nLab rational map

Idea

Given an irreducible variety $X$ and a variety $Y$ a rational map $f: X\dashrightarrow Y$ (notice dashed arrow notation) is an equivalence class of partially defined maps, namely the pairs $(U, f_U)$ where $f_U$ is a regular map $f_U: U\to Y$ defined on dense Zariski open subvarieties $U\subset X$ and the equivalence is the agreement on the common intersection.

The notion of an image of a rational map is nontrivially defined, see that entry. A rational map $f: X\dashrightarrow Y$ is dominant if its image as a rational map is the whole of $Y$.

The composition of rational maps $g\circ f$ where $f: X\dashrightarrow Y$ and $g: Y\dashrightarrow Z$ is not always defined, namely it is even possible that the image of $f$ lies out of any dense open subset in $Y$, where $g$ is defined as a regular map. The composition is defined as the class of equivalence of pairs $(g_V\circ f_U|, f_U^{-1}(V))$ where $U\subset X$ and $V\subset Z$ are open dense subsets and $f_U^{-1}(V)\neq \emptyset$ if such exist and undefined otherwise.

If $f$ is dominant then in this situation is the composition $g\circ f$ is always defined.

Related concepts

• rational function

• birational map

• birational geometry

• rational variety

• unirational variety

• homotopy of rational maps

• $\mathbb{C}P^N$ sigma-model

References

General

Textbook account:

• Igor Shafarevich, Section 1.4 of: Basic Algebraic Geometry 1 – Varieties in Projective Space, Springer 1977, 1994, 2013 (pdf, doi:10.1007/978-3-642-57908-0)

Review:

• János Kollár, Section 3 of: Algebraic hypersurfaces, Bull. Amer. Math. Soc. 56 (2019), 543-568 (arXiv:1810.02861, doi:10.1090/bull/1663)

Lecture notes:

• Daniel Plaumann, Rational Functions and Maps (pdf, pdf), Lecture 5 in Classical algebraic geometry 2015

See also:

• Wikipedia, Rational mapping

Exposition for the case of maps from the Riemann sphere to a complex projective space:

• Sheldon Katz, Sections 2,3 of: Enumerative Geometry and String Theory, Student Mathematical Library 32 (2006) (ISBN:978-1-4704-2143-4, spire:739788)

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)

Skyrmions from rational maps

The following is a list of references on the construction of Skyrmion-solutions of the Yang-Mills field via rational maps from the complex plane, hence holomorphic maps from the Riemann sphere, to itself, akin to the Donaldson-construction of Yang-Mills monopoles.

The original idea:

• Conor J. Houghton, Nicholas Manton, Paul Sutcliffe, Rational Maps, Monopoles and Skyrmions, Nucl. Phys. B510 (1998) 507-537 (arXiv:hep-th/9705151, doi:10.1016/S0550-3213(97)00619-6)

Further discussion:

• Steffen Krusch, $S^3$ Skyrmions and the Rational Map Ansatz, Nonlinearity 13:2163, 2000 (arXiv:hep-th/0006147)

• Nicholas S. Manton, Bernard M.A.G. Piette, Understanding Skyrmions using Rational Maps, in: Casacuberta C., Miró-Roig R.M., Verdera J., Xambó-Descamps S. (eds.) European Congress of Mathematics. Progress in Mathematics, vol 201. Birkhäuser, Basel. 2001 (doi:10.1007/978-3-0348-8268-2_27, arXiv:hep-th/0008110)

• Richard Battye, Paul Sutcliffe, Skyrmions, Fullerenes and Rational Maps, Rev. Math. Phys. 14 (2002) 29-86 (arXiv:hep-th/0103026)

• W.T. Lin, Bernard M.A.G. Piette, Skyrmion Vibration Modes within the Rational Map Ansatz, Phys. Rev. D77:125028, 2008 (arXiv:0804.4786, doi:10.1103/PhysRevD.77.125028)

On quantization of Skyrmions informed by homotopy of rational maps:

• Steffen Krusch, Homotopy of rational maps and the quantization of Skyrmions, Annals of Physics Volume 304, Issue 2, April 2003, Pages 103-127 (doi:10.1016/S0003-4916(03)00014-9, arXiv:hep-th/0210310)

• Steffen Krusch, Skyrmions and Rational Maps, talk at KIAS 2004 (pdf, pdf)

• Steffen Krusch, Quantization of Skyrmions (arXiv:hep-th/0610176)

the impact of which, on the computation of atomic nuclei, is highlighted in:

• Richard A. Battye, Nicholas Manton, Paul Sutcliffe, p. 23 of: Skyrmions and Nuclei, pp. 3-39 (2010) (doi:10.1142/9789814280709_0001) in: Mannque Rho, Ismail Zahed (eds.) The Multifaceted Skyrmion, World Scientific 2016 (doi:10.1142/9710)

See also:

• Derek Harland, Paul Sutcliffe, Rational Skyrmions $[$arXiv:2307.09355$]$