Morita equivalence and quantization of singular coadjoint orbits

Determine what the Morita equivalence induced by the Marsden–Weinstein and Donaldson dual pairs gives for the coadjoint orbits of singular elements and for their quantization.

Background

The paper recalls dual-pair constructions associated with the Marsden–Weinstein and Donaldson frameworks and states that the corresponding manifolds are Morita equivalent. Its constructions identify symplectic structures on spaces of maps, vortex membranes, vortex sheets, weighted isotropic submanifolds, and singular 1-form densities.

The unresolved question concerns the consequences of this Morita equivalence specifically for coadjoint orbits represented by singular elements and for the quantization of those orbits.

References

Here are several open questions related to the introduction of symplectic structures on the augmented triples.

  1. What could be an interesting (and possible integrable in lower dimensions) analog of the binormal equation for membranes in this compressible setting?
  2. Is it possible to obtain such an analog of the binormal equation by means of a reduction or ansatz from the compressible Euler equation on $Q$, similar to the localized induction approximation in the incompressible setting?
  3. What could be an extension of the above symplectic structures to the semidirect product groups ${\rm Diff}_\sigma(S)\ltimes C\infty(S)$ and ${\rm Diff} (Q)\ltimes C\infty(Q)$.
  4. The dual pairs discussed by Marsden--Weinstein and (implicitly) by Donaldson imply the Morita equivalence of the corresponding manifolds. What would this equivalence give for the coadjoint orbits of the ``singular elements'' and for their quantization?
— Generalized Marsden-Weinstein symplectic structures  (2610.01401 - Khesin et al., 1 Oct 2026) in Section "Dynamics on membranes and open questions", subsection "The binormal equation and its analogs" (Section 5.1)