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Monopole harmonics on $\mathbb{CP}^{n-1}$
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 $\mathbb{CP}^{n-1}$ 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 1) 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 $\mathbb{CP}^n$ 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 $\mathbb{CP}^n$ 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.