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June 14, 2007

Polyvector Super-Poincaré Algebras

Posted by Urs Schreiber

Just heard a talk on the work

D. V. Alekseevsky, V. Cortés, C. Devchand, A. Van Proeyen
Polyvector Super-Poincare Algebras

which is about classification of extensions of Poincaré Lie algebras of a vector space with scalar product of signature (p,q)(p,q) iso(V)iso(p,q) \mathrm{iso}(V) \simeq \mathrm{iso}(p,q) to super Lie algebras iso(p,q)W 0 evenS odd. \underbrace{iso(p,q) \oplus W_0}_{\mathrm{even}} \oplus \underbrace{S}_{\mathrm{odd}} \,.

At least parts of this is ancient knowledge in physics, but I am being told that to get this coherent, comprehensive and rigorous form quite a bit of work was required.

One reason why these super Poincaré algebras are very interesting is the following:

as is well known, it turns out that the parts of the super extension of the Poincaré algebra called W 0W_0 above consists of various copies of exterior powers pV \wedge^p V (called “polyvector spaces”) of the underlying vector space VV.

Now, like ordinary Einstein gravity may be conceived as a gauge theory for iso(3,1)\mathrm{iso}(3,1), theories of supergravity come from the respective super extensions of that.

Like flat Minkowski space is a special solution to Einstein’s equations, characterized by the fact that it exhibits globally the symmetry of iso(3,1)\mathrm{iso}(3,1), supergravity theories have special solutions which globally respect parts of the super Poincaré symmetry.

Strikingly, for each power pV \wedge^p V that appears in the super extension of the Poincaré algebra these solutions may feature (p+1)(p+1)-dimensional hypersurfaces that behave much like charged particles – only that instead of being 0-dimensional and coupling to a connection, they are pp-dimensional and couple to a (p+1)(p+1)-connection!

A review of these structures – called (solitonic/BPS)pp-branes – may be found for instance in

K.S. Stelle
BPS Branes in Supergravity

Now, this is especially interesting for us, because on the other hand, as nn-Café-regulars have heard us say before, at least some of these pp-branes should really correspond to certain (p+1)(p+1)-functors (p+1)Cob(p+1)Hilb. (p+1)\mathrm{Cob} \to (p+1)\mathrm{Hilb} \,. To indicate the categorification step, I like to speak of (n=p+1)(n =p+1)-particles.

There is some tantalizing interaction between supersymmetrization and categorification – many of the details of which still escape me.

The most direct hint, so far, concerning what is really going on, is Castellani’s observation:

Castellani remarks (not in these words, though, but I think these words are part of the clue) that with the super Lie 3-algebra sugra(10,1)3sLie \mathrm{sugra}(10,1) \in 3\mathrm{sLie} which D’Auria and Fré once found to be the structure governing 11-dimensional supergravity (as discussed at length in SuGra 3-Connection Reloaded) comes a certain Lie 1-algebra of derivations of the Lie 3-algebra, and that this is the polyvector super extension of iso(10,1)\mathrm{iso}(10,1).

So it seems that there is a close relation between

a) super Lie nn-algebras g (n)g_{(n)} extending the Poincaré Lie 1-algebra

b) polyvector super Lie 1-algebras extending the Poincaré Lie 1-algebra

and apparently b) is part of the Lie (n+1)(n+1)-algebra DER(g (n))\mathrm{DER}(g_{(n)}).

(I am being careful with saying “part of” etc, since the derivations considered in Derivation Lie 1-Algebras of Lie n-Algebras and What is a Lie Derivative, really? are closely related but not exactly the derivations that Castellani considers.

Membranes and 5-Branes

Using the results of the above paper by D. V. Alekseevsky, V. Cortés, C. Devchand, A. Van Proeyen, there is a quick way to see that 11-dimensional supergravity has a 2-brane (a 3-particle) and a 5-brane (a 6-particle) as follows:

Proposition 1 on p. 391 says, essentially, that in odd dimensions there is (up to isomorphism, of course – these authors tend not to mention isomorphisms when they are obvious) a unique super extension of the Poincaré algebra which is of maximal size, i.e. where the polyvector part

W 0= p pV W_0 = \sum_p \wedge^p V

is as large as possible: namely this is the case when

W 0SS, W_0 \simeq S \vee S \,, where SS is the irreducible spinor module and SSS \vee S its symmetric second power. And the isomorphism here is precisely the super Lie bracket

[,]:SSW 0. [\cdot,\cdot] : S \vee S \stackrel{\simeq}{\to} W_0 \,.

Then all that remains to be done is to decompose SSS \vee S. For the case where we work over the complex numbers, the result is given in theorem 2, on p. 396:

SS i=0 [m/4] m4iV i=0 [(m3)/4] m34iV, S \vee S \simeq \sum_{i=0}^{[m/4]} \wedge^{m-4i} V \oplus \sum_{i=0}^{[(m-3)/4]} \wedge^{m-3-4i} V \,, where mm is half the dimension of VV dim V=p+q=2m+1. \mathrm{dim}_\mathbb{C} V = p + q = 2 m +1 \,.

So for d=11d = 11 we find that SSV 2V 5V. S \vee S \simeq V \oplus \wedge^2 V \oplus \wedge^5 V \,.

This says that there are membranes (= 2-branes) and 5-branes in the game.

Recall that, by D’Auria-Fré-Castellani, this is a direct consequence of the fact that in (p+q)=(10+1)(p+q) = (10+1) dimensions, the ordinary unextended super Poincaré Lie algebra happens to have a 4-cocycle ψ¯Γ abψe ae bH 4(siso(10,1)), \bar \psi \wedge \Gamma^{a b} \psi \wedge e_a \wedge e_b \in H^4(\mathrm{siso}(10,1)) \,, which in turn implies that there is a super Lie 3-algebra extension of Baez-Crans type.

(By the way, I have the impression that by intregrating the 3-connection with values in the Lie 3-algebra over an S 1S^1-fiber, thereby “contracting away one leg” of this 4-cocycle, we get something related to the siso(10,1)\mathrm{siso}(10,1) Chern-Simons 3-form. If anyone knows anything about this, please drop me a note!)

Repository of useful Formulas

One of the general nice things about this paper is that it coherently and comprehensively lists lots of useful data that helps not to get lost in the jungle of superstudies, especially when comparing notatin and terminology used in math and physics, respectively.

For one, table 1 on p. 401 summarizes basic facts about Clifford algebras, their spinor modules and relates them to the physics terminology.

Then, the entire appendix B reformulates the entire paper in physicist’s notation, listing all these matrices that appear there (charge conjugation etc.) and relating them to the abstract formulation. Very useful. I tend to always forget this stuff after a while.

Posted at June 14, 2007 3:17 PM UTC

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5 Comments & 4 Trackbacks

Re: Polyvector Super-Poincaré Algebras

Obviously, the symmetric tensor product of two so(n) spinors can be decomposed into so(n) irreps. To see that you get 2- and 5-branes follows directly from the dimensions:

32*33/2 = (11 choose 1) + (11 choose 2) + (11 choose 5)

528 = 11 + 55 + 462

Is there more to it that this? In d dimensions, you probably want a term (d choose 1) because otherwise you don’t have translations, although Bars’ two-time physics seems to get away with a decomposition starting with (d choose 2).

Something else which has occurred to me is the following: is there a reason why the polyvector fields have to commute with the spinors? It is certainly possible to cook up nilpotent Lie superalgebras with a grading of depth > 2. This would mean that in addition to the relations

{Qα, Qβ} = γαβμ Pμ + more

one would have e.g.

[Qα, Pμ] != 0.

IIRC, the analogous thing (with Q bosonic) happens if you consider the e7 grading with g0 = so(10) + su(2) + u(1).

Posted by: Thomas Larsson on June 15, 2007 3:07 AM | Permalink | Reply to this

Re: Polyvector Super-Poincaré Algebras


Yes, as I said, much of this is well known. I am not expert enough to tell at which points the paper I mentioned genuinely fills gaps that have been left open before.

Posted by: urs on June 15, 2007 9:27 AM | Permalink | Reply to this

Re: Polyvector Super-Poincaré Algebras

All I am saying is that the symmetrized tensor product of two spinor irreps can be decomposed into irreps. That is pretty obvious.

As for my second comment, I just realized that Nicolai’s e10 stuff, and covariantly West’s e11, does that for the bosonic subalgebra. You first embed iso(11) into igl(11) because you have gravity, and then embed igl(11) into e11, whose level decomposition certainly does not stop at level 1. Then you need to fill in the fermonic operators at half-integer levels, but I don’t think there has been much progress on that.

Posted by: Thomas Larsson on June 15, 2007 9:36 AM | Permalink | Reply to this

Re: Polyvector Super-Poincaré Algebras

You first embed iso(1)\mathrm{iso}(1) into igl(11)\mathrm{igl}(11) because you have gravity

Yeah, this is a point I was wondering about last time, already (very end of Nicolai on E10 and Supergravity):

is there a way to understand the e 10e_{10} description of supergravity with the others we are talking about here?

In all cases we are dealing with gravity, so “beacause we have gravity” cannot quite be what underlies the transition from iso(11)\mathrm{iso}(11) to igl(11)\mathrm{igl}(11), I ‘d think.

Something is going on here, which involves at least three different aspects of one underlying structure:

polyvector super Lie 1-algebra ? ? super Lie 3-algebra ? supere 10 \array{ && \href{}{ \text{polyvector super Lie 1-algebra} } \\ & {}^?\swarrow && \searrow^? \\ \href{}{\text{super Lie 3-algebra}} &&\stackrel{?}{\leftrightarrow}&& \href{}{\text{super} \;\;e_{10}} }

Posted by: urs on June 15, 2007 10:42 AM | Permalink | Reply to this

Re: Polyvector Super-Poincaré Algebras

In all cases we are dealing with gravity, so beacause we have gravity cannot quite be what underlies the transition from iso(11) to igl(11), I’d think.

In gravity, you treat spinors with vielbeins, so the relevant Lie algebra is the semidirect product vect(d) (x map(d, so(d)) (diffeomorphisms and local Lorentz transformations). gl(d) is the zero degree subalgebra of vect(d), and there is a 1-1 correspondence between their irreps; to the gl(d) irrep R corresponds a tensor field of type R.

Posted by: Thomas Larsson on June 15, 2007 12:17 PM | Permalink | Reply to this
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