Gauge theory for embedded surfaces, II

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14 P. B. Kronheimer and T. S. Mrowka (In [KrM], we considered only the case that α lay in the interior of the interval, but the formula is valid at the end-points too.) The monopole number l is equal to αΣ·Σ, whichismabove, by Proposition 5.9 of [KrM]. Substituting α = m/n gives the formula in (c). The cases (a) and(b) follow from the same formulae when α = 0 (a trivial case) and α = 1 2 respectively. See also [MMR]. ⊓⊔ ForacomponentR⊂R(Y) and a real number κ equal to C.S.(R) mod1, we write Mκ(X o ; R) for the moduli space of anti-self-dual connections which have action κ and are asymptotic to a flat connection in R. As another point of notation for the representation variety, we write R s for what we informally call the smooth part of the component R. For the components Rm, this is the whole component, while for R+ and R− it is the irreducible part (which may be the whole of R− in the last case when n is odd). More generally, R s is that part where the obstruction map in the Kuranishi description is zero. At a point ρ ∈R s , the tangent space to the representation variety is the first cohomology of Y with twisted coefficients in the adjoint representation of SU(2), a space which we denote by H 1 (Y,ρ). (ii) The dimension formulae. Let ρ be a flat connection in the smooth part R s of a component R⊂R(Y)and let [A] ∈ Mκ(X o ; ρ) be an anti-self-dual connection asymptotic to ρ. The deformations of [A] within Mκ(X o ; ρ) are described by a Kuranishi-type deformation theory, based on a space of connections which decay exponentially to ρ on the end of the manifold; there is a deformation complex (the “δ-decay” complex of [T] and [MMR]) Ω 0 δ → Ω1 δ → Ω+ δ , which is appropriate for the problem when δ is sufficiently small. We shall write Hi A (X o ) for the cohomology groups of this complex. The full moduli space Mκ(X o ; R) involves also the possible deformations of ρ. Inthecasethatρ∈Rsand [A] is irreducible, the appropriate Fredholm deformation complex can be cast in the form Ω 0 δ e1 −→ Ω 1 δ ⊕ H 1 (Y ; ρ) e2 −→ Ω + , (3.5) This is essentially the complex Eδ(A) of [MMR]. A local model for Mκ(X o ; R) at [A] is given, as usual, by the zero set of a smooth map ϕ from the first to the second cohomology groups of the complex. We shall say that [A] is a regular point of the moduli space if e2 is surjective. In this case the moduli space is a smooth manifold and its tangent space is the first cohomology kere2/ im e1. What we need next is an expression for the formal dimension of Mκ(X o ; R s ): that is, the difference in dimensions of the spaces in (3.5), which will be the true dimension of the moduli space at a regular point. The formulae depend a little δ
Gauge theory for embedded surfaces, II 15 on the sign of the self-intersection number n; we will assume henceforth that n is positive and will write the formula as dim Mκ(X o ; R s )=8κ−3(b + (X) − b 1 (X)+1)+η(R). (3.6) We choose to express this in terms of the invariants of the closed manifold X. Note also that 8κ is not necessarily an integer. The term η(R) involvesanηinvariant of the boundary data as well as the genus and self-intersection number of the surface. To use the dimension formula, we need to know this term: Proposition 3.7. For positive self-intersection number n, thetermη(R)in the dimension formula is given by (a) η(R+)=0; (b) η(R−)=0; (c) η(Rm)=−(8m2 /n − 4m +2g−3). ⊓⊔ Remarks. These formulae, or rather, the equivalent calculations for the index of the Kuranishi complex with exponential decay, are essentially worked out in [T] (section 12), though the results are stated only modulo 8. The mod 8 ambiguity can be resolved by going through the proofs, though in the case of (c) the proof is only briefly discussed. We shall give an independent verification of (c) in subsection (vi) by using the gluing theorem (described below) to relate the dimension formula here to the dimension formula for the twisted moduli spaces M α k,l . An alternative treatment is given in [MMR]. It is helpful to think of the term η(R) as the ‘efficiency’ of the boundary condition R, in that a larger value of η means a larger dimension of the moduli space for given action. In these terms, the next lemma gives the condition under which the components R+ and R− are more efficient than all the components Rm. Lemma 3.8. If the genus g and self-intersection number n satisfy the inequality 2g − 2 > 1 2n +1,withn>0,thenη(Rm)is negative for all m. Ifthestronger inequality 2g − 2 > 1 2 n +2 holds, then η(Rm) is always less than −1. Proof. We only have to look at the expression in part (c) of (3.7) as a quadratic function of m and calculate the discriminant b 2 − 4ac, 4 2 − 4·(8/n)(2g − 3) = 16 (n − 4g +6) n = 32 n 1 ( 2n +1−(2g − 2)), to see that η(Rm) is always negative under the stated condition. The rider to the lemma is an immediate corollary. Note that η(Rm) achieves its maximum when m = n/4, which is in the middle of its range of validity. ⊓⊔
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