Gauge theory for embedded surfaces, II

22 P. B. Kronheimer and T. S. Mrowka
(vi) The index formula.
As promised, we shall end this section with a proof of the dimension formula
(3.7)(c). Let X o = X\Σ be a manifold with cylindrical end Y × R + and
let A be a connection in Mκ(X o ; Rm). Let ρ = r(A) be the limiting flat connection
in Rm. As in the previous section let W o be the line bundle over Σ,
with cylindrical end Y ′ × R + , containing Σ as the zero-section. Because W o \Σ
contracts onto Y ′ , which is diffeomorphic to Y with reversed orientation, the flat
connection ρ defines a flat connection B on W o \Σ, and this connection defines
point in the moduli space M α (W o ,Σ; Rm)ofα-twisted anti-self-dual connections
asymptotic to the component Rm. Hereαis m/n, the monopole number
l for the connection B is m (see [KrM], Proposition 5.9 again), and since B has
zero action the formula (3.24) gives τ = −m 2 /n, so
B∈M α −m 2 /n,m (W o ,Σ; Rm). (3.25)
Lemma 3.26. The formal dimension of the moduli space (3.25) is 2g − 2 if the
self-intersection number n is positive, and 2g − 1 if n is negative.
Proof of 3.7(c), assuming the lemma. The manifolds X o and W o can be joined on
their cylindrical ends, and the gluing construction provides a grafted connection
A#B. (We are not really interested in whether this approximate anti-self-dual
connection can be deformed to an anti-self-dual one; the question is not material
for the index calculation, in which we work quite formally.) Since the closed manifold
formed from X o and W o is diffeomorphic to the original X, we can regard
A#B as an α-twisted connection on the closed pair (X,Σ). Its instanton and
monopole numbers are κ+τ = κ−m 2 /n and m respectively. From the dimension
formula for the α-twisted moduli spaces, we obtain the formal dimension of the
moduli space corresponding to the topological type of A#B as
dim Mk,l(X,Σ)=8k+4l−3(b + − b 1 +1)−(2g − 2)
=8κ−(8m 2 /n − 4m) − 3(b + − b 1 +1)−(2g − 2).
Combining this formula with the previous lemma, we obtain from the gluing
result (3.22)
dim Mκ(X o ; Rm) = dim Mk,l(X,Σ) − dimM α τ,l(W o ,Σ)+2g−1
=8κ−3(b + − b 1 +1)−(8(m 2 /n) − 4m +2g−3)
if n is positive. This is the formula predicted in (3.7). ⊓⊔
Proofof3.26.The formal dimension of this moduli space at B can be written
as
2g − 1+Iδ,
where the 2g comes from the dimension of Rm, the 1 is the dimension of H 0 B
since B is reducible to S 1 ,andIδ is the index of an exponentially weighted
deformation complex, for a small positive weight δ (see [T], [MMR]).