Open Problems in Integrable Systems — Integrability and Quantisation

Definitions and mathematical statement

For a circle action $S^1\curvearrowright M$, its fixed-point set is \[ M^{S^1}=\{x\in M:t\cdot x=x\text{ for every }t\in S^1\}. \] A finite number $N=|M^{S^1}|$ requires hypotheses ensuring finitely many isolated fixed points; a general fixed set may have positive-dimensional components. The supplied sentence gives a map $\mu:M\to S^1$, but does not explicitly specify the action whose fixed points are to be counted.

In the generalized moment-map interpretation, a symplectic circle action with infinitesimal generator $X$ can have a circle-valued moment map whose local real lifts satisfy $d\mu=\iota_X\omega$, up to the chosen sign convention. This differs from an ordinary Hamiltonian circle action with a globally real-valued Hamiltonian $H:M\to\mathbb R$. The research summary's Hamiltonian terminology must not silently replace the source's circle-valued target.

Question (Á. Pelayo). Can studying the spectrum of a quantisation of the circle-valued map $\mu:M\to S^1$ give progress in counting the fixed points of the intended action? More concretely, after fixing the action, a quantisation model, and its spectral data, seek spectral information that determines or constrains the fixed-point count. The original request allows progress rather than demanding an exact reconstruction theorem in a specified class. It does not define the quantisation of a circle-valued observable, the spectral parameter or multiplicities, or whether point counts or counts of components are intended. The supplied summary reports no verified result specifically settling this spectral-counting request.

Short English statement

Use the spectrum of a quantised circle-valued moment map to obtain information about fixed-point counts. The action, finiteness assumptions, and circle-valued quantisation model need specification.

Sources

• Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018), Problem 7.9, attributed to Á. Pelayo, https://arxiv.org/abs/1804.03737; source-TeX extraction, accessed 2026-07-29.
• The supplied summary mentions Atiyah--Bott equivariant localization and the Duistermaat--Heckman formula as context, without publication or theorem identifiers connecting them to this spectral question.
• Local registry data/unsolvedmath/problems.json, permanent ID $2100709$, source code catalogue entry, Open Problems in Integrable Systems --- Integrability and Quantisation. The record labels this open and OPEN-TRIAGE, with NEEDS_REVIEW source-status and rights qualifications.