nLab Yang-Mills monopoles as rational maps -- references

Identification of Yang-Mills monopoles with rational maps

The following lists references concerned with the identification of the (extended) moduli space of Yang-Mills monopoles (in the BPS limit, i.e. for vanishing Higgs potential) with a mapping space of complex rational maps from the complex plane, equivalently holomorphic maps from the Riemann sphere $\mathbb{C}P^1$ (at infinity in $\mathbb{R}^3$) to itself (for gauge group SU(2)) or generally to a complex flag variety such as (see Ionnadou & Sutcliffe 1999a for review) to a coset space by the maximal torus (for maximal symmetry breaking) or to complex projective space $\mathbb{C}P^{n-1}$ (for gauge group SU(n) and minimal symmetry breaking).

The identification was conjectured (following an analogous result for Yang-Mills instantons) in:

• Michael Atiyah, Section 5 of: Instantons in two and four dimensions, Commun. Math. Phys. 93, 437–451 (1984) (doi:10.1007/BF01212288)

Full understanding of the rational map involved as “scattering data” of the monopole is due to:

• Jacques Hurtubise, Monopoles and rational maps: a note on a theorem of Donaldson, Comm. Math. Phys. 100(2): 191-196 (1985) (euclid:cmp/1103943443)

The identification with (pointed) holomorphic functions out of $\mathbb{C}P^1$ was proven…

…for the case of gauge group $SU(2)$ (maps to $\mathbb{C}P^1$ itself) in

• Simon Donaldson, Nahm’s Equations and the Classification of Monopoles, Comm. Math. Phys., Volume 96, Number 3 (1984), 387-407, (euclid:cmp.1103941858)

…for the more general case of classical gauge group with maximal symmetry breaking (maps to the coset space by the maximal torus) in:

• Jacques Hurtubise, The classification of monopoles for the classical groups, Commun. Math. Phys. 120, 613–641 (1989) (doi:10.1007/BF01260389)

• Jacques Hurtubise, Michael K. Murray, On the construction of monopoles for the classical groups, Comm. Math. Phys. 122(1): 35-89 (1989) (euclid:cmp/1104178316)

• Michael Murray, Stratifying monopoles and rational maps, Commun. Math. Phys. 125, 661–674 (1989) (doi:10.1007/BF01228347)

• Jacques Hurtubise, Michael K. Murray, Monopoles and their spectral data, Comm. Math. Phys. 133(3): 487-508 (1990) (euclid:cmp/1104201504)

… for the fully general case of semisimple gauge groups with any symmetry breaking (maps to any flag varieties) in

• Stuart Jarvis, Euclidian Monopoles and Rational Maps, Proceedings of the London Mathematical Society 77 1 (1998) 170-192 (doi:10.1112/S0024611598000434)

• Stuart Jarvis, Construction of Euclidian Monopoles, Proceedings of the London Mathematical Society, 77 1 (1998) (doi:10.1112/S0024611598000446)

and for un-pointed maps in

• Stuart Jarvis, A rational map of Euclidean monopoles via radial scattering, J. Reine angew. Math. 524 (2000) 17-41(doi:10.1515/crll.2000.055)

Further discussion:

• Charles P. Boyer, B. M. Mann, Monopoles, non-linear $\sigma$-models, and two-fold loop spaces, Commun. Math. Phys. 115, 571–594 (1988) (arXiv:10.1007/BF01224128)

• Theodora Ioannidou, Paul Sutcliffe, Monopoles and Harmonic Maps, J. Math. Phys. 40:5440-5455 (1999) (arXiv:hep-th/9903183)

• Theodora Ioannidou, Paul Sutcliffe, Monopoles from Rational Maps, Phys. Lett. B457 (1999) 133-138 (arXiv:hep-th/9905066)

• Max Schult, Nahm’s Equations and Rational Maps from $\mathbb{C}P^1$ to $\mathbb{C}P^n$ [arXiv:2310.18058]

• L. A. Ferreira, L. R. Livramento: Harmonic, Holomorphic and Rational Maps from Self-Duality [arXiv:2412.02636]

Review:

• Alexander B. Atanasov, Magnetic monopoles and the equations of Bogomolny and Nahm (pdf), chapter 5 in: Magnetic Monopoles, ‘t Hooft Lines, and the Geometric Langlands Correspondence, 2018 (pdf, slides)

On the relevant homotopy of rational maps (see there for more references):

• Graeme Segal, The topology of spaces of rational functions, Acta Math. Volume 143 (1979), 39-72 (euclid:1485890033)