Given a symplectic manifold $M$, one can consider its deformation quantization, i.e. an associative multiplication law on functions on $M$ that depends on a formal parameter $h$. When $M$ is the cotangent bundle of a manifold $X$, one essentially recovers the algebra of differential operators on $X$, or rather of $h$-differential operators $P(x, hd/dx)$. Modules over differential operators are well known to have interesting applications to PDE and other topics, so it is natural to hope that modules over deformation quantization would be interesting as well.

However, what one really encounters in PDE and quantum mechanics is some sort of modules where two formal parameters are involved: $h$ and $\exp(1/h)$. What we present in the talk is a construction of modules over a bigger algebra that contains not only expressions $P(q,p,h)$ as in deformation quantization but also $\exp(f(q,p)/h)$. Here $q$ and $p$ are local Darboux coordinates on $M$.

Note that from a module over differential operators on $X$ one can pass to a sheaf on $X$ (the module is a generalized bundle with a flat connection, and the sheaf is its De Rham complex). Since our construction (for a cotangent bundle) is the enlarged algebra of differential operators, one can ask whether its sheaf-theoretic counterpart exists. This is indeed the case. This sheaf-theoretical counterpart is given by Tamarkin's category of sheaves on $X \times \mathbb{R}$ ($t$ on $\mathbb{R}$ roughly corresponds to $\exp(t/h)$). This construction was recently generalized by Tamarkin to a general symplectic manifold.

The above constructions are conjecturally connected by some version of a Riemann-Hilbert correspondence (as $D$-modules and sheaves are). They have some of the features possessed by the Fukaya category of a symplectic manifold. We do not know of any functor going in either direction. The talk will give a broad overview of the topic, not assuming any prior knowledge and using the example of the plane for much of the construction.

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