In [7], Hitchin showed that the data (∇, Φ),
comprising an SU(2) Yang–Mills–Higgs monopole in the Prasad–Sommerfeld limit on
ℝ3, encodes faithfully into an auxiliary rank 2 holomorphic vector bundle
E˜ over T, the total space of the holomorphic tangent bundle of ℙ1.
In this construction ℝ3 is viewed as a subset of
H0(ℙ1, [Oscr ](T)) ≅ [Copf ]3.
Generically, the restriction of E˜ to a line is trivial. (The image of a global section
ℙz ⊂ T, for z ∈ [Copf ]3, is referred
to here as a line on T.) Hence c1(E˜) = 0 and, for all
z ∈ [Copf ]3, there exists m ∈ {0} ∪ ℕ such that
E˜[mid ]ℙz ≅ [Oscr ](m) [oplus ] [Oscr ](−m). If
m [ges ] 1 then ℙz is a jumping line of E˜ of height m. The jumping lines are parameterized by an analytic set J ⊂ [Copf ]3,
which is stratified by height. When the monopole has charge k,
the height is bounded above by k. In this case we write
J = J1 ∪ … ∪ Jk, where
Ji parameterizes jumping lines of height i.
A priori, some Ji may be empty.
The analytic continuation of the monopole to [Copf ]3 has singularities over J. To see
this recall how the monopole data are recovered from E˜: very briefly, E˜ induces a
sheaf [Escr ] = π2*ε*E˜ over [Copf ]3 which is locally free away from
J2 ∪ … ∪ Jk, (π2 and ε
are defined in Section 2). A holomorphic connection and Higgs field are defined in
[Escr ] over [Copf ]3 null planes that cut out a given direction (see
[1, 7, 9]). On restriction to ℝ3, [Escr ]
gives a rank 2, SU(2) bundle and the holomorphic connection and Higgs
field give the monopole data. It is easy to see that the flat connections are singular
at points of J: for example, an analogous situation is described in [10].