nLab
rational map

Contents

Contents

Idea

Given an irreducible variety XX and a variety YY a rational map f:XYf: X\dashrightarrow Y (notice dashed arrow notation) is an equivalence class of partially defined maps, namely the pairs (U,f U)(U, f_U) where f Uf_U is a regular map f U:UYf_U: U\to Y defined on dense Zariski open subvarieties UXU\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:XYf: X\dashrightarrow Y is dominant if its image as a rational map is the whole of YY.

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

If ff is dominant then in this situation is the composition gfg\circ f is always defined.

References

General

Textbook account:

Review:

Lecture notes:

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

See also:

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

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 P 1\mathbb{C}P^1 (at infinity in 3\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 P n1\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:

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

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

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

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

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

and for un-pointed maps in

Further discussion:

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):

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:

Further discussion:

On quantization of Skyrmions informed by homotopy of rational maps:

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

Last revised on August 13, 2021 at 15:10:55. See the history of this page for a list of all contributions to it.