Degeneracy loci of families of Dirac operators

Generalizing some results from R. Leung's thesis, we compute, in rational cohomology, the Poincare dual of the degeneracy locus of the family of Dirac operators parameterized by the moduli space of projectively anti-self-dual $\SO(3)$ connections. This is the first step in a program to derive a relation between the Donaldson and spin invariants.


Introduction
The definition of the spin invariants of a smooth four-manifold, due to V. Pidstrigach and A. Tyurin in [25], is sufficiently similar to that of the Donaldson invariant to suggest that there must a formula relating the two. In this note, we perform a computation which will be useful in deriving such a relation. This relation, together with relations between the Donaldson and Seiberg-Witten invariants and between the spin and Seiberg-Witten invariants given by the SO(3) monopole cobordism, [10,9,8], should be an important tool for deriving more explicit forms of these relations and possibly constraints on these invariants.
For E w κ → X a complex, rank-two vector bundle over a smooth, closed, oriented, fourmanifold, the Donaldson invariants, [3,18], are defined by integrating µ-classes over the moduli space M w κ of projectively anti-self-dual connections on E w κ . If s = (ρ, W ) is a spin c structure on X, then each unitary connection A on E w κ defines a Dirac operator on the spin u structure t = (ρ⊗id E , W ⊗E) of index n a (t). If n a (t) ≤ 0, then by [5,27] the subspace J Λ,w κ ⊂ M w κ , defined in 2.3, of connections whose Dirac operator has one-dimensional kernel is, for generic perturbations, a smooth submanifold of codimension 2(1 − n a (t)). In this note, we generalize some results of [20] by computing the Poincare dual of J Λ,w κ .
Theorem 1.1. Let E w κ → X be a complex, rank-two vector bundle over a smooth, closed, oriented, four-manifold with b 1 (X) = 0. Let M w κ be the moduli space of projectively anti-selfdual connections on E w κ and let K w κ ⊂ M w κ be any compact, codimension-zero submanifold. Let t be a spin u structure on X with p 1 (t) = κ, c 1 (t) = Λ, and n a (t) ≤ 0. Let J Λ,w κ ⊂ M w κ be the degeneracy locus of the spin u structure t. Then, as an element of rational cohomology, the Poincare dual of J Λ,w κ in K w κ is where µ(t) and Ω are the µ-classes defined in (3.5) and the coefficients f i,j,k are determined by the recursion relation (3.13) and are given as where the functions J i (z) are given by . The spin invariants of [25] are defined by integrating µ-classes over the Uhlenbeck closure of J Λ,w κ : . Theorem 1.1 implies that the Poincare dual of J Λ,w κ can be written as an expression in µ-classes, which we denote as µ(T Λ,w κ ). One might thus expect the Donaldson and spin invariants to be related by . However, R. Leung's thesis, [20], shows that (1.2) only holds when n a (t) = 0. If V(T Λ,w κ ) is the geometric representative of µ(T Λ,w κ ) used to define the Donaldson invariant, then Theorem 1.1 shows that the intersections of V(T Λ,w κ ) and J Λ,w κ with any compact subset of M w κ are cobordant but the same is not true for their closures in the Uhlenbeck compactification. Thus, while Theorem 1.1 does not give a relation between the Donaldson and spin invariants, it does allow one to localize the error in (1.2) near the lower strata of the Uhlenbeck compactification. In §4, we discuss the role this localization plays in producing a formula for the error in (1.2). We do not expect such a formula for the error to be explicit but rather one of the type appearing in the Kotschick-Morgan conjecture, [16], or in the SO(3)-monopole cobordism formula, [8]. The SO(3)-monopole cobordism formula gives relations between both the Donaldson and Seiberg-Witten invariants and between the spin and Seiberg-Witten invariants but these relations are given by unknown polynomials with universal coefficients depending on topological invariants of X. These coefficients have been determined in some cases but not all. Having a direct relation between the Donaldson and spin invariants would give us additional leverage for determining these coefficients. Indeed, it is possible that this additional leverage will reveal new, topological constraints of the type appearing in [6] on these invariants of the smooth structure.

Preliminaries
2.1. ASD moduli space. Let E w κ → X be a complex, rank-two, Hermitian vector bundle over a smooth, closed, oriented four-manifold with b 1 (X) = 0, w = c 1 (E w κ ) and κ = . Fix a connection A w on the line bundle det(E w κ ). For k ≥ 2, define A w κ to be the L 2 k completion of the space of unitary connections A on E w κ with A det = A w . Let A w, * κ ⊂ A w κ be the irreducible connections. Define G w κ to be the L 2 k+1 completion of the group of special-unitary gauge transformations of E w κ . Define quotient spaces, κ , let F A be the curvature of A, let (F + A ) 0 be the self-dual and trace-free component of the curvature, (F + A ) 0 ∈ Ω + (su(E)). Then define the moduli space of projectively anti-self-dual connections on E w κ by The following result is well-known.
Theorem 2.1. [13,4] For generic choices of metric g on X, the moduli space M w κ is a smooth, orientable manifold of dimension where χ(X) is the Euler characteristic and σ(X) the signature of X.

2.2.
Spin u structures and Dirac operators. Let (W, ρ W ) be a spin c structure on a four-dimensional, Riemannian manifold (X, g) as defined in [23,17,10]. Let E → X be a rank-two, complex vector bundle. A spin u structure on X is the pair . A more intrinsic definition of these concepts appears in [10]. We define characteristic classes of t by A spin connection A W determines a bijection between A w κ and connections on V respecting the Clifford multiplication map, A → A ⊗ A W . This map associates a Dirac operator The index of such a Dirac operator is given by The degeneracy locus. For t a spin u structure on X with characteristic classes κ = − 1 4 p 1 (t), Λ = c 1 (t), and w an integer lift of w 2 (t), define the degeneracy locus Hence, Further discussions and applications of (2.6) appear in [1, p. 110].

Chern classes of the index bundle
To apply (2.6) to prove Theorem 1.1, we use the following version of the Atiyah-Singer index theorem for families.
The first obstruction to applying Theorem 3.1 to the family of Dirac operators D A for [A] ∈ M w κ is that the bundle playing the role of E for this family, As discussed [20, p.34-35], the approach to this problem in [20] only works when the universal SO(3) bundle defined in (3.3) admits a U(2) lifting. To overcome this difficulty in general, we introduce a larger space with the same rational homotopy type.
is a K(Z 2 , 1)-bundle and hence defines an isomorphism in rational homotopy.
We define the following subspaces of B w, * κ (S) analogously, The following is immediate from Theorem 2.2 and the surjectivity of π S .
For K w κ ⊂ M w κ a compact subset, consider the family of Fredholm operators, parameterized by ( . This is not a compact family, but it admits a stabilization in the sense that there is a surjective map from a finite-dimensional, trivial bundle onto the cokernels of these operators. The construction of the stabilization follows immediately from the independence of D A from Φ and the gauge equivariance of the Dirac operator (see [9,Thm. 3.19] for an example of this type of argument). This stabilization allows the following definition of an index bundle for this family of operators.
To apply Theorem 3.1 to compute ch(D Λ,w κ ), we observe that the index bundle D Λ,w κ is defined as the index bundle of a family of Dirac operators obtained by twisting the spin c structure (ρ, W ± ) by Thus we must compute ch(E w κ (S)). To this end, we introduce the following cohomology classes. Recall that for β ∈ H • (X; R), µ(β) ∈ H 4−• (B w, * κ ; R) was defined by Lemma 3.6. Assume X is a smooth, closed four-manifold with b 1 (X) = 0. Let E w κ → X be a complex, rank-two vector bundle with c 1 (E w κ ) = w and c 2 (E w κ ) = κ + 1 4 w 2 . Let β 1 , . . . , β d be a basis for H 2 (X)/ Tor and let x ∈ H 0 (X) be a generator. Let β * i = PD[β i ]. Let Q ij = Q X (β i , β j ) and let P ij be the inverse matrix of Q ij . Let µ i = π * S µ(β i ) and ℘ = π * S µ(x). Then, for E w κ (S) and F w κ (S) as defined in (3.2) and (3.4) respectively, as elements of rational cohomology.
Proof. The first equality follows from observing that . The second equality then follows from [20, 5.4.1] which we now review. By Lemma 3.2, π S is an isomorphism in real cohomology so it suffices to compute p 1 (F w κ ). By [4, Prop. 5.1.15], H • (B w, * κ ; R) is a polynomial algebra in µ(β i ) and µ(x) so we can write To compute the coefficient a 0 , observe that Finally, observe that The linear independence of µ(β i ) then implies that a i,j = −4P i,j as required.
Lemma 3.6 and the following will yield the Chern character of E w κ (S). Lemma 3.7. Let E → Y be a rank two, complex Hermitian vector bundle. Let c 1 = c 1 (E) and p 1 = p 1 (su(E)). Then Proof. By the splitting principle, we may assume that The lemma then follows from the observation that x + y = c 1 (E) and (x − y) 2 = p 1 (su(E)).
Corollary 3.8. Continue the notation of Lemma 3.6. Then, Our computation of ch(D Λ,w κ )) requires the following algebraic result.
Lemma 3.9. Continue the notation of Lemma 3.6. Let h ∈ H 2 (X; R) satisfy h = d k=1 h k β * k . Then, Proof. The first equality follows from and the equality k P ij Q jk = δ i k where δ i k is the Kronecker delta. The second equality follows from computing , and the definition of P ij as the inverse of the matrix Q jℓ .
Proposition 3.10. Continue the notation of Lemma 3.6. For c 1 (t) = i λ i β * i define Then Proof. Applying Theorem 3.1, Corollary 3.8 and c The second equality of Lemma 3.9 implies that  Using the preceding, we expand the factor in (3.7): Then the definition of µ(t) and Ω in (3.5), applying (3.8), and the first equality of Lemma 3.9 to (3.7) yield − ch(D Λ,w κ ) = 2 1 + 1 2 1 × c 1 (t) + If we observe that n a (t) + κ = (c 1 (t) 2 − σ(X))/4, then we can write This completes the proof of Proposition 3.10 Proof of Theorem 1.1. Because π * S is an isomorphism on rational cohomology, it suffices to compute the pullback by π * S of the Chern class. After Proposition 3.10 and Lemma 3.5, the only remaining step in the proof of Theorem 1.1 is to compute the Chern class from the Chern character.

Comparing the spin and Donaldson invariants
We now discuss the sources of the error in the equality (1.2) and the role of Theorem 1.1 in deriving a correct formula for the relation between the spin and Donaldson invariants. Define Then the µ-map of (3.3) extends to The error in the equality , arises from the difference in the geometric representatives V(T Λ,w κ ) and J Λ,w κ for the cohomology class µ(T Λ,w κ ) defined following Theorem 1.1. The intersections which define D w X (T Λ,w κ z) and P Λ,w X (z) respectively both have compact support in the interior of a compact subset N w κ (z) ⊂ V(z)∩M w κ . While there is a cobordism between the geometric representatives V(T Λ,w κ ) and J Λ,w κ , this cobordism need not be supported in N w κ (z). To understand the error (4.1), one must therefore study the ends of V(z) ∩ M w κ .
The lower strata ofM w κ have the form M w κ−ℓ ×Σ where Σ ⊂ Sym ℓ (X) is a smooth stratum. BecauseV(z) is, roughly, transverse to the lower strata, the gluing maps of Taubes, [26] (see also [7] or [14,§III.3.4]), present the ends of V(z) ∩ M w κ as a fiber bundle, where the fiber M is a cone, given by the product of moduli spaces of framed, centered, anti-self-dual connections on S 4 .
In [20], Leung computes the error (4.1) assuming that n a (t) = −1. He constructs an Ozsvathian "cap" (in the sense of [24]) C Λ,w κ ⊂ B w, * κ , for the end of V(z) ∩ M w κ using the description in (4.2). If W Λ,w κ is the intersection of V(z) ∩ M w κ with the end (4.2), then Leung then argues that V(T Λ,w κ ) ∩ C Λ,w κ is empty so the right-hand-side of (4.3) is given by D w X (zT Λ,w κ ) while the left-hand-side of (4.3) is given by Some partial computations for the case n a (t) = −2 also appear in [20]. However, computing the error (4.1) for general n a (t) presents some technical challenges. For n a (t) < −2, the closureV(z) ∩M w κ intersects more than one lower stratum ofM w κ and thus more than one open set of the form (4.2) is needed to cover the ends of V(z) ∩ M w κ . Constructing caps in the general case is thus far more challenging not only because of the greater topological complexity of the picture (4.2) for lower strata but also because the open sets covering the ends overlap. Some approaches to this type of problem involving multiple open sets have appeared in [16,12,8,19].
Thus, while Theorem 1.1 does not give a general relation between the Donaldson and spin invariants, this result does allow one to localize the error (4.1) near the lower strata of M w κ . The examples computed by Leung in [20] and similar computations for the Kotschick-Morgan conjecture, [19], and for the SO(3) monopole cobordism, [11], suggest that the error (4.1) has the form described in the following conjecture. Conjecture 4.2. Continue the notation and hypotheses of Theorem 1.1. The error E Λ,w X (z) of (4.1) is given by a polynomial in D w X (zO Λ,w κ ), Q X , and Λ, where O Λ,w κ is an expression in the characteristic classes of D Λ,w κ−i where i > 0. The coefficients of this polynomial depend only on topological invariants of X, κ, and Λ 2 . .