# Is there much theory of superalgebras acting on manifolds by alternating polyvector fields?

Usual story: vector fields on $M$, with their Lie bracket, form a Lie algebra. We can consider "actions" of some other Lie algebra ${\mathfrak g}$ on $M$ by looking at Lie homomorphisms ${\mathfrak g}\to Vec(M)$.

Now soup up $Vec(M)$ to alternating multivector fields (i.e. sections of $\Lambda^\bullet TM$), and extend the Lie bracket to the Schouten bracket, making it into a graded Lie superalgebra. (Careful: for the grading to work right, a section of $\Lambda^k TM$ must have degree $k-1$.)

My question: Has there been much theory developed of graded Lie superalgebras acting on (non-super) manifolds?

For example, if ${\mathfrak g}$ is generated by a single generator $\pi$ of degree $1$, with $[\pi,\pi]=0$ (which one would ordinarily only expect in even degree), then ${\mathfrak g}$-manifolds are Poisson manifolds.

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I think this is Markdown's fault, but the link, at least on my computer, is bad. It should point to en.wikipedia.org/wiki/… – Theo Johnson-Freyd Oct 30 '10 at 21:52
So, if I understand right, you're suggesting: a classical manifold $M$, a graded Lie superalgebra $\mathfrak g$, and a homomorphism of graded Lie superalgebras $\mathfrak g^\bullet \to \Gamma(\Lambda^{\bullet+1}TM)$, and your question is: are such things studied? – Theo Johnson-Freyd Oct 30 '10 at 21:57
Fixed, and yes, that's the question. – Allen Knutson Oct 31 '10 at 15:34

## 1 Answer

I am aware of one context in which something similar to this has been studied. I apologise in advance for "tooting my own horn" since this is based on a paper I co-authored in 2008 and some other work of mine.

The context is that of the geometric construction of Lie superalgebras, which arises quite naturally in the study of supersymmetric supergravity background. No Physics is necessary to understand the ideas, though -- so do read on!

The geometric data depends on the supergravity theory in question, but it will invariably include a lorentzian spin manifold $(M,g)$ and a connection $D$ on the spinor bundle. (There may be other fields and the spinor bundle might be twisted by a line bundle with its own connection, but such details are peripheral to the main story.) Spinor fields in the kernel of $D$ (subject, perhaps, to additional algebraic conditions) are called (supergravity) Killing spinors and they form a vector space we will denote $\mathfrak{k}_1$.

The symmetric square of the spinor bundle can be decomposed in terms of exterior powers of the tangent bundle and hence one can "square" Killing spinors to obtain polyvector fields satisfying some differential conditions. For example, in so-called Freund-Rubin backgrounds, these polyvectors are dual to so-called special Killing forms.

Let $\mathfrak{k}_0$ denote the image of the Killing spinors under the squaring map. Then in many cases, on the super-vector space $\mathfrak{k} = \mathfrak{k}_0 \oplus \mathfrak{k}_1$ there is a Lie superalgebra structure, called the Killing superalgebra of the background, where the $\mathfrak{k}_1 \otimes \mathfrak{k}_1 \to \mathfrak{k}_0$ component of the Lie bracket is precisely the squaring map of spinor fields. And in some cases, but not in general, the $\mathfrak{k}_0 \otimes \mathfrak{k}_0 \to \mathfrak{k}_0$ component of the bracket is induced from the Schouten bracket of polyvectors. The polyvectors which appear have odd degree, whence they are in the even subspace of the Lie superalgebra.

A similar construction works also in riemannian geometry and is related to a geometric construction of certain exceptional Lie superalgebras such as $F_4$ and $E_8$. Although in those constructions only vector fields were considered in $\mathfrak{k}_0$ and it would be interesting if there were an extension involving higher-rank polyvector fields.

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No apology necessary -- that's what it means to get an answer from an expert! – Allen Knutson Nov 2 '10 at 1:48