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

by Dmitri Bykov, Andrei Smilga

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Submission summary

Authors (as registered SciPost users): Dmitri Bykov
Submission information
Preprint Link:  (pdf)
Date accepted: 2023-09-15
Date submitted: 2023-04-19 23:29
Submitted by: Bykov, Dmitri
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
  • Mathematical Physics
Approach: Theoretical


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)

Submission & Refereeing History

Reports on this Submission

Anonymous Report 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


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.


No significant weaknesses.


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.

  • validity: top
  • significance: good
  • originality: good
  • clarity: high
  • formatting: excellent
  • grammar: excellent

Anonymous Report 1 on 2023-8-15 (Invited Report)


The paper is well-written with a clearly stated results.

  • validity: top
  • significance: good
  • originality: high
  • clarity: top
  • formatting: excellent
  • grammar: excellent

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