Gauge theory for embedded surfaces, II

26 P. B. Kronheimer and T. S. Mrowka
(ii) An invariant for W o .
As in section 3(v), let W o be the total space of a complex line bundle of positive
degree n over Σ, equipped with a metric with a cylindrical end at infinity. We will
also suppose that the metric has an orbifold singularity along the zero-section
(a copy of Σ which we henceforth identify with Σ itself) so that the cone-angle
in the transverse directions is 2π/ν for some large integer ν. Let I ⊂ (0, 1
2 )be
an interval of compactness for ν (in the sense of section 2(ii)), and α an element
of I. Some restrictions on α, I and ν will be imposed shortly.
We shall suppose that the genus of Σ is odd and at least three, and we shall
consider the moduli space of twisted connections
for the special value of l given by
M α 0,l(W o ,Σ; R s +) (4.9)
l = 1
4 (2g − 2). (4.10)
As explained in the introduction, the significance of this value of l is that the
formal dimension of the moduli space of twisted connections then coincides with
the formal dimension of the corresponding ordinary moduli space, which in this
case is the space M0(W o ; R s + ) ∼ = R s +
. The formal dimension of this ordinary
moduli space coincides with its actual dimension (the obstruction space in the
deformation complex is zero), so the formal dimension of (4.9) is equal to 6g −6,
the dimension of R+. We will also be concerned with the lower moduli spaces
for l ′ ≤ l.
M α 0,l ′(W o ,Σ; R s + ) (4.11)
Lemma 4.12. The moduli space (4.11) contains no flat or reducible connections,
and is empty unless l ′ is strictly positive.
Proof. There are no non-flat reductions to S 1 because there are no anti-self-dual
L 2 harmonic forms on W o when n is positive. There are no flat connections
because the holonomy on the circle fibres is supposed to be non-trivial near to
Σ but trivial at the end of the manifold, to meet the component R+. The last
clause follows from the Chern-Weil formula, which tells us that the topological
action of any solution in the moduli space is 2αl ′ − α 2 n. ⊓⊔
Lemma 4.13. There exists an α0, explicitly calculated in terms of g and n below,
such that if α is less than α0 and ν is sufficiently large (depending on α), then
any sequence of connections [Am] in the moduli space
M α 0,l (W o ,Σ; R+)
has a subsequence converging weakly to a connection A in
M α 0,l ′(W o ,Σ; R+)