The n-Category Café

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
arXiv:hep-th/0311107v2

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

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 $W_0$ above consists of various copies of exterior powers $$\wedge^p V$$ (called “polyvector spaces”) of the underlying vector space $V$.

Now, like ordinary Einstein gravity may be conceived as a gauge theory for $\mathrm{iso}(3,1)$, 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 $\mathrm{iso}(3,1)$, supergravity theories have special solutions which globally respect parts of the super Poincaré symmetry.

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

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

K.S. Stelle
BPS Branes in Supergravity
arXiv:hep-th/9803116v2.

Now, this is especially interesting for us, because on the other hand, as $n$-Café-regulars have heard us say before, at least some of these $p$-branes should really correspond to certain $(p+1)$-functors $$(p+1)\mathrm{Cob} \to (p+1)\mathrm{Hilb} \,.$$ To indicate the categorification step, I like to speak of $(n =p+1)$-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 $$\mathrm{sugra}(10,1) \in 3\mathrm{sLie}$$ 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 $\mathrm{iso}(10,1)$.

So it seems that there is a close relation between

a) super Lie $n$-algebras $g_{(n)}$ 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 $(n+1)$-algebra $\mathrm{DER}(g_{(n)})$.

(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

$$W_0 = \sum_p \wedge^p V$$

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

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

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

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

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

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

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 $(p+q) = (10+1)$ dimensions, the ordinary unextended super Poincaré Lie algebra happens to have a 4-cocycle $$\bar \psi \wedge \Gamma^{a b} \psi \wedge e_a \wedge e_b \in H^4(\mathrm{siso}(10,1)) \,,$$ 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 $S^1$-fiber, thereby “contracting away one leg” of this 4-cocycle, we get something related to the $\mathrm{siso}(10,1)$ 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.