# Is there a higher homotopical spinor theory?

Gabriele Vezzosi and I have been musing on the following. Consider the standard double cover $\mathop{\rm Pin}_{n} \to \mathrm{O}_{n}$, whose kernel is $\mathbb Z/2\mathbb Z$. This allows to associate with every $\mathrm{O}_{n}$-principal bundle $E$ over a space $S$ a class in $\mathrm H^{2}(X, \mathbb Z/2\mathbb Z)$, which is well known to be the second Stiefel--Whitney class of $E$.

Now, suppose that we have given a simplicial group $G$, with a central simplicial subgroup $Z$ such that the quotient $G/Z$ is homotopy equivalent to a topological group $H$. Assume that $Z$ is homotopy equivalent, as a simplicial group, to an Eilenberg--MacLane group $K(A, m)$, where $m$ is a non-negative integer and $A$ an abelian group. Then, if we are not mistaken, we should obtain a map from $H$-principal bundles on $S$ to $\mathrm H^{m+2}(S, A)$, and thus a characteristic class for $H$-principal bundles: the group $G$ gives a homomorphism from $H$ to the classifying simplicial group $B\,K(A,m) = K(A, m + 1)$, which yields a homorphism $B\,H \to B\,K(A,m+1) = K(A, m + 2)$, and this should give the characteristic class.

Now, the question is: for each $m ≥ 0$, is there a simplicial group $\mathop{\rm Pin}_{n}^{(m)}$, with a surjective homomorphism $\mathop{\rm Pin}_{n}^{(m)} \to \mathrm{O}_{n}$, whose kernel is a central simplicial subgroup $K(\mathbb Z/2\mathbb Z, m)$, such that the associated characteristic class is the $(m+2)^{\rm nd}$ Stiefel--Whitney class?

If there isn't, is there some analogue in which $G$ is something less than a simplicial group (maybe an $H$-group, or something along that line)? We think this should be possible, but we'd be interested in an explicit construction, more than in an abstract existence theorem.

Ultimately, we'd like to have an analogous construction for simplicial schemes over a field, in order to study higher Hasse--Witt classes, in the sense of Jardine; but we were wondering if such a construction, in the simpler case of simplicial sets, is known to topologists.

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I edited the post to make clear I was asking for a central extension. – Angelo Nov 30 '10 at 9:34

Certainly, such a gadget exists. Let $\newcommand{\Pg}{B\mathrm{Pin}_n^{(m)}}\Pg$ be the homotopy fiber of the map $BO_n\to K(\mathbb{Z}/2, m+2)$ classifying the Stiefel-Whitney class. If we realize the homotopy fiber as the path-space, then the map $\Pg\to BO_n$ is a Serre fibration. Apply the Kan loop group construction to get a surjective map of simplicial groups which does what you want, I think. (You don't ask for a central extension, and I don't think my construction gives one; the kernel of the map of simplicial groups is not going to be abelian, though it models $K(\mathbb{Z}/2,m)$.)

As for an explicit construction, there's quite a bit of work on giving explicit constructions of $\mathrm{String}_n$, which sits in a central extension $K(\mathbb{Z},2)\to \mathrm{String}_n \to \mathrm{Spin}_n$. I particularly like Chris Schommer-Pries's paper "Central extensions of smooth $2$-groups and a finite dimensional string $2$-group", which produces an extension of $2$-groups, from which one can extract an extension of topological or simplicial groups.

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Thanks, I'll think about it (simplicial homotopy theory is really not my field, I need time to digest about it). I meant to ask for a central extension, though. – Angelo Nov 29 '10 at 17:01
Should be easy enough to make it central: form the fiber product in simplicial groups, where you can model K(Z/2, m+1) and its path space by simplicial abelian groups. More generally, I believe that simplicial objects in "groups with a central extension" is a model for the homotopy theory of triples (X,Y,f) where f is a pointed connected space, Y is a 1-connected HZ-module spectrum, and f is a map from X to the zeroth space of Y. – Jacob Lurie Nov 29 '10 at 18:33
By the way, if "higher spinor" is interpreted in the sense of quantum field theory, then it is not (at least not directly) the Stiefel-Whitney classes that correspond to higher spin structures, but the higher fractional Pontryagin classes: a 1-dimensional supersymmetric sigma model QFT exists on a target space that has spin structure, a 2d susy QFT on a target that has in addition a string-structure, obstructed by p1/2, then a 6d SQFT on a target that has what was called a fivebrane structure, obstructed by p2/6. And in fact a 0d susy QFT needs an orientation. So we are lifting here through... – Urs Schreiber Nov 29 '10 at 22:34
... the Whitehead tower of the orthogonal group fivebrane -> string -> spin -> so -> o . There is a refinement in differential cohomology of the obstructions classes to these higher lifts, and these yield the famous "anomaly poloynomials" in heterotic string theory and in electric-magnetic dual heterotic (= fivebrane) theory. – Urs Schreiber Nov 29 '10 at 22:37


Note that this goes more or less backwards in the following observations: If $1\rightarrow Z\rightarrow G\rightarrow H\rightarrow1$ is a central extension, then as multiplication $Z\times G\rightarrow G$ is a group homomorphism we get a map $\hw Z\times\hw G\rightarrow\hw G$ which is an action of the simplicial group $\hw Z$ on $\hw G$ making $\hw G\rightarrow\hw H$ a $\hw Z$-torsor giving a classifying map $\hw H\rightarrow\hw\hw Z$.

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