Local Nahm transform and singularity formation of ASD connections

This paper develops a local analogue of the ADHM construction, which characterises ASD instantons defined over smooth bounded domains inside Euclidean $\mathbb{R}^4$ diffeomorphic to the 4-ball, in terms of infinite dimensional Hilbert spaces and bounded Hermitian linear operators satisfying an analogue of the ADHM equation. Morever, we describe the degeneration of this construction when a family of instantons develops a curvature singularity at the origin.


Introduction
This paper studies two related problems for ASD connections A defined on a bounded region B ⊂ R 4 with the standard Euclidean metric: local Nahm transform, and singularity formation. The unifying theme is to understand ASD connections by studying solutions to the coupled Dirac equation (i.e. 'Dirac fields').
The Nahm transform in various contexts is extensively studied (cf. e.g. [9][4][2] [5]). It resembles the Fourier transform in many ways, and is also closely related to the celebrated ADHM construction. The prototype Nahm transform [4] constructs ASD connections over a given 4-torus, starting from coupled Dirac fields attached to ASD connections over the dual torus. It enjoys remarkable properties, such as • One-irreducible ASD connections are preserved under the transform.
• The inverse Nahm transform reproduces the original ASD connection.
It is desirable to establish a local version of Nahm transform; some motivations from the twistor space perspective are discussed in [14]. Our main result in this direction, which borrows many techniques from [4], is Theorem 1.1. (Local Nahm transform, cf. Chapter 2 and 3) Let B be a bounded region inside the Euclidean R 4 , such thatB is diffeomorphic to the unit 4-ball with boundary. For a smooth ASD connection on a Hermitian vector bundle (E, A) overB, there is a bundle of Hilbert spaces with an ASD connection (Ê,Â) over the dual vector spaceR 4 , called the local Nahm tranform. Morever, the inverse Nahm transform is canonically isomorphic to (E, A).
The local Nahm transform admits an alternative description analogous to the ADHM construction [9]. The information of the bundleÊ is equivalent to the Hilbert space of solutions to the coupled Dirac equations , which we call the 'Bergmann space'. The information of the connectionÂ is equivalent to 4 bounded Hermitian operators on H 2 D , defined bŷ x µ = P 0 ○ x µ , µ = 1, 2, 3, 4, where x µ means multiplication by the coordinate function, and P 0 is the orthogonal projection from L 2 (B, E ⊗ S − ) to H 2 D . The ASD condition onÂ is equivalent to analgous to the ADHM equation. Thus (Ê,Â) is analogous to the ADHM data, albeit H 2 D is infinite dimensional, which brings forth some nuanced functional analytic features.
The inverse Nahm transform can also be equivalently described in this ADHM language. Given the ADHM data as above, the inverse Nahm transform bundleÊ is the subbundle of the trivial bundle H 2 D ⊗ S − , defined aŝ and we equipÊ ⊂ H 2 D ⊗ S − with the subbundle connectionÂ, which turns out to be again ASD. The main content of Theorem 1.1 is that (Ê,Â) reconstructs (E, A). Thus we have related ASD connections on domains to operator theory.
M. Atiyah suggests to the author that this picture may be related to Alain Connes's noncommutative geometry [7].
The second problem studied in this paper is singularity formation. Problem 1.2. Given a 1-parameter family {A t } t>0 of smooth ASD connections A t on the Hermitian bundle E over the closed ballB = B(R), such that as t → ∞, the connections converge smoothly away from the origin to A ∞ , and near the origin they are allowed to develop a curvature singularity. Describe the limiting behaviour of the Bergmann spaces H 2 D A t of Dirac fields for A t as t → ∞.
Remark. The singularity formation problem is local in nature, so we assume B is a ball for convenience. Morever, by possibly shrinking the ball, it makes sense to assume A ∞ has small L 2 curvature.
We develop a convergence theory in Chapter 4 and 5 (which can be read independent of Chapter 2 and 3). The Hamiltonian of the harmonic oscillator induces a Hermitian operator with discrete and non-negative spectrum. The spectral theory is our key tool to understand the spaces H 2 There is a positive number called the spectral gap, such that for sufficiently concentrated ASD connections, the eigenvalues are either greater than the gap, which form the so called large spectrum, or very close to zero, which form the so called small spectrum, for which the eigenstates are localised near the origin.
The dimension of the small spectrum is constant for large t. To describe the limiting behaviour of the large spectrum, we recall that Uhlenbeck's removable singularity theorem allows us to smoothly extend the ASD connection A ∞ across the origin to a smooth connection on a different bundleẼ over B. Then the large spectrum for A t converges in a natural sense to the spectrum of A ∞ . Morever, Theorem 1.3. (Convergence theory) There is a natural topological bundle over 0 < t ≤ ∞, whose fibres over 0 < t < ∞ are H 2 D A t and the fibre over ∞ is the orthogonal direct sum H 2 D A∞ ⊕ V , where V has the dimension of the small spectrum. Morever, certain natural operators on the spaces H 2 D A t can be extended continuously to the limit H 2 D A∞ ⊕ V . Remark. From the Nahm transform perspective, the fact that the bundle of Hilbert spaces splits into a direct sum, is analogous to the familiar picture of reducible connections.
An important quantity describing the curvature singularity formation is the instanton number, i.e. the amount of energy loss This is well known to have a more topological interpretation: for any 0 < r ≤ R, where CS(A) = 1 8π 2 Tr(dA∧A+ 2 3 A∧A∧A) is the Chern Simons form, defined up to an integer, and becomes uniquely defined by demanding continuity at t = ∞: We combine our theory of the local Nahm transform and our analytic theory of singularity formation, together with some rudiments of K-theory, to show This work was supported by the Engineering and Physical Sciences Research Council [EP/L015234/1], the EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London. The author is also funded by Imperial College London for his PhD studies.

The local Nahm transform
We define the local Nahm transform, and explain two different perspectives, which exhibit its analogy with the Nahm transform on the 4-torus and the ADHM construction of instantons on R 4 .

Nahm transform perspective
Suppose we are given a Hermitian bundle E with a smooth ASD connection A over the closure of a bounded connected domain B ⊂ R 4 with smooth boundary, then we can produce an ASD connection on some (infinite rank) Hilbert bundlê E →R 4 over the dual vector spaceR 4 as follows. First, we recall the Poincaré bundle P over R 4 ×R 4 , given by equipping the trivial line bundle with the connection where x µ are the standard coordinates on R 4 and z µ are the dual coordinates. For each fixed z ∈R 4 , the bundle E couples to the flat line bundle P z . Equivalently, one thinks of the coupled connection as a modified connection on E These A z are all gauge equivalent, but we keep them separate to make apparent the analogy with the Nahm transform on the 4-torus [4]. We introduce the coupled Dirac operators: The convention in this paper is that the Clifford operators are anti-self-adjoint, with Clifford relations so the Dirac operator is formally self adjoint.
The Weitzenböck formula implies that ker D + Az is zero. We define a bundlê E over R 4 with fibres given by all the L 2 solutions to the Dirac equation This sits inside the trivial Hilbert bundle overR 4 with fibre L 2 (B, E ⊗ S − ). Let P z be the projections to the kernels according to the orthogonal splitting The Sobolev space W 2,1 0 here imposes zero boundary condition; the subspace D + Az W 2,1 0 is closed. The projection operator P z admits the formula where G z is the Green's operator solving the Dirichlet problem for the coupled Laplacian . ThenÊ inherits the subbundle connectionÂ, with covariant derivative given bŷ ∇ = P zd , whered is the trivial connection. Proof. (Sketch) The same proof in [4], Theorem 1.5 works (but their curvature formula is off by a sign). This goes by taking a local orthonormal framing ofÊ, denotedf j (z) = ψ j z (x). Then one computes the connection coefficientŝ where the inner product comes from L 2 integration. The curvature components areF where The curvature formula shows the connectionÂ to be ASD.

ADHM perspective
The local Nahm transform can also be cast in a form which makes apparent the analogy with the ADHM data [9].
By gauge equivalence, it is easy to seeÊ z = exp(−2πiz)Ê 0 . This means we can alternatively think about the underlying bundle ofÊ as the trivial bundle with fibre isomorphic to H 2 D =Ê 0 = ker D − A ; this assigns a canonical trivialisation toÊ. Notice that for s ∈ H 2 D , so after identifying all fibres with H 2 D , the information of the connectionÂ is up to some constant given by the operatorsx µ = P 0 ○ x µ , defined in essentially the same way as the matrices appearing in the ADHM data, except that these operators act on infinite dimensional spaces.
Proof. By the above discussions, the connection matrix ofÂ in the canonical trivialisation is the constant matrix valued 1-formÂ = −2πi ∑ µxµ dz µ onR 4 . Thus the curvatureF =Â ∧Â, and the result is clear after expanding this.

Remark.
A useful intuition when one tackles analytical questions concerning the space H 2 D of solutions to the coupled Dirac equation, is the analogy between the Dirac equation and the Cauchy-Riemann equation. The space of L 2 solutions to the Cauchy-Riemann equation is sometimes called the Bergmann space, which will be our terminology for H 2 D .
Remark. In the next Chapter we will be concerned with reconstructing the original ASD connection from these ADHM data. We give some physical intuitions here about why such reconstructions are possible. The connection A can be thought of as some gauge field, and physically this is detected by observing the motion of fermions inside the gauge field. The space H 2 D , i.e. solutions to the coupled Dirac equation, can be viewed as the Hilbert space for the fermion, and operators such as P 0 ○x µ are the natural ingredients to specify the quantum mechanics for the fermion.

The inverse construction
We show how to reconstruct the original ASD connection from its local Nahm transform. We proceed by defining the inverse Nahm transform, interpret it in the frameworks of both the Nahm transform and the ADHM construction, and then define a canonical comparison map between the original bundle and the inverse Nahm transform bundle, which turns out to preserve both the Hermitian structure and the connection matrix.

Inverse Nahm transform
We wish to reconstruct the original bundle by an analogue of the inverse Nahm transform in [4]. This suggests us to look at solutions to the Dirac equation coupled to the infinite rank bundleÊ. To set up, we take the dual Poincaré bundleP overR 4 × R 4 , with connection x µ dz µ , and for every x ∈ R 4 , we coupleÊ toP x . Alternatively, one thinks of the coupled connection asÂ x =Â + 2πi ∑ 4 µ=1 x µ dz µ . We identify the spinors on R 4 with those onR 4 .
The sections ofÊ which are constant under the canonical trivialisation E ≃ H 2 D are called invariant. One can think of this condition as the analogue of the periodicity condition over the dual 4-torus, when the size of the dual torus has shrunk to zero. It is clear that the covariant derivative onÊ preserves the invariance condition.
Then we define the fibres of the inverse Nahm transform bundleÊ over x ∈ R 4 to be the space of invariant solutions to the coupled Dirac operatorŝ D − Ax . A priori the inverse Nahm transform bundle may have singularities. An invariant section s in Γ(R 4 ,Ê ⊗ S − ) has constant modulus for all fibres z ∈R 4 , so the Hermitian metric on any fibre can be taken to give a norm on s. This equipsÊ with a Hermitian structure. There is also a natural connection onÊ, given by the projection of the natural differentiation in the x variable, where we view a section ofÊ as a function from R 4 to the invariant part of Γ(R 4 ,Ê ⊗ S − ).

A formal comparison with ADHM construction
The Nahm transform formulation emphasizes the analogy with the torus case, and retains as much symmetry between the Nahm transform with its inverse as is possible. We can also view the inverse Nahm transform from the perspective of the ADHM construction. The dictionary for this interpretation, is that the invariant quantities onR 4 are replaced by the information on the zero fibre. The correspondence is: • The space of invariant sections ofÊ corresponds to H 2 D .
• The connectionÂ correspond to four operators on H 2 D : the covariant derivatives∇ µ become −2πix µ = −2πiP 0 ○ x µ , the projection of the multiplication by x µ .
• The fibre of the inverse Nahm transformÊ y becomes • The Hermitian metric on the inverse Nahm transform bundle is always inherited from the metric on H 2 D .
• The connection on the inverse Nahm transform is the subbundle connec- is invertible, thenÊ is a nonsingular Hermitian bundle over B, and the subbun- For a sketch proof of this fact in the Nahm transform language, see Corollary 3.9 below. The Proposition does not depend on knowing that the ADHM data arise from (E, A), although the invertibility of∆Â y turns out to be automatic in that situation.

Construction of Dirac fields
We proceed as in [4]. We aim to produce coupled Dirac fields forÂ using the Green's operator G z in order to define a canonical comparison map E →Ê.
Notice there is a tautological element Ψ ∈ Γ(B ×R 4 ,Ê * ⊗ E ⊗ S − ), given by evaluating any element ofÊ on E ⊗ S − . If we further compose with the Green's operator G z , we obtain a section wheref j * stands for the dual basis off j = ψ j z . Now suppose we have f ∈ E * x , then evaluating against f produces an element in Γ({x} ×R 4 ,Ê * ⊗ S − ); denote it as GΨ(f ). We notice that for each z ∈R 4 , the value of GΨ(f ) at the point z is finite. This is because if we contract by a unit spinor η, is in fact the norm square of the linear functional ⟨f ○ G z , η⟩ on the Hilbert spaceÊ z . Since the connection A is smooth, this linear functional is bounded by basic elliptic analysis. Proof. (Sketch) The formal calculation for the Dirac equation is the same as in [4], Proposition 2.1, which uses explicit expressions for the connection coefficient ofÂ, and reduces the proof to properties of the Green's operator.
x µĉµ , the first claim is clear. We consider the z dependence of the construction.
where we used the fact the f ∈ E * x is located above the point x ∈ B. This means the element is invariant.
Recall there is a complex antilinear automorphism of the spin bundle, denoted ǫ ∶ S → S, which preserves the chiral splitting S = S + ⊕ S − and the Clifford multiplication. Tensoring this with the canonical isomorphism E ≃ E * , one obtains an antilinear isomorphism which is checked to preserve the Dirac equation.
This allows us to define a canonical comparison map α ∶ E →Ê by The adjoint operator of this canonical map E y →Ê y ⊂ H 2 D ⊗ S − can be described as follows. Identify the space of invariant sections ofÊ with H 2 D , then for ζ ∈ H 2 D ⊗ S − , the adjoint operator maps it into an element of E y , given by where the ǫ ○ G 0 sends the H 2 D factor to Γ(E) ⊗ S − , so we can then contract the two copies of S − to get some element in Γ(E), and finally evaluate at the point y ∈ B.

An example: the trivial line bundle
Let us work out explicitly what this construction means for (E, A) being the trivial flat line bundle, and B = B(R) ⊂ R 4 is a ball. Then we can pick ξ to be a unit length basis vector, and henceforth suppress it. Modulo the twisting ǫ, the canonical map E → H 2 D ⊗S − essentially comes down to the following description: calculate the composition and then write this (spinor-valued) functional on H 2 D using Riesz representation. The Green's function for the Euclidean Laplacian on the ball is well known. We recall the formula We treat p as the independent variable, and q is a parameter. In the spinorial situation, the Green's function is merely the above tensored by a parallel spinor field η. We calculate the projection of the Green's function to the space This means for s ∈ H 2 D , Geometrically, the Dirac field It is interesting to observe the limiting behaviour of this formula near the boundary of B(R). For y ∼ R, write y = Rl cos θ, where θ ∼ 0, and l is a unit vector. Write also x = lR + 1 2 x ′ Rθ 2 , then the formula has the asymptote If we view this naïvely as a function of x, then the pointwise limit will be zero, because the Dirac field concentrates near lR with distance scale θ 2 . if we extract an appropriately rescaled limit, we obtain a Dirac field in some half space, generated by a point charge placed at unit distance outside the boundary plane.

Nonsingularity
This Section studies the Laplacian∆Â y =∇ * Ay∇Ây on the local Nahm transform bundleÊ, where y ∈ R 4 . The importance of this operator for our construction is based on the observation that sinceÂ y is ASD, on the positive spin E ⊗ S + , we have∇ * We start with a matrix description of this Laplacian acting on invariant sections ofÊ. Recall that s ∈ H 2 D is canonically identified with the invariant section se −2πiz .
Lemma 3.4. Under the identification, the Laplacian can be identified with the operator Proof. Let s ∈ H 2 D and τ = G 0 s. Then by Green's formula We consider the L 2 decomposition with τ µ ∈ W 2,1 0 and Dσ µ = 0. Then we observe Then we compute This equation has the interpretation Under the identification, the directional derivatives ∇ μ Ay can be identified as −2πiP 0 ○ (x µ − y µ ) (cf. Section 3.2). Thus Since this holds for all s, the self adjoint operators 4π 2 P 0 ○ { x − y 2 − 4G 0 } and ∆Â y must be the same. Remark. One can also derive the Laplacian formula by a calculation similar to Lemma 2.3, Lemma 2.4 in [4].
These preparations lead to Proof. An invariant section is the same data as an element s ∈ H 2 D . Then the kernel of∆Â y translates into the condition Comparing with the proof of Lemma 3.4, this implies (cf. (20)) and inside H 2 D (by equation (21)), and therefore x − y 2 s = 4τ.
In particular s has zero boundary data. Substituting this into (17), we get So for any µ, In other words, where ρ is covariant constant in E ⊗ S + .
This forces s = 0, by the zero boundary condition.
Remark. This proof suggests the mechanism of singularity formation: if the discussion is extended to singular connections, then the kernel of∆Â y can be nonzero, and (22) is expected to describe the asymptotic profile of the kernel elements near the singularity.
Our next aim is to justify the invertibility of∆Â y , acting on the space of invariant sections ofÊ, identified as H 2 D . We recall the Laplacian is always self-adjoint and semipositive. In our context it is also bounded. So it is enough to prove Lemma 3.6. For y ∈ ∂B, there is a coercive estimate on the space of invariant sections Here the norm of an invariant section is the norm on any fibreÊ z . The constant can be taken to be uniform for y in any compact subset of B.
Proof. Since we already know the kernel is zero, the strategy for the coercive estimate is a compactness argument, which is delicate due to norm collapsing issues. Suppose we have a sequence s m ∈ H 2 D , with unit norms, satisfying By interior regularity, we may assume s m converges smoothly to s inside B.
We associate τ m = G 0 s m as in the previous proposition. Then τ m satisfies the elliptic equation ∆ A τ m = s m , with the zero boundary condition, so for any large Lebesgue exponent p of our choice, In general, if a higher Lebesgue norm of a function can be controlled by a lower Lebesgue norm, then the support of the function can not have arbitrarily small volume, i.e. the function cannot be concentrated in region of small measure. If we replace the Lebesgue norm by a weighted version, then the function cannot be concentrated where the weight is bounded positively below. Hence the norm of s m cannot be lost entirely to the boundary: where C depends only on R and the connection A. In particular the limit s is nontrivial. Now we look at the equation where t m is just some element in W 2,1 0 . When we take the weak limit, we get Thus despite the possiblity of partial norm collapsing, the limit s still satisfies or in other words,∆Â y (se −2πiz ) = 0. which implies s = 0, a contradiction.
Hence we finally achieved Proposition 3.7. For y ∉ ∂B, the Laplacian∆Â y is invertible on the space of invariant sections ofÊ.
Remark. The subtle argument in the lemma involving norm collapsing reflects the genuine distinction between interior points and the boundary. The coercivity estimate is expected to fail on the boundary, so thatÊ would not extend to a smooth bundle on R 4 ; otherwise the original ASD connection A would extend to a global connection on R 4 , which would be very surprising.
We can therefore define the inverse operatorĜ y = (∆Â y ) −1 , for y ∉ ∂B, on the space of invariant sections ofÊ. We collect a few formal consequences: , when restricted to the invariant sections, is a surjective map. Morever, the orthogonal projection from H 2 D ⊗ S − to the kernelÊ y is given bŷ Proof. SinceÂ y is ASD, we have∆Â y =D − AyD + Ay . The surjectivity follows from the explicit formula for a preimage, given by the operatorD + AyĜ y . To see the orthogonal projection formula, one also needs to checkD + Ay andD − Ay are adjoint on the invariant sections.
Proof. The nonsingularity is a formal consequence of a smooth formula for the orthogonal projectionP y ; here the word bundle is interpreted as a Hermitian vector bundle with fibres possibly being Hilbert spaces.
The ASD condition is implied by the following curvature formula forÂ analogous to (10), which relies on formula (24) for the projection operator. At whereP yĜy acts on the Γ(Ê) factor ofÊ y , andĉ µĉν acts on the spin factor.
We next wish to show the inverse Nahm transform vanishes in the exterior of B. Proof. Since we know the bundle is nonsingular in the exterior region, the rank is constant, so it is only necessary to showÊ y = 0 for y >> 1. But we knoŵ andD − A acting on invariant sections is a bounded operator, so for large y , the Dirac operator is merely a small perturbation of the invertible operator 2πi ∑ y µĉµ , hence itself invertible, implying the kernelÊ y = 0.

The canonical map
To study the canonical map α ∶ E →Ê, we look at where x ∈ B is a parameter. For x 1 , x 2 ∈ B, we consider the correlator function ⟨v x1 (z), v x2 (z)⟩Ê z ⊗S− , meaning we contract theÊ z and the S − factor, to get a matrix in Hom(E x2 , E x1 ). We aim to derive a formula for the correlators, by adapting arguments in [4].
which is independent of z.
Here the bra-ket notation indicates can be viewed as the Schwartz kernel of an operator acting on Γ(B, E ⊗ S − ), which converts a section h with independent variable x 2 to a section with independent variable x 1 . This can be factorised into several steps. The first step sends h to Here the equality uses the self-adjointness of G z , and the decomposition formula for the projection P z = ∑f k ⟨f k , ⋅⟩. The second step contracts theÊ z factor witĥ f j to get a number, multiplies the result by G z ψ j z (x 1 ), and sums over j. Using again the decomposition formula for the projection, the result is Now we simplify the formula and take the trace over the spinor factor S − . The projection can be written as Hence using that the Clifford multiplication c µ c ν can contribute to the spinor trace if and only if µ = ν, we see that (using summation convention) ∂Gz ∂zµ (see [4], Lemma 2.2) to simplify: From this we see Here we need to take into account the fact that S − is 2-dimensional, so contributes twice to the trace.
Combining the above, we get the formula for the correlator by comparing the Schwartz kernel of operators: The claim follows.
Given ξ ∈ E x , the image under the canonical map α ∶ E →Ê is precisely the invariant section ⟨v x , ξ⟩ ∈ Γ(R 4 ,Ê ⊗ S − ). We show that the canonical map α preserves the structures (cf. [4], Theorem 2.8 and 2.9). Proof. The Green's function has the short distance asymptotic expansion Thus the norm square of ⟨v x , ξ⟩ can be calculated by first extending ξ to be part of a local orthonormal frame, and then taking the limit y → x. This yields ⟨v x , ξ⟩ 2 = ⟨ξ I ξ⟩ = ξ 2 , by looking at the lowest order in the expansion.
Proposition 3.13. The canonical map α preserves the connection matrix.
Proof. Let ξ and ξ ′ be two local sections of E, fitting into a local orthonormal frame of E. These map to v ξ x = ⟨v x , ξ⟩ and v ξ ′ x = ⟨v x , ξ ′ ⟩. The connection matrix onÊ at the point x ∈ B is specified by knowing ⟨v ξ ′ x , d dy x v ξ y ⟩(x). But this can be calculated from the expansion of the Green's function above, which implies the required

The reconstruction theorem
We derive the main reconstruction theorem 1.1 assuming the following Lemma on Fredholm index, which we shall prove later in Section 6.1.  Proof. By Corollary 3.10, the inverse Nahm transform vanishes in the exterior region. By Corollary 3.8, the operatorD − Ay is surjective, so its kernel dimension is equal to the index. For y ∈ B, by the Lemma above the index is rank (E), so rank (Ê) = rank (E). By Proposition 3.12, the canonical map α is an injective isometry, so must be an isomorphism. The connection A agrees withÂ by Proposition 3.13.
We now discuss the result from a number of perspectives.
Remark. The reconstruction theorem is anologous to the Cauchy integration formula in complex analysis, which says the contour integral vanishes for z in the exterior region of the contour, and reproduces the holomorphic function f in the interior domain.
Remark. Viewed in another way, the reconstrction theorem means there is a canonical embedding of E inside the trivial Hilbert bundle H 2 D ⊗ S − . Then (27) has the interesting interpretation: Corollary 3.16. The Green's function G 0 (x, y) on the original bundle is related to the embedding data by Similar results are known in the context of the usual ADHM construction (cf. equation (29) in [6]).
Remark. Theorem 1.1 links ASD connections to operator theory via the ADHM interpretation. Donaldson proved in [8] a Runge approximation type theorem, which says that ASD connections on domains can be C ∞ approximated on compact subsets up to gauge, by restrictions of global ASD connections over S 4 for arbitrarily large second Chern class. This suggests that on the operator theory side, one may approximatex µ in some sense by finite rank Hermitian operators satisfying the ADHM equation. It is interesting to ask how this may be proved using purely operator theoretic techniques.
Remark. The ADHM data (H 2 D ,x µ ) resembles the key concept of 'spectral triple' in noncommutative geometry (NG) [7], and the ADHM type equation (11) fits into the basic philosophy of NG, namely to encode geometry by operator algebras.

The spectral problem
For a smooth ASD connection A on B = B(R) ⊂ R 4 , we wish to understand the Bergmann space , whose elements are called Dirac fields, by studying the spectrum of a natural operator on H 2 D , which physically is just the Hamiltonian of a harmonic oscillator (cf. Section 4.1).
We are interested in the behaviour of H 2 D A as A develops a curvature singularity at the origin. This means A is a member of a sequence A i (or a 1parameter family A t ), which is uniformly bounded to all orders on the complement of any given neighbourhood of the origin. We derive uniform estimates to control the eigenstates associated to the spectral problem. These are based on the Weitzenböck formula. The picture emerging from the analysis is the following: The spectrum is divided into 3 characteristic ranges: λ << 1, the intermediate range, and λ ∼ 1. • If the eigenvalue is bounded above away from 1, then the solution has good bounds away from the origin (cf. Proposition 4.4 and its ensuing Remark).
• If the eigenvalue is small, then the density of the eigenstate is concentrated near the origin (cf. (36)).
• If the eigenvalue is close to 1, then the density of the eigenstate is concentrated near the boundary (cf. (37)).

The spectral problem
Let A be an ASD connection on B = B(R), which is smooth up to the boundary. We introduce a functional on the Bergmann space H 2 D , Physically this is the Hamiltonian of a harmonic oscillator.
Now H defines a Hermitian form, which in the presence of the L 2 inner product defines a bounded self-adjoint operator L acting on H 2 D . We observe Lemma 4.1. The operator L = 1 2 + K where K is a compact operator.
Proof. We can write L − 1 2 as the operator norm limit of a sequence of linear operators L i corresponding to the Hermitian forms But each L i is a compact operator becasuse the L 2 norm of a Dirac field controlls all interior higher order derivatives. Compact operators are closed under norm limits, so L − 1 2 must be compact as well.
Standard functional analysis implies that L has discrete spectrum; this amounts to the simultaneous diagonalisation of the the Hermitian form H and the L 2 inner product, and is computable by the Rayleigh-Ritz method.

Remark. We have by definition
so 0 < λ < 1. We make an elementary observation that when λ << 1, the eigenstate s is concentrated near the origin: for any fixed 0 < r < R, Heuristically, the small eigenvalue eigenstates describe particles trapped in the potential well, analogous to bound states in physics. If A has very concentrated curvature, we expect the characteristic length scale λ of the eigenstates corresponding to the small eigenvalues to be roughly the same as the length scale of the curvature of A.
On the opposite extreme, if the eigenvalue is close to 1, then the eigenstate s is concentrated near the boundary: for any fixed 0 < r < R, We now characterise the eigenstates s of L: Proof. If s ∈ H 2 D is an eigenstate of L with eigenvalue λ 2 2 , then ( x 2 2R 2 − λ 2 2 )s is orthogonal to all elements in the space H 2 D in the L 2 sense, by linear algebra. This implies the second equation above by using the decomposition Here ζ is unique because by a standard Weitzenböck formula, there is no coupled Dirac field with positive spin and zero boundary condition.

The Weitzenböck formula
The well known Weitzenböck formula says that because A is ASD, acting on the coupled positive spinor ζ; the curvature effect is not directly visible. Thus ζ enjoys more favourable analytic properties compared to the negative spinors, so we shall mainly focus on ζ when deriving the estimates.

Notice equation (38) implies
which, by an application of Weitzenböck formula, gives where our convention of the Hodge Laplacian is ∆ = − ∑ i ∇ i ∇ i . Integrating the identity, we obtain: Here the boundary term does not appear because ζ 2 vanishes to second order.
As a consequence, Here C is an absolute constant.
Proof. By the Poincaré inequality, The inequality (41) follows from Hardy's inequality, by noticing that ζ has zero boundary condition.
Proposition 4.4. Given constants 0 < r < R and 0 < C ′′ < 1, in the annulus region r ≤ x ≤ R, if we normalise s to have unit L 2 norm, then for eigenstates with eigenvalues satisfying λ < C ′′ , we have smooth estimates on s and ζ to all orders, which are uniform in A in the setup of this Chapter.
Proof. We can rewrite (38) as This equation is elliptic with uniformly bounded coefficients away from the locus { x = λ} and the origin, so by the L 2 estimate and the zero boundary condition on ζ , we obtain uniform smooth estimates of ζ in such region.
Near the locus { x = λ} we can use the local elliptic regularity of the Dirac equation to estimate s to all orders, which implies the uniform elliptic estimates on ζ to all orders.
Remark. If we drop the condition λ < C ′′ , the arguments above still imply uniform smooth bounds on s away from both the origin and ∂B.

Large eigenvalues imply interior control
We study the situation where the eigenvalue is positively bounded below.
Lemma 4.5. We have the estimate where C is an absolute constant.
Proof. We interpret (39) as a Poisson equation for ζ 2 , with zero boundary data. The Green representation formula gives By rearranging terms and applying Cauchy-Schwartz inequality, Now we use the consequence of Hardy's inequality (41) to bound the RHS by an absolute constant. We then use the L 2 gradient estimate on ζ (cf. (40)) to drop ∫ B 1 R 2 ∇ζ 2 from LHS, and drop the term involving ζ 2 (0) to see the claim. Proposition 4.6. (Interior Morrey estimate for large eigenvalues) We have where C is an absolute constant. In particular, if λ is bounded positively below, then for any 0 < r < R, we have the Morrey decay estimate Proof. We apply (38) and the above Lemma to see The result is then clear.
Remark. This estimate, ultimately due to the Weitzenböck formula, is highly non-perturbative because it holds even when there can be an arbitrarily large amount of curvature concentrating to the origin. The intuition is that if λ is bounded below by some positive constant, then the characteristic length scale of the Dirac field is much larger than the scale of the concentrated curvature, hence the curvature singularity is not very visible to the solution.

Convergence theory for Bergmann spaces
Given a sequence A i or a one-paramter family {A t } t>0 of smooth ASD connections onB forming a curvature singularity at the origin (cf. the setup in Chapter 4). Away from the origin, we assume the connections converge in C ∞ loc to a connection A ∞ on E B∖{0} , which is necessarily smooth and ASD, so by the removable singularity theorem (E B∖{0} , A ∞ ) extends to (Ẽ, A ∞ ). HereẼ is conceptually a different topological bundle from E, although their L 2 sections can be identified. We may assume A ∞ has small L 2 curvature, by possibly shrinking B. The convergence problem asks for a limiting description of the corresponding Bergmann spaces H 2 D A . The basic picture is that the part of the spectrum for A above a threshold value converges to the spectrum for A ∞ (cf. Theorem 5.3, Proposition 5.4); the 1-parameter family of Bergmann spaces H 2 D A t converge in a natural way to H 2 D A∞ ⊕ V where V is a finite dimensional space (cf. Theorem 5.15); the natural operators on Bergmann spaces extend naturally to the limit space (cf. Proposition 5.16, 5.17).
Remark. If we think of the Dirac equation as the analogue of the∂ equation, then this convergence theory is somewhat analogous to the picture in [11].

Convergence of eigenstates
We consider a sequence of eigenstates s i associated to the connections A i , with L 2 norms equal to 1, the eigenvalues 0 < λ i < 1 and corresponding coupled positive spinor fields ζ i solving (38), and we ask for a convergence theory of s i . Without loss of generality λ i → λ ∞ converges, after taking subsequence. The basic picture of Chapter 4 implies: • If λ i → 0, then the density of s i is concentrated to the origin, so s i converges to zero weakly.
• If λ i → 1, then the density of s i is concentrated to the boundary, so s i converges to zero weakly. Now let us assume the uniform two sided eigenvalue bound 0 < C ′ < λ i < C ′′ < 1. By the main results of Chapter 4, in any given annulus region 0 < r ≤ x ≤ R, we have uniform smooth estimates on all the data λ i , s i , ζ i , so we can extract a subsequence to ensure smooth convergence, by standard compactness arguments. A standard diagonal argument implies we can assume C ∞ loc convergence away from the origin, to the limiting data λ ∞ , s ∞ and ζ ∞ , which satisfy the limiting version of (38) on B ∖ {0}.
Since we have uniform estimates s i L 2 = 1 and ζ i W 2,1 ≤ C (cf. (40)), and the norms cannot increase in the limit, so the same estimates hold for s ∞ and ζ ∞ . By elliptic regularity of this limiting PDE system, the limiting data extend smoothly across the origin to give a solution of (38).
Crucially, we claim strong convergence of s i to s ∞ inside L 2 . The only possible issue is to lose L 2 mass to the origin. But this cannot happen thanks to the uniform interior Morrey estimate (44).
In particular, the norm of s ∞ does not collapse to zero, so s ∞ ∈ H 2 D A∞ is an eigenstate with eigenvalue λ ∞ .

The spectral gap
A rather striking consequence of Section 5.1 is Corollary 5.1. (Spectral gap) Suppose the minimal eigenvalue for the limiting connection A ∞ is λ 0 > 0. Then either λ ∞ = 0 or λ ∞ ≥ λ 0 .
Proof. If the limit of eigenvalues λ ∞ ≠ 0, then we can assume a positive lower bound on λ i . Unless λ ∞ = 1, we can also assume an upper bound smaller than 1. So we are in the situation above and we see the result from the good convergence theory.
Lemma 5.2. In the special case that A ∞ is the trivial flat connection, the minimal eigenvalue is 1 2 λ 2 = 1 3 , attained precisely for for parallel Dirac fields.
Proof. Finding the minimal eigenvalue is the same as minimising the functional x 2 2R 2 s 2 subject to Ds = 0 and s L 2 = 1.
But we know by Weitzenböck formula that ∇ * ∇s = 0, so s 2 is subharmonic, with forcing term given by 2 ∇s 2 . This means the spherical average is a non-decreasing function in r. From this it is clear that the only way to minimise the functional is for the spherical averages to be constant, which implies ∇s = 0.
We may also record a quantitative version of the spectral gap: then any eigenvalue 1 2 λ 2 must satisfy the dichotomy Proof. This is the rigid version of the previous spectral gap result. The proof is a compactness argument. More precisely, assume a counterexample sequence (A i , s i , λ i ) on B, then standard compactness theory implies A i → A ∞ on a shrinked punctured disk B ∖{0}. Here A ∞ must be flat because its L 2 curvature vanishes. By the previous discussions, after taking subsequence λ i → λ ∞ with λ ∞ = 0 or λ ∞ ≥ 2 3, which would give a contradiction if λ i fails the dichotomy.

Remark.
It is curious what the physical interpretation of the spectral gap should be.
Since we assume in this Chapter that A ∞ has small L 2 curvature, the spectral gap theorem suggests us to separate the spectrum associated to A i into two parts: the large spectrum with λ > 2 3 − δ 2 , and the small spectrum with λ < δ 1 . Correspondingly, the Bergmann spaces decompose as

Convergence in the large spectrum
The convergence theory for eigenstates can be rather formally extended to a convergence theory for the large spectrum. For any given A i , let s j i be an orthonormal basis of eigenstates belonging to the large spectrum, where j is arranged in increasing order of eigenvalues. We associate the data ζ j i and λ j i in a self explanatory way. We will show Proposition 5.4. (Spectral convergence) The large spectrum for A i converges to the spectrum of the smooth limit A ∞ , i.e. if we index the eigenvalues for A ∞ in increasing order as λ k 0 , then Remark. We will henceforth often suppress mentioning taking subsequences, and we shall tacitly use diagonal arguments. A posteriori we shall see that this is not necessary due to the uniqueness of limit.
We first introduce an algorithm. By previous work in this Chapter, either λ 1 i → 1, or they satisfy two sided bounds so that s 1 i converges strongly to s 1 ∞ , which is some eigenstate in H 2 A∞ with eigenvalue λ 1 ∞ . In the first case, we terminate and define λ k ∞ = 1 for all k. In the second case, we proceed with λ 2 i . We either terminate after a finite stage (which a posteriori does not happen), or continue indefinitely to achieve a sequence of limiting eigenstates s k ∞ with eigenvalues λ k ∞ , which must be orthonormal by strong convergence. The algorithm implies The equality is achieved precisely if every eigenvalue λ k 0 for A ∞ arises as subsequential limits of eigenvalues, including multiplicity.

Our next step is to show
Lemma 5.6. If s ∈ H 2 D A∞ , then it is a strong L 2 limit of sections of H 2 Proof. As preparation, we study the decomposition of Notice by orthogonality Now since τ has positive spin and zero boundary condition, using Weitzenböck, one has These preliminary remarks establish that the L 2 decomposition in this context respects the natural norms.
By the norm control, on every annulus 0 < r ≤ x ≤ R we can extract smoothly convergent subsequences for τ i and σ i . Hence we have smooth convergence on 0 < x ≤ R to the limiting data (τ ∞ , σ ∞ ), for which the limiting version of (48) holds, and morever τ ∞ W 2,1 0 ≤ C, σ ∞ L 2 ≤ C. Notice the equation implies that τ ∞ extends to a smooth section, so σ ∞ is also smooth. It is clear that the norm identity (49) holds in the limit, namely But Fatou's lemma implies that So the only possibility is for equalities to be achieved everywhere, i.e.
The non-collapsing of norms imply that σ i converges strongly to σ ∞ in L 2 . Notice also that since the subsequential limit is unique, in fact the whole sequence has to converge.
Now by an approximation argument on s, it is clear that the smoothness of s is not essential. We specialise to the case s ∈ H 2 D A∞ . Then s = σ ∞ and the claim follows.
Corollary 5.7. If morever s is an eigenstate for A ∞ , for which the eigenvalue λ m 0 has multiplicity one, then we may assume the sequence consists of eigenstates as well. If the eigenvalue is degenerate, we need to take linear combinations of eigenstates with approximately the same eigenvalue. In particular λ m 0 arises as a limit of eigenvalues.
Proof. We argue in the nondegenerate case (the degenerate case has only a little more combinatorial complexity). The crucial point is that the spectrum is discrete. There are only finitely many eigenvalues below λ m 0 :

Consider the spectral decomposition
By the generalised spectral gap lemma, there are essentially at most m − 1 eigenvalues λ k i bounded above by λ m−1 0 + ǫ < λ m 0 . For these eigenvalues the eigenstates are almost orthogonal to s when i is large, because they converge to eigenstates with lower eigenvalues. So their corresponding Fourier coeffients a j in the spectral decomposition must go to zero. But once we are not allowed to have contributions from low eigenvalues, then for overall L 2 mass reasons, neither are we allowed to have contributions from eigenvalues larger than λ m+1 Once we know λ k 0 arises as limits of eigenvalues, Proposition 5.4 follows from the generalised spectral gap Lemma. We leave the reader to ponder the issue of multiplicity. As remarked earlier, the limit of λ k i turns out a posteriori to be independent of the subsequence.

The limit of the Bergmann spaces
From last section, one has the appealing picture that H 2 D A∞ is the limit of (H 2 D A i ) large , while the small spectrum is lost in the naïve smooth limit. In this Section we give a more operator theoretic perspective; the main result is Proposition 5.15.
The first step is to study a comparison map between (H 2 To save some writing, we will often suppress sequential indices i. Recall D A and D A∞ induce two decompositions of L 2 . In particular, the two Bergmann spaces H 2 D A and H 2 D A∞ project to each other, via the operator P ∶ H 2 and its adjoint P † . Thus there are canonical maps and its adjoint in the operator norm, as i → ∞.
Proof. Pick any s ∈ H 2 D A∞ , which is normalised to s L 2 = 1. We have the decomposition Here σ is the image of the canonical map. Then In particular, τ is bounded in L 2 . We also have so D 2 A τ is L 2 small away from the origin, whereby τ is W 2,2 loc (B ∖ {0}) bounded. We can therefore assume τ to converge strongly in W 2,1 loc (B ∖{0}) as i → ∞. The limit τ ∞ is globally bounded in L 2 , has zero boundary condition, and satisfies the Laplace equation, so must be zero. Thus in the sequence, τ must be W 2,1 close to 0 away from the origin, with bounds independent of s. In particular, D A τ is L 2 small away from the origin.

Notice also
Therefore s is L 2 close to σ away from the origin. In particular, since the Dirac equation implies s has pointwise bound near the origin, we have for fixed δ << 1, Comparing with the partition of norm (53), The constants are independent of s. Since δ is arbitrary, this proves the result.
Lemma 5.9. The canonical comparison map πP for A i is surjective for i >> 1.
Proof. Since the above Lemma implies the coercivity of πP, the image is closed. The cokernel of πP is the kernel of (51), which means s ∈ (H 2 D A ) large , and s = D A∞ τ , for τ ∈ W 2,1 0 . If the cokernel does not vanish for a subsequence of connections A i , we normalise s to s L 2 = 1, and derive a contradiction by a compactness argument as follows.
First, A ∞ is ASD, so ∇ A∞ τ L 2 = D A∞ τ L 2 = 1, hence τ L 2 is controlled. Now τ satisfies D Ai D A∞ τ = 0, with the zero boundary condition, hence it is controlled to any order away from the origin. Any subsequential limit of τ must be zero, by the same argument as in last Lemma. This implies τ converges to zero smoothly in the punctured disc; so must s, which means the L 2 mass of s is concentrated at the origin. But s lives in the large spectrum, so ∫ B(R) x 2 R −2 s 2 ≥ C s 2 L 2 , contradiction. Corollary 5.10. For large i, the canonical map πP is an isomorphism. Morever, it is close to being a unitary equivalence: We next study the small spectrum H 2 D A . Here it is more convenient to set up the problem in terms of a one-parameter family (A t ) t>0 of ASD connections converging smoothly away from the origin. This is more suited to continuity arguments.
Proposition 5.11. For large t so that the curvature is sufficiently concentrated, the dimension of the small spectrum is constant in the family.
Proof. Imagine t to flow from a large number to ∞. By general functional analysis, the eigenvalues flow continuously. But spectral gap (cf. Theorem 5.3) prevents the spectral flow between the large spectrum and the small spectrum, so the dimension of the small spectrum must be a constant finite number.
Remark. The concept of small spectrum is only meaningful in the large t limit.
By examining the comparison map πP in the limit t → ∞, we see is Fredholm, with index being the negative of the dimension of the small spectrum. Proof. Consider t near t 0 , and the natural projection operator H 2 This is an isomorphism by the argument proving πP to be an isomorphism.
Lemma 5.14. For large t < ∞, the subspaces (H 2 D A t ) small fit together into a vector bundle.
Proof. As a general fact, consider a family of operators depending smoothly on a parameter t, with discrete spectrum consisting of eigenvalues. If we fix a spectral domain Ω ⊂ C, and consider the projection operator to the the span of the eigenspaces for all the eigenvalues inside Ω, then as long as no eigenvalue crosses the boundary of Ω, the projection operator depends smoothly on t, and the image has constant dimension. (cf. Appendix of [13]).
Theorem 5.15. (Natural limit of Bergmann spaces) Let V be an inner product space with the dimension of the small spectrum. Then there is a natural topological bundle over 0 < t ≤ ∞, whose fibres over 0 < t < ∞ agree with the Bergmann space H 2 Proof. It suffices to assign a trivialisation near ∞. One can use the isomorphism provided by the canonical comparison map (50) to deal with the large spectrum part. To trivialise the small spectrum part, one notices there is a canonical connection on the bundle ⊔ ∞>t>t0>>1 (H 2 ) small coming from the embedding into L 2 . Over the one dimensional base (t 0 , ∞), this connection gives a parallelisation of this finite rank bundle compatible with the Hermitian structure, which can be extended to the ∞ fibre by formally adding a copy of V .

Natural operators on the Bergmann space
In the setup of Theorem 5.15, we can ask whether natural operators on H 2 D A t extend continuously to operators on the limit Bergmann space H 2 D A∞ ⊕ V . Here we use the trivialisation near ∞ in Theorem 5.15 to regard H 2 D A t as a fixed Hilbert space, and the convergence of operators is reduced to the usual definition of norm convergence.
Given a smooth function f onB, the Toeplitz operatorf = P A ○f on H 2 D A means multiplying by the function f , composed with the orthogonal projection of L 2 to H 2 D A . Physically, the Bergmann space is the state space of a fermion, and then these correspond to quantum observables of the shape ⟨s f s ′ ⟩. The Toeplitz operators are Hermitian.
on the limit Bergmann space, wheref also denotes the corresponding Toeplitz operator on H 2 D A∞ .
Proof. We use the trivialisation near ∞ described in the proof of 5.15. The limit operator is zero on the V factor, because on the small spectrum for A t , f s 2 Hence we only need to consider the large spectrum for A t . Then the result follows from the norm convergence πP → 1 as t → ∞.
Remark. It is unclear to the author how to extend the matrix describing Clifford multiplications ⟨s c j c k s ′ ⟩ for s, s ′ ∈ (H 2 D A t ) small to the limit. This question is intimately tied to compactification of ASD moduli spaces. It is interesting to extend the local Nahm transform of Chapter 2, 3 to singularities. On the ASD side, the Uhlenbeck compactification involves ideal instantons. On the operator theory side, the ideal instanton corresponds to (H 2 encodes the smooth limit A ∞ and dim V encodes the delta mass of curvature. But if we enlarge the operator algebra to include secondary operators induced by ⟨s c j c k s ′ ⟩ and ask how they converge, then we may obtain a more refined compactification.
Next we consider the Green operator by solving the Laplace equation ∆ A s = 0 with zero boundary condition. For a point x inside the annulus 0 < r < x < r ′ < R, elliptic regularity gives the pointwise estimate Thus the evaluation map Now we vary the connection, so the Green's operator depends on the parameter t, and we study its limit.
Proposition 5.17. (limit Green operator) On the annulus 0 < r < x < r ′ < R, as t → ∞, the evaluation maps for the Green operators converge strongly and uniformly in x to some E x ⊗ S − valued bounded linear functional on the limit Bergmann space H 2 D A∞ ⊕V . The limit functional vanishes on V and agrees with ev x ○ G A∞ on the H 2 D A∞ factor.
Proof. We first consider any s ∈ (H 2 where C is an absolute constant. Thus the norm of the evaluation functional on the small spectrum converges to zero.
On the large spectrum, we use a compactness argument. Take some L 2 normalised counterexample sequence s i ∈ H 2 D A∞ , and let s ′ i = P i s i ∈ H 2 D A t i be their corresponding elements via the trivialisation. We have points x i inside the annulus, such that We will freely pass to subsequences.
Due to uniform W 2,1 0 bounds, we can extract a weak limit for G At i s ′ i . By elliptic estimates in the annulus region, we can assume the convergence to be uniform in 0 < r < x < r ′ < R, and s i converges weakly to s ∞ ∈ H 2 D A∞ . Since P i → 1, the sequence s ′ i has the same limit s ∞ . Then using the weak equation for the limit, one sees the weak limit of G At i s ′ i must be G A∞ s ∞ . The same discussions apply to G A∞ s i . Thus the difference G At i s ′ i − G A∞ s i has weak limit zero, the convergence is uniform in the annulus, but Remark. These linear functionals can be Riesz represented as elements of , which also converge strongly as t → ∞.
6 Index theory and Chern numbers

The rank ofÊ and index computation
We prove Lemma 3.14, which amounts to computing the index of an operator. It is convenient to work in the ADHM formulation, which links more easily to operator theory. The discussions below are self contained, although the main results are likely to be known in the literature of Toeplitz operators and index theory.
The index problem fits into a more general picture. Consider the operator whereĉ i is the Clifford multiplication on the spin factor, and the Toeplitz operatorf i = P 0 ○ f i is the projection of the multiplication by a smooth real valued function f i onB, i = 1, 2, 3, 4. We put on the assumption that the quaternion valued function f T = f 1 + if 2 + jf 3 + kf 4 , which we think of as the symbol of the operator, does not vanish anywhere on ∂B. In the generality of this Section, the connection A needs not to be ASD.
Lemma 6.1. The operator T is Fredholm.
Proof. The operator is clearly bounded. We show the finite dimensionality of the kernel. First write down the L 2 decomposition where s ∈ H 2 D ⊗ S − , τ i ∈ W 2,1 0 ⊗ S − , and Dσ i = 0. Thus using Ds = 0, This means by elliptic regularity, τ i can be chosen to satisfy As a remark, for general connections A, we need to impose τ i ∈ (ker D) ⊥ to make τ i satisfy the required estimates. Combining the above, The nonvanishing of the matrix ∑ f iĉi near the boundary is equivalent to its invertibility. Thus in a small neighbourhood of the boundary But we also have s W 2,1 (B∖B δ ) ≤ C s L 2 by the interior regularity of the Dirac equation, so s W 2,1 (B) ≤ C s L 2 , which forces the kernel to be finite dimensional by compactness.
But the degree is additive with respect to symbol multiplication, and the index is additive with respect to composition, so Lemma 6.2. For a fixed connection A, the index is proportional to the degree.

Independence of connection A.
We wish to remove the dependence on the background connection A. For this, introduce the Bergmann space H 2 D corresponding to a trivial flat connectionĀ on the original vector bundle E, which exists because B is contractible. This induces another orthogonal decomposition of L 2 . The two Bergmann spaces project onto each other, via operators P ∶ H 2 D → H 2 D and its adjoint P † . Another viewpoint is that H 2 D and H 2 D have a natural L 2 pairing, so induce two linear operators P and P † . It is clear that P and P † are Fredholm; in fact, inside L 2 they are compact perturbations of the identity operator. For example, We can then consider the composition This has the same index as T , because the index of P cancels with that of its adjoint. The composite operator is a compact perturbation of This shows the index is the same as that of the corresponding problem in the flat case.
6. In the case of the trivial connection, it is clear that H 2 D decomposes according to the rank of the vector bundle, so the index is proportional to the rank of E. Now we treat the standard case of the trivial flat line bundle, with T = ∑xµĉµ. We assume without loss of generality that the origin is an interior point of B, so the symbol function f T is non-vanishing on ∂B, and morever f T ∶ ∂B → H ∖ 0 has degree 1.
The cokernel of T vanishes, thanks to our non-singularity discussion (cf. Corollary 3.8). The condition for the kernel is where s ∈ H 2 D ⊗ S − and τ ∈ W 2,1 0 (E ⊗ S + ) ⊗ S + , and E is the trivial flat line bundle. We observe ∆τ = D 2 τ = c µĉµ s.
The significance of this comes from representation theory. Notice is a Spin(4)−equivariant map. We can decompose the representations By Schur's lemma, the map factors through the one dimensional representation Λ 2 S − , and the image lands inside the line spanned by This means ). On the other hand, direct differentiation shows , hence using the defining equation (56), ).
Comparing the above, ∆ρ = − 4 x 2 ⟨∇ρ, x⟩, or Recall that ρ is a smooth function with zero boundary condition, so ρ x 2 has zero boundary condition, and the only possible singularity is a pole of order 2 at the origin, which by our assumption is an interior point of B. It has to be proportional to the Dirichlet Green's function with delta mass placed at the origin. This proves the kernel dimension is 1, so the index is 1 in the special case.
To summarise the results of this Section, Proposition 6.4. The index of T equals rank(E) deg(f T ).
Remark. Compare this with the non-singularity discussion (cf. Corollary 3.8).
This shows the inverse Nahm transform bundleÊ has rank equal to rank(E) in the interior of B and vanishes in the exterior, thus resolving Lemma 3.14. In particular there is a jump of index when we cross the boundary, so the operator T = ∑ µ (x µ − y µ )ĉ µ fails to be Fredholm when y ∈ ∂B.

Singularity formation and small spectrum
The aim of this Section is to show Theorem 1.4. The key input is the following result, which combines our local Nahm transform theory with our analytic convergence theory for singularity formation.
Lemma 6.5. In the setup of a 1-parameter family of ASD connections A t developing a curvature singularity at the origin, we have the exact sequence In the limit t → ∞, the operator ∑ µ (x µ − y µ )ĉ µ tends to where convergence holds on the whole ball, and α t tends to where convergence holds on {0 < x < R}, and is uniform on compact subsets.
Proof. The existence of the exact sequence is rephrasing the reconstruction theorem 3.15.
The convergence of (x µ − y µ )ĉ µ follows from the convergence theory of Toeplitz operators (cf. Proposition 5.16), applied to f = x µ . The convergence of the canonical comparison maps α t follows from Proposition 5.17, because α t is essentially defined as the evaluation map of the Green operator.
Remark. The heuristic idea of Theorem 1.4 is that, by (2) we can think of the instanton number as a 'Chern class'. Since we have the exact sequence (58), we think of the 'Chern class' of E as a 'difference of the Chern classes for infinite rank bundles', and compute it by topological manipulations. This idea involves several difficulties: • This involves infinite rank bundles.
• The domain is not a closed manifold.
• In the limit the exactness fails.
The main idea to remedy these, is to replace vector bundles by relative K-theory classes, and work in the Fredholm setting.
We sketch the following relative version of the index bundle construction, using only first principles in K-theory. Lemma 6.6. Let X be a compact connected manifold with boundary Y . Given the data • A family of Fredholm maps between (possibly finite rank) Hilbert bundles, defined over X: • A trivial finite rank vector bundleẼ Y over Y , with a morphism α into ker F Y , such that the complex is a short exact sequence.
Then we obtain a natural class in the relative K-theory K(X, Y ). Denote this as Morever, this class is invariant under small collective deformations of all the defining data, and the construction is naturally additive.
Proof. (Sketch) We follow the following steps: 1. Since X is connected, the index of F over each point in X is constant. So if F is fibrewise surjective, then the kernel bundle is a finite rank vector bundle over X with a trivialisation over Y , so defines a relative K-theory class. Morever, the construction is invariant under small deformations of F away from Y .
2. The surjectivity of F is satisfied near Y . If it fails somewhere away from Y , we can replace H 1 by H 1 ⊕C N , where C N is a trivial vector bundle with sufficiently large rank. Now perturb F by adding a morphism p ∶ C N → H 2 . Generically this will force surjectivity to hold. We require p to vanish near Y ; this uses the existence of local cutoff functions. We define the class by 3. We check the well definition of the above construction. Clearly Morever any two choices of perturbations p and p ′ are connected by some path. This path may pass through some elements p ′′ which do not make F ⊕ p ′′ surjective. But there is some N ′ such that p ⊕ 0 ∶ C N +N ′ → H 2 is connected to p ′ ⊕ 0 by some non-singular path, because we can always use the extra degrees of freedom to remove any failure of transversality. The upshot is that by homotopy invariance the class is independent of the perturbation p, and is clearly independent of N .
4. Suppose we have a one parameter family of the data (H 1 parametrised by t ∈ [0, 1], with constraints as given in the theorem. We claim they define the same class. This is because without loss of generality F is surjective, and then the claim reduces to the usual homotopy invariance property.
5. The additivity is obvious.
Back to our context, the characteristic number description (2) gives Lemma 6.7. The instanton number is equal to the second Chern class of the relative K-theory class defined by the pair (E,Ẽ ∂B(R 2) ), under the Chern-Weil homomorphism ).
Lemma 6.8. The class defined by the pair (E,Ẽ ∂B(R 2) ) inside the relative K-theory K(B( R 2 ), ∂B( R 2 )) is Proof. Using the exact sequence (58), this class is which by continuity is By the additivity of the index bundle construction, this is Now by the analogue of the exact sequence (58) applied to A ∞ , the first summand is the relative K-theory class defined by the pair (Ẽ,Ẽ ∂B(R 2) ), which is To make sense of this, we need to apply a perturbation to the map to make it surjective. A particular choice is given by the map appearing in the ADHM construction of the standard 1-instanton (cf. [9], Section 3.4.1). Thus the relative K-theory class is given by the class of the 1-instanton, together with a trivialisation near infinity. This has charge c 2 = 1.
Combining these Lemmas give Theorem 1.4.