Resurgence and Quantization (6/6)
Cours des Professeurs permanents de l'IHES
There are two canonical « quantizations'' of symplectic manifolds:
begin{itemize}
item Deformation quantization, associating with any ($C^infty$, analytic, algebraic over field of characteristic zero)
symplectic manifold $(M,omega)$ a sheaf of catgeories, which is locally equivalent to categories of modules over quantized algebras
$mathcal{O}_M[[hbar]]$ where the « Planck constant'' $hbar$ is formal parameter.
item Fukaya category $mathcal{F}(M,omega)$ associated to a emph{real} $C^infty$ symplectic manifold,
with the morphism space between objects corrsponding to Lagrangian subvarieties $L_1,L_2subset M$ given by Floer homology $HF(L_1,L_2)$.
This is an $A_infty$-category (an analog of triangulated category) linear over the Novikov field consisting of formal sums
[c_1 e^{-frac{A_1}{hbar}}+ c_2 e^{-frac{A_2}{hbar}}+dots, quad text{ where } c_iin mathbb{Q},A_iin mathbb{R},lim_i A_i=+infty]
end{itemize}
The goal of my course is to unify these two quantizations, proposing the following conjecture, a generalization of Riemann-Hilbert correspondence (joint work with Y.Soibelman):
{it For a symplectic algebraic variety $(M,omega)$ over $mathbb{C}$ together with an approriate data at infinity, the formal deformation quantization gives an analytic in $hbar$ family of categories of holonomic modules over the quantized space, and this family of categories for $hbarne 0$ is equivalent
to the Fukaya category of $M$ considerd as a $C^infty$ manifold, endowed with the symplectic form $Re(omega/hbar)$
and $B$-field $Im(omega/hbar)$.}
The general construction is a mixture of Fukaya categories, deformation quantization and of wall-crossing formalism.
As a corollary we obtain
the resurgence properties of WKB solutions, conjectured long time ago. Exponentially small corrections coming from pseudo-holomorphic discs, upgrade a divergent formal power series in $hbar$ to a holomorphic function.
