that seem to get in the way of this story: Deligne’s theorem speaks of finite dimensional representations, while the irreps of relevance in the Wigner classification are infinite-dimensional.
I am wondering if there is a way to nevertheless complete this story. Intuitively it seems clear: one wants to separate the infinite-dimensional piece coming from representations of translations from a finite dimensional piece correponding to representations of rotations (-represetation).
So the ’super’ aspect need not already be there in A? Say we deliberately start with the (non-super) Poincaré group. Does B mean its category of reps has an alternative definition as reps of some supergroup? Isn’t the passing to the super-group what gives rise to super multiplets?
Or do we have to start in A with a certain kind of collection of particles?
About the “technical details”, I see in the lengthy blog discussion John points out:
many physically interesting irreps of the Poincaré group are closely related to finite-dimensional, nonunitary irreducible representations of , using Mackey’s technology of ‘induced representations’.
So the ’super’ aspect need not already be there in A?
Right. Ordinarily we have a non-super spacetime symmetry group and find the particle content in its irreps. The spacetime symmetry need not be Poincaré symmetry, it could also be (anti-)de Sitter symmery, or conformal symmetry, etc. So generally we want to consider different symmetry groups and their corresponding elementary particles (or rather their possible “quantum numbers”).
But then we may turn this around and ask: given some tensor category whose simple objects we identify with some elementary particle species, then is there a symmetry group of which these are the irreps? Once we ask the question in this generality, then if something like Deligne’s theorem still holds of the infinite dimensional reps of relevance, we would discover that the most general groups induced this way are supergroups. Which would be interesting.
many physically interesting irreps of the Poincaré group are closely related to finite-dimensional, nonunitary irreducible representations of , using Mackey’s technology of ‘induced representations’.
That’s a good point, thanks.
Okay, I suppose that’s the correct hint: unitary irreps of the Poincare group at a fixed mass are precisely the induced representations of finite dimensional unitary irreps of the stabilizer group (Wigner’s “little group”) of a vector of norm given by that mass. A concise statement of this is for instance on p. 2 (left column) of Dragon 16. I gather the classification of supermultiplets (unitary irreps of the super-Poincare group) works the same.
For the purpose of the argument regarding motivation of supersymmetry it is sufficient to restrict to massless particles. I suppose then the tensor product on the reps of the little group should be compatible with the tensor product on the induced representations. (Is this right?)
If so, then this would complete the story I am after: apply Deligne’s theorem to those little groups.
I am writing an informal exposition of the relation of Deligne’s theorem and supersymmetry for PhysicsForums Insights. It is not published yet. There is a preview here. Is this visible to anyone besides me?
“You do not have permission to preview drafts.”
I see the same as fastlane.
A couple of vague questions:
(1) What becomes of this story when we move to string theory? E.g., do branes relate to representations? Perhaps I should be following this class.
(2) We has a discussion about Deligne and Kapranov on super algebra. Could there be something similar to Deligne’s result on bicategories of 2-reps of 2-groups?
What becomes of this story when we move to string theory?
Once we have any supersymmetry at all, then The brane bouquet (schreiber) gives string theory. That’s why it’s interesting to see which abstracts resons there may be for supersymmetry in the first place.
Perhaps I should be following this class.
Thanks for this pointer. I have added this to the references at quantization via the A-model.
This relates to the general story of string theory. One observes that the boundary theory of a 2d topological string theory is quantum mechanics. This was first established by Kontsevich-Cattaneo-Felder, who solved the problem of deformation quantization of any Poisson manifolds by perturbatively quantizing the Poisson-sigma-model topological string with ends fixed on the Poisson manifold “brane”. Joost Nuiten in his thesis showed that a non-perturvative quantization of the boundary theory of the Poisson sigma-model string yields non-perturbative geometric quantization of symplectic manifolds.
The A-model topological string is a particular gauge fixed version of the Poisson sigma-model topological string. I suspect that under this translation the Gukov-Witten approach is related to the PSM approach.
We has a discussion about Deligne and Kapranov on super algebra. Could there be something similar to Deligne’s result on bicategories of 2-reps of 2-groups?
It is natural to wonder whether there is a generalization of Deligne’s theorem from tensor 1-categories to monoidal -linear stable -categories, yes. And it is natural to speculate that for these then the -graded geometry that appears in Deligne’s theorem is lifted to -graded geometry. Yes. But presently I have no clue how to go about seeing this in any detail.
Whatever happened to those attempts to understand 2-representations, including Urs’s? There was Martin Neuchl’s thesis, Representation Theory of Hopf Categories, which showed that
a Hopf category has a monoidal bicategory of 2-representations,
and then Baez et al, Infinite-Dimensional Representations of 2-Groups, (arxiv).
But things seem to have gone quiet. Let’s look for some papers citing the latter in the last year:
Aleksandar Mikovic, Miguel A. Oliveira, Marko Vojinovic, Hamiltonian analysis of the BFCG theory for the Poincare 2-group, arXiv:1508.05635 [Spin-foams]
Dmitriy Rumynin, Alex Wendland, 2-Groups, 2-Characters, and Burnside Rings, arXiv:1604.02926
Benjamín Alarcón Heredia, Josep Elgueta, On the representations of 2-groups in {Baez-Crans} 2-vector spaces, arXiv:1607.04986 [seems a negative result - maybe because not the broadest 2-vector spaces].
Samuel Monnier, Topological field theories on manifolds with Wu structures, arXiv:1607.01396 [Hmm, talk of a ’Heisenberg 2-group’. Is that in sense of Heisenberg n-group?}
Paul Martin, Volodymyr Mazorchuk, Fiat categorification of the symmetric inverse semigroup IS_n and the semigroup F^*_n, arXiv:1605.03880
Anyway, no clear sign of people looking at reconstruction theorems for 2-reps.
There was a talk at Sheffield by Mark Penney on constructing Hopf categories from a suitably higher categorical pov. He sent me his slides, but they aren’t generally available.
An abstract of a talk here:
Mark Penney: Hopf categories from categorified Hall algebras.
Abstract: It has been known since the early ’90s that Hopf algebras determine 3D TFTs. In that same decade Crane and Frenkel proposed that 4D TFTs may be determined by so-called Hopf categories, which are linear categories having compatible monoidal and comonoidal structures. Unfortunately this proposal faltered due to the lack of examples of such categories, a problem which has not been remedied until recently. In this talk I will present a construction which takes as input an abelian category (satisfying certain finiteness properties) and produces a Hopf category. It is based on a categorification of the Hall algebra of an abelian category. I will conclude the talk by describing the Hopf category resulting from the abelian category of finite dimensional vector spaces over a finite field.
and a talk here
4D TQFTs, Hopf categories and 2-Segal groupoids
Here is now the public version of the article for PhysicsForums Insights.
Regarding 2-representation theory and the quest for 4D TQFTs, there is a large body of work which stems from that of Raphaël Rouquier (who was my PhD supervisor). What Raphaël has made publically available so far in written up form is much less than what he understands, and a fraction of his overall vision; but what has been made available has already been very influential. I just discovered some slides here which give something of the flavour. But this is not naïve stuff with 2-groups; there is a lot of deep Lie theory and geometric representation theory involved.
Looks good. Did Deligne have any interest in a possible physics interpretation of his result?
I don’t have any further information on Deligne’s motivation. I suspect that he was simply (“simply”) motivated by completing Tannaka reconstruction for linear tensor categories. And this is part of what makes the result so interesting: even without any a priori interest in supergroups, one is bound to discover them in the classification of something as fundamental and non-controversial as tensor categories.
So nothing about graded vector spaces goes into the hypotheses? That’s rather impressive that they just come out.
How precisely do all the conditions of Deligne’s theorem relate to the physics?
Symmetric monoidal: exchanging two particles twice leaves the system unchanged?
Rigid: each particle type has an anti-particle?
Finitely generated: why does the condition there specify a single generator ? Wouldn’t that suggest that all particle types can arise from a single particle type?
That odd condition on the Schur functor: ??
I see from the definition of super-representation:
A super-representation of a supergroup , def. \ref{Supergroup}, with inner parity , def.\ref{InnerParity}, is a superrepresentation of on a finite dimensional super vector space over such that is taken to the parity endomorphism of (which is the identity on even graded vectors and multiplication with on odd-graded vectors).
that one could recover the grading from an involution in the centre of , by splitting the ordinary vector space into eigenspaces for . I guess then part of Deligne’s theorem shows that the supercommutative Hopf algebra has its parity involution coming from an inner parity (giving then the grading on ) – probably in the ’resultat clef’.
How precisely do all the conditions of Deligne’s theorem relate to the physics?
Good point, I should add more on that to the entry.
Symmetric monoidal: exchanging two particles twice leaves the system unchanged?
Yes.
Rigid: each particle type has an anti-particle?
Yes.
Finitely generated: why does the condition there specify a single generator ? Wouldn’t that suggest that all particle types can arise from a single particle type?
Actually, that’s no real extra condition. For an algebraic group, then 1. there exists a finite dimensional faithful representation of and 2. every other finite dimensional representation is a subquotient of that. So any serves as a generator in the required sense. (I have added these facts to faithful representation – Properties).
That odd condition on the Schur functor: ??
I presently don’t have any insight on that condition, would be interested in hearing more about it from anyone who does. All I hear is that this is satisfied unless the category is “insanely huge”. Arguably, I should try to extract a more precise statement.
So nothing about graded vector spaces goes into the hypotheses? That’s rather impressive that they just come out.
Yes, I think that’s the point.
Re Urs #10:
It is natural to wonder whether there is a generalization of Deligne’s theorem from tensor 1-categories to monoidal -linear stable -categories, yes.
Iwanari in Tannaka duality and stable infinity-categories has a characterisation of ’fine’ -linear symmetric monoidal stable presentable -categories as quasicoherent sheaves over some derived quotient stack (theorem 1.4, p. 2).
Thanks for the pointer!
Hm, so where is the super-geometry? Probably it is ruled out by the fine-ness condition, as for super-vector spaces then there no longer is a top exterior power.
I find the definition if super-representation a bit self-referential, as currently given. Also, the example of an ordinary group doesn’t seem quite right, because surely the data of the grading on the vector space doesn’t just go away?
I find the definition if super-representation a bit self-referential, as currently given.
You mean as given in the entry? I have slightly re-phrased by replacing the second “superrepresentation” by “even/odd elements of act with even/odd parity on ”.
the example of an ordinary group doesn’t seem quite right, because surely the data of the grading on the vector space doesn’t just go away?
Not for general , but for chosen trivially it does, by the condition that is the parity endomorphism of .
All this is straight from Deligne 02, section 0.
Sure. I was thinking that it would be better to observe that the grading on is not extra data, but is in fact defined by the action of , and thus we don’t need to ask that acts as even/odd, merely that the rest of the group act compatibly with . Then in the non-super case, since is trivial, the grading is trivial as you point out.
I agree. I admit that I am investing less energy into optimizing the entry than one might hope, given how I have been announcing it.
That’s cool. Do you mind if I make the change? I can keep Deligne’s definition there in the form of a remark.
Do you mind if I make the change?
Please edit to your heart’s content!
I’ve made the edit to the definition of super representation, with scaffolding to ensure Deligne’s definition is referenced but gently put aside.
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