Does the $(\mathbb Z/2)$-graded isomorphism $E_n \cong E_{n+2}$ have any nice properties?

This question assumes everything is dg. Let's decide to work over the "field" $\mathbb Q[\mu,\mu^{-1}]$ where $\mu$ has homological degree $+2$. Then chain complexes are just $\mathbb Z/2$-graded. In the sequel I will write "$X$" for $X\otimes_{\mathbb Q}\mathbb Q[\mu,\mu^{-1}]$ if the object $X$ was more standardly defined over $\mathbb Q$. If in fact the object $X$ was originally defined in terms of topological spaces, then I will write $X$ for $\mathrm C_\bullet(X)$, always with coefficients in $\mathbb Q[\mu,\mu^{-1}]$.

Let $n \geq 2$ be an integer. There are (zig-zags of quasi)isomorphisms of dg operads:

$$E_n \cong P_n \cong P_{n+2} \cong E_{n+2}.$$

The operad $E_n$ is the operad of little $n$-dimensional rectangles inside a large $n$-dimensional rectangle. The operad $P_n$ is the operad controlling $(n-1)$-shifted Poisson algebras, i.e. commutative algebras $A$ equipped with a Lie algebra structure on $A[n-1]$ which is a derivation in each variable.

The middle isomorphism $P_n \cong P_{n+2}$ is totally canonical: the data of a Poisson bracket $\{,\}$ is no different from the data of the Poisson bracket $\mu\{,\}$, since $\mu$ is invertible. The first and last isomorphisms are pretty horrible, and exist only because of the deep theorem called "Kontsevich formality". In particular, there is no canonical such isomorphism.

Is there anything that can be said about the composition $E_n \overset\sim\to E_{n+2}$? More precisely, is it possible to choose the two formality morphisms so that this composition has any nice property at all? Here are the types of "nice properties" that I could imagine:

• Perhaps this isomorphism is somewhat canonical? At least, perhaps a choice of it can be written down without going through all the machinery of Kontsevich formality?
• My (probably faulty) understanding is that you can get all formality morphisms uniformly once you choose a Drinfeld associator. Perhaps you need to use the same one?
• Both $E_n$ and $E_{n+2}$ are "Hopf operads", meaning there's a natural way to tensor together algebras for either one. Perhaps this isomorphism can be chosen to respect this structure? Actually, they are cocommutative Hopf algebras; perhaps that makes the question easier or harder?
• Perhaps the isomorphism (if both the above work out well) can be explained in terms of some topological construction?
• Kontsevich formality has an interpretation in terms of path integrals and quantum field theory (and gauge fixing and renormalization and counterterms and all that sort of stuff). Perhaps even if the isomorphism itself is not canonical, it can be interpreted as a "partial path integral" for a "relative" or "compactified" or "equivariant" or ... quantum field theory?
• A warm-up question: I don't really know what $\mathbb Q[\mu,\mu^{-1}]$ "means" in terms of derived geometry, but I'd love to. It's something like the functions on $\mathrm B \mathbb G_m$, or $\mathrm B^2 \mathbb G_m$, or something. I mean, $\mathrm{Spec}(\mathbb Q[\mu,\mu^{-1}])$ is a "(derived) abelian group scheme" that integrates the line in degree $2$, thought of as an abelian Lie algebra, and that line is something like $\mathrm B^2$(usual 1-dimensional Lie algebra). What is $\mathbb Q[\mu,\mu^{-1}]$ really? – Theo Johnson-Freyd Dec 13 '14 at 0:42
• In physics, I've been told that $\mu$ is a "mass", or perhaps "mass squared", but I don't claim to understand why. I've also been told that $\mu = \hbar$, a claim that I understand even less. – Theo Johnson-Freyd Dec 13 '14 at 0:48
• Here's a comment-question. The passage from a space $X$ to its $2$-periodic rational homology $H_{\bullet}(X) \otimes \mathbb{Q}[\mu, \mu^{-1}]$ is quite destructive. It factors through a less destructive thing, namely complex K-homology $KU_{\bullet}(X)$, and this in turn factors through an even less destructive thing, namely the free KU-module spectrum $KU \wedge \Sigma_{+}^{\infty} X$. – Qiaochu Yuan Dec 13 '14 at 8:02
• In particular, the passage to $2$-periodic rational homology that really lets us identify the $E_n$ and $E_{n+2}$ operads factors through a passage to free KU-module spectra, getting us a pair of operads in KU-module spectra which are isomorphic after passing to rational homotopy. But are they already isomorphic before doing that? (And can we write down an isomorphism having anything to do with complex Bott periodicity?) – Qiaochu Yuan Dec 13 '14 at 8:03
• @QiaochuYuan Unfortunately it seems very unlikely that those two operads in $KU$-modules are equivalent. – Tyler Lawson Dec 13 '14 at 19:09