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# Collection Maxim Kontsevich - Resurgence and Quantization

There are two canonical quantizations'' of symplectic manifolds:

-- 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.

-- 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 $L_1,L_2\subset M$ 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

$$c_1 e^{-\frac{A_1}{\hbar}}+ c_2 e^{-\frac{A_2}{\hbar}}+\dots, \quad \text{ where } c_i\in \mathbb{Q},A_i\in \mathbb{R},\lim_i A_i=+\infty$$

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 $\hbar\ne 0$ 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.

Appears in collections : Cours de l'IHES, Fields medallists - 1998

Organizer(s) Emmanuel Ullmo
Date(s) 4/25/17 - 5/12/17
• 01:49:21
published on April 26, 2017

## 1/6 Resurgence and Quantization

By Maxim Kontsevich

02:03:00
published on April 29, 2017

## 2/6 Resurgence and Quantization

By Maxim Kontsevich

01:52:33
published on May 3, 2017

## 3/6 Resurgence and Quantization

By Maxim Kontsevich

01:57:28
published on May 7, 2017

## 4/6 Resurgence and Quantization

By Maxim Kontsevich

02:04:31
published on May 10, 2017

## 5/6 Resurgence and Quantization

By Maxim Kontsevich

01:12:53
published on May 12, 2017

## 6/6 Resurgence and Quantization

By Maxim Kontsevich

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