Resurgence and Quantization (4/6)

par Prof. Maxim KONTSEVICH (IHES)

vendredi 5 mai 2017, 10:30 → 12:30 Europe/Paris

Centre de conférences Marilyn et James Simons (IHES)

Cours des Professeurs permanents de l'IHES

 

There are two canonical ``quantizations'' of symplectic manifolds:

\begin{itemize}

\item Deformation quantization, associating with any ($C^{\mathrm{\infty }}$, 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

${\mathcal{O}}_M[[\hslash ]]$ where the ``Planck constant'' $\hslash$ is formal parameter.

\item Fukaya category $\mathcal{F}(M,\omega )$ associated to a \emph{real} $C^{\mathrm{\infty }}$ 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 $A_{\mathrm{\infty }}$-category (an analog of triangulated category) linear over the Novikov field consisting of formal sums

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

\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 $\hslash$ family of categories of holonomic modules over the quantized space, and this family of categories for $\hslash \ne 0$ is equivalent

to the Fukaya category of $M$ considerd as a $C^{\mathrm{\infty }}$ manifold, endowed with the symplectic form $\mathrm{\Re }(\omega /\hslash )$

and $B$-field $\mathrm{\Im }(\omega /\hslash )$.}

 

 

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 $\hslash$ to a holomorphic function.

 

Video https://youtu.be/m3chNHYXN9A?list=PLx5f8IelFRgHnsNrcsc-n16Mtpui3qDLn