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Monopole harmonics on CPn−1
by Dmitri Bykov, Andrei Smilga
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Dmitri Bykov |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2302.11691v2 (pdf) |
| Date accepted: | 2023-09-15 |
| Date submitted: | 2023-04-19 23:29 |
| Submitted by: | Bykov, Dmitri |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We find the spectra and eigenfunctions of both ordinary and supersymmetric quantum-mechanical models describing the motion of a charged particle over the CPn−1 manifold in the presence of a background monopole-like gauge field. The states form degenerate SU(n) multiplets and their wave functions acquire a very simple form being expressed via homogeneous coordinates. Their relationship to multidimensional orthogonal polynomials of a special kind is discussed. By the well-known isomorphism between the twisted Dolbeault and Dirac complexes, our construction also gives the eigenfunctions and eigenvalues of the Dirac operator on complex projective spaces in a monopole background.
Published as SciPost Phys. 15, 195 (2023)
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2023-8-30 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2302.11691v2, delivered 2023-08-30, doi: 10.21468/SciPost.Report.7742
Strengths
1. Clear discussion of solving the Schrödinger equation on CPn with and without monopoles.
2. Lucid description of using homogeneous vs inhomogeneous coordinates.
3. Re-derives classic result of Kuwabara for eigenfunctions in presence of monopoles using an explicit parametrization and relation to Young tableaux.
4. Extends calculation of spectrum to the supersymmetric CPn model and relates it to the spectrum of a certain Dirac operator (which captures the Dolbeault complex on a Kähler manifold) and the Witten index.
Weaknesses
No significant weaknesses.
Report
This is a beautifully written paper which solves an interesting, concrete problem in quantum mechanics. It easily meets the requirements to be published in SciPost.