Gauge theory for embedded surfaces, II

Gauge theory for embedded surfaces, II 21
W \(Σ ∪ Σ ′ ) a metric which is incomplete at Σ (extending either smoothly or
with an orbifold singularity) and has a cylindrical end modeled on Y ′ × R + ,
where Y ′ is the circle bundle over Σ ′ . In this situation we shall write W o for
the manifold W \Σ ′ and we shall write M α (W o ,Σ; R) for the moduli space of
anti-self-dual connections which are twisted with holonomy α at the surface Σ
and which are asymptotic to the flat connections of the component R⊂R(Y ′ )
on the end of the manifold.
The basic results on transversality and compactness, valid in either of the two
situations, are still applicable to moduli spaces M α (W o ,Σ; R) which combine
both features. For the compactness theorem, it is necessary to account for loss
of action from three sources: bubbling off of action at points of Σ, during which
both k and l may change; bubbling off of action at points of W o away from
the surface Σ; and finally loss of action on the end of the manifold. The gluing
results described in the previous subsection will also extend without essential
change to incorporate the additional embedded surface.
The sort of moduli space just discussed will be considered in this paper only
for one particular type of manifold W ,aruled surface over Σ. IfV is a complex
line bundle of degree n over Σ, then the ruled surface W is the manifold obtained
by adding a single point at infinity to each fibre of V ;thusW is sphere bundle
over Σ. The ruled surface contains two distinguished copies of Σ, namelythe
zero section of the bundle V and the section at infinity. We shall refer to these
usually as Σ and Σ ′ ; their self-intersection numbers are n and −n respectively.
The manifold W o = W \Σ ′ can be identified with the original line bundle V and
will be equipped with a metric which has a cylindrical end at infinity and has
an orbifold singularity along the zero-section Σ with cone-angle 2π/ν.
Since the monopole number l of a twisted connection is determined by the
data in the neighbourhood of Σ, there is no problem in distinguishing the
monopole number for the moduli spaces M α (W o ,Σ; R) on the cylindrical-end
manifold. For the instanton number, we have to make a choice of some convention
for our notation. In the cylindrical-end setup described in the previous
subsection, we indexed the moduli spaces by the total action κ, with the understanding
that κ =C.S.(R) (mod 1) if the moduli space Mκ(X o ; R) istobe
non-empty. This is not a good choice when there is an embedded surface with
non-trivial holonomy, since the twisting contributes a quantity 2αl + α 2 Σ·Σ to
the total action. We shall therefore adopt the convention that M α τ,l (Wo ,Σ; R)
denotes the moduli space of twisted connections with total action
κ(A)=τ+2αl − α 2 Σ·Σ. (3.24)
This convention still ensures that τ =C.S.(R) (mod 1), which is independent of
α. In particular, when R = R+ (the case we are most concerned with), τ will be
an integer, and in this case we shall usually write k instead. The need for these
notations will not arise sufficiently often to cause much confusion.