Appears in collections : Algebraic Analysis in honor of Masaki Kashiwara's 70th birthday, Fields medallists - 1998, Maxim Kontsevich

For complex number $q, 0 < |q| < 1$, denote by $A_q:=\mathbb C\langle X^{\pm 1},Y^{\pm 1}\rangle / (relation $Y X=qXY$) the corresponding quantum torus algebra. By the q-version of Riemann-Hilbert correspondence (essentially due to Ramis, Sauloy and Zhang, 2009), the category of holonomic $A_q$-modules is naturally equivalent to the abelian category of coherent sheaves on elliptic curve $E_q:=\mathbb C^{\times}/q^{\mathbb Z}$ endowed with two anti-Harder-Narasimhan filtrations. I propose a generalization of this correspondence to the higher-dimensional case $A_q^{\otimes n}$, $n>1$ (joint work in progress with Y. Soibelman). The hypothetical description of the “Betti side” of RH-correspondence combines the categories of constructible sheaves with a given microlocal support, and the bounded derived category of coherent sheaves on the abelian variety Enq.