Gauge theory for embedded surfaces, II

Gauge theory for embedded surfaces, II 23
We are free to make a convenient choice of metric for the index calculation, so
we choose a cone-like singularity with cone-angle 2π/ν, whereνis some multiple
of |n|. We can then pass to the |n|-fold branched cover of W o , branched along
Σ. We call the covering space ˜ W o ; it is a line bundle of degree ±1, according to
the sign of n.
The connection B can be regarded as an orbifold connection on W o , and lifts
to a trivial bundle on ˜ W o . We can write the adjoint bundle of this trivial bundle
as R ⊕ C, so that the covering transformation acts trivially on the R summand
andwithweightmon the C summand. To calculate the index Iδ, we need only
calculate the invariant part of the cohomology of the exponentially weighted
deformation complex with these trivial coefficients on ˜ W o . With coefficients R,
this complex has trivial H 0 and H 1 , while H 2 is the space of self-dual, harmonic,
L 2 2-forms, which has dimension 0 or 1 according as n is negative or positive.
The covering transformation clearly acts trivially on this space, so we find that
Iδ is 0 or −1, depending on the sign of n. This is the desired result. ⊓⊔
Remark. The dimension formula can, of course, be arrived at from many directions.
We have chosen the above route because it turns up an interesting point
concerning the comparison of the dimension formulae for the cylindrical-end and
twisted moduli spaces. In the argument given, we started with a connection A
on the cylindrical-end space X o , and formed a twisted connection A#B on the
pair (X,Σ) by attaching a flat connection B. What we have done is to compare
the formal dimensions for A and A#B. The proof shows that the corresponding
moduli spaces differ in dimension by the quantity Iδ; so when the self-intersection
number is negative, the two dimensions are equal, while in the positive case the
dimension of the twisted moduli space is one less, due to the self-dual harmonic
form on W o .
4. Invariants from cylindrical end moduli spaces
The Yang-Mills moduli spaces associated to manifolds with cylindrical ends are
also the source of possible invariants. We describe the two simple cases which
we shall use. Since these are only needed as stepping stones in our proof of
Theorem 1.1, the fact that the invariants we define really are independent of,
for example, the Riemannian metric is not of importance. Indeed, we shall later
prove (5.10) that the invariants we construct here can be expressed in terms of
the ordinary invariants, and their ‘invariance’ can be deduced from this, at the
same time showing that we have defined nothing new.