Let X be a smooth scheme, then an infinity enchancement of QCoh(X) has an E(infinity) structure and in particular an E(n) structure for any n. In this paper, http://arxiv.org/abs/0805.0157 Ben-Zvi, Francis, and Nadler compute E(n) Hochschild cohomology of this category as $T_X[-n]$. I believe there is also separate work of Francis explaining how E(n) Hochschild cohomology (QCoh(X)) gives deformations of these categories. I've been trying to get some kind of idea as to what this result means. The case n=1, I understand pretty well, but I am curious about all of the other odd cases in particular. The thing that I find strange is that the vector space and Lie-structure of the deformation space are very similar for all of the odd n and yet I believe from degree considerations, it must be the case that more often than not, deformations of the E(2k+1) structure are trivial as deformations of the E(2k-1) structure(the notable exception coming from commutative deformations of the underlying scheme).
Just to keep things concrete(I hope), let's just assume X is Spec(A) and n=3. Can these deformations be made reasonably explicit in the same way the n=1 case is say by deforming some sort of associator map? What about in some special case such as that of a function f(these will be some kind of derived deformations in the sense that will break the Z-grading)?