Open Problems in Integrable Systems — Integrability and Quantisation

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $(M^{2n},\omega)$, $n\geq2$ be a compact prequantizable Kähler manifold with a positive holomorphic line bundle $L$ of curvature $-i\omega$. For $k\to\infty$, put $\hbar=k^{-1}$ and $\mathcal H_k=H^0(M,L^{\otimes k})$. A Berezin--Toeplitz operator family has an expansion $P_k\sim\sum_{j\geq0}k^{-j}\Pi_k M_{p_j}\Pi_k$, where $\Pi_k$ is the orthogonal projection to $\mathcal H_k$ and $M_{p_j}$ multiplies by the smooth symbol $p_j$. Its principal symbol is $p_0$.

Let $P_{1,k},\ldots,P_{n,k}$ be commuting self-adjoint Toeplitz families with principal-symbol map $F=(p_1,\ldots,p_n)$, where all $\{p_i,p_j\}=0$ and $dp_1,\ldots,dp_n$ are independent almost everywhere on $M$. A focus-focus block at a nondegenerate singular point is a four-dimensional transverse symplectic block whose commuting quadratic normal forms, after an invertible linear change of components, are \[q_1=x_1\xi_1+x_2\xi_2,\qquad q_2=x_1\xi_2-x_2\xi_1.\] Here $(x_1,x_2,\xi_1,\xi_2)$ are Darboux coordinates on that block. In dimension four these are the rank-zero focus-focus model; in higher dimensions they occur alongside the remaining regular directions and nondegenerate transverse blocks. Focus-focus singularities in the question include these higher-dimensional cases. The joint spectrum consists of $n$-tuples of simultaneous eigenvalues. An error $O(k^{-\infty})$ means an error $O(k^{-A})$ for every $A>0$.

Statement recorded in the supplied source.

Construct complete Bohr--Sommerfeld quantization rules near focus-focus singular fibers in Berezin--Toeplitz quantization: geometric integrality conditions, including corrections to every order in $k^{-1}$, that are necessary and sufficient for $E\in\mathbb R^n$ to belong to the joint spectrum modulo $O(k^{-\infty})$. [S2]

Short English statement

Give all-order geometric quantization conditions describing the joint spectrum near a focus-focus singularity for commuting Toeplitz operators.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2100703 (AMR-020-0703). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Alexey Bolsinov, Vladimir S. Matveev, Eva Miranda and Serge Tabachnikov, Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018), Problem 7.3 and its complete preceding definitions. Original question and full discussion. https://arxiv.org/abs/1804.03737.