Why does deformation quantisation have anything to do with $\mathbf{E}_2$/little disks?

Question

Kontsevich proved that any Poisson manifold $M$ has a quantisation $\mathcal{O}_\hbar(M)$: an associative algebra recovering the $\mathcal{O}(M)$ with its Poisson bracket by taking $\hbar=0$. Later he and Tamarkin realised that the crucial step in his proof was secretly saying that the little disks $\mathbf{E}_2$ was formal, i.e. $P_2:=\text{H}^*(\mathbf{E}_2)\simeq \mathbf{E}_2$ as operads. This was applied to the $P_2$-algebra of polyvector fields on $M$.

From nlab, a physics heuristic for why 2d things show up is:

it was shown that [...] the Poisson model of a [...] Poisson manifold computes the star product in the deformation quantization [...]. One may think of this relation between the 2d Poisson sigma-model and quantum mechanics = 1d quantum field theory as an example of the Chern-Simons type holographic principle.

Is it currently known how to make this mathematically precise, or another way that gives a satisfying explanation why the construction of $\mathcal{O}_\hbar(M)$ should pass through $\mathbf{E}_2$?

Answer 1

On page 3 of Operads and Motives in Deformation Quantization, Kontsevich outlines how formality of $E_n$ interacts with deformation quantization. It seems step 1 is the answer to your question:

"STEP 1) On the cohomological Hochschild complex of any associative algebra acts the operad of chains in the little discs operad (Deligne’s conjecture). This result is purely topological/combinatorial."

In other words, although $C_*(E_2)$ is equivalent to $\mathrm{Pois}$ through the formality theorem, they do not automatically act on the same objects. In the case of Deligne's conjecture, we get a $C_*(E_2)$ action on the Hochschild complex, which is something like a homotopy coherent Poisson algebra structure. The Hochschild complex comes up in the discussion of deformation quantization because in some precise way it controls the deformations of the associative algebra.

I would also recommend reading the entire 4 step summary of the paper because it is very well written.

Answer 2

I don't know if this is helpful, but one common way of constructing $(C^*)$ algebras is by using twisted convolutions algebras of groupoids $G\rightrightarrows X.$ These look like \begin{equation*} (f\ast h)(g)=\int_{g_1g_2=g}f(g_1)h(g_2)e^{i\Omega(g_1,g_2)}\,d\mu\;, \end{equation*} where $f,h$ are functions on the space of arrows, $\Omega$ is a 2-cocycle on $G$ and $d\mu$ is a Haar measure. This construction (at least formally) makes sense on Lie 2-groupoids, with the domain of integration replaced by $g_1g_2\sim g$ (since composition isn't uniquely defined on a 2-groupoid). What this means is that we integrate over all 2-morphisms in the nerve of the 2-groupoid whose corresponding 1-morphisms are $g_1,g_2,g.$

So far this may not look like it has anything to do with disks, but a Poisson manifold $(M,\Pi)$ has a canonical Lie 2-groupoid associated to it, as does any Lie algebroid. Its space of arrows are Lie algebroid morpisms $$T[0,1]\to T^*M\,,$$ and its 2-morphisms are Lie algebroid morphisms $$X:TD\to T^*M$$ up to homotopy, relative to the boundary. Here, $D$ is the disk. There is a canonical 2-cocycle given by $$ X\mapsto\int_D X^*\Pi\;. $$ We can form the twisted convolution algebra now. The integral is over disks, which correspond to 2-morphisms in the groupoid. You might think that this twisted convolution algebra is equal to the deformation quantization, and this seems to be true in many cases. This construction is related to the Poisson sigma model you mentioned.