On counting associative submanifolds and Seiberg-Witten monopoles
aa r X i v : . [ m a t h . DG ] O c t On counting associative submanifolds and Seiberg–Wittenmonopoles
Aleksander Doan Thomas Walpuski2018-09-17
Dedicated to Simon Donaldson on the occasion of his 60 th birthday Abstract
Building on ideas from [DT98; DS11; Wal17; Hay17], we outline a proposal for constructingFloer homology groups associated with a G –manifold. These groups are generated by asso-ciative submanifolds and solutions of the ADHM Seiberg–Witten equations. The constructionis motivated by the analysis of various transitions which can change the number of associa-tive submanifolds. We discuss the relation of our proposal to Pandharipande and Thomas’stable pair invariant of Calabi–Yau –folds. Donaldson and Thomas [DT98, Section
3] put forward the idea of constructing enumerative in-variants of G –manifolds by counting G –instantons. The principal difficulty in pursuing thisprogram stems from non-compactness issues in higher-dimensional gauge theory [Tia00; TT04].In particular, G –instantons can degenerate by bubbling along associative submanifolds. Don-aldson and Segal [DS11] realized that this phenomenon can occur along –parameter families of G –metrics. Therefore, a naive count of G –instantons cannot lead to a deformation invariant of G –metrics; see also [Wal17]. Donaldson and Segal proposed to compensate for this phenomenonwith a counter-term consisting of a weighted count of associative submanifolds. However, theydid not elaborate on how to construct a suitable coherent system of weights. Haydys and Walpuskiproposed to define such weights by counting solutions to the Seiberg–Witten equations associ-ated with the ADHM construction of instantons on R [HW15, paragraphs following Remark 1.7;Hay17; DW17, Introduction; DW18, Appendix B].The construction of these weights depends on the choice of the structure group of G –instantons,an obvious choice being SU ( r ) . If one specializes to r = , that is, to trivial line bundles, then thereare no non-trivial G –instantons and their naive count is, trivially, an invariant. However, accord-ing to the Haydys–Walpuski proposal one should still count associatives weighted by the count1f solutions to the Seiberg–Witten equation on them. It is known that counting associatives bythemselves does not lead to an invariant, because the following situations may arise along a –parameter family of G –metrics:1. An embedded associative submanifold develops a self-intersection. Out of this self-intersectiona new associative submanifold is created, as shown by Nordström [Nor13]. Topologically,this submanifold is a connected sum.2. By analogy with special Lagrangians in Calabi–Yau –folds [Joy02, Section 3], it has beenconjectured that it is possible for three distinct associative submanifolds to degenerate intoa singular associative submanifold with an isolated singularity modeled on the cone over T [Wal13, p.154; Joy17, Conjecture 5.3]. Topologically, these three submanifolds form asurgery triad.We will argue that known vanishing results and surgery formulae for the Seiberg–Witten invari-ants of –manifolds [MT96, Proposition 4.1 and Theorem 5.3], show that the count of associativesweighted by solutions to the Seiberg–Witten equation is invariant under transitions (1) and (2),assuming that all connected components of the associative submanifolds in question have b > .This restriction is needed in order to be able to avoid reducible solutions and obtain a well-definedSeiberg–Witten invariant as an integer . We know of no natural assumption that would ensurethat this restriction holds for all relevant associative submanifolds. Hence, the Haydys–Walpuskiproposal cannot yield an invariant which is just an integer.One can define a topological invariant using the Seiberg–Witten equation for any compact,oriented –manifold. This invariant, however, is not a number but rather a homology group,called monopole Floer homology [MW01; Man03; KM07; Frø10]. The behavior of monopole Floerhomology under connected sum and in surgery triads is well-understood [KMOS07, Theorem 2.4;BMO; Lin15, Theorem 5]. We will explain how to construct a chain complex associated with a G –manifold using the monopole chain complexes of associative submanifolds. The homology ofthis chain complex might be invariant under transitions (1) and (2).The discussion so far only involved the classical Seiberg–Witten equation. There is a furthertransition that might spoil the invariance of the proposed homology group:3. Along generic –parameter families of G –metrics, somewhere injective immersed associa-tive submanifolds can degenerate by converging to a multiple cover.We will explain why this phenomenon occurs and that it can change the number of associatives,even when weighted by counts of solutions to the Seiberg–Witten equation. This is where ADHMmonopoles , solutions to the Seiberg–Witten equations related to the ADHM construction, enterthe picture. Counting ADHM monopoles does not lead to a topological invariant of –manifolds.We will provide evidence for the conjecture that the change in the count of ADHM monopoles Using spectral counter-terms, Chen [Che97; Che98] and Lim [Lim00] were able to define Seiberg–Witten invari-ants of 3–manifolds with b
1. These, however, are rational and cannot satisfy the necessary vanishing theorem. G –manifolds: a homology group generated by associatives and ADHM monopoles.This paper is organized as follows. After reviewing in Section 2 the basics of G –geometry, wediscuss in Section 3 and Section 4 the three problems with counting associatives described above.The core of the paper are: Section 5 where we introduce ADHM monopoles and relate them tomultiple covers of associatives, and Section 6 where we outline a construction of a Floer homologygroup associated with a G –manifold. In Section 7 we argue that a dimensional reduction ofour proposal should lead to a symplectic analogue of Pandariphande and Thomas’ stable pairinvariant known in algebraic geometry [PT09]. Appendix A contains the proof of a transversalitytheorem for somewhere injective associative immersions. Appendix B and Appendix C developa general theory of the Haydys correspondence with stabilizers for Seiberg–Witten equationsassociated with quaternionic representations. Appendix D summarizes the linear algebra of theADHM representation.Finally, we would like to point out that an alternative approach to counting associative sub-manifolds has been proposed recently by Joyce [Joy17]. His proposal does not lead to a number ora homology group, but rather a more complicated object: a super-potential up to quasi-identitymorphisms. Acknowledgements
We are grateful to Simon Donaldson for his generosity, kindness, and op-timism which have inspired and motivated us over the years. We thank Tomasz Mrowka forpointing out [BMO] and advocating the idea of incorporating the monopole chain complex, Os-car García–Prada for a helpful discussion on vortex equations, and Richard Thomas for answeringour questions about stable pairs.This material is based upon work supported by the National Science Foundation under GrantNo. 1754967 and the Simons Collaboration Grant on “Special Holonomy in Geometry, Analysisand Physics”.
We begin with a review of G –manifolds and associative submanifolds with a focus towards ex-plaining what we mean by “counting associative submanifolds”. G –manifolds The exceptional Lie group G is the automorphism group of the octonions O , the unique –dimensional normed division algebra: G = Aut ( O ) . Since any automorphism of O preserves the unit ∈ O and its –dimensional orthogonal com-plement Im O ⊂ O , we can think of G as a subgroup of SO ( ) .3 efinition 2.1. A G –structure on a –dimensional manifold Y is a reduction of the structuregroup of the frame bundle of Y from GL ( ) to G . An almost G –manifold is a –dimensionalmanifold Y equipped with a G –structure.The multiplication on O endows Im O with: • an inner product , д : S Im O → R satisfying д ( u , v ) = − Re ( uv ) , • a cross-product · × · : Λ Im O → Im O defined by ( u , v ) 7→ u × v ≔ Im ( uv ) and a corresponding –form ϕ ∈ Λ Im O ∗ defined by ϕ ( u , v , w ) ≔ д ( u × v , w ) , as well as • an associator [· , · , ·] : Λ Im O → Im O defined by(2.2) [ u , v , w ] ≔ ( u × v ) × w + h v , w i u − h u , w i v and a corresponding –form ψ ∈ Λ Im O ∗ defined by ψ ( u , v , w , z ) ≔ д ([ u , v , w ] , z ) . These are related by the identities i ( u ) ϕ ∧ i ( v ) ϕ ∧ ϕ = д ( u , v ) vol д and ∗ д ϕ = ψ (2.3)for a unique choice of an orientation on Im O . We refer the reader to [HL82, Chapter IV; SW17]for a more detailed discussion.A G –structure on Y endows TY with analogous structures: • a Riemannian metric д , • a cross-product · × · : Λ TY → TY , • a –form ϕ ∈ Ω ( Y ) , • an associator [· , · , ·] : Λ TY → TY , and • a –form ψ ∈ Ω ( Y ) , 4atisfying the same relations as above. From (2.3) it is apparent that from ϕ one can reconstruct д and thus also ψ , the cross-product, and the associator. Similarly, one can reconstruct д from ψ together with the orientation. The condition for a –form ϕ or a –form ψ to arise from a G –structure is that the form be definite; see [Hit01, Section 8.3; Bry06, Section 2.8]. We say that a –form ϕ is definite if the bilinear form G ϕ ∈ Γ ( S T ∗ Y ⊗ Λ T ∗ Y ) defined by G ϕ ( u , v ) ≔ i ( u ) ϕ ∧ i ( v ) ϕ ∧ ϕ is definite. We say that a –form ψ is definite if the bilinear form G ∗ ψ ∈ Γ (cid:0) S TY ⊗ ( Λ T ∗ Y ) ⊗ (cid:1) defined by G ∗ ψ ( α , β ) ≔ i ( α ) ψ ∧ i ( β ) ψ ∧ ψ is definite. Here we identify Λ T ∗ Y (cid:27) Λ TY ⊗ Λ T ∗ Y . Therefore, a G –structure can be specifiedeither by a definite –form ϕ , or by a definite –form ψ together with an orientation.A G –structure on a –manifold induces a spin structure through the inclusion G ⊂ Spin ( ) .In fact, a –manifold admits a G –structure if and only if it is spin, see [Gra69, Theorems 3.1 and3.2] and [LM89, p. 321]. This means that the existence of a G –structure is a soft, topologicalcondition. More rigid notions are obtained by imposing conditions on the torsion of the G –structure, in the sense of G –structures, see [Joy00, Section 2.6]. The most stringent and mostinteresting condition to impose is that the torsion vanishes. Definition 2.4. A G –manifold is a –manifold equipped with a torsion-free G –structure. Theorem 2.5 (Fernández and Gray [FG82, Theorem 5.2]) . A G –structure on a –manifold Y istorsion-free if and only the associated –form ϕ as well as the associated –form ψ are closed: d ϕ = and d ψ = . The Riemannian metric induced by a torsion-free G –structure has holonomy contained in G —one of two exceptional holonomy groups in Berger’s classification [Ber55, Theorem 3]. If Y is compact, then equality holds if and only if π ( Y ) is finite [Joy00, Proposition 10.2.2]. We referthe reader to [Joy00, Section 10] for a thorough discussion of the properties of G –manifolds. Example 2.6. If Z is a Calabi–Yau –fold with a Kähler form ω and a holomorphic volume form Ω , and if t denotes the coordinate on S , then S × Z is a G –manifold with ϕ = d t ∧ ω + Re Ω and ψ = ω ∧ ω + d t ∧ Im Ω . In this case the holonomy group is contained in SU ( ) ⊂ G . Example 2.7.
The first local, complete, and compact examples of manifolds with holonomy equalto G are due to Bryant [Bry87], Bryant and Salamon [BS89], and Joyce [Joy96a; Joy96b; Joy00]respectively. Joyce’s examples arise from a generalized Kummer construction based on smooth-ing flat G –orbifolds of the form T / Γ where Γ is a finite group of isometries of the –torus. This5ethod has recently been extended to more general G –orbifolds by Joyce and Karigiannis [JK17].The most fruitful construction method for G –manifolds to this day is the twisted connected sumconstruction, which was pioneered by Kovalev [Kov03] and improved by Kovalev and Lee [KL11]and Corti, Haskins, Nordström, and Pacini [CHNP13; CHNP15]. It is based on gluing, in a twistedfashion, a pair of asymptotically cylindrical G –manifolds which are products of S with asymp-totically cylindrical Calabi–Yau –folds. Using this construction, Corti, Haskins, Nordström, andPacini [CHNP15] produced tens of millions of examples of compact G –manifolds. Definition 2.8.
Let Y be an almost G –manifold, let P be an oriented –manifold, and let ι : P → Y be an immersion. We say that ι is associative if(2.9) ι ∗ [· , · , ·] = ∈ Ω ( P , ι ∗ TY ) and ι ∗ ϕ is positive . An immersed associative submanifold is an equivalence class [ ι ] of an associative immer-sion ι ∈ Imm ( P , Y )/ Diff + ( P ) for some oriented –manifold P . Here Imm ( P , Y ) is the space ofimmersions P → Y and Diff + ( P ) is the group of orientation-preserving diffeomorphisms of P .Harvey and Lawson [HL82, Chapter IV, Theorem 1.6] proved the identity(2.10) ϕ ( u , v , w ) + |[ u , v , w ]| = | u ∧ v ∧ w | . This shows that ϕ is a semi-calibration and that associative submanifolds are calibrated by ϕ . Werefer to [HL82, Introduction] and [Joy00, Section 3.7] for an introduction to calibrated geometry;we recall only the following simple but fundamental fact. Proposition 2.11. If ι : P → Y is associative, then ι ∗ ϕ = vol ι ∗ д . In particular, if ϕ is closed and P is compact, then the immersed submanifold ι ( P ) is volume-minimizingin the homology class ι ∗ [ P ] and vol ( P , ι ∗ д ) = h[ ϕ ] , ι ∗ [ P ]i . Proposition 2.12 (see, e.g., [SW17, Lemma 4.7]) . If ι : P → Y is an immersion, then the followingare equivalent:1. ι ∗ [· , · , ·] = ,2. for all u , v ∈ ι ∗ T x P , u × v ∈ ι ∗ T P , and3. for all u ∈ ι ∗ T x P and v ∈ ( ι ∗ T x P ) ⊥ , u × v ∈ ( ι ∗ T x P ) ⊥ . Example 2.13.
Let Z be a Calabi–Yau –fold. Equip S × Z with the G –structure from Example 2.6.If Σ ⊂ Z is a holomorphic curve, then S × Σ is associative. If L ⊂ Z is a special Lagrangiansubmanifold, then, for any t ∈ S , { t } × L is associative.6 xample 2.14. Examples of associative submanifolds which arise as fixed points of involutionshave been given by Joyce [Joy96b, Section 4.2]. Examples of associative submanifolds arisingfrom holomorphic curves and special Lagrangians in asymptotically cylindrical Calabi–Yau –folds were constructed by Corti, Haskins, Nordström, and Pacini [CHNP15, Section 5] L functional Associative submanifolds can be formally thought of as critical points of a functional L on theinfinite-dimensional space of submanifolds. In contrast to many other functionals studied in dif-ferential geometry (for example, the Dirichlet functional), the Hessian of L at a critical point isnot positive definite. As we will see, it is a first order elliptic operator whose spectrum is dis-crete and unbounded in both positive and negative directions. Morse theory of functionals withthis property, most notably the Chern–Simons functional in gauge theory, was first developedby Floer [Flo88; Don02]. The existence of such L already hints at the possibility of constructing Floer homology groups from a chain complex formally generated by associative submanifolds.
Definition 2.15.
Define the –form δ L = δ L ψ ∈ Ω ( Imm ( P , Y )) by δ ι L ( n ) = ˆ P ι ∗ i ( n ) ψ = ˆ P h ι ∗ [· , · , ·] , n i for n ∈ T ι Imm ( P , Y ) = Γ ( P , ι ∗ TY ) . Proposition 2.16. ι is associative if and only if δ ι L = and ι ∗ ϕ is positive.2. δ L is Diff + ( P ) –invariant.3. If d ψ = , then δ L is a closed –form. In fact, there is a Diff + ( P ) –equivariant covering space π : g Imm ( P , Y ) → Imm ( P , Y ) and a Diff + ( P ) –equivariant function ˜ L : g Imm ( P , Y ) → R whosedifferential is π ∗ δ L . Proof.
Assertions (1) and (2) are both trivial. For β ∈ H ( Y , R ) , let Imm β ( P , Y ) denote the set ofimmersions ι : P → Y such that ι ∗ [ P ] = β . Fix P ∈ Imm β ( P , Y ) and denote by g Imm β ( P , Y ) thespace of pairs ( ι , [ Q ]) with ι ∈ Imm β ( P ) and [ Q ] an equivalence class of –chains in Y such that ∂ Q = P − P with [ Q ] = [ Q ′ ] if and only if [ Q − Q ′ ] = ∈ H ( Y , Z ) . Define ˜ L : g Imm β ( P , Y ) → R by ˜ L ( ι , [ Q ]) = ˆ Q ψ . The function ˜ L has the desired properties; see also [DT98, Section 8]. (cid:3) Although n is not a vector field on Y , by slight abuse of notation we denote by ι ∗ i ( n ) ψ the 3–form on P given by ( u , v , w ) 7→ ψ ( ι ∗ u , ι ∗ v , ι ∗ w , n ) . This justifies the notation δ L since locally it is the differential of a function. .4 The moduli space of associatives Definition 2.17.
Let P be a compact, oriented –manifold and let β ∈ H ( Y , Z ) . Denote by Imm β ( P , Y ) the space of immersions ι : P → Y with ι ∗ [ P ] = β . The group Diff + ( P ) acts on Imm β ( P , Y ) . The moduli space of immersed associative submanifolds is A ( ψ ) = Þ β ∈ H ( Y , Z ) A β ( ψ ) = Þ β ∈ H ( Y , Z ) Þ P A P , β ( ψ ) with A P , β ( ψ ) ≔ (cid:8) [ ι ] ∈ Imm β ( P , Y )/ Diff + ( Y ) : ( . ) (cid:9) . Here P ranges over all diffeomorphism types of compact, oriented –manifolds.Denote by D ( Y ) the space of definite –forms on Y . If P is a subspace of D ( Y ) , then the P –universal moduli space is A ( P ) = Ø ψ ∈ P A ( ψ ) . The moduli space A ( P ) inherits a topology from the C ∞ –topology on Imm β ( P , Y ) . As wewill explain in the following, the infinitesimal deformation theory of associatives submanifolds iscontrolled by a first-order elliptic operator and A ( P ) admits corresponding Kuranishi models. Definition 2.18.
Let ι : P → Y be an associative immersion. Denote by N ι ≔ ι ∗ TY / T P (cid:27)
T P ⊥ ⊂ ι ∗ TY its normal bundle and by ∇ the connection on N ι induced by the Levi-Civita connection. The
Fueter operator associated with ι is the first order differential operator F ι = F ι , ψ : Γ ( N ι ) → Γ ( N ι ) defined by F ι ( m ) ≔ Õ i = ι ∗ e i × ∇ e i m . Here ( e , e , e ) is an orthonormal frame of P .This operator arises as follows. Identify N ι with
T P ⊥ ⊂ ι ∗ TY and, given a normal vector field m ∈ Γ ( N ι ) , define ι m : P → Y by ι m ( x ) ≔ exp ( m ( x )) . The condition for ι εm to be associative to first order in ε is that = dd ε (cid:12)(cid:12)(cid:12)(cid:12) ε = [( ι εm ) ∗ e , ( ι εm ) ∗ e , ( ι εm ) ∗ e ] = ( ι ∗ e × ι ∗ e ) × ∇ e m + cyclic permutations = Õ i = ι ∗ e i × ∇ e i m . ι : P → Y is associativeso we have ι ∗ e × ι ∗ e = ι ∗ e (as well as all of its cyclic permutations). Proposition 2.19 (Joyce [Joy17, paragraph after Theorem 2.12]) . If d ψ = , then Hess ˜ L ( n , m ) = ˆ P h n , F ι m i with ˜ L as in Proposition 2.16 (3) . In particular, F ι is self-adjoint. Theorem 2.20 (McLean [McL98] and Joyce [Joy17, Theorem 2.12]) . Let [ ι : P → Y ] ∈ A β ( ψ ) .Denote by Aut ( ι ) the stabilizer of ι in Diff + ( P ) .The group Aut ( ι ) is finite. The Fueter operator F ι is equivariant with respect to the action of Aut ( ι ) on Γ ( N ι ) . If P is a submanifold of the space of definite –forms containing ψ , then there are:• an Aut ( ι ) –invariant open subset U ⊂ P × ker F ι ,• a smooth Aut ( ι ) –equivariant map ob : P × U → coker F ι with ob ( ψ , ·) and its derivativevanishing at ,• an open neighborhood V of ([ ι ] , ψ ) in A β ( P ) , and• a homeomorphism x : ob − ( )/ Aut ( ι ) → V .Moreover, if ( p , n ) ∈ ob − ( ) , then the stabilizer of any immersion representing x ( p , n ) is the stabilizerof n in Aut ( ι ) . Definition 2.21.
We say that an associative immersion ι : P → Y is unobstructed (or rigid ) if F ι is invertible. It follows from Theorem 2.20 that if all associative immersions are rigid, then the moduli space A β ( ψ ) is a collection of isolated points—in other words, the functional L is a Morse function.While this is not always true, below we show that it does hold for a large class of immersions andfor a generic choice of a closed positive –form ψ . Definition 2.22.
An immersion ι : P → Y is called somewhere injective if each connected com-ponent of P contains a point x such that ι − ( ι ( x )) = { x } . Denote by A si β ( ψ ) the open subset of somewhere injective immersions with respect to ψ . Given a submanifold P ofthe space of definite –forms, set A si β ( P ) = Ø ψ ∈ P A si β ( ψ ) . roposition 2.23. Denote by D c ( Y ) the space of closed, definite –forms.1. There is a residual subset D c , reg ⊂ D c ( Y ) such that for every ψ ∈ D c , reg (a) the moduli space A si β ( ψ ) is a –dimensional manifold and consists only of unobstructedassociative submanifolds, and(b) A si β ( ψ ) consists only of embedded associative submanifolds.2. If ψ , ψ ∈ D c , reg ( Y ) , then there is a residual subset D c , reg ( ψ , ψ ) in the space of paths from ψ to ψ in D c ( Y ) such that for every ( ψ t ) t ∈[ , ] ∈ D c , reg ( ψ , ψ ) (a) the universal moduli space A si β ({ ψ t : t ∈ [ , ]}) is a –dimensional manifold, and(b) for each component {( ψ t , [ ι t ]) : t ∈ J } with J ⊂ [ , ] an interval, there is a discrete set J × ⊂ J such that:i. for t ∈ J \ J × the map ι t is an embedding andii. for t × ∈ J × there is a T > and with the property that P ≔ Ø | t − t × | < T { t } × ι t ( P ) ⊂ R × Y has a unique self-intersection and this intersection is transverse. The proof of this result is deferred to Appendix A. It is similar to that of analogous resultsabout pseudo-holomorphic curves in symplectic manifolds, cf. McDuff and Salamon [MS12, Sec-tions 3.2 and 3.4]. In fact, our situation is simpler because we assume from the outset that ι is animmersion. As we have seen, transversality for associative embeddings can be achieved by perturbing ψ .However, even if the moduli space A β ( ψ ) consists of isolated points, the number of points canbe infinite. Indeed, for an arbitrary definite –form ψ there is no reason to expect A β ( ψ ) tobe compact. The situation is better when one considers a special class of tamed –forms. Thisis analogous to the notion of a tamed almost complex structure in symplectic topology, whichguarantees area bounds for pseudo-holomorphic curves. Definition 2.24 (Donaldson and Segal [DS11, Section 3.2], Joyce [Joy17, Definition 2.6]) . Let Y bean almost G –manifold with –form ϕ , –form ψ , and associator [· , · , ·] . We say that τ ∈ Ω ( Y ) tames ψ if d τ = and for all x ∈ Y and u , v , w ∈ T x Y with [ u , v , w ] = and ϕ ( u , v , w ) > , wehave τ ( u , v , w ) > . Example 2.25. If ψ corresponds to a torsion-free G –structure, then ψ , as well as any nearby –form, is tamed by ϕ = ∗ ψ . 10ne should think of tamed, closed, definite –forms as a softening of the notion of a definite –form giving rise to a torsion-free G –structure. The advantage of working with tamed formsis that the volume of any associative submanifold in A β ( ψ ) is bounded and one can, in principle,use geometric measure theory to compactify A β ( ψ ) . Proposition 2.26 (Donaldson and Segal [DS11, Section 3.2], Joyce [Joy17, Section 2.5]) . Let Y bea compact almost G –manifold with –form ψ . If ψ is tamed by a closed –form τ , then there is aconstant c > such that for every associative immersion ι : P → Y with P compact vol ( P , ι ∗ д ) c · h[ τ ] , ι ∗ [ P ]i . Question 2.27.
Is there a residual subset of tamed, closed, definite –forms for which A β ( ψ ) is acompact –dimensional manifold (or orbifold)?If the answer to this question is yes, then for every ψ from this residual subset we can define(2.28) n β ( ψ ) ≔ A β ( ψ ) . Question 2.29. Is n β ( ψ ) , or some modification of it, invariant under deforming ψ ?If the answer to this question is also yes, then n β would give rise to a deformation invariant of G –manifolds by defining its value on a torsion-free G –structure ψ to be that on a nearby tamed,closed, definite –form.It is easy to see that a naive interpretation of A β ( ψ ) as the cardinality of A β ( ψ ) does not lead toan invariant. Suppose that P = { ψ t : t ∈ (− , )} is –parameter family of tamed, closed, definite –form and [ ι : P → Y ] ∈ A β ( ψ ) with dim ker F ι , ψ = . By Theorem 2.20, a neighborhood of ([ ι ] , ψ ) ∈ A β ( P ) is given by ob − ( ) with ob a smooth map satisfying ob ( t , δ ) = λt + cδ + higher order terms . For a generic –parameter family we will have λ , c , . For simplicity, let us assume that λ = c = .In this situation for − ≪ t < , there are two associative submanifolds [ ι ± t : P → Y ] with respectto ψ t near [ ι ] . As t tends to , [ ι ± t ] tends to [ ι ] . For t > there are no associatives near [ ι ] . Thismeans that n β ( ϕ ) as defined in (2.28) changes by − as t passes through .The origin of this problem is that A β ( ψ ) should be an oriented manifold and we should countassociative immersions [ ι ] ∈ A β ( ψ ) with signs ε ([ ι ] , ψ ) ∈ {± } . These signs should be such that if { ι t : P → Y : t ∈ [ , ]} is a –parameter family of associative immersions along a –parameterfamily of closed, definite –forms, then(2.30) ε ([ ι ] , ψ ) = (− ) SF ( F ιt , ψt : t ∈[ , ] ) · ε ([ ι ] , ψ ) . In the above situation we have ε ([ ι + t ] , ψ t ) = − ε ([ ι − t ] , ψ t ) . n β ( ψ ) will be be invariant as t passes though if we interpret as as signed count, thatis,(2.31) n β ( ψ ) ≔ Õ [ ι ]∈ A β ( ψ ) ε ([ ι ] , ψ ) with some choice of ε satisfying (2.30). An almost canonical method for determining ε was re-cently discovered by Joyce [Joy17, Section 3]. We refer the reader to Joyce’s article for a carefuland detailed discussion. [ ι + ] + [ ι − ] − [ ι ] ψ t Figure 1: Two associatives submanifold with opposite signs annihilating in an obstructed associa-tive submanifold. T –singularities, and the Seiberg–Witten invariant In what follows we describe in more detail transitions (1) and (2) from Section 1, and explain whythey spoil the deformation invariance of n β ( ψ ) . We then argue that the Seiberg–Witten equationon –manifolds might play a role in repairing the deformation invariance. There is, however, aprice to pay: one has to give up on defining a numerical invariant and instead work with morecomplicated algebraic objects: chain complexes and homology groups. Let ( ψ t ) t ∈(− T , T ) be a –parameter family of closed, tamed, definite –forms on Y and let ( ι t : P → Y ) t ∈(− T , T ) be a –parameter family of somewhere injective unobstructed associative immersions.By Proposition 2.23, if ( ψ t ) is generic, then we can assume that ι t is an embedding for all t , and ι has a unique self-intersection as in Proposition 2.23(2b). This intersection is locally modeled onthe intersection of two transverse associative subspaces of R . Given any pair of transverse asso-ciative subspaces of R , there is a smooth associative submanifold asymptotic to these subspacesat infinity, called the Lawlor neck. Nordström proved that out of the unique self-intersection of ι a new –parameter family of associative submanifolds is created in Y by gluing in a Lawlor neck.12 heorem 3.1 (Nordström [Nor13]) . Let Y be a compact –manifold and let ( ψ t ) t ∈(− T , T ) be a familyof closed, definite –forms on Y . Let P be a compact, oriented –manifold. Suppose that ( ι t : P → Y ) t ∈(− T , T ) is a –parameter family of unobstructed associative immersions such that P ≔ Ø t ∈(− T , T ) { t } × ι t ( P ) ⊂ R × Y has a unique self-intersection which occurs for t = and is transverse. Let x ± denote the preimagesin P of the intersection in Y and denote by P ♯ the connected sum of P at x + and x − .There is a constant ε > , a continuous function t : [ , ε ] → (− T , T ) , and a –parameter familyof immersions ( ι ♯ ε : P ♯ → Y ) ε ∈( , ε ] such that, for each ε ∈ ( , ε ] , ι ♯ ε is an unobstructed associativeimmersion with respect to ψ t ( ε ) . Moreover, as ε tends to zero the images of ι ♯ ε converge to the imageof ι in the sense of integral currents.Remark . The paper [Nor13] has not yet been made available to a wider audience. A part ofwhat goes into proving Theorem 3.1 can be found in [Joy17, Section 4.2]. There it is also arguedthat for a generic choice of ( ψ t ) t ∈(− T , T ) the function t is expected to be of the form t ( ε ) = δε + O ( ε ) with a non-zero coefficient δ whose geometric meaning is also explained therein. Remark . Denote by P , . . . , P n the connected components of P . Let j ± be such that x ± ∈ P j ± .We have P ♯ (cid:27) (Ý j , j ± P j ⊔ ( P j + ♯ P j − ) for j + , j − and Ý j , j + P j ⊔ ( P j + ♯ S × S ) for j + = j − . [ ι t ] ± [ ι ♯ ε ] ± ψ t Figure 2: An associative being born out of an intersection another associative.In the situation described in Theorem 3.1 and depicted in Figure 2, n β ( ψ t ) as defined in (2.31)changes by ± as t crosses . In particular, n β is not invariant. T –singularities Suppose that ˆ P is an associative submanifold in ( Y , ψ ) with a point singularity at x ∈ ˆ P modelledon the following cone over T : ˆ L = (cid:8) ( , z , z , z ) ∈ R ⊕ C : | z | = | z | = | z | , z z z ∈ [ , ∞) ∈ C (cid:9) = n r · ( , e iθ , e iθ , e − iθ − iθ ) : r ∈ [ , ∞) , θ , θ ∈ S o . − . That is: we should expect them not to exist for ageneric choice of ψ but to appear along generic –parameter families ( ψ t ) .The singularity model ˆ L can be resolved in ways: L λ = (cid:8) ( , z , z , z ) ∈ R ⊕ C : | z | − λ = | z | = | z | , z z z ∈ [ , ∞) ∈ C (cid:9) , L λ = (cid:8) ( , z , z , z ) ∈ R ⊕ C : | z | = | z | − λ = | z | , z z z ∈ [ , ∞) ∈ C (cid:9) , and L λ = (cid:8) ( , z , z , z ) ∈ R ⊕ C : | z | = | z | = | z | − λ , z z z ∈ [ , ∞) ∈ C (cid:9) . These are asymptotic to ˆ L at infinity and smooth, which can be seen by identifying L iλ with S × C via S × C → L λ , ( e iθ , z ) 7→ (cid:16) , e iθ p | z | + λ , z , e − iθ ¯ z (cid:17) , S × C → L λ , ( e iθ , z ) 7→ (cid:16) , e − iθ ¯ z , e iθ p | z | + λ , z (cid:17) , and S × C → L λ , ( e iθ , z ) 7→ (cid:16) , z , e − iθ ¯ z , e iθ p | z | + λ (cid:17) . (3.4)Topologically, L iλ can be obtained from ˆ L via Dehn surgery. Definition 3.5.
Let P ◦ be a –manifold with ¯ ∂ P ◦ = T . Let µ be a simple closed curve in T . The Dehn filling of P ◦ along µ , denoted by P ◦ µ , is the –manifold obtained by attaching S × D to P ◦ in such a way that {∗} × S is identified with µ . Remark . Up to diffeomorphism, P ◦ µ depends only on the homotopy class of µ ⊂ T ; moreover,it does not depend on the orientation of µ .We can identify the boundary of ˆ L ◦ ≔ ˆ L \ B with T via ( e iθ , e iθ ) 7→ √ (cid:16) , e iθ , e iθ , e − iθ − iθ (cid:17) . Comparing the maps introduced in (3.4) restricted to {∗} × S with the above identification, wesee that L iλ is obtained by Dehn filling ˆ L ◦ along loops representing the homology classes(3.7) µ = ( , ) , µ = (− , ) , and µ = ( , − ) where ( , ) and ( , ) are the generators of H ( T , Z ) corresponding to the loops θ
7→ ( e iθ , ) and θ
7→ ( , e iθ ) .We expect that ˆ P can be resolved in three ways as well. Conjecture 3.8 (cf. Joyce [Joy17, Conjecture 5.3]) . Let ( ψ t ) t ∈(− T , T ) be a –parameter family of closed,tamed, definite –forms on Y . Let ˆ P be an unobstructed singular associative submanifold in ( Y , ψ ) ith a unique singularity at x which is modeled on ˆ L . Associated to this data there are constants δ , δ , γ ∈ R . For a generic –parameter family ( ψ t ) t ∈(− T , T ) , δ , , δ , , δ , δ and γ , . Ifthis holds, then there is ε > and, for i = , , , there are functions t i : [ , ε ] → (− T , T ) , compact,oriented –manifolds P i , and –parameter families of immersions ( ι iε : P i → Y ) ε ∈( , ε ] such that:1. ι iε is an unobstructed associative immersion with respect to ψ t i ( ε ) .2. ι iε ( P i ) is close to ˆ P away from x and close to L iε near x .3. P i is diffeomorphic to the manifold obtained by Dehn filling ˆ P ◦ = ˆ P \ B σ ( x ) along µ i where µ i ∈ H ( ∂ ˆ P ◦ ) = H ( T ) is as in (3.7) .4. We have t ( ε ) = − δ γ ε + O ( ε ) , t ( ε ) = δ γ ε + O ( ε ) , and t ( ε ) = δ − δ γ ε + O ( ε ) . [ ι ] ± [ ι ] ± [ ι ] ± ˆ P ψ t Figure 3: Three associatives emerging out of a singular associative for δ > δ > .In the situation described in Conjecture 3.8 and depicted in Figure 3, n β ( ψ t ) as defined in (2.31)changes as t crosses . Again, the occurrence of the phenomenon described above would preclude n β from being a deformation invariant. –manifolds If there were a topological invariant w ( P ) ∈ Z defined for every compact, oriented –manifoldand satisfying w ( P ♯ P ) = and ε w ( P ◦ µ ) + ε w ( P ◦ µ ) + ε w ( P ◦ µ ) = (3.9)with µ , µ , µ as in (3.7) and some choice of ε , ε , ε ∈ {± } , then(3.10) n β ( ψ ) ≔ Õ [ ι ]∈ A β ( ψ ) ε ([ ι ] , ψ ) w ( P ) –manifolds is trivial since w ( P ) = w ( P ♯ S ) = for all oriented –manifolds P . However, for those –manifolds P for which b ( P j ) > for all connected components P j , there are non-trivial invariants satisfying (3.9). One example ofsuch an invariant is the Seiberg–Witten invariant SW ( P ) . We refer the reader to [MT96, Section2] for a detailed discussion of the construction of SW ( P ) . For the moment, it shall suffice to thinkof SW ( P ) as the signed count of all gauge-equivalence classes of solutions to the Seiberg–Wittenequation; that is, pairs of ( Ψ , A ) ∈ Γ ( W ) × A ( det ( W )) satisfying / D A Ψ = and F A = µ ( Ψ ) . (3.11)Here W is the spinor bundle of a spin c structure w on P , / D A is the twisted Dirac operator, and µ ( Ψ ) = ΨΨ ∗ − | Ψ | id W is identified with an imaginary-valued –form using the Clifford multi-plication. Remark . The actual definition of SW ( P ) involves perturbing (3.11) by a closed –form η inorder to ensure that the moduli space of solutions is cut-out transversely and contains no reduciblesolutions. The necessity to choose η and the fact that H ( P , Z ) has codimension b ( P ) in H ( P , R ) ,where the cohomology class of η lies, is responsible for the restriction b ( P ) > . Remark . SW ( P ) has a refinement SW ( P ) defined for oriented –manifolds P with b ( P ) > ;roughly speaking, it is an integer-valued function on the set of the isomorphism classes of spin c structures w on P . When b > , it is zero for all but finitely many w and we can take SW ( P ) tobe the sum of the invariants over all spin c structures. We come back to this point in Section 7.2. Theorem 3.14 (Meng and Taubes [MT96, Proposition 4.1]) . If P , P are two compact, connected,oriented –manifolds with b ( P i ) > , then SW ( P ♯ P ) = . Theorem 3.15 (Meng and Taubes [MT96, Theorem 5.3]) . Let P ◦ be a compact, connected, oriented –manifold with ∂ P ◦ = T . If µ , µ , µ ∈ H ( ∂ P ◦ ) are such that µ · µ = µ · µ = µ · µ = − (with T = ∂ P ◦ oriented as the boundary of P ◦ ), then ε · SW ( P ◦ µ ) + ε · SW ( P ◦ µ ) + ε · SW ( P ◦ µ ) = for suitable choices of ε , ε , ε ∈ {± } , provided b ( P ◦ µ i ) > for all i = , , . emark . The formulation of [MT96, Theorem 5.3] is in terms of p / q –surgery on a link L which is rationally trivial in homology. The discussion in [KM07, Section 42.1] explains how thisis related to Dehn filling, and from this it is clear that the surgery formula given by Meng andTaubes implies the above theorem. Remark . The Seiberg–Witten invariant is often defined only for compact, connected, oriented –manifolds P . If P has connected components P , . . . , P m , then SW ( P ) ≔ Î mj = SW ( P j ) .Let us temporarily assume that all associative immersions ι : P → Y with ι ∗ [ P ] = β happento be such that all connected components P j satisfy b ( P j ) > . If we defined n β by (3.10) withthe weight w = SW , then n β would be invariant in the situations described in Section 3.1 andSection 3.2, at least if the signs work out correctly, or modulo . Defining n β in this way reallyamounts to counting a much larger moduli space than A β ( ψ ) , namely: A SW β ( ψ ) = Þ P Þ w A SW P , β , w ( ψ ) with A SW P , β , w ( ψ ) ≔ ( ι , Ψ , A ) ∈ Imm β ( P , Y ) × Γ ( W ) × A ( det W ) : ι satisfies (2.9) and ( Ψ , A ) satisfies (3.11)with respect to ι ∗ д ψ Diff + ( P ) ⋉ C ∞ ( P , U ( )) . Here w ranges over all isomorphism classes of spin c structures on P and W denotes the spinorbundle. The non-invariance of n β as defined in (2.31) can be traced back to the completion of A β ({ ψ t }) not being a –manifold. The moduli space A SW β ({ ψ t }) smooths out the singularities inthe completion of A β ({ ψ t }) encountered in the situations described in Section 3.1 and Section 3.2;see Figure 4. [ ι , Ψ , , A , ] [ ι , Ψ , , A , ][ ι , Ψ , , A , ] [ ι , Ψ , , A , ][ ι , Ψ , , A , ] [ ι , Ψ , A ] ψ t Figure 4: An example of how counting with Seiberg–Witten solutions can smooth out the situa-tion depicted in Figure 3.To sum up: the issue with defining a topological invariant w ( P ) ∈ Z with the propertiesdescribed in (3.9) means that there is indeed no invariant n β ( ψ ) ∈ Z defined by a formula ofthe form (3.10). If it happens that for all associatives with ι ∗ [ P ] = β all connected components17 j satisfy b ( P j ) > , then the invariance of n β ( ψ ) can be rescued by setting w ( P ) = SW ( P ) .Unfortunately, there is no reason to believe that this holds for any reasonable class of closed,tamed, definite –forms ψ or choice of β . (The situation is somewhat better for associatives arisingfrom holomorphic curves in Calabi–Yau –folds. We discuss this case in Section 7.) However,Seiberg–Witten theory of –manifolds suggests an alternative approach to defining an invariantof G –manifolds. Although there is no topological invariant w ( P ) ∈ Z defined for all closed, oriented –manifolds,satisfying the properties described in (3.9), there are Seiberg–Witten–Floer homology theories sat-isfying analogues of (3.9), see Marcolli and Wang [MW01], Manolescu [Man03], Kronheimer andMrowka [KM07], and Frøyshov [Frø10]. We focus on one of the variants defined by Kronheimerand Mrowka. To each closed, oriented –manifold P they assign a homology group d HM ( P ) = H (cid:0)d CM ( P , ♣) , ˆ ∂ (cid:1) . Very roughly, the chain complexes d CM ( P , ♣) are the C ∞ ( P , U ( )) –equivariant Morse complexes ofthe Chern–Simons–Dirac functional
CSD : Γ ( W ) × A ( det W ) → R defined by(3.18) CSD ( Ψ , A ) = ˆ P ( A − A ) ∧ F A + ˆ P (cid:10) / D A Ψ , Ψ (cid:11) vol on the configuration space C ( P ) = Þ w C ( P , w ) with C ( P , w ) = Γ ( W ) × A ( det W ) . (The fact that C ∞ ( P , U ( )) does not act freely is a significant problem, which Kronheimer andMrowka resolve by blowing up C ( P ) to a manifold with boundary and defining correspondingMorse complexes adapted to this situation.) The chain complexes d CM ( P , ♣) depend on choicesof additional data ♣ , in particular, a Riemannian metric on P as well as the choice of a suitableperturbation of the equation). Different choices of ♣ , however, lead to quasi-isomorphic chaincomplexes. We denote by d CM ( P ) quasi-isomorphism class of d CM ( P , ♣) , or rather its isomorphismclass in the derived category of chain complexes. If Q is a –dimensional cobordism with ∂ Q = P − P , then Kronheimer and Mrowka define an induced chain map d CM ( Q ) : d CM ( P ) → d CM ( P ) . If Q = [ , ] × P , then d CM ( Q ) is simply the differential ˆ ∂ on d CM ( P ) . The construction of d HM involves a choice of coefficients. For the upcoming results to hold one needs to work with Z coefficients (or suitable local systems). The monopole homology groups are then Z J U K –modules.Here one should think U as the same U as in H • ( B U ( )) = Z [ U ] .The following results are the analogues of the vanishing result from Theorem 3.14 and thesurgery formula from Theorem 3.15. 18 heorem 3.19 (Bloom, Mrowka, and Ozsváth [BMO]; Lin [Lin15, Theorem 5]) . Let P + and P − betwo compact, connected, oriented –manifolds. Denote by P + ♯ P − their connected sum and by Q thesurgery cobordism from P + ⊔ P − to P + ♯ P − . Then there is an exact triangle d CM ( P + ⊔ P − ) d CM ( Q ) −−−−−→ d CM ( P + ♯ P − ) → d CM ( P + ⊔ P − ) → d CM ( P + ⊔ P − )[− ] ; in particular, (3.20) d HM ( P + ⊔ P − )) (cid:27) H (cid:0) cone (cid:0)d CM ( P + ⊔ P − ) d CM ( Q ) −−−−−→ d CM ( P + ♯ P − ) (cid:1) (cid:1) . Remark . In [Lin15, Theorem 5], Theorem 3.19 is stated and proved as an isomorphism d HM ( P + ♯ P − ) (cid:27) H (cid:0) cone (cid:0)d CM ( P + ) ⊗ d CM ( P − )[ ] id ⊗ U + U ⊗ id −−−−−−−−−→ d CM ( P + ) ⊗ d CM ( P − ) (cid:1) (cid:1) induced by the cobordism Q . This formulation is much more useful for actual computations of d HM ( P + ♯ P − ) , but we need (3.20) for our purposes. The equivalence of these statements follows byobserving that once we identify d CM ( P + ⊔ P − ) = d CM ( P + ) ⊗ d CM ( P − ) the map d CM ( P + ⊔ P − ) → d CM ( P + ⊔ P − )[− ] is given by id ⊗ U + U ⊗ id and rotating the aboveexact triangle. Remark . More generally, if P ♯ is obtained by performing a connected sum at two points x ± in P and Q denotes the surgery cobordism from P to P ♯ , then we expect there to be an exact triangle d CM ( P ) d CM ( Q ) −−−−−→ d CM ( P ♯ ) → d CM ( P ) → d CM ( P )[− ] . Theorem 3.19 asserts that this is holds if the points x ± lie in different connected components of P . Theorem 3.23 (Kronheimer, Mrowka, Ozsváth, and Szabó [KMOS07, Theorem 2.4]; see also [KM07,Theorem 42.2.1]) . Let P ◦ be a compact, connected, oriented –manifold with ∂ P ◦ = T . Let µ , µ , µ ∈ H ( ∂ P ◦ ) be such that µ · µ = µ · µ = µ · µ = − (with T = ∂ P ◦ oriented as the boundary of P ◦ .) Denote by Q ij the surgery cobordism from P ◦ µ i to P ◦ µ j . There is an exact triangle d CM ( P ◦ µ ) d CM ( Q ) −−−−−−→ d CM ( P ◦ µ ) → d CM ( P ◦ µ ) → d CM ( P ◦ µ )[− ] ; in particular, (3.24) d HM ( P ◦ µ ) (cid:27) H (cid:0) cone (cid:0)d CM ( P ◦ µ ) d CM ( Q ) −−−−−−→ d CM ( P ◦ µ ) (cid:1) (cid:1) . We use square brackets to denote the translation C [ p ] n = C p + n , see [W ei94, Translation 1.2.8]. emark . While Theorem 3.23 holds for all three version of monopole homology defined byKronheimer and Mrowka, Theorem 3.19 only holds form d HM ; see [Lin15, paragraph after (13)].This is why we restricted ourselves to this version from the outset.Associative submanifolds are critical points of the functional L defined in Proposition 2.16.Gradient flow lines of the functional L can naturally be identified with immersions ι : R × P → R × Y such that ι ∗ ( ψ + d t ∧ ϕ ) = vol ι ∗ д and π R ◦ ι ( t , x ) = t ; see, e.g., [SW17, Lemma 12.6]. Definition 3.26.
Let ι ± : P ± → Y be associative immersions with respect to ψ . A Cayley cobor-dism in R × Y from ι − to ι + is an oriented –manifold Q together with an immersion ι : Q → R × Y such that ι ∗ ( ψ + d t ∧ ϕ ) = vol ι ∗ д and there are two open subsets U ± ⊂ Q such that Q \( U + ∪ U − ) is compact, constants T ± and c > ,and diffeomorphisms ϕ + : ( T + , ∞) × P + → U + and ϕ − : (−∞ , T − ) × P − → U − such that dist ( ι ◦ ϕ ± ( t , x ) , ( t , ι ± ( x ))) = O ( e − c | t | ) as t → ±∞ . The truncation of a Cayley cobordism is (the diffeomorphism type of) ¯ Q ≔ Q \ ( ϕ − (−∞ , T − − ) ∪ ϕ + ( T + + , ∞)) . The functorial behavior of Seiberg–Witten Floer homology groups under cobordisms leads tothe following questions about the existence of Cayley cobordisms.
Question 3.27.
In the situation of Theorem 3.1, does there exist a Cayley cobordism ι : Q → R × Y from ι t ( ε ) to ι ♯ ε , for all ε ∈ ( , ε ) , whose truncation ¯ Q is the surgery cobordism from P to P ♯ ? Question 3.28.
In the situation of Conjecture 3.8, if δ > δ > , does there exist a Cayleycobordism ι : Q → R × Y from ι t to ι t with ¯ Q being the surgery cobordism from P ◦ µ to P ◦ µ foreach t ∈ ( , T ) ? (Similarly for the cases δ > δ > , δ < δ < , and δ < δ < .)We hope that the answer to these questions is yes. For the sake of argument, let us assumethat this is indeed the case. Define(3.29) CMA β ( ψ ) ≔ Ê P Ê [ ι ]∈ A P , β ( ψ ) CMA β , [ ι ] ( ψ ) with CMA β , [ ι ] ( ψ ) ≔ d CM ( P ) and define a differential on CMA β ( ψ ) by declaring (cid:0) ∂ : CMA β , [ ι − ] ( ψ ) → CMA β , [ ι + ] ( ψ ) (cid:1) ≔ Õ [ ι ] d CM ( ¯ Q ) [ ι : Q → R × Y ] ranges over all equivalence classes of Cayley cobordisms from [ ι − ] to [ ι + ] .Since d CM ([ , ] × P ) is just the differential ˆ ∂ on d CM ( P ) , in the situation of Theorem 3.1 with δ > as in Remark 3.2 (and assuming that there no other Cayley cobordism involving [ ι t ] or [ ι ♯ t ] ),for t < , the chain complex CMA β ( ψ t ) contains the contribution CMA × β ( ψ t ) = d CM ( P ) with ∂ = ˆ ∂ ; for t > this changes to CMA × β ( ψ t ) = d CM ( P ) ⊕ d CM ( P ♯ ) with ∂ = (cid:18) ˆ ∂ d CM ( Q ) ˆ ∂ (cid:19) with Q the surgery cobordism from P to P ♯ . The latter is simply the mapping cone cone (cid:0)d CM ( P ) d CM ( Q ) −−−−−→ d CM ( P ♯ ) (cid:1) . Therefore, it follows from Theorem 3.19, that the homology group H ( CMA × β ( ψ t ) , ∂ ) does not change as t passes through zero. Similarly, in the situation of Conjecture 3.8, by Theo-rem 3.23, the relevant contribution to H ( CMA β ( ψ t ) , ∂ ) does not change as t passes through zero.To conclude: while there seem to be no way of making the weighted count of associatives n β ( ψ ) invariant under transitions (1) and (2) described in Section 1, we conjecture that a morerefined object, the homology group H ( CMA × β ( ψ )) is invariant under both of these transitions. A further problem with counting associatives arises from multiple covers; namely, transition (3)from Section 1. This section is concerned with describing the nature of this phenomenon and itsconsequences for counting associative submanifolds. In the following we explain how this issuemight be rectified using the ADHM Seiberg–Witten equations, in a similar way that the issuesdescribed in the previous sections were dealt with using the classical Seiberg–Witten equation.We have already established that, most likely, one cannot guarantee the number n β ( ψ ) , orsome other weighted count of associatives, to be invariant under deformations. However, theproblem with multiple covers is independent of the phenomena discussed earlier. Thus, for thesake of simplicity we will only discuss how multiple covers affect n β ( ψ ) rather than the homologygroup H ( CMA × β ( ψ )) ; see also Remark 4.8 below.21 .1 Collapsing of immersions of multiple covers Consider the following situation. Let ι : P → Y be an associative immersion with respect to ψ ∈ D c ( Y ) and with ( ι ) ∗ [ P ] = β ∈ H ( Y ) . Let π : ˜ P → P be an orientation preserving k –foldunbranched normal cover with deck transformation group Aut ( π ) . The composition κ ≔ ι ◦ π : ˜ P → Y is an associative immersion with ( κ ) ∗ [ ˜ P ] = k · β and Aut ( π ) ⊂ Aut ( κ ) . Suppose that [ ι ] is unobstructed but ker F κ = R h n i ⊂ Γ ( N κ ) . We expect that this situation can arise along generic paths ( ψ t ) t ∈(− T , T ) in D c ( Y ) . A neighbor-hood of ([ κ ] , ψ ) in the –parameter family of moduli spaces Ð t M k · β ( ψ t ) can be analyzed usingTheorem 2.20.The stabilizer of κ plays an important role in this analysis. Since Aut ( κ ) acts on N κ and F κ is Aut ( κ ) –equivariant, Aut ( κ ) acts on ker F κ . This yields a homomorphism sign : Aut ( κ ) →{± } such that(4.1) f · n = sign ( f ) n for all f ∈ Aut ( κ ) . The homomorphism sign must be non-trivial, for otherwise n would be Aut ( π ) –invariant and descend to a non-trivial element of ker F ι .To summarize, κ : P → Y is an associative immersion with respect to ψ ∈ D c ( Y ) such that:1. Aut ( κ ) is non-trivial,2. [ κ ] is obstructed; more precisely: ker F κ = R h n i , and3. the homomorphism sign : Aut ( κ ) → {± } defined by (4.1) is non-trivial.In this situation, if ( ψ t ) t ∈(− T , T ) is generic, then the obstruction map ob from Theorem 2.20, whosezero set models a neighborhood of ([ κ ] , ψ ) in Ð t M k · β ( ψ t ) , will be of the form ob ( t , δ ) = λtδ + cδ + higher order terms . We can assume that λ = c = . Ignoring the higher order terms, ob − ( ) consists of the line { δ = } and the parabola { t + δ = } . Since [ ι ] is unobstructed, for each | t | ≪ , there is anassociative immersion ι t : P → Y with respect to ψ t near ι . The line { δ = } corresponds tothe unobstructed associative immersions [ κ t ] ≔ [ ι t ◦ π ] for | t | ≪ . By Theorem 2.20, for each − ≪ t < there are also associative immersions [ κ ± t : ˜ P → Y ] with respect to ψ t near [ κ ] .22hese correspond to the two branches of the parabola { t + δ = } . As t tends to , [ κ ± t ] tendsto [ κ ] ; and Aut ( κ ± t ) is the stabilizer of n in Aut ( κ ) . Since sign : Aut ( κ ) → {± } is non-trivial,there is an f ∈ Aut ( κ ) such that f ∗ n = − n . Therefore, κ + t and κ − t differ by a diffeomorphism of ˜ P and give rise to the same element in themoduli space of associatives: [ ˜ ι t ] ≔ [ κ + t ] = [ κ − t ] . Thus, the neighborhood ob − ( )/ Aut ( κ ) of ([ κ ] , ψ ) in Ð t M k · β ( ψ t ) is homeomorphic to the fig-ure depicted in Figure 5. Consequently, n k · β ( ψ t ) as in (3.10) with the weight w = SW changes by ± SW ( ˜ P ) as t crosses zero. Similarly, if one were to adopt the approach described in Section 3.4,part of the chain complex CMA k · β ( ψ t ) would disappear as t crosses zero. [ ˜ ι ][ κ ] ψ t Figure 5: An family of associative immersions collapsing to a multiple cover.
The standard way to deal with the issue of multiple covers is to count the immersions [ κ ] and [ ˜ ι ] described before as orbifold points in the moduli space; that is, to define(4.2) n β ( ψ ) ≔ Õ [ ι ]∈ M β ( ψ ) ε ([ ι ] , ψ ) w ( P )| Aut ( ι )| . Since [ κ ] is obstructed, more precisely, since the Fueter operator associated with κ has a –dimensional kernel, (2.30) implies that the sign ε ([ κ t ] , ψ t ) ∈ {± } flips as t passes through .Moreover, Aut ( ˜ ι ) = ker sign ⊂ Aut ( κ ) , where sign : Aut ( κ ) → {± } is the homomorphism introduced above, and thus | Aut ( κ )| = · | Aut ( ˜ ι )| . Consequently, for < t ≪ , we have ε ([ κ − t ] , ψ − t ) w ( ˜ P )| Aut ( κ − t )| + ε ([ ˜ ι − t ] , ψ − t ) w ( ˜ P )| Aut ( ˜ ι )| = ε ([ κ + t ] , ψ + t ) w ( ˜ P )| Aut ( κ + t )| ∈ Q . π : ˜ P → P . If π is a branched cover (with non-empty branching locus), then κ ≔ ι ◦ π is not an immersion and thus the theory from Section 2 does not apply. What exactly replacesthis theory is unclear to us; the work of Smith [Smi11] might be a starting point. Nevertheless,one would need to count [ κ ] to be able to compensate the jump. The crucial point is that, for anygiven –manifold P and k ∈ N , infinitely many diffeomorphism types of –manifolds might berealized as k –fold branched covers of P . This is illustrated by the following result. Theorem 4.3 (Hilden [Hil74; Hil76] and Montesinos [Mon74]) . Every compact, connected, ori-entable –manifold is a –fold branched cover of S . Therefore, if ι : S → Y is an associative immersion in ( Y , ψ ) , then, for every compact, con-nected, oriented –manifold ˜ P , there is a –fold branched cover π : ˜ P → P , and [ ι ◦ π ] would haveto contribute to (4.2). This would lead to an infinite contribution from branched covers. We believe that the origin of the problem is that all the associative submanifolds [ ι ◦ π ] representthe same geometric object, namely, “ k times im ( ι ) ”. Instead of trying to count immersions and theircompositions with branched covers with weights, we should count embeddings with multiplicity.Embeddings with with multiplicity one should be weighted by the Seiberg–Wittten invariant, asin Section 3.3 or Section 3.4. Below we briefly outline an approach for defining the weights withwhich to count embeddings with multiplicity k larger than one. More details are given in Section 5and Section 6. Remark . Our approach should be compared with holomorphic curve counting via Donaldson–Thomas/Pandharipande–Thomas theory in algebraic geometry where one counts embedded sub-schemes, including contributions from thickened subschemes, rather than images of maps. Weelaborate on the relationship of this approach with Pandharipande–Thomas theory in Section 7.To set the stage, let us go back to the situation described at the beginning of this section;that is, we have an unobstructed associative embedding ι : P → Y and an orientation preserving k –fold unbranched cover π : ˜ P → P such that κ ≔ ι ◦ π : ˜ P → Y is an obstructed associative immersion with dim ker F κ = . Denote by ˜ ι : ˜ P → Y the associativeimmersion which is the deformation of κ that does not come from deforming ι . (For simplicity’ssake, we dropped the subscripts t from the notation.) Consider the bundle of stratified spaces Sym k N ι ≔ SO ( N ι ) × SO ( ) Sym k H = ( N ι ) k / S k . Here H = R is the space of quaternions and S k is the symmetric group on k elements. To everynormal vector field n ∈ Γ ( N κ ) we assign a corresponding section ˜ n ∈ Γ ( Sym k N ι ) defined by ˜ n ( x ) ≔ [ n ( ˜ x ) , . . . , n ( ˜ x k )] ˜ x , . . . , ˜ x k denoting the preimages of x with multiplicity. Given such a section ˜ n ∈ Γ ( Sym k N ι ) ,set P ˜ n ≔ {( x , v ) ∈ N ι : v ∈ ˜ n ( x )} . If n ∈ Γ ( N κ ) is a normal vector field spanning ker F κ , then P ˜ n is a model for im ( ˜ ι ) . In particular, im ( ˜ ι ) and P ˜ n are diffeomorphic in case they are smooth, which we conjecture be true genericallyif π is unbranched.We can decompose im ( ˜ ι ) into components P , . . . , P m such that P j is an ℓ j –fold cover of P and,for each ˜ x ∈ P j corresponding to ( x , v ) ∈ P ˜ n , v appears in ˜ n ( x ) with multiplicity k j . Geometrically, [ ˜ ι ] represents(4.5) k · P + · · · + k m · P m . Clearly, we have(4.6) m Õ j = ℓ j k j = k . Henceforth, let us assume that im ( ˜ ι ) is smooth. In the simplest case, we have m = and k = k .In this case, ˜ n is a section of Sym k reg N ι ≔ n ( x , [ v , . . . , v k ]) ∈ Sym k N ι : v , . . . , v k are pairwise distinct o , the top stratum of Sym k N ι . In general, ˜ n will be a section of a stratum Sym kλ N ι ⊂ Sym k N ι determined by λ , the partition of the natural number k given by (4.6). Each of the strata Sym kλ N ι is a smooth fibre bundle, which is naturally equipped with a connection ∇ and and a Cliffordmultiplication γ on its vertical tangent bundle V Sym kλ N ι . These can be used to define a Fueteroperator, which assigns to each section ˜ n ∈ Γ ( Sym kλ N ι ) an element F ˜ n ∈ Γ ( ˜ n ∗ V Sym kλ N ι ) . The condition that n ∈ Γ ( N κ ) is in the kernel of F κ means that F ˜ n ≔ γ (∇ ˜ n ) = that is, ˜ n is a Fueter section of Sym kλ N ι .The above discussion show that what causes k · P + · · · + k m · P m to collapse to k · im ( ˜ ι ) is precisely a Fueter section ˜ n of Sym kλ N ι . For simplicity, let us specialize to the case m = and k = k ; that is: 25 for t < there are two embedded associative submanifolds of interest, namely, [ ˜ ι t : ˜ P → Y ] and [ ι t : P → Y ] ; • as t tends to zero, ˜ ι t converges to the associative immersion κ , the k –fold covering of ι ,and then ceases to exist; and • for t > we only have the embedded associative submanifold [ ι t : P → Y ] .Extending the approach of Section 3.3, we would like to define weights w such that(4.7) w ( ˜ P , ψ − t ) + w ( k · P , ψ − t ) = w ( k · P , ψ + t ) for < t ≪ . From the discussion in Section 3.3 we learn that w ( ˜ P , ψ t ) should be ε ( ˜ P , ψ − t )· SW ( ˜ P ) with ε ( P , ψ − t ) ∈ {± } as in Section 2.7 and SW ( ˜ P ) ∈ Z being the Seiberg–Witten invariant of ˜ P .Thus (4.7) means that the weight w ( k · P , ψ t ) must jump by ± SW ( ˜ P ) as t passes through zero.We propose that w ( k · P , ψ t ) should be defined as the signed count of solutions to the ADHM , k Seiberg–Witten equation on P . This is the Seiberg–Witten equation associated with the ADHMconstruction of Sym k H . Unlike in the case of the classical Seiberg–Witten equation, compactnessfails for the ADHM , k Seiberg–Witten equation. As a consequence, the number of solutions canjump as the geometric background varies. According to the
Haydys correspondence , those jumpsoccur precisely when (possibly singular) Fueter sections of
Sym k N ι appear. We will argue thatin the above situation the jumps should be precisely by ± SW ( ˜ P ) .The next section is concerned with introducing the ADHM , k Seiberg–Witten equation, stat-ing and proving the Haydys correspondence with stabilizers, and formally analyzing the failureof non-compactness for the ADHM , k Seiberg–Witten equation. After this discussion we willalso explain what replaces (4.7), in general, and why defining w via the ADHM , k Seiberg–Wittenequation should be consistent with that.
Remark . Of course, instead of a weighted count of embedded associatives with multiplicities,one should really try to define a Floer homology generalizing the discussion in Section 3.4. SuchADHM , k Seiberg–Witten–Floer homology groups are yet to be defined. It will become clear fromthe discussion in the following sections that these groups could only be expected to yield topo-logical invariants of –manifolds in the case k = (classical Seiberg–Witten–Floer homology). Ingeneral, they will depend on various parameters of the equation such as the Riemannian metric. Remark . We believe that this approach is also capable of dealing with branched covers. Theseshould correspond to singular Fueter sections, that is, sections of
Sym kλ N ι defined outside a subsetof codimension at most one (which corresponds to the branching locus) and extend a continuoussection of the closure of
Sym kλ N ι in Sym k N ι . It is known that singular Fueter sections appear inthe compactifications of moduli spaces of solutions to Seiberg–Witten equations, cf. [DW18].
The purpose of this section is to introduce ADHM monopoles and to relate their degenerationsto the phenomenon of collapsing of associatives to multiple covers.26 .1 The ADHM Seiberg–Witten equations
There is a general construction, summarized in Appendix B, which associates with every quater-nionic representation of a Lie group a generalization of the Seiberg–Witten equation on –manifolds.In a nutshell, the ADHM Seiberg–Witten equations arise from this construction by choosing par-ticular quaternionic representations which appear in the famous ADHM construction of instan-tons on R ; see Example B.5. However, below we introduce the ADHM Seiberg–Witten equationsdirectly, without assuming that the reader is familiar with the general construction. Definition 5.1.
Let M be an oriented Riemannian –manifold. Consider the Lie group Spin U ( k ) ( n ) ≔ ( Spin ( n ) × U ( k ))/ Z . A spin U ( k ) structure on M is a principal Spin U ( k ) ( ) –bundle together with an isomorphism(5.2) w × Spin U ( k ) ( ) SO ( ) (cid:27) SO ( T M ) . The spinor bundle and the adjoint bundle associated with a spin U ( k ) structure w are W ≔ w × Spin U ( k ) ( ) H ⊗ C C k and g H ≔ w × Spin U ( k ) ( ) u ( k ) respectively. The left multiplication by Im H on H ⊗ C k induces a Clifford multiplication γ : T M → End ( W ) .A spin connection on w is a connection A inducing the Levi-Civita connection on T M . Asso-ciated with each spin connection A there is a Dirac operator / D A : Γ ( W ) → Γ ( W ) .Denote by A s ( w ) the space of spin connections on w , and by G s ( w ) the restricted gaugegroup , consisting of those gauge transformations which act trivially on T M . Let ϖ : Ad ( w ) → g H be the map induced by the projection spin U ( k ) ( ) → u ( k ) . Definition 5.3.
Let M be an oriented –manifold. The geometric data needed to formulate theADHM r , k Seiberg–Witten equation are: • a Riemannian metric д , • a spin U ( k ) structure w , • a Hermitian vector bundle E of rank r with a fixed trivialization Λ r E = C and an SU ( r ) –connection B , • an oriented Euclidean vector bundle V of rank together with an isomorphism(5.4) SO ( Λ + V ) (cid:27) SO ( T M ) and an SO ( ) –connection C on V with respect to which this isomorphism is parallel.27 emark . If ι : P → Y is an associative immersion, then the normal bundle V = N ι admits anatural isomorphism (5.4) by Proposition 2.12 and we can take C to be the connection inducedby the Levi-Civita connection. In this context, the bundle E should be the restriction to P of abundle on the ambient G –manifold and B should be the restriction of a G –instanton. Soon wewill specialize to the case r = , in which E is trivial and B is the trivial connection.The above data makes both Hom ( E , W ) and V ⊗ g H into Clifford bundles over M ; hence, thereare Dirac operators / D A , B : Γ ( Hom ( E , W )) → Γ ( Hom ( E , W )) and / D A , C : Γ ( V ⊗ g ) → Γ ( V ⊗ g ) . TheADHM r , k Seiberg–Witten equation involves also two quadratic moment maps defined as follows.If Ψ ∈ Hom ( E , W ) , then ΨΨ ∗ ∈ End ( W ) . Since Λ T ∗ M ⊗ g H acts on W , there is an adjoint map (·) : End ( W ) → Λ T ∗ M ⊗ g H . Define µ : Hom ( E , W ) → Λ T ∗ M ⊗ g H by µ ( Ψ ) ≔ ( ΨΨ ∗ ) . If ξ ∈ V ⊗ g , then [ ξ ∧ ξ ] ∈ Λ V ⊗ g H . Denote its projection to Λ + V ⊗ g H by [ ξ ∧ ξ ] + . Identifying Λ + V (cid:27) Λ T ∗ M via the isomorphism (5.4), we define µ : V ⊗ g → Λ T ∗ M ⊗ g H by µ ( ξ ) ≔ [ ξ ∧ ξ ] + Definition 5.6.
Given a choice of geometric data as in Definition 5.3, the
ADHM r , k Seiberg–Witten equation is the following partial differential equation for ( Ψ , ξ , A ) ∈ Γ ( Hom ( E , W )) × Γ ( V ⊗ g H ) × A s ( w ) : / D A , B Ψ = , / D A , C ξ = , and ϖF A = µ ( Ψ ) + µ ( ξ ) . (5.7)A solution of this equation is called an ADHM r , k monopole .The moduli space of ADHM r , k monopoles might be non-compact. The reason is that the L norm of the pair ( Ψ , ξ ) is not a priori bounded and can diverge to infinity for a sequence ofsolutions. To understand this phenomenon, one blows-up the equation by multiplying ( Ψ , ξ ) by ε − and studies the equation obtained by taking the formal limit ε → . This is explained ingreater detail in Appendix B. Definition 5.8.
The limiting ADHM r , k Seiberg–Witten equation the following partial differentialequation for ( Ψ , ξ , A ) ∈ Γ ( Hom ( E , W )) × Γ ( V ⊗ g H ) × A s ( w )/ D A , B Ψ = , / D A , C ξ = , and µ ( Ψ ) + µ ( ξ ) = . (5.9)together with the normalization k( Ψ , ξ )k L = .28he ADHM r , k Seiberg–Witten equation (5.7) and the corresponding limiting equation arepreserved by the action of the restricted gauge group G s ( w ) . Remark . Suppose that r = k = . A spin U ( ) structure is simply a spin c structure and ϖF A = F det A . Also, g H = i R ; hence, / D A , C is independent of A and µ ( ξ ) = . The ADHM , Seiberg–Wittenequation is thus simply / D A Ψ = and F det A = µ ( Ψ ) , the classical Seiberg–Witten equation (3.11) for ( Ψ , A ) , together with the Dirac equation / D C ξ = . If ι : P → Y is an associative immersion and M = P and V = N ι , then / D C is essentially theFueter operator F ι from Definition 2.18. In particular, ξ must vanish if ι is unobstructed. (Thereis a variant of (5.7) in which ξ is taken to be a section of V ⊗ g ◦ H with g ◦ H denoting the trace-free component of g H . For r = k = , this equation is identical to the classical Seiberg–Wittenequation. However, working with this equation somewhat complicates the upcoming discussionof the following sections.) , k Seiberg–Witten equation
In what follows, we specialize to the case r = and analyze solutions of the limiting ADHM , k Seiberg–Witten equation (5.9). This will lead to a conjectural compactification of the moduli spaceof ADHM , k monopoles. Our analysis is based on the general framework of the Haydys corre-spondence with stabilizers developed in Appendix C. We will also make use of several algebraicfacts proved in Appendix D. It is helpful but not necessary have read the appendices to understandthe results stated in this section.Assume the situation of Section 5.1; that is: w is a spin U ( k ) structure on M with spinor bundle W and adjoint bundle g H , and V is a Dirac bundle of rank over M with connection C . The limiting ADHM , k Seiberg–Witten equation for a triple ( Ψ , ξ , A ) ∈ Γ ( W ) × Γ ( V ⊗ g H ) × A s ( w ) is / D A Ψ = , / D A , C ξ = , and µ ( Ψ ) + µ ( ξ ) = (5.11)as well as k( Ψ , ξ )k L = .It follows from the third equation that if ( Ψ , ξ , A ) is a solution of (5.11), then1. Ψ = , and 29. ξ induces a section ˜ n of the bundle Sym k V over M whose fiber is Sym k H .The first statement is the content of Proposition D.4 and the second statement follows from aspecial case of the Haydys correspondence discussed in Appendix C, combined with the observa-tion that Sym k H is the hyperkähler quotient of the ADHM , k representation; see Theorem D.2.Furthermore, the section ˜ n satisfies the Fueter equation, as explained in Section C.3.A more difficult part of the Haydys correspondence deals with the converse problem: givena section ˜ n of Sym k V which satisfies the Fueter equation, can we lift it to a solution ( Ψ , ξ , A ) of(5.11)? If yes, what is the space of all such lifts up to the action of the gauge group?A technical difficulty that one has to overcome is that ˜ n takes values in the symmetric product Sym k H which is not a manifold. Rather, it is a stratified space whose strata correspond to thepartitions of k . Definition 5.12. A partition of k ∈ N is a non-increasing sequence of non-negative integers λ = ( λ , λ , . . . ) which sums to k . The length of a partition is | λ | ≔ min { n ∈ N : λ n = } − . For every n ∈ N , denote by S n the permutation group on n elements. With each partition λ we associate the groups G λ ≔ (cid:8) σ ∈ S | λ | : λ σ ( n ) = λ n for all n ∈ { . . . , | λ |} (cid:9) and the generalized diagonal ∆ | λ | = { v , . . . , v | λ | ∈ H | λ | : v i = v j for some i , j } . There is an embedding ( H | λ | \ ∆ | λ | )/ G λ ֒ → Sym k H defined by [ v , . . . , v | λ | ] 7→ [ v , . . . , v | {z } λ times , · · · , v | λ | , . . . , v | λ | | {z } λ | λ | times ] . The image of this inclusion is denoted by
Sym kλ H .Each stratum Sym kλ H is a smooth manifold. Let us assume that ˜ n takes values in such astratum: ˜ n ∈ Γ ( Sym kλ V ) , for some partition λ of k . This is familiar from Section 4.3.The next result summarizes the Haydys correspondence for solutions of (5.11). On first reading,the reader might assume that λ = ( , . . . , ) , the partition yielding the top stratum of Sym k H , sincethis simplifies the situation considerably. For j = , . . . , m , denote by k j the j –th largest positivenumber appearing in the partition λ and by ℓ j the multiplicity with which it appears.30 roposition 5.13. Given ˜ n ∈ Γ ( Sym kλ V ) , set ˜ M ≔ {( x , v ) ∈ V : x ∈ M and v ∈ ˜ n ( x )} and denote by π : ˜ M → M by the projection map.1. The map π is a | λ | –fold unbranched cover of M . Moreover, we can decompose ˜ M into compo-nents ˜ M , . . . , ˜ M m such that π j ≔ π | ˜ M j restricts to a ℓ j –fold cover on ˜ M j .2. There is a natural bijective correspondence between(a) gauge equivalence classes of solutions ( Ψ , ξ , A ) of (5.11) for which the corresponding sec-tion of Sym k V takes values in the stratum Sym kλ V , and(b) Fueter sections ˜ n ∈ Γ ( Sym kλ V ) together with a spin U ( k j ) structure w j on ˜ M j and a spinconnection A j on w j for each j = , . . . , m .Remark . If λ = ( , . . . , ) , then m = and w is simply a spin c structure on ˜ M . Proof.
Part (1) follows from the definitions of
Sym kλ V and ˜ M . It is part (2) which requires aproof. This statement is a special case of the Haydys correspondence with stabilizer proved inAppendix C; in particular, we will use the notation introduced in there.We require the following pieces of notation. For every n ∈ N , denote by [ n ] the set { , . . . , n } ,and let S n be the permutation groups on n elements. Denote by Q ⋄ the principal Î mj = S ℓ j –bundleover M , denoted whose fibre over x is(5.15) Q ⋄ x = m Ö j = Bij (cid:0) [ l j ] , π − j ( x ) (cid:1) . Tautologically, ˜ M is the fiber bundle with fiber [ l ] × · · · × [ l m ] associated with Q ⋄ using the actionof Î mj = S ℓ j on [ l ] × · · · × [ l m ] . Define T λ ≔ | λ | Ö n = U ( λ n ) ⊂ U ( k ) , W ˆ H ( T λ ) ≔ m Ö j = S ℓ j ! × SO ( ) , and N ˆ H ( T λ ) ≔ Spin ( ) × Z m Ö j = S ℓ j ⋉ U ( k j ) ℓ j ! = Spin ( ) × Z m Ö j = S ℓ j ⋉ U ( k j ) ℓ j !! × SO ( ) SO ( ) . With this notation the following summarizes the discussion in Appendix C.31 roposition 5.16.
Let Q ⋄ be the principal Î mj = S ℓ j –bundle defined by (5.15). Define a principal W ˆ H ( T λ ) –bundle ˆ Q ⋄ associated with ˜ n by ˆ Q ⋄ = Q ⋄ × SO ( V ) .
1. The choice of a N ˆ H ( T λ ) –bundle ˆ Q ◦ lifting ˆ Q ⋄ is equivalent to the choice of a spin U ( k j ) structure w j on ˜ M j for each j = , . . . , m .2. Given a spin U ( k j ) structure w j on ˜ M j for each j = , . . . , m , there exists a lift ( Ψ , ξ ) of ˜ n . Thespace of connections A Ψ , ξ C ( ˆ Q ) , defined in (C.24) , is identified with the space m Ö j = A s ( w j ) and t P , defined in (C.14) , is identified with the sum of the push-forward bundles m Ê j = ( π j ) ∗ g H j . Proof of Proposition 5.16.
We prove part (1). Given a spin U ( k j ) ( ) structure w j on ˜ M j for each j = , . . . , m , denote by ˜ w j the corresponding spin U ( k j ) ( ) structure on π ∗ j V . The principal N ˆ H ( T λ ) –bundle ˆ Q ◦ with fibre over x given by ˆ Q ◦ x = m Ö j = n ( f , д , . . . , д ℓ j ) ∈ Bij (cid:0) { , . . . , ℓ j } , π − j ( x ) (cid:1) × ˜ w ℓ j j : д i ∈ ( ˜ w j ) f ( i ) o lifts ˆ Q ⋄ . Conversely, given principal N ˆ H ( T λ ) –bundle ˆ Q ◦ lifting ˆ Q ⋄ its pullback to ˜ M j contains aprincipal Spin U ( k j ) ( ) –bundle ˜ w j which yields a spin U ( k j ) structure on π ∗ j V and thus on ˜ M j . Withthis discussion in mind and the discussion in Appendix C, part (2) of this proposition becomesapparent. (cid:3) Once Proposition 5.16 is established, part (2) of Proposition 5.13 follows from the discussionin Section C.2 and Section C.3 together with Theorem D.2. (cid:3)
Proposition 5.13 imposes very weak conditions on a connection A ∈ A s ( w ) which is part of asolution of the limiting equation (5.11). Indeed, given ( ξ , Ψ ) and one such connection, all otherchoices of A are parametrized by choices of spin connections A j on w j , for every j , and the spacesof these spin connections are infinite-dimensional. However, we are only interested in thosesolutions of (5.11) which are obtained as limits of rescaled ADHM , k monopoles. To determine32urther constraints for such limits, let ( Ψ = , ξ , A ) be a solution of (5.11) with ˜ n ∈ Γ ( Sym kλ V ) for some partition λ of k , and suppose that Ψ ε = ∞ Õ i = ε i Ψ i , ξ ε = ∞ Õ i = ε i ξ i , and A ε = A + ∞ Õ i = i ε i a i is a formal power series solution of the rescaled ADHM , k Seiberg–Witten equation: / D A ε Ψ ε = , / D A ε , C ξ ε = , and ε ϖF A ε = µ ( Ψ ε ) + µ ( ξ ε ) . (5.17)Moreover, we can assume the gauge fixing condition ξ ⊥ ρ ( g P ) ξ , that is, R ∗ ξ ξ = in the notation of Proposition D.6. The next proposition imposes constraints on the terms of order ε in the power series expansions.Let W j and g H j be, respectively, the spinor bundle and adjoint bundle associated with thespin U ( k j ) structure w j on the total space of the covering map π j : ˜ M j → M . Proposition 5.18.
In the above situation, there exist ˜ Ψ , j ∈ Γ ( W j ) and ˜ ξ j ∈ Γ ( V ⊗ g H j ) such that (5.19) Ψ = m Ê j = ( π j ) ∗ ˜ Ψ , j and ξ = m Ê j = ( π j ) ∗ ˜ ξ , j . Furthermore, A arises from a collection of spin connections A , j ∈ A s ( w j ) , and each triple ( A , j , ξ , j , ˜ Ψ , j ) satisfies the ADHM , k j equation / D A , j ˜ Ψ , j = , / D A , j , C ξ , j = , and ϖF A , j = µ ( ˜ Ψ , j ) + µ ( ˜ ξ , j ) (5.20) on ˜ M j for j = , . . . , m .Proof. From Proposition 5.16, we know that ξ = ( ξ , , · · · , ξ , m ) ∈ Γ ( V ⊗ t P ) with t P = m Ê j = ( π j ) ∗ g H j and A arises from spin connections A , j ∈ A s ( w j ) .33he coefficient in front of ε on the right-hand side of the third equation of (5.17) must vanish;hence, ( d ξ µ ) ξ = . By Proposition D.6 it follows that [ ξ ∧ ξ ] = . Therefore, µ ( ξ ) ∈ Ω ( M , [ t P , t P ]) by the following self-evident observation combined with Theorem D.2. Proposition 5.21. If ξ , ξ ∈ H ⊗ g , [ ξ ∧ ξ ] = , and the stabilizer of ξ ∈ U ( k ) is precisely T λ = Î | λ | n = U ( λ n ) , then ξ ∈ H ⊗ t λ with t λ = É | λ | n = u ( λ n ) . In particular, [ ξ ∧ ξ ] ∈ H ⊗ [ t λ , t λ ] ⊂ H ⊗ t λ . Remark . If λ = ( , . . . , ) , then [ t P , t P ] = ; cf. Remark 5.10.The third equation in (5.17) to order ε is thus equivalent to(5.23) ϖF A = µ ( ξ ) + ( d ξ µ ) ξ + µ ( Ψ ) . In terms of the spin connections A , j ∈ A s ( w j ) , we have ϖF A = m Ê j = ( π j ) ∗ ϖF A , j ∈ Ω ( M , t P ) . By (C.9), we have ( d ξ µ ) ξ ∈ Ω ( M , t ⊥ P ) . Thus, if we denote by µ q ( Ψ ) the component of µ ( Ψ ) in t P and by µ ⊥ ( Ψ ) the component of µ ( Ψ ) in t ⊥ P ⊂ g P , then (5.23) is equivalent to ϖF A = µ ( ξ ) + µ q ( Ψ ) and ( d ξ µ ) ξ = − µ ⊥ ( Ψ ) . (5.24)Since t P is parallel with respect to A and V ⊗ t P is perpendicular to ¯ γ ( T ∗ M ⊗ g P ) ξ , the first andthe second equation of (5.17) to order ε are equivalent to / D A Ψ = , / D A , C ξ = , and γ ( a ) ξ = . (5.25)Let ˜ Ψ , j ∈ Γ ( W j ) and ˜ ξ j ∈ Γ ( V ⊗ g H j ) be such that (5.19) holds. The first equation of (5.24)and the first two equations of (5.25) are precisely equivalent to the ADHM , k j Seiberg–Wittenequation (5.20) for the triple ( A , j , ξ , j , ˜ Ψ , j ) . (cid:3) .4 A compactness conjecture for ADHM , k monopoles The discussion in the preceding sections together with known compactness results for Seiberg–Witten equations [Tau13a; Tau13b; HW15; Tau16; Tau17] lead to the following conjecture.
Conjecture 5.26.
Let ( ε i , Ψ i , ξ i , A i ) be a sequence of solutions of the blown-up ADHM , k Seiberg–Witten equation / D A i Ψ i = , / D A i , C ξ i = , ε i ϖF A i = µ ( Ψ i ) + µ ( ξ i ) , and k( Ψ i , ξ i )k L = with ε i → . After passing to a subsequence the following hold:1. There is a closed subset Z ⊂ M of Hausdorff dimension at most one, such that outside of Z andup to gauge transformations ( Ψ i , ξ i , A i ) converges to a limit ( , ξ ∞ , A ∞ ) and ε − i ( Ψ i , ξ i − ξ ∞ ) converges to a limit ( Ψ ∞ , ξ ∞ ) .2. The triple ( , ξ ∞ , A ∞ ) is a solution of the limiting ADHM , k Seiberg–Witten equation (5.11) .3. There is a section ˜ n ∈ Γ ( M \ Z , Sym kλ V ) for some partition λ of k induced by ξ ∞ . The section ˜ n extends to to a continuous section of Sym k V on all of M .4. Denote by ˜ M \ ˜ Z the unbranched cover of M \ Z induced by ˜ n . If k j , ˜ M j \ ˜ Z j , w j are as in Propo-sition 5.13 and A , j ∈ A s ( w j ) denote the spin connections giving rise to A ∞ , and ˜ Ψ , j and ˜ ξ , j are such that Ψ ∞ = m Ê j = ( π j ) ∗ ˜ Ψ , j and ξ ∞ = m Ê j = ( π j ) ∗ ˜ ξ , j , then, for each j = , . . . , m , ( ˜ Ψ , j , ˜ ξ , j , A , j ) is a solution of the ADHM , k j Seiberg–Wittenequation on ˜ M j \ ˜ Z j .Remark . The reader should observe that while ˜ M j in ˜ M j \ ˜ Z j does exist, it need not be a smoothmanifold. Remark . If Ψ = , V = T M ⊕ R and ( a , ξ ) ∈ Ω ( M , g H ) ⊕ Ω ( M , g H ) = Γ ( V ⊗ g H ) , then theADHM , k Seiberg–Witten equation becomes the equation F A + ia − ∗[ ξ , a ] + ∗ i d A ξ = and d ∗ A a = (5.29)with F A + ia = F A − [ a ∧ a ] + i d A a . ( a , ξ , A ) is a solution of (5.29) and M is closed, then a simple integration by parts argumentshows that d A ξ = ; hence, F A + ia = . That is, (5.29) is effectively the condition that conditionthat A + ia is a flat GL k ( C ) –connection together with the moment map equation d ∗ A a = .Conjecture 5.26 thus predicts that as limits of flat GL k ( C ) –connections we should see dataconsisting of a closed subset Z ⊂ M of Hausdorff dimension at most one, m ∈ N and, for each j = , . . . , m , a ℓ j –fold cover ˜ M \ ˜ Z j of M \ Z , and solutions of (5.29) on ˜ M j \ ˜ Z j such that Í mj = ℓ j k j = k . We are ready to outline how ADHM monopoles can be used to deal with the problem of multiplecovers described in Section 4.Let ψ be a tamed, closed, definite –form, let P be a compact, connected, oriented –manifold,let P ⊂ Y be an unobstructed associative embedding. Set M , k ( P , ψ ) ≔ Þ w M , k w ( P , ψ ) with the disjoint union taken over all spin U ( k ) structures w on P and M , k w ( P , ψ ) ≔ (cid:26) ( Ψ , ξ , A ) ∈ Γ ( W ) × Γ ( N P ⊗ g H ) × A s ( w ) : ( Ψ , ξ , A ) satisfies ( . ) with respect to д ψ | P (cid:27) G s ( w ) . Ignoring issues to do with reducible solutions, one should be able to extract a number w ( kP , ψ ) ∈ Z by counting M , k ( P , ψ ) , at least, for generic ψ and possibly after slightly perturbing the ADHMSeiberg–Witten equation (5.7). More generally, if P has connected components P , . . . , P m and k , . . . , k m ∈ N , we set w ( k · P + · · · + k m · P m , ψ ) ≔ k Ö j = w ( k j · P j , ψ ) . For k = , this number is the Seiberg–Witten invariant SW ( P ) ∈ Z mentioned in Section 3.3For k > , this number should be independent of the choice of perturbation but it will depend on ψ . Assume the situation of Section 4.1; that is, we have: • a generic –parameter family of tamed, closed, definite –forms ( ψ t ) t ∈(− T , T ) , • a –parameter family of compact, connected, unobstructed embedded associative subman-ifolds ( P t ) t ∈(− T , T ) with respect to ( ψ t ) t ∈(− T , T ) , and36 for every j = , . . . , m a –parameter family of compact, connected, unobstructed embeddedassociative submanifolds ( P jt ) t ∈(− T , ) with respect to ( ψ t ) t ∈(− T , ) such that P jt → ℓ j · P as integral currents as t tends to zero for some ℓ j ∈ { , , . . . } .Given k , . . . , k m , set k ≔ m Õ j = ℓ j k j . From the discussion in preceding three sections we expect that, for < t ≪ ,(6.1) w ( k · P − t , ψ − t ) + w ( k · P − t + · · · + k m · P m − t , ψ − t ) = w ( k · P + t , ψ + t ) because Conjecture 5.26 suggests that as t passes through zero w ( k · P + · · · + k m · P , ψ ) ADHM , k monopoles on P t degenerate and disappear (if counted with the correct sign).Suppose that one can indeed define a weight w as above satisfying (6.1) as well as analoguesof (3.9). Define(6.2) n β ( ψ ) = Õ w ( k · P + · · · + k m · P m , ψ ) with the summation ranging over all m ∈ N , k , . . . , k m ∈ N and all compact, connected, unob-structed embedded associative submanifolds P , . . . , P m ⊂ Y such that m Õ j = k j [ P j ] = β . This number would be invariant under the transitions described in Section 3.1, Section 3.2, andSection 4.1.From Section 3.3 we know that reducible solutions will prevent us from defining w in general.However, the above can serve as a first approximation. To deal with reducibles one likely has todevelop ADHM , k analogues of Kronheimer and Mrowka’s monopole homology and construct achain complex extending (3.29) which does depend on ψ but whose homology does not. Remark . By analogy with monopole Floer homology, one can envision also a corresponding –dimensional version of the invariant proposed in this article. Such an invariant would be ob-tained by counting Cayley submanifolds inside a closed Spin ( ) –manifold, weighted by solutionsof the –dimensional ADHM Seiberg–Witten equations. A relative version of this theory wouldassociate with every cylindrical Spin ( ) –manifold X whose end is asymptotic to a compact G –manifold Y a distinguished element of the Floer homology group associated with Y . In order todevelop such a + dimensional theory, one has to deal with higher-dimensional moduli spacesof Cayley submanifolds and ADHM monopoles, which poses additional technical complications.Note that in order to define G Floer homology, one has to consider only
Spin ( ) –manifolds ofthe form Y × (−∞ , ∞) , and only zero-dimensional moduli spaces.37 Counting holomorphic curves in Calabi–Yau –folds Let Z be a Calabi–Yau –fold with Kähler form ω and holomorphic volume form Ω . The product S × Z is naturally a G –manifold with the G –structure given by ϕ = dt ∧ ω + Re Ω . Every holomorphic curve Σ ⊂ Z gives rise to an associative submanifold S × Σ ⊂ S × Z . Proposition 7.1.
Let β ∈ H ( Z ) be a homology class. Every associative submanifold in S × Z representing the class [ S ] × β is necessarily of the form S × Σ with Σ ⊂ Z a holomorphic curve.Proof. The argument is similar to the one used to prove an analogous statement for instantons[Lew98, Section 3.2]. Let P ⊂ S × Z be an associative submanifold representing [ S ] × β . Since ϕ | P = ( d t ∧ ω + Re Ω )| P = vol P , there is a smooth function f on P such that d t ∧ ω | P = f vol P and Re Ω | P = ( − f ) vol P By Wirtinger’s inequality [Wir36], f . We need to prove that f = , since this implies that ∂ t is tangent to P and, therefore, P is of the form S × Σ , with Σ ⊂ Z calibrated by ω .One the one hand we have ˆ P vol P = h[ ϕ ] , [ P ]i = h[ d t ∧ ω ] + [ Re ω ] , [ S ] × β i = h[ d t ∧ ω ] , [ S ] × β i , while on the other hand ˆ P f vol P = ˆ P d t ∧ ω = h[ d t ∧ ω ] , [ P ]i = h[ d t ∧ ω ] , [ S ] × β i . It follows that f has mean-value and thus f = because f . (cid:3) The deformation theory of the associative submanifold S × Σ in S × Z coincides with thatof the holomorphic curve Σ in Y [CHNP15, Lemma 5.11]. In particular, the putative enumerativetheory for associative submanifolds discussed in this paper should give rise to an enumerativetheory for holomorphic curves in Calabi–Yau –folds. Algebraic geometry abounds in such the-ories and various interplays between them; see [PT14] for an introduction to this rich subject.Our approach is closer in spirit to the original proposal by Donaldson and Thomas [DT98]. Wewill argue that it should lead to a symplectic analogue of a theory already known to algebraicgeometers. 38 .1 The Seiberg–Witten invariants of Riemann surfaces In the naive approach of Section 3.3 each associative submanifold is counted with its total Seiberg–Witten invariant. The Seiberg–Witten equation (3.11) over the –manifold M = S × Σ was studiedextensively [MST96; MOY97; MW05]. The equation admits irreducible solutions only for thespin c –structures pulled-back from Σ . Such a spin c structure corresponds to a Hermitian linebundle L → Σ ; the induced spinor bundle is W = L ⊕ T ∗ Σ , ⊗ L . Up to gauge transformations, allirreducible solutions of the Seiberg–Witten equation are pulled-back from triples ( A , ψ , ψ ) on Σ ,where ( ψ , ¯ ψ ) ∈ Γ ( L ) ⊕ Ω , ( Σ , L ) , A ∈ A ( det ( W )) and ¯ ∂ A ψ = , ¯ ∂ ∗ A ¯ ψ = , h ψ , ¯ ψ i = , and i ∗ F A + | ψ | − | ¯ ψ | = . (7.2)Here h ψ , ¯ ψ i is the ( , ) –form obtained from pairing ψ and ¯ ψ using the Hermitian inner product.The second equation implies that either ψ or ¯ ψ must vanish identically—which one, dependson the sign of the degree d ≔ h c ( W ) , Σ i . Since det ( W ) = L ⊗ K − Σ , we have deg ( L ) = д − + d . Suppose that d < . It follows from integrating the third equation that ψ , and so ¯ ψ = .The pair ( A , ψ ) corresponds to an effective divisor of degree д − + d on Σ : the zero set of ψ countedwith multiplicities. This corresponds to an element of the symmetric product Sym д − + d Σ . If d > ,then a similar argument and Serre duality associates with every solution of (7.2) an element of Sym д − − d Σ . The above correspondence, in fact, goes both ways: Theorem 7.3 (Noguchi [Nog87], Bradlow [Bra90], and García-Prada [Gar93]) . Let λ ∈ R \{ d } . Themoduli space of solutions to the perturbed vortex equation ¯ ∂ A ψ = , ¯ ∂ ∗ A ¯ ψ = , h ψ , ¯ ψ i = , and i ∗ F A + | ψ | − | ¯ ψ | = π vol ( Σ ) · λ (7.4) is homeomorphic to ( Sym д − + d ( Σ ) if d − λ < and Sym д − − d ( Σ ) if d − λ > . Σ , S ,then the total Seiberg–Witten invariant is SW ( S × Σ ) = Õ d ∈ Z (− ) д − + d χ ( Sym д − + d Σ ) . Here we can sum over all d ∈ Z since for | d | > д − we have χ ( Sym д − + d Σ ) = . For Σ = S , the above series is not summable. This is consistent with the general theory alludedto in Section 3.3: we have b ( S × S ) = and, due to the appearance of reducible solutions, thetotal Seiberg–Witten invariant is defined only for –manifolds with b > . In full generality, thisproblem can be solved within the framework of Floer homology. However, if one considers onlyclosed, oriented –manifolds with b > there is also a middle ground approach due to Mengand Taubes [MT96]. For every such a –manifold M they define an invariant SW ( M ) ∈ Z J H K / H . Here H is the torsion-free part of H ( M , Z ) , Z J H K is the set of Z –valued functions on H , and H acts on Z J H K by pull-back.The Meng–Taubes invariant takes a particularly simple form for M = S × Σ . In this case,there is a distinguished spin c structure, corresponding to the line bundle L being trivial, and theinvariant can be naturally lifted to an element SW ( M ) ∈ Z J H K . Moreover, the support of SW ( M ) is Z = H ( Σ , Z ) ⊂ H , reflecting the fact that the Seiberg–Witten equation has solutions only forthe spin c structures pulled-back from Σ . Thus, SW ( M ) can be interpreted as an element of thering of formal Laurent series in a single variable, q say, SW ( M ) ∈ Z (( q )) . For д > , this is the Laurent polynomial whose coefficients are the Seiberg–Witten invariants:(7.5) SW ( S × Σ ) = Õ d ∈ Z (− ) д − + d χ ( Sym д − + d Σ ) q d and we see that SW ( S × Σ ) is obtained by evaluating SW ( S × Σ ) at q = . It is easy to see fromthe definition of the Meng–Taubes invariant that the same formula is true for Σ = S , althoughnow the series has infinitely many non-zero terms. One cannot evaluate SW ( S × S ) at q = andis forced to work with the refined invariant. 40 .3 Stable pair invariants of Calabi–Yau –folds Pandharipande and Thomas introduced a numerical invariant counting holomorphic curves inCalabi–Yau –folds together with points on them; see [PT14, Section ] for a brief introductionand [PT09; PT10] for more technical accounts. Since the space of curves and points on themis not necessarily compact, one considers the larger moduli space of stable pairs , consisting of acoherent sheaf F on Z together with a section s ∈ H ( Z , F ) which, thought of as a sheaf morphism s : O Z → F , is surjective outside a zero-dimensional subset of Z . The sheaf is required to besupported on a (possibly singular and thickened) holomorphic curve Σ ⊂ Z . Example 7.6.
The simplest examples arise when Σ is smooth and ( F , s ) is the pushforward of apair ( L , ψ ) on Σ consisting of a holomorphic line bundle and a non-zero section. Conversely, allstable pairs whose support is a smooth, unobstructed curve are of this form [PT09, Section 4.2].The topological invariants of a stable pair are the homology class [ Σ ] ∈ H ( Z ) and the Eulercharacteristic χ ( X , F ) ∈ Z . For instance, in Example 7.6, with Σ of genus д , we have(7.7) χ ( X , F ) = − д + deg ( F ) . For every β ∈ H ( Z ) and d ∈ Z , Pandharipande and Thomas use virtual fundamental class tech-niques to define an integer PT d , β which counts stable pairs with homology class β and Eulercharacteristic d . These numbers for different values of d can be conveniently packaged into thegenerating function PT β = Õ d PT β , d q d . For a holomorphic curve Σ ⊂ Z with [ Σ ] = β , denote by PT Σ ( q ) the contribution to PT β ( q ) comingfrom stable pairs whose support is Σ . (It makes sense to talk about such a contribution even fornon-isolated curves [PT10, Section 3.1].)In the situation of Example 7.6, the moduli space of stable pairs with support on Σ and Eulercharacteristic d is simply the space of effective divisors whose degree, computed using (7.7), is д − + d . From the deformation theory of such stable pairs one concludes that in this case,(7.8) PT Σ ( q ) = Õ d (− ) д − + d χ ( Sym д − + d Σ ) q d ; see [PT09, Equation (4.4)] for details. As a result, we obtain the following. Proposition 7.9. If Σ ⊂ Z is a smooth, unobstructed holomorphic curve, then PT Σ = SW ( S × Σ ) . More precisely, F is pure of dimension one and s has zero-dimensional cokernel. emark . From the –dimensional perspective, the symmetry between d and − d is a specialcase of the involution in Seiberg–Witten theory induced from the involution on the space ofspin c structures [Mor96, Section 6.8]; from the –dimensional viewpoint, it is a manifestation ofthe Serre duality between H ( L ) and H ( K Σ ⊗ L ∗ ) . Remark . The fact that the stable pair invariant is partitioned into an integers worth of in-variants corresponding to the degrees of the spin c structures on curves suggests that somethingsimilar could be true for associative submanifolds. However, unlike in the dimensionally reducedsetting, where a spin c corresponds in a natural way to an integer, for two distinct associatives P and P we are not aware of any way to relate the spin c structures on them.In general, the stable pair invariant includes also more complicated contributions from singu-lar and obstructed curves representing the given homology class. For irreducible classes, Pand-haripande and Thomas proved that such a contribution is a finite sum of Laurent series of theform (7.8) [PT10, Theorem 3 and Section 3]. The stable pair invariant includes also contributions from thickened curves. If a homology class β ∈ H ( Z , Z ) is divisible by k and β / k is represented by a holomorphic curve Σ ⊂ Z , then thereexist stable pairs having k Σ as their support. Thinking of S × k Σ as a multiple cover of theassociative S × Σ in S × Z , we are led by the discussion of Section 4.3 to the conclusion thatthe contribution of such a thickened curve should be in some way related to the solutions of theADHM , k Seiberg–Witten equation on the –manifold S × Σ . We will argue that this is indeedthe case.Consider the more general ADHM r , k Seiberg–Witten equation introduced in Section 5.1 underthe following assumptions:
Hypothesis 7.12.
Let Σ be a closed Riemann surface and M = S × Σ with the geometric data as inDefinition 5.3 such that:1. д is a product Riemannian metric,2. E and the connection B are pulled-back from Σ , and3. V and the connection C are pulled-back from a U ( ) –bundle with a connection on Σ such that Λ C V (cid:27) K Σ as bundles with connections. Proposition 7.13.
If Hypothesis 7.12 holds and ( Ψ , ξ , A ) is an irreducible solution of the ADHM r , k Seiberg–Witten equation (5.7), then the spin U ( k ) structure w is pulled-back from a spin U ( k ) structureon Σ and ( Ψ , ξ , A ) is gauge-equivalent to a configuration pulled-back from Σ , unique up to gaugeequivalence on Σ . This is a special case of [Doa17, Theorem 3.8]. In the situation of Proposition 7.13, (5.7) reducesto a non-abelian vortex equation on Σ . Recall that a choice of a spin U ( k ) structure on Σ is equivalent42o a choice of a U ( k ) –bundle H → Σ . Consequently, A can be seen as a connection on H . Thecorresponding spinor bundles are g H = u ( H ) and W = H ⊕ T ∗ Σ , ⊗ H . Proposition 7.14.
Let ( A , Ψ , ξ ) be a configuration pulled-back from Σ . Under the splitting W = H ⊕ T ∗ Σ , ⊗ H we have Ψ = ( ψ , ψ ∗ ) where ψ ∈ Γ ( Σ , Hom ( E , H )) , ψ ∈ Ω , ( Σ , Hom ( H , E )) , and ξ ∈ Γ ( Σ , V ⊗ End ( H )) . Equation (5.7) for ( A , Ψ , ξ ) is equivalent to ¯ ∂ A , B ψ = , ¯ ∂ A , B ψ = , ¯ ∂ A , C ξ = , [ ξ ∧ ξ ] + ψ ψ = , and i ∗ F A + [ ξ ∧ ξ ∗ ] + ψ ψ ∗ − ∗ ψ ∗ ψ = . (7.15) In the second equation we use the isomorphism Λ C V (cid:27) K Σ so that the left-hand side is a section of Ω , ( Σ , End ( H )) . In the third equation we contract V with V ∗ so that the left-hand side is a sectionof i u ( H ) . This follows from [Doa17, Proposition 3.6, Remark 3.7] and the complex description (D.7) ofthe hyperkähler moment map appearing in the ADHM construction.We can also perturb (7.15) by τ ∈ R and θ ∈ H ( Σ , K Σ ) : ¯ ∂ A , B ψ = , ¯ ∂ A , B ψ = , ¯ ∂ A , C ξ = , [ ξ ∧ ξ ] + ψ ψ = θ ⊗ id , and i ∗ F A + [ ξ ∧ ξ ∗ ] − ψ ψ ∗ + ∗ ψ ∗ ψ = τ id . (7.16)There is a Hitchin–Kobayashi correspondence between gauge-equivalence classes of solutionsof (7.16) and isomorphism classes of certain holomorphic data on Σ . Let E = ( E , ¯ ∂ B ) and V = ( V , ¯ ∂ C ) be the holomorphic bundles induced from the unitary connections on E and V . Definition 7.17. An ADHM bundle with respect to ( E , V , θ ) is a quadruple ( H , ψ , ψ , ξ ) consisting of: • a rank k holomorphic vector bundle H → Σ , • ψ ∈ H ( Σ , Hom ( E , H )) , 43 ψ ∈ H ( Σ , K Σ ⊗ Hom ( H , E )) , and • ξ ∈ H ( Σ , V ⊗ End ( H )) such that [ ξ ∧ ξ ] + ψ ψ = θ ⊗ id ∈ H ( Σ , K Σ ⊗ End ( H )) . Definition 7.18.
For δ ∈ R , the δ – slope of an ADHM bundle ( H , ψ , ψ , ξ ) is µ δ ( H ) ≔ π vol ( Σ ) deg H rk H + δ rk H . The slope of H is µ ( H ) ≔ µ ( H ) . Definition 7.19.
Let δ ∈ R . An ADHM bundle ( H , ψ , ψ , ξ ) is δ stable if it satisfies the followingconditions:1. If δ > , then ψ , and if δ < , then ψ , .2. If G ⊂ H is a proper ξ –invariant holomorphic subbundle such that im ψ ⊂ G , then µ δ ( G ) < µ δ ( H ) .3. If G ⊂ H is a proper ξ –invariant holomorphic subbundle such that G ⊂ ker ψ , then µ ( G ) < µ δ ( H ) .We say that ( H , ψ , ψ , ξ ) is δ –polystable if there exists a ξ –invariant decomposition H = É i G i É j I j such that:1. µ δ ( G i ) = µ δ ( H ) for every i and the restrictions of ( ψ , ψ , ξ ) to each G i define a δ stableADHM bundle, and2. µ ( I j ) = µ δ ( H ) for every j , the restrictions of ψ , ψ to each I j are zero, and there exist no ξ –invariant proper subbundle J ⊂ I i with µ ( J ) < µ ( I j ) .In the proposition below we fix δ and the topological type of H , and set τ = µ δ ( H ) . Proposition 7.20.
Let ( A , ψ , ψ , ξ ) be a solution of (7.16) . Denote by H the holomorphic vectorbundle ( H , ¯ ∂ A ) . Then ( H , ψ , ψ , ξ ) is a δ –polystable ADHM bundle. Conversely, every δ –polystableADHM bundle arises in this way from a solution to (7.16) which is unique up to gauge equivalence.Proof. A standard calculation going back to [Don83] shows that (7.16) implies δ –polystability.The difficult part is showing that every δ –polystable ADHM bundle admits a compatible unitaryconnection solving the third equation of (7.16), unique up to gauge equivalence. This is a specialcase of the main result of [ÁG03, Theorem 31], with the minor difference that the connections onthe bundles E and V are fixed and not part of a solution. The necessary adjustment in the proofis discussed in a similar setting in [BGM03]. (cid:3) n = , E is a trivial line bundle, and V is the direct sum of two linebundles. Thus, we have a splitting ξ = ( ξ , ξ ) and [ ξ ∧ ξ ] = [ ξ , ξ ] , so the holomorphic equation [ ξ ∧ ξ ] + ψ ψ = θ ⊗ id is preserved by the C ∗ –action t ( ψ , ψ , ξ , ξ ) = ( tψ , t − ψ , tξ , t − ξ ) . Moreover, if the perturb-ing form θ is chosen to be zero, there is an additional C ∗ symmetry given by rescaling everythe sections. Assuming that the stability parameter δ is sufficiently large, Diaconescu shows thefixed-point locus of the resulting C ∗ × C ∗ –action on the moduli space of δ stable ADHM bundles iscompact. Furthermore, the moduli space is equipped with a C ∗ × C ∗ –equivariant perfect obstruc-tion theory. This can be used to define a numerical invariant via equivariant virtual integration.This number is then shown to be equal to the local stable pair invariant of the non-compactCalabi–Yau –fold V . This invariant counts, in the equivariant and virtual sense, stable pairswhose support is a k –fold thickening of the zero section Σ ⊂ V . Here k is the rank of H sothat the stable ADHM bundles in question correspond, by Proposition 7.20, to solutions to theADHM , k Seiberg–Witten equation on S × Σ . This suggests that the relation between Seiberg–Witten monopoles and stable pairs discussed in the previous section could extend to the case ofmultiple covers. Due to the appearance of reducible solutions, one does not expect to be able to count solutions tothe ADHM , k Seiberg–Witten equation on a general –manifold. Instead, the enumerative theoryfor associatives in tamed almost G –manifolds should incorporate a version of equivariant Floerhomology, as explained in Section 3.4 and Section 6. However, the existence of the stable pairinvariant and the discussion of the previous sections indicate that we can hope for a differential-geometric invariant counting pseudo-holomorphic curves in a symplectic Calabi–Yau –manifold Z which in the projective case would recover the stable pair invariant. It is expected that suchan invariant would encode the same symplectic information as the Gromov–Witten invariantsby the conjectural GW/PT correspondence, known also as the MNOP conjecture [PT14, Sections and ]. The algebro-geometric version of this conjecture is at present widely open. Like theGromov–Witten invariant, the putative symplectic stable pair invariant is given by a weightedcount of simple J –holomorphic maps. Thus, we expect that a symplectic definition of the stablepair invariant will shed new light on the MNOP conjecture.For a homology class β ∈ H ( Z , Z ) the invariant would take values in the ring of Laurentseries Z (( q )) and be defined by n β ( Z ) = Õ Σ , ..., Σ m m Ö j = SW , k j ( S × Σ j ) sign ( Σ j ) . Some explanation is in order: 45. The sum is taken over all collections of embedded, connected pseudo-holomorphic curves Σ , . . . , Σ m such that m Õ j = k j [ Σ j ] = β . We assume here that we can choose a generic tamed almost-complex structure such thatthere are finitely many such curves and all of them are unobstructed.2. sign ( Σ ) = ± comes from an orientation on the moduli space of pseudo-holomorphiccurves.3. SW , k ( S × Σ ) is a generalization of the Meng–Taubes invariant defined using the modulispaces of solutions to the ADHM , k Seiberg–Witten equation on S × Σ j . This is yet to be de-fined, but if it exists, it should be naturally an element of Z (( q )) because of the identificationof the set of the spin U ( k ) structures on Σ with the integers, as in Section 7.2.4. We use here crucially that b ( S × Σ ) > ; otherwise even the classical Meng–Taubes in-variant SW , is ill-defined. For k > , the ADHM , k Seiberg–Witten equation, admits ingeneral, reducible solutions: for example, flat connections or solutions to the ADHM , k − Seiberg–Witten equation. A good feature of the dimensionally-reduced setting is that ifthe perturbing holomorphic –form θ in (7.16) is non-zero, then we automatically avoidreducible solutions. Indeed, a simple algebraic argument shows that in this case the triple ( ξ , ψ , ψ ) has trivial stabilizer in U ( k ) at every point where θ is non-zero. A Transversality for associative embeddings
The goal of this section is to prove Proposition 2.23. The proof relies on the following observations.The tangent space T ψ D c ( Y ) ⊂ Ω ( Y ) is the space of closed –forms. Define X ι , ψ : T ψ D c ( Y ) → Γ ( N ι ) by(A.1) h X ι , ψ η , n i L ≔ dd ε (cid:12)(cid:12)(cid:12)(cid:12) ε = δ L ψ + εη ( n ) = ˆ P ι ∗ i ( n ) η for every closed –form η on Y . Proposition A.2. If ι : P → Y is a somewhere injective associative immersion, then for every non-zero n ∈ ker F ι ⊂ Γ ( N ι ) , there exists α ∈ Ω ( Y ) such that h X ι , ψ d α , n i , . roof. We can assume that P is connected. Pick a point x such that ι − ( ι ( x )) = { x } . Since P is compact, there is a neighborhood U of x ∈ P which is embedded via ι and satisfies ι ( U ) ∩ ι ( P \ U ) = œ . Choose a tubular neighborhood V of ι ( U ) and ρ > such that B ρ ( N ι ( U )) exp −−→ V isa diffeomorphism. By unique continuation, n cannot vanish identically on U . Thus we can find afunction f supported in V such that d f ( n ) > and d f ( n ) > somewhere. Let ν be a –form on Y with ν | U = ( vol P )| U and i ( n ) d ν | V = . With α = f ν we have ˆ P ι ∗ ( i ( n ) d α ) = ˆ P d f ( n ) vol P > . (cid:3) For a somewhere injective immersed associative [ ι : P → Y ] , Aut ( ι ) must be trivial. Denoteby π Imm : Imm β ( P , Y ) × D c ( Y ) → Imm β ( P , Y ) the canonical projection. By Proposition A.2, thelinearization of the section δ L ∈ Γ ( π ∗ Imm T ∗ Imm β ( P , Y )) is surjective. Hence, it follows from the Regular Value Theorem, and the fact that there are onlycountably many diffeomorphism types of –manifolds [CK70], that the universal moduli space ofimmersed associatives A si β = A si β ( D c ( Y )) is a smooth manifold. This directly implies (1a) and (2a) by the Sard–Smale Theorem.Consider the moduli space of immersed associative submanifolds with n marked points A si β , n ( ψ ) ≔ Þ P n ( ι , x , . . . , x n ) ∈ Imm β ( P , Y ) × P n : [ ι ] ∈ A si β ( ψ ) o . Diff + ( P ) as well as the corresponding universal moduli space A si β , n ≔ Ø ψ ∈ D c ( Y ) A si β , n ( ψ ) . Define the map ev : A si β , n → Y n by ev ([ ι , x , . . . , x n ] , ψ ) ≔ ( ι ( x ) , . . . , ι ( x n )) . Proposition A.3.
For each ([ ι , x , . . . , x n ] , ψ ) ∈ A si β , n , the derivative of ev , d ([ ι , x , ..., x n ] , ψ ) ev : T ([ ι , x , ..., x n ] , ψ ) A si β , n → n Ê i = T ι ( x i ) Y , is surjective. roof. We will show that if ( v , . . . , v n ) ∈ É ni = N x i ι , then there exist n ∈ Γ ( N ι ) and η ∈ T ψ D c ( Y ) such that n ( x i ) = v i and ( n , η ) ∈ T [ ι ] , ψ A si β . This immediately implies the assertion.Denote by ev x , ..., x n : Γ ( N ι ) → É ni = N x i ι the evaluation map and define F k ≔ (cid:16) F ι ⊕ X ι , ψ : W k , ker ev x , ..., x n ⊕ T ψ D c ( Y ) → W k − , Γ ( N ι ) (cid:17) , where F ι is the Fueter operator and X ι , ψ is defined in (A.1). We prove that the operator F is surjec-tive, cf. McDuff and Salamon [MS12, Proof of Lemma 3.4.3]. To see this note that its image is closedand thus we need to show only that if ν ⊥ im F , then ν = . Since ν ⊥ F ι ( W , ker ev x , ..., x n ) , on P \{ x , . . . , x n } , ν is smooth and satisfies F ι ν = . We also know that ν ⊥ im X ι , ψ . The argumentfrom the Proof of Proposition A.2 shows that ν = , because the set of points x ∈ P satisfying ι − ( ι ( x )) = { x } is open in P so we can choose such a point x belonging to P \{ x , . . . , x n } ). That F k is surjective follows from the fact that F is surjective by elliptic regularity.Pick n ∈ Γ ( N ι ) with n ( x i ) = v i and pick ( n , η ) ∈ ker ev x , ..., x n ⊕ T ψ D c ( Y ) such that F ι n + X ι , ψ ( η ) = − F ι n . The pair ( n + n , η ) ∈ T [ ι ] , ψ A si β has the desired properties. (cid:3) Finally, we are in a position to prove (1b) and (2b) of Proposition 2.23. Denote by π : A si β , → A si β the forgetful map and denote by ∆ = {( x , x ) ∈ Y × Y : x ∈ Y } the diagonal in Y . Proposition A.3The universal moduli space of non-injective but somewhere injective immersed associatives isprecisely π ( ev − ( ∆ )) . By Proposition A.3, ev − ( ∆ ) ⊂ A si β is a codimension submanifold. Since π is a Fredholm mapof index and ρ : A si → D c ( Y ) is a Fredholm map of index , it follows that ρ ( π ( ev − ( ∆ ))) ⊂ D c ( Y ) is residual. This proves (1b) because an injective immersion of a compact manifold is anembedding. The proof of (2b) is similar. This completes the proof of Proposition 2.23. (cid:3) B Seiberg–Witten equations in dimension three
We very briefly review how to associate a Seiberg–Witten equation to a quaternionic represen-tation. More detailed discussions can be found in [Tau99; Pid04; Hay08; Sal13, Section 6; Nak16,Section 6(i)]; we follow [DW17, Section 1] closely. The first ingredient is a choice of algebraicdata. 48 efinition B.1. A quaternionic Hermitian vector space is a real vector space S together with alinear map γ : Im H → End ( S ) and an inner product h· , ·i , such that γ makes S into a left moduleover the quaternions H = R h , i , j , k i , and i , j , k act by isometries. The unitary symplectic group Sp ( S ) is the subgroup of GL ( S ) preserving γ and h· , ·i . A quaternionic representation of a Liegroup G on S is a homomorphism ρ : G → Sp ( S ) .Let ρ : G → Sp ( S ) be a quaternionic representation. Denote by g the Lie algebra of G . Thereis a canonical hyperkähler moment map µ : S → ( g ⊗ Im H ) ∗ defined as follows. By slight abuseof notation denote by ρ : g → sp ( S ) the Lie algebra homomorphism induced by ρ . Combine ρ and γ into the map ¯ γ : g ⊗ Im H → End ( S ) given by ¯ γ ( ξ ⊗ v ) Φ ≔ ρ ( ξ ) γ ( v ) Φ . The map ¯ γ takes values in the space of symmetric endomorphisms of S . Denote by ¯ γ ∗ : End ( S ) →( g ⊗ Im H ) ∗ the adjoint of ¯ γ . Define µ ( Φ ) ≔
12 ¯ γ ∗ ( ΦΦ ∗ ) . Definition B.2.
The canonical permuting action θ : Sp ( ) → O ( S ) is defined by left-multiplicationby unit quaternions. It satisfies θ ( q ) γ ( v ) Φ = γ ( Ad ( q ) v ) θ ( q ) Φ for all q ∈ Sp ( ) = { q ∈ H : | q | = } , v ∈ Im H , and Φ ∈ S . Definition B.3.
A set of algebraic data consists of: • a quaternionic Hermitian vector space ( S , γ , h· , ·i) , • a compact, connected Lie group H , an injective homomorphism Z → Z ( H ) , an Ad –invariant inner product on Lie ( H ) , • a closed, connected, normal subgroup G ⊳ H , and • a quaternionic representation ρ : H → Sp ( S ) such that − ∈ Z ⊂ Z ( H ) acts as − id S . Definition B.4.
Given a set of algebraic data, set ˆ H ≔ ( Sp ( ) × H )/ Z , K ≔ H / G , and ˆ K ≔ ( Sp ( ) × K )/ Z . The group K is called the flavor group . Example B.5.
The ADHM r , k Seiberg–Witten equation arise by choosing S = S r , k ≔ Hom C ( C r , H ⊗ C C k ) ⊕ H ⊗ R u ( k ) G = U ( k ) ⊳ H = SU ( r ) × Sp ( ) × U ( k ) where SU ( r ) acts on C r in the obvious way, U ( k ) acts on C k in the obvious way and on u ( k ) by the adjoint representation, and Sp ( ) acts on the first copy of H trivially and on the secondcopy by right-multiplication with the conjugate. The homomorphism Z → Z ( H ) is defined by −
7→ ( id C r , − id H , − id C k ) . In particular, ˆ H = SU ( r ) × Spin U ( k ) ( ) with Spin U ( k ) ( n ) ≔ ( Spin ( n ) × U ( k ))/ Z . Although notationally cumbersome, we usually prefer to think of ˆ H as ˆ H = SU ( r ) × Spin U ( k ) ( ) × SO ( ) SO ( ) . Here the second factor is the fiber product of
Spin U ( k ) ( ) with SO ( ) with respect to the obvioushomomorphism Spin U ( k ) ( ) → SO ( ) and the homomorphism SO ( ) → SO ( ) is given by theaction on Λ + R .In addition to a set of algebraic data has been chosen one also needs to fix the geometric datafor which the Seiberg–Witten equation will be defined. Definition B.6.
Let M be a closed, connected, oriented –manifold. A set of geometric data on M compatible with a set of algebraic data as in Definition B.3 consists of: • a Riemannian metric д on M , • a principal ˆ H –bundle ˆ Q → M together with an isomorphism(B.7) ˆ Q × ˆ H SO ( ) (cid:27) SO ( T M ) , and • a connection B on the principal K –bundle R ≔ ˆ Q × ˆ H K . Definition B.8.
Given a choice of geometric data, the spinor bundle and the adjoint bundle arethe vector bundles S ≔ ˆ Q × θ × ρ S and g P ≔ ˆ Q × Ad g . Because of (B.7) the maps γ and µ induce maps γ : T ∗ M → End ( S ) and µ : S → Λ T ∗ M ⊗ g P . Here we take µ to be the moment map corresponding to the action of G ⊳ H . If H = G × K , then there is a principal G –bundle P → M associated with ˆ Q and g P is the adjoint bundle of P . Ingeneral, P might not exist but traces of it remain, e.g., its adjoint bundle g P and its gauge group G ( P ) . efinition B.9. Set A B ( ˆ Q ) ≔ (cid:26) A ∈ A ( ˆ Q ) : A induces B on R and theLevi-Civita connection on T M (cid:27) . Any A ∈ A B ( ˆ Q ) defines a covariant derivative ∇ A : Γ ( S ) → Ω ( M , S ) . The Dirac operator associ-ated with A is the linear map / D A : Γ ( S ) → Γ ( S ) defined by / D A Φ ≔ γ (∇ A Φ ) . A B ( ˆ Q ) is an affine space modeled on Ω ( M , g P ) . Denote by ϖ : Ad ( ˆ Q ) → g P the projectioninduced by Lie ( ˆ H ) → Lie ( G ) .Finally, we are in a position to define the Seiberg–Witten equation. Definition B.10.
The
Seiberg–Witten equation associated with the chosen algebraic and geomet-ric data is the following system of partial differential equations for ( Φ , A ) ∈ Γ ( S ) × A B ( ˆ Q ) : / D A Φ = and ϖF A = µ ( Φ ) . (B.11)The Seiberg–Witten equation is invariant with respect to gauge transformations which pre-serve the flavor bundle R and SO ( T ∗ M ) . Definition B.12.
The group of restricted gauge transformations is G ( P ) ≔ (cid:8) u ∈ G ( ˆ Q ) : u acts trivially on R and SO ( T M ) (cid:9) . G ( P ) can be identified with the space of sections of ˆ Q × ˆ H G with ˆ H acting on G via [( q , h )] · д = hдh − .If µ − ( ) = { } , then one proves in the same way as for the classical Seiberg–Witten equationthat solutions of (B.11) obey a priori bounds on Φ . In many cases of interest µ − ( ) , { } andin these cases a priori bounds fail to hold. Anticipating this, we blow-up the Seiberg–Wittenequation. Definition B.13.
The blown-up Seiberg–Witten equation is the following partial differentialequation for ( ε , Φ , A ) ∈ [ , ∞) × Γ ( S ) × A B ( ˆ Q ) : / D A Φ = , ε ϖF A = µ ( Φ ) , and k Φ k L = . (B.14)The limiting Seiberg–Witten equation is the following partial differential equation for ( Φ , A ) ∈[ , ∞) ∈ Γ ( S ) × A B ( ˆ Q ) : / D A Φ = and µ ( Φ ) = (B.15)as well as k Φ k L = . 51he phenomenon of Φ tending to infinity for (B.11) corresponds to ε tending to zero for (B.14)Formally, the compactifiction of the moduli space of solutions of (B.11) should thus be given byadding solution of the limiting equation. Taubes [Tau13a] and Haydys and Walpuski [HW15]proved that—up to allowing for codimension on singularities in the limiting solutions—this istrue for the flat P SL ( , C ) –connections and the Seiberg–Witten equation with multiple spinors,which are particular instances of equation (B.11). Although one might initially hope that it isunnecessary to allow for singularities in solutions of the limiting equation, it has been shown in[DW18] that this phenomenon cannot be avoided. C The Haydys correspondence with stabilizers
Throughout this appendix we assume that algebraic data and geometric data as in Definition B.3and Definition B.6 have been chosen. Denote by X ≔ S /// G = µ − ( )/ G the hyperkähler quotient of X by G , and denote by p : µ − ( ) → X the canonical projection. Theaction of ˆ H on S induces an action of ˆ K = ˆ H / G on X . Set X ≔ ˆ R × ˆ K X . If Φ ∈ Γ ( S ) satisfies µ ( Φ ) = , then(C.1) s ≔ p ◦ Φ ∈ Γ ( X ) . The Haydys correspondence [Hay12, Section 4.1] relates solutions of the limiting Seiberg–Wittenequation (B.15) with certain sections of X . The discussions of the Haydys correspondence availablein the literature so far [Hay12, Section 4.1; DW17, Section 3] assume that the action of G on µ − ( ) is generically free. This hypothesis does not hold in Example B.5 with r = , which leads tothe ADHM , k Seiberg–Witten equation. This appendix is concerned with extending the Haydyscorrespondence to the case when G acts on µ − ( ) with a non-trivial generic stabilizer. C.1 Decomposition of hyperkähler quotients
Denote by S { e } the subset of S on which G acts freely. By [HKLR87, Section 3(D)], the quotient ( S { e } ∩ µ − ( ))/ G can be given the structure of hyperkähler manifold of dimension ( dim H S − dim G ) such that, for Φ ∈ S { e } ∩ µ − ( ) ,(C.2) p ∗ : ( ρ ( g ) Φ ) ⊥ ∩ T Φ µ − ( ) → T [ Φ ] X is a quaternionic isometry. If G acts on µ − ( ) with trivial generic stabilizer (that is: S { e } is denseand open), then this makes an dense open subset of X into a hyperkähler manifold. In general, X can be decomposed as a union of hyperkähler manifolds according to orbit type as follows.52 efinition C.3. For Φ ∈ S , denote by G Φ the stabilizer of Φ in G . Let T < G be a subgroup. Set S T ≔ { Φ ∈ S : G Φ = T } and S ( T ) ≔ { Φ ∈ S : дG Φ д − = T for some д ∈ G } . Definition C.4.
Given a subgroup T < G , set W G ( T ) ≔ N G ( T )/ T . Here N G ( T ) denotes the normalizer of T in G . Remark
C.5 . This notation is motivated by the example S = H ⊗ g , with G acting via the adjointrepresentation. In this case, the stabilizer T of a generic point in µ − ( ) is a maximal torus and W G ( T ) is the Weyl group of G ; cf. Appendix D for the case G = U ( k ) . Theorem C.6 (Dancer and Swann [DS97, Theorem 2.1]; Sjamaar and Lerman [SL91], Nakajima[Nak94, Section 6]) . For each T < G , the quotient X ( T ) ≔ ( µ − ( ) ∩ S ( T ) )/ G is a hyperkähler manifold, and (C.7) X = Ø ( T ) X ( T ) where ( T ) runs through all conjugacy classes of subgroups of G . More precisely, for each T < G :1. S T is a hyperkähler submanifold of S and S ( T ) is a submanifold of S .2. We have ( µ − ( ) ∩ S ( T ) )/ G = ( µ − ( ) ∩ S T )/ W G ( T ) .
3. Denote by S T denotes the union of the components of S T intersecting µ − ( ) . Then W G ( T ) actsfreely on S T and µ ( S T ) ⊂ ( w ⊗ Im H ) ∗ with w ≔ Lie ( W G ( T )) . In particular, the restriction of µ to S T induces a hyperkähler momentmap on S T for the action of W G ( T ) .4. X ( T ) can be given the structure of a hyperkähler manifold such that, for each Φ ∈ µ − ( ) ∩ S ( T ) , p ∗ : ( ρ ( g ) Φ ) ⊥ ∩ ker d Φ µ ∩ T Φ S ( T ) → T [ Φ ] X ( T ) is a quaternionic isometry. There can be subgroups T < G with S ( T ) , , but µ − ( ) ∩ S ( T ) = œ . roof. We recall Dancer and Swann’s argument, since some aspects of it will play a role later on.To prove (1), denote by S T ≔ { Φ ∈ S : G Φ ⊃ T } the fixed-point set of the action of T . S T is an H –linear subspace of S and S T is an open subsetof S T (by the Slice Theorem). Therefore, S T is a hyperkähler submanifold of S . The group actioninduces a bijection S T × W G ( T ) G / T (cid:27) S ( T ) , [ Φ , дT ] 7→ ρ ( д ) Φ . This shows that S ( T ) is a submanifold of S . For future reference, we also observe that(C.8) T Φ S T = S T and T Φ S ( T ) = S T + ρ ( g ) Φ (cid:27) S T ⊕ ρ ( g ) Φ ρ ( w ) Φ . The assertion made in (2) follows directly from the definitions.To prove (3), observe that by the definition of S T , the group W G ( T ) = N G ( T )/ T acts freely on S T . Since µ is G –equivariant, µ ( S T ) ⊂ ( g ∗ ) T ⊂ n ∗ with n ≔ Lie ( N G ( T )) . Let t = Lie ( T ) . If Φ ∈ S T ,then(C.9) d Φ µ ∈ Ann g ∗ t ⊗ ( Im H ) ∗ , because, for ξ ∈ t , v ∈ Im H , and ϕ ∈ S , we have h( d Φ µ ) ϕ , ξ ⊗ v i = h γ ( v ) ρ ( ξ ) Φ , ϕ i = . Since w ∗ = n ∗ ∩ Ann g ∗ t , we have µ ( S T ) ⊂ ( w ⊗ Im H ) ∗ . This proves (3).Finally, we prove (4). Since X ( T ) = ( µ − ( ) ∩ S ( T ) )/ G = ( µ − ( ) ∩ S ( T ) )/ W G ( T ) = S T /// W G ( T ) , X ( T ) can be given a hyperkähler structure by the construction in [HKLR87, Section 3(D)]. If Φ ∈ S T ,then ( ρ ( g ) Φ ) ⊥ ∩ T Φ S ( T ) = ( ρ ( w ) Φ ) ⊥ ∩ T Φ S T by (C.8); hence, by the discussion before Definition C.3, p ∗ : ( ρ ( g ) Φ ) ⊥ ∩ ker d Φ µ ∩ T Φ S ( T ) = ( ρ ( w ) Φ ) ⊥ ∩ ker d Φ µ ∩ T Φ S T → T [ Φ ] X ( T ) is a quaternionic isometry. This finishes the proof of (4). (cid:3) In general, the action of ˆ K = ˆ H / G need not preserve the strata X ( T ) . The following hypothesis,which holds for all the examples considered in this article, guarantees that the action of ˆ H on S preserves S ( T ) and that the action of ˆ K on X preserves X ( T ) ⊂ X . Hypothesis C.10.
Given T < G , assume that, for all h ∈ H , there is a д ∈ G such that hTh − = дTд − . roposition C.11. If Hypothesis C.10 holds for T < G , then the action of ˆ H on S preserves thesubmanifold S ( T ) and the action of ˆ K on X preserves X ( T ) . (cid:3) Proof.
For h ∈ H and Φ ∈ S ( T ) , we have G ρ ( h ) Φ = hG Φ h − = hTh − = дTд − for some д ∈ G .Thus, ρ ( h ) Φ ∈ S ( T ) and the action of H preserves S ( T ) . The action of Sp ( ) commutes with thatof H and so it also preserves S ( T ) . We conclude that S ( T ) is preserved by the action of ˆ H . Since X ( T ) = ( µ − ( ) ∩ S ( T ) )/ G , the action of ˆ K preserves X ( T ) . (cid:3) Proposition C.12.
For any T < G , N G ( T ) is a normal subgroup of N H ( T ) , and the identity K = H / G induces an injective homomorphism N H ( T )/ N G ( T ) ֒ → K . If Hypothesis C.10 holds for T < G , thenthis map is an isomorphism N H ( T )/ N G ( T ) (cid:27) K . Proof. If д ∈ N G ( T ) and h ∈ N H ( T ) , then ˜ д ≔ hдh − ∈ G since G ⊳ H ; hence, ˜ д ∈ N G ( T ) . Since N H ( T ) ∩ G = N G ( T ) , we have an injective homomorphism N H ( T )/ N G ( T ) ֒ → K .Assuming Hypothesis C.10 and given k = hG ∈ K , there is a д ∈ G such that hTh − = дTд − . It follows that ˜ h ≔ д − h ∈ N H ( T ) and ˜ hG = k ; hence, N H ( T )/ N G ( T ) ֒ → K is an isomorphism. (cid:3) Assuming Hypothesis C.10 for T < G , we can define fiber bundles over M whose fibers arethe strata S ( T ) and X ( T ) : S ( T ) ≔ ˆ Q × ˆ H S ( T ) and X ( T ) ≔ ˆ R × ˆ K X ( T ) . If it holds for all T < G with non-empty S T , we decompose S and X as S = Ø ( T ) S ( T ) and X ( T ) = Ø ( T ) X ( T ) . C.2 Lifting sections of X ( T ) For the remainder of this section we will assume Hypothesis C.10 for T < G . The first part of theHaydys correspondence is concerned with the questions:When can a section s ∈ Γ ( X ( T ) ) be lifted a section of Φ ∈ Γ ( S ( T ) ) with µ ( Φ ) = forsome choice of ˆ Q ?and To what extend is the principal ˆ H –bundle ˆ Q determined by s ?55 roposition C.13. If Φ ∈ Γ ( S ( T ) ) , then ˆ Q ◦ = ˆ Q ◦ Φ ≔ (cid:8) q ∈ ˆ Q : Φ ( q ) ∈ S T (cid:9) is a principal N ˆ H ( T ) –bundle over M whose associated principal ˆ H –bundle is isomorphic to ˆ Q . More-over, the stabilizer of Φ in G ( P ) = Γ ( ˆ Q × ˆ H G ) is Γ ( ˆ Q ◦ × N ˆ H ( T ) T ) ⊂ G ( P ) , and the kernel of ρ (·) Φ : g P → S is (C.14) t P ≔ ˆ Q ◦ × N ˆ H ( T ) Lie ( T ) ⊂ g P . Proof. If Φ ∈ S T , ˆ h = [( q , h )] ∈ ˆ H = ( Sp ( ) × H )/ Z and Ψ ≔ θ ( q ) ρ ( h ) Φ , then G Ψ = hG Φ h − = hTh − . Therefore, Ψ ∈ S T if and only if ˆ h ∈ N ˆ H ( T ) = ( Sp ( ) × N H ( T ))/ Z . Moreover, for each Φ ∈ S ( T ) there is a д ∈ G ⊂ ˆ H such that ρ ( д ) Φ ( q ) ∈ S T . This implies that ˆ Q ◦ is a principal N ˆ H ( T ) –bundle.The isomorphism ˆ Q ◦ × N ˆ H ( T ) ˆ H (cid:27) ˆ Q is given by [( ˆ q , ˆ h )] 7→ ˆ q · ˆ h . In particular, G ( P ) (cid:27) Γ ( ˆ Q ◦ × N ˆ H ( T ) G ) where N ˆ H ( T ) acts on G by conjugation. The last two assertions follow from the fact that, for every q ∈ ˆ Q ◦ , the G –stabilizer of Φ ( q ) is T . (cid:3) Definition C.15.
Given any Φ ∈ Γ ( S ( T ) ) , the Weyl group bundle associated with Φ is ˆ Q ⋄ = ˆ Q ⋄ Φ ≔ ˆ Q ◦ Φ / T . Proposition C.16.
Suppose that two choices of geometric data have been made such that ˆ R = ˆ R .Suppose that Φ i ∈ Γ ( S ˆ Q i , ( T ) ) satisfy µ ( Φ i ) = . Denote by ˆ Q ⋄ i the associated Weyl group bundles.If p ◦ Φ = p ◦ Φ ∈ Γ ( X ( T ) ) , then there is an isomorphism ˆ Q ⋄ (cid:27) ˆ Q ⋄ compatible with theisomorphism ˆ Q ⋄ / W G ( T ) (cid:27) ˆ R = ˆ R (cid:27) ˆ Q ⋄ / W G ( T ) . Remark
C.17 . The principal N G ( T ) –bundles ˆ Q ◦ and ˆ Q ◦ need not be isomorphic. Here we think of Φ as a ˆ H –equivariant map Φ : ˆ Q → S . Hypothesis C.10 ensures that hTh − ⊂ G . roof of Proposition C.16. Since ˆ Q i / G (cid:27) ˆ R i , we have ˆ Q ◦ i / N G ( T ) (cid:27) ˆ R i . The sections Φ i restrict to N G ( T ) –equivariant maps Φ ◦ i : ˆ Q ◦ i → µ − ( ) ∩ S T , which in turn induce W G ( T ) –equivariant maps Φ ⋄ i : ˆ Q ⋄ i = ˆ Q ◦ i / T → µ − ( ) ∩ S T . The resulting commutative diagrams ˆ Q ⋄ i µ − ( ) ∩ S T ˆ R i X ( T ) q ⋄ i Φ ⋄ i ps are pullback diagrams; hence, the assertion follows from the universal property of pullbacks. (cid:3) Proposition C.18.
Let ˆ R be a principal ˆ K –bundle. Given s ∈ Γ ( X ( T ) ) , there exists a principal W ˆ H ( T ) –bundle ˆ Q ⋄ together with an isomorphism ˆ Q ⋄ / W G ( T ) (cid:27) ˆ R and a section Φ ⋄ ∈ Γ ( ˆ Q ⋄ × W ˆ H ( T ) S T ) satisfying µ ( Φ ⋄ ) = and p ◦ Φ ⋄ = s . The section Φ ⋄ is unique up to the action of the restricted gauge group Γ ( ˆ Q ⋄ × W ˆ H ( T ) W G ( T )) .Proof. We can think of the section s as a ˆ K –equivariant map s : ˆ R → X ( T ) . The quotient map p : µ − ( ) ∩ S T → X ( T ) defines a principal W G ( T ) –bundle. Set ˆ Q ⋄ ≔ s ∗ ( µ − ( ) ∩ S T ) = {( r , Φ ) ∈ R × ( µ − ( ) ∩ S T ) : s ( r ) = W G ( T ) · Φ } and denote by Φ ⋄ : ˆ Q ⋄ → µ − ( ) ∩ S T the projection to the second factor. The projection to thefirst factor q ⋄ : ˆ Q ⋄ → R makes ˆ Q ⋄ into a principal W G ( T ) –bundle over ˆ R . We have the followingdiagram with the square being a pullback: ˆ Q ⋄ µ − ( ) ∩ S T ˆ R X ( T ) M . q ⋄ Φ ⋄ ps Q ⋄ can be given the structure of a principal W ˆ H ( T ) –bundle over M as follows. By Proposi-tion C.12 we have a short exact sequence W G ( T ) W ˆ H ( T ) ˆ K . π Define an right-action of W ˆ H ( T ) on ˆ Q ⋄ by ( r , Φ ) · [ ˆ h ] ≔ ( r · π ([ ˆ h ]) , ( θ × ρ )( ˆ h − ) Φ ) for [ ˆ h ] ∈ W ˆ H ( T ) and ( r , Φ ) ∈ ˆ Q ⋄ and with θ as in Definition B.2. A moment’s thought shows thatthis action is free and ˆ Q ⋄ / W ˆ H ( T ) = ( ˆ Q ⋄ / W G ( T ))/ ˆ K = ˆ R / ˆ K = M . Since s is ˆ K –equivariant, Φ ⋄ is W ˆ H ( T ) –equivariant and thus defines the desired section. Theassertion about the uniqueness of Φ ⋄ is clear. (cid:3) Proposition C.19.
Assume the situation of Proposition C.18. Suppose that ˆ Q ◦ is a principal N ˆ H ( T ) –bundle with an isomorphism ˆ Q ◦ / T (cid:27) ˆ Q ⋄ ; that is: ˆ Q ◦ is a lift of the structure group from W ˆ H ( T ) to N ˆ H ( T ) . Set ˆ Q ≔ ˆ Q ◦ × N ˆ H ( T ) ˆ H . In this situation, there is a section Φ of S ( T ) ≔ ˆ Q × ˆ H S ( T ) satisfying µ ( Φ ) = and p ◦ Φ = s ; moreover, there is an isomorphism ˆ Q ◦ Φ (cid:27) ˆ Q ◦ . Any other section satisfying these conditions is related to Φ by the action of G ( P ) .Proof. With Φ ⋄ as in Proposition C.18 define Φ : ˆ Q → µ − ( ) ⊂ S by Φ ([ q , ˆ h ]) ≔ ( θ × ρ )( ˆ h − ) Φ ⋄ ( qT ) . This is well-defined because Φ ⋄ ( qT ) is T –invariant; moreover, Φ is manifestly ˆ H –equivariant and,hence, defines the desired section. The assertion about the uniqueness of Φ is clear. (cid:3) To summarize the preceeding discussion and answer the questions raised at the beginning ofthis section:1. s determines the Weyl group bundle ˆ Q ⋄ uniquely,2. every s lifts to a section Φ ⋄ of ˆ Q ⋄ × W ˆ H ( T ) S T , and3. if ˆ Q ◦ is a lift of the structure group of ˆ Q ⋄ from W ˆ H ( T ) to N ˆ H ( T ) and we set ˆ Q ≔ ˆ Q ◦ × N ˆ H ( T ) ˆ H ,then Φ ⋄ induces a section Φ of S ( T ) = ˆ Q × ˆ H S ( T ) lifting s .58 .3 Projecting the Dirac equation The second part of the Haydys correspondence is concerned with the questionTo what extend is the Dirac equation for a section Φ ∈ Γ ( S ( T ) ) equivalent to adifferential equation for s ≔ p ◦ Φ ∈ Γ ( X ( T ) ) ? Definition C.20.
The vertical tangent bundle of X ( T ) π −→ M is V X ( T ) ≔ ˆ R × ˆ K T X ( T ) . The hyperkähler structure on X ( T ) induces a Clifford multiplication γ : π ∗ Im H → End ( V X ( T ) ) . Given B ∈ A ( ˆ R ) we can assign to each s ∈ Γ ( S ) its covariant derivative ∇ B s ∈ Ω ( M , s ∗ V X ) .A section s ∈ Γ ( X ) is called a Fueter section if it satisfies the
Fueter equation (C.21) F ( s ) = F B ( s ) ≔ γ (∇ B s ) = ∈ Γ ( s ∗ V X ( T ) ) . The map s F ( s ) is called the Fueter operator . Proposition C.22.
Given Φ ∈ Γ ( S ( T ) ) satisfying µ ( Φ ) = , set s ≔ p ◦ Φ ∈ Γ ( X ( T ) ) . The following hold:1. A ∈ A B ( ˆ Q ) satisfies / D A Φ = if and only if (C.23) F B ( s ) = and ∇ A Φ ⊥ ρ ( g P ) Φ .
2. Let t P be as in (C.14) . The space of connections (C.24) A Φ B ( ˆ Q ) ≔ (cid:8) A ∈ A B ( ˆ Q ) : ∇ A Φ ⊥ ρ ( g P ) Φ (cid:9) is an affine space modeled on Ω ( M , t P ) with t P as in (C.14) . In particular, if F B ( s ) = , thereexists an A ∈ A B ( ˆ Q ) such that / D A Φ = ; A is unique up to Ω ( M , t P ) .3. Any connection A ∈ A Φ B ( ˆ Q ) reduces to a connection on ˆ Q ◦ . Conversely, any connection on ˆ Q ◦ induces a connection in A Φ B ( ˆ Q ) .4. The subbundle t P ⊂ g P is parallel with respect to any A ∈ A Φ B ( ˆ Q ) . roof. We prove (1). If / D A Φ = , then it follows from p ∗ (∇ A Φ ) = ∇ B s that F B ( s ) = . Let ( e , e , e ) be an orthonormal basis of T ∗ x M . The equations / D A Φ = and ∇ A µ ( Φ ) = can be written as ∇ A , e i Φ = − ε kij γ ( e j )∇ A , e k Φ and h γ ( e j )∇ A , e k Φ , ρ ( ξ ) Φ i = for all ξ ∈ g P , x . This proves that ∇ A Φ ⊥ ρ ( g P ) Φ . By Theorem C.6(4), (C.23) implies / D A Φ = We prove (2). If A ∈ A Φ B ( ˆ Q ) and a ∈ Ω ( M , g P ) are such that A + a ∈ A Φ B ( ˆ Q ) , then ρ ( a ) Φ ⊥ ρ ( g P ) Φ ; hence, ρ ( a ) Φ = and it follows that a ∈ Ω ( M , t P ) by Proposition C.13. It remains to show that A Φ B ( ˆ Q ) is non-empty. To see this, note that if A ∈ A Φ B ( ˆ Q ) , then one can find a ∈ Ω ( M , g P ) suchthat ∇ A Φ + ρ ( a ) Φ is perpendicular to ρ ( g P ) Φ .We prove (3). If A ∈ A Φ B ( ˆ Q ) and H A denote its horizontal distribution, then we need to showthat for q ∈ ˆ Q ◦ we have H A , q ⊂ T q ˆ Q ◦ . This, however, is an immediate consequence of thedefinitions of A Φ B ( ˆ Q ) and ˆ Q ◦ .We prove (4). Suppose τ ∈ Γ ( t P ) , that is, ρ ( τ ) Φ = . Differentiating this identity along v yields ρ (∇ A , v τ ) Φ = − ρ ( τ )∇ A , v Φ ; Set σ = ∇ A , v τ . We need to show that ρ ( σ ) Φ = . We compute | ρ ( σ ) Φ | = −h ρ ( σ ) Φ , ρ ( τ )∇ A , v Φ i = h ρ ( τ ) ρ ( σ ) Φ , ∇ A , v Φ i = h ρ ([ τ , σ ]) Φ , ∇ A , v Φ i = because ∇ A Φ ⊥ ρ ( g P ) Φ . (cid:3) To summarize:1. The Dirac equation / D A Φ = implies the Fueter equation F B s = .2. Given a solution s of the Fueter equation and ˆ Q ◦ as at the end of the last subsection, thereis a connection A ∈ A B ( ˆ Q ) such that the lift Φ satisfies / D A Φ = .3. A is unique up to Ω ( M , t P ) with t P as in (C.14). D The ADHM representation
We now focus on the case r = in Example B.5. We will see that in this case the hyperkählerquotient of the representation is the symmetric product Sym k H . This fact is the basis of therelationship between multiple covers of associatives and ADHM monopoles.60dentifying H ⊗ C C r = Hom C ( C k , H ) , we can write the quaternionic vector space S fromExample B.5 with r = as S = Hom C ( C k , H ) ⊕ H ⊗ R u ( k ) . The group U ( k ) acts on S via ρ ( д )( Ψ , ξ ) ≔ ( Ψ д − , Ad ( д ) ξ ) preserving the hyperkähler structure. We will now determine the hyperkähler quotient S /// U ( k ) and its decomposition into hyperkähler manifolds described in Theorem C.6. Definition D.1. A partition of k ∈ N is a non-increasing sequence of non-negative integers λ = ( λ , λ , . . . ) which sums to k . The length of a partition is | λ | ≔ min { n ∈ N : λ n = } − . With each partition λ we associate the groups G λ ≔ (cid:8) σ ∈ S | λ | : λ σ ( n ) = λ n for all n ∈ { . . . , | λ |} (cid:9) and T λ ≔ | λ | Ö n = U ( λ n ) ⊂ U ( k ) . For each partition λ of k , consider the generalized diagonal ∆ | λ | = { v , . . . , v | λ | ∈ H | λ | : v i = v j for some i , j } There is an embedding ( H | λ | \ ∆ | λ | )/ G λ ֒ → Sym k H defined by [ v , . . . , v | λ | ] 7→ [ v , . . . , v | {z } λ times , · · · , v | λ | , . . . , v | λ | | {z } λ | λ | times ] . The image of this inclusion is denoted by
Sym kλ H . Theorem D.2 (Nakajima [Nak99, Proposition 2.9]) . We have S /// G = Ø λ S T λ /// W U ( k ) ( T λ ) = Ø λ Sym kλ H = Sym k H . Here we take the union over all partitions λ of k . The proof of Theorem D.2 occupies the remaining part of this section. Various algebraicidentities derived in the course of proving the theorem are also used in the discussion of theHaydys correspondence for ADHM monopoles.61 roposition D.3.
The canonical moment map µ : S → ( u ( k ) ⊗ Im H ) ∗ for the action ρ : U ( k ) → Sp ( S ) is given by µ ( Ψ , ξ ) ≔ µ ( Ψ ) + µ ( ξ ) with µ ( Ψ ) ≔ (cid:0) ( Ψ ∗ i Ψ ) ⊗ i + ( Ψ ∗ j Ψ ) ⊗ j + ( Ψ ∗ k Ψ ) ⊗ k (cid:1) and µ ( ξ ) ≔ ([ ξ , ξ ] + [ ξ , ξ ]) ⊗ i + ([ ξ , ξ ] + [ ξ , ξ ]) ⊗ j + ([ ξ , ξ ] + [ ξ , ξ ]) ⊗ k . Proof.
We can compute the moment maps for the action of U ( k ) on Hom ( C k , H ) and H ⊗ u ( k ) separately. If v = v i + v j + v k ∈ Im H and η ∈ u ( k ) , then h µ ( Ψ ) , v ⊗ η i = h Ψ , γ ( v ) ρ ( η ) Ψ i = −h Ψ , γ ( v ) Ψ ◦ η i = h Ψ ∗ γ ( v ) Ψ , η i and h µ ( ξ ) , v ⊗ η i = h ξ , γ ( v ) ρ ( η ) ξ i = v (−h ξ , [ η , ξ ]i + h ξ , [ η , ξ ]i − h ξ , [ η , ξ ]i + h ξ , [ η , ξ ]i) + v (−h ξ , [ η , ξ ]i + h ξ , [ η , ξ ]i + h ξ , [ η , ξ ]i − h ξ , [ η , ξ ]i) + v (−h ξ , [ η , ξ ]i − h ξ , [ η , ξ ]i − h ξ , [ η , ξ ]i + h ξ , [ η , ξ ]i) = v h[ ξ , ξ ] + [ ξ , ξ ] , η i + v h[ ξ , ξ ] + [ ξ , ξ ] , η i + v h[ ξ , ξ ] + [ ξ , ξ ] , η i using that h ξ , [ η , ζ ]i = −h[ ξ , ζ ] , η i for ξ , η , ζ ∈ u ( k ) . (cid:3) The key to proving Theorem D.2 is the following result.
Proposition D.4. If µ ( Ψ , ξ ) = , then Ψ = . One can derive this result using Geometric Invariant Theory [Nak99, Section 2.2]. We providea proof at the end of this section. It essentially follows Nakajima’s reasoning but avoids the useof GIT and comparison results between GIT and Kähler quotients.It follows from Proposition D.4 that S /// U ( k ) = H ⊗ g /// U ( k ) . The latter can be computed in a straight-forward fashion using the following observation. We identify ( u ( k ) ⊗ Im H ) ∗ = u ( k ) ⊗ Im H . roposition D.5. We have | µ ( ξ )| = Õ α , β = |[ ξ α , ξ β ]| . Proof.
A direct computation shows that | µ ( ξ )| − Õ α , β = |[ ξ α , ξ β ]| = − h ξ , [ ξ , [ ξ , ξ ]] + [ ξ , [ ξ , ξ ]] + [ ξ , [ ξ , ξ ]]i . This expression vanishes by the Jacobi Identity. (cid:3)
Proof of Theorem D.2.
From Proposition D.4 and Proposition D.5 it follows that we have µ ( Ψ , ξ ) = if and only if Ψ = and ξ ∈ H ⊗ t for some maximal torus t ⊂ u ( k ) . Therefore, for a fixedmaximal torus T ⊂ U ( k ) and t ≔ Lie ( T ), S /// G = ( H ⊗ t )/ W U ( k ) ( T ) (cid:27) H k / S k = Sym k H , using that the Weyl group of U ( k ) is the permutation group S k .The map S /// G → Sym k H can be described more directly as the joint spectrum. Since µ ( ξ ) = implies [ ξ , ξ ] = ∈ Λ H ⊗ g , we can find a basis e , . . . , e k of C k and elements v , . . . , v k ∈ H such that ξ ( e i ) = v i ⊗ e i . Up to ordering, the v i are independent of the choice of basis e i . The isomorphism S /// G → Sym k H is the map ξ spec ( ξ ) ≔ { v , . . . , v k } . From this description the decomposition of
Sym k H into its strata Sym kλ H is clear. (cid:3) The following result, which can be viewed as the linearization of Proposition D.5, plays animportant role in Section 5.3.
Proposition D.6.
Denote by R ξ : u ( k ) → H ⊗ u ( k ) the linearization of the action of U ( k ) on H ⊗ u ( k ) at ξ and by R ∗ ξ : H ⊗ u ( k ) → u ( k ) its adjoint. If µ ( ξ ) = , then |( d ξ µ ) η | + | R ∗ ξ η | = Õ α , β = |[ ξ α , η β ]| + Õ α = |[ ξ α , η α ]| . Proof. If µ ( ξ ) = , then on the one hand | µ ( ξ + t η )| = t |( d ξ µ ) η | + O ( t ) ; | µ ( ξ + t η )| = Õ α , β = |[ ξ α + t η α , ξ β + t η β ]| = Õ α , β = |[ ξ α , t η β ] + [ t η α , ξ β ]| + O ( t ) = t Õ α , β = |[ ξ α , η β ]| + h[ ξ α , η β ] , [ η α , ξ β ]i + O ( t ) . We also have | R ∗ ξ η | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Õ α = [ ξ α , η α ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = Õ α , β = h[ ξ α , η α ] , [ ξ β , η β ]i + Õ α = |[ ξ α , η α ]| . By the Jacobi identity h[ ξ α , η β ] , [ η α , ξ β ]i = −h η β , [ ξ α , [ η α , ξ β ]]i = h η β , [ η α , [ ξ β , ξ α ]]i + h η β , [ ξ β , [ ξ α , η α ]]i = h η β , [ ξ β , [ ξ α , η α ]]i = −h[ ξ β , η β ] , [ ξ α , η α ]i . Putting everything together, yields the asserted identity. (cid:3)
Proof of Proposition D.4.
For the proof it is convenient to write S as C k ⊕ j C k ⊕ End ( C k )⊕ j End ( C k ) .A direct computation shows that with respect to this identification the moment map is given by(D.7) µ ( v , w , A ∗ , B ) = ( vv ∗ − ww ∗ − [ A , A ∗ ] − [ B , B ∗ ]) + j ( wv ∗ − [ A , B ]) . Therefore, if ( Ψ , ξ ) = ( v + jw , A ∗ + jB ) ∈ µ − ( ) , then(D.8) vv ∗ − ww ∗ = [ A , A ∗ ] + [ B , B ∗ ] and wv ∗ = [ A , B ] . Set T ≔ [ A , A ∗ ] + [ B , B ∗ ] . Taking traces and inner products with v and w , (D.8) implies | v | = | w | ≕ λ , h v , w i = , (D.9) h Tv , v i = λ , and h Tw , w i = − λ . (D.10) 64 roposition D.11 ([Nak99, Lemma 2.8]) . Denote by V the smallest subspace of C k which contains w and is preserved by both A and B . We have v ⊥ V .Proof. Let C be a product of A s and B s. We need to show that h v , Cw i = . The proof is byinduction on k , the number of factors of C . If k = , then C = id and we have h v , w i = by (D.9).By induction we can assume that h v , ˜ Cw i = for all ˜ C with fewer than k factors. If C = C l BAC r , then Cw = C l BAC r w = C l ABC r w − C l [ A , B ] C r w = C l ABC r w − C l wv ∗ C r w = C l ABC r w because v ∗ C r w = h v , C r w i and C r has fewer than k factors. Henceforth, we can assume that C = A k B k . For such C , we have h v , A k B k w i = tr ( A k B k wv ∗ ) = tr ( A k B k [ A , B ]) = tr ([ A k B k , A ] B ) = tr ( A k [ B k , A ] B ) = k − Õ ℓ = tr ( A k B ℓ [ B , A ] B k − ℓ ) = k − Õ ℓ = tr ( B k − ℓ A k B ℓ [ B , A ]) = − k − Õ ℓ = h v , B k − ℓ A k B ℓ w i = − k h v , A k B k w i . This concludes the proof. (cid:3)
As a warm up consider the case k = . If λ > , then ( w / λ , v / λ ) is an orthonormal basis for C . With respect to this basis A and B are given by matrices of theform A = (cid:18) a a a (cid:19) and B = (cid:18) b b b (cid:19) . Consequently, the first diagonal entry of T = [ A , A ∗ ] + [ B , B ∗ ] is T = | a | + | b | > . However, since h Tw , w i = − λ according to (D.10), we have T = − λ < . It follows that λ = ; that is, Ψ = v + jw = . 65n general, let V be as in Proposition D.11 and set V ≔ V ⊥ . With respect to the splitting C k = V ⊕ V , we have A = (cid:18) A A A (cid:19) and B = (cid:18) B B B (cid:19) . It follows from wv ∗ = [ A , B ] and v ∈ V , that [ A , B ] = [ A , B ] = Moreover, we have T = ([ A , A ∗ ] + [ B , B ∗ ]) = [ A , A ∗ ] + [ B , B ∗ ] + A A ∗ + B B ∗ ; hence, [ A , A ∗ ] + [ B , B ∗ ] + A A ∗ + B B ∗ + ww ∗ = . Thus [ A , A ∗ ] + [ B , B ∗ ] . By Proposition D.12, it follows that [ A , A ∗ ] = [ B , B ∗ ] = .Since A A ∗ + B B ∗ + ww ∗ is a sum of non-negative definite matrices, we must have | w | = ;hence, Ψ = v + jw = by (D.9). Proposition D.12. If [ A , B ] = and [ A , A ∗ ] + [ B , B ∗ ] , then A and B can be simultaneouslydiagonalized and [ A , A ∗ ] = [ B , B ∗ ] = .Proof. Since A and B commute, we can simultaneously upper triagonalize them; that is, afterconjugating A and B with a unitary matrix we can assume that A = Λ + U and B = M + V where Λ , M are diagonal and U , V are strictly upper triangular. We have [ A , A ∗ ] = [ Λ , Λ ∗ ] + [ Λ , U ∗ ] − [ Λ ∗ , U ] + [ U , U ∗ ] . The first term vanishes, and the second and third terms have vanishing diagonal entries. Writing U = ( u mn ) , the m –th diagonal of [ A , A ∗ ] is k Õ n = | u mn | − | u nm | ; and similarly for B with V = ( v mn ) .The first diagonal entry of [ A , A ∗ ] + [ B , B ∗ ] is k Õ n = | u n | + | v n | . k Õ n = | u n | + | v n | − | u | − | v | = k Õ n = | u n | + | v n | Being non-positive, this term vanishes as well. Repeating this argument eventually shows that U = V = . (cid:3) This completes the proof of Proposition D.4. (cid:3)
References [ÁG03] L. Álvarez-Cónsul and O. García-Prada.
Hitchin–Kobayashi correspondence, quivers,and vortices . Communications in Mathematical Physics doi : . MR: . Zbl: (cit. on p. 44).[Ber55] M. Berger. Sur les groupes d’holonomie homogène des variétés à connexion affine etdes variétés riemanniennes . Bulletin de la Société Mathématique de France
83 (1955),pp. 279–330. MR: . Zbl: (cit. on p. 5).[BGM03] S. B. Bradlow, Oscar García-Prada, and I. Mundet i Riera.
Relative Hitchin–Kobayashicorrespondences for principal pairs . The Quarterly Journal of Mathematics doi : . MR: . Zbl: (cit. on p. 44).[BMO] J. Bloom, T. Mrowka, and P. Ozsváth. Connected sums in monopole Floer homology .unpublished work in progress (cit. on pp. 2, 3, 19).[Bra90] S.B. Bradlow.
Vortices in holomorphic line bundles over closed Kähler manifolds . Com-munications in Mathematical Physics doi : .MR: . Zbl: (cit. on p. 39).[Bry06] R. L. Bryant. Some remarks on G –structures . Proceedings of Gökova Geometry-Topology Conference 2005 . 2006, pp. 75–109. arXiv: math/0305124 . MR: .Zbl: (cit. on p. 5).[Bry87] R. L. Bryant.
Metrics with exceptional holonomy . Annals of Mathematics doi : . MR: . Zbl: (cit. on p. 5).[BS89] R. L. Bryant and S. M. Salamon. On the construction of some complete metrics withexceptional holonomy . Duke Mathematical Journal doi : . MR: (cit. on p. 5).[Che97] W. Chen. Casson’s invariant and Seiberg–Witten gauge theory. Turkish Journal ofMathematics . Zbl: (cit. on p. 2).67Che98] W. Chen.
The Seiberg–Witten theory of homology three-spheres . 1998, p. 112. arXiv: dg-ga/9703009 . MR: . url : http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:9909273 (cit. on p. 2).[CHNP13] A. Corti, M. Haskins, J. Nordström, and T. Pacini. Asymptotically cylindrical Calabi–Yau –folds from weak Fano –folds . Geometry and Topology doi : . MR: . Zbl: (cit. onp. 6).[CHNP15] A. Corti, M. Haskins, J. Nordström, and T. Pacini. G –manifolds and associative sub-manifolds via semi-Fano –folds . Duke Mathematical Journal doi : . MR: . Zbl: (cit. on pp. 6,7, 38).[CK70] J. Cheeger and J. M. Kister. Counting topological manifolds . Topology doi : . MR: . Zbl: (cit. onp. 47).[Dia12a] D.-E. Diaconescu. Chamber structure and wallcrossing in the ADHM theory of curves, I . Journal of Geometry and Physics doi : .arXiv: . MR: . Zbl: (cit. on p. 45).[Dia12b] D.-E. Diaconescu. Moduli of ADHM sheaves and the local Donaldson–Thomas theory . Journal of Geometry and Physics doi : .arXiv: . MR: . Zbl: (cit. on p. 45).[Doa17] A. Doan. Seiberg–Witten monopoles with multiple spinors on a surface times a circle .2017. arXiv: (cit. on pp. 42, 43).[Don02] S. K. Donaldson.
Floer homology groups in Yang–Mills theory . Vol. 147. CambridgeTracts in Mathematics. With the assistance of M. Furuta and D. Kotschick. Cam-bridge, 2002, pp. viii+236. doi : . MR: . Zbl: (cit. on p. 7).[Don83] S. K. Donaldson. An application of gauge theory to four-dimensional topology . Journalof Differential Geometry . url : http://projecteuclid.org/getRecord?id=euclid.jdg/1214437665 (cit. on p. 44).[DS11] S. K. Donaldson and E. P. Segal. Gauge theory in higher dimensions, II . Surveys indifferential geometry. Volume XVI. Geometry of special holonomy and related topics .Vol. 16. 2011, pp. 1–41. arXiv: . MR: . Zbl: (cit. onpp. 1, 10, 11).[DS97] A. Dancer and A. Swann.
The geometry of singular quaternionic Kähler quotients . Internat. J. Math. doi : . MR: . Zbl: (cit. on pp. 53, 54).68DT98] S. K. Donaldson and R. P. Thomas. Gauge theory in higher dimensions . The geometricuniverse (Oxford, 1996) . Oxford, 1998, pp. 31–47. MR:
MR1634503 . Zbl: . url : (cit. on pp. 1, 7, 38).[DW17] A. Doan and T. Walpuski. Deformation theory of the blown-up Seiberg–Witten equa-tion in dimension three . 2017. arXiv: (cit. on pp. 1, 48, 52).[DW18] A. Doan and T. Walpuski.
On the existence of harmonic Z spinors . Journal of Differ-ential Geometry (2018). arXiv: . to appear (cit. on pp. 1, 26, 52).[FG82] M. Fernández and A. Gray.
Riemannian manifolds with structure group G . Ann. Mat.Pura Appl. (4)
132 (1982), 19–45 (1983). doi : . MR: . Zbl: (cit. on p. 5).[Flo88] A. Floer. An instanton-invariant for –manifolds . Communications in MathematicalPhysics . url : http://projecteuclid.org/getRecord?id=euclid.cmp/1104161987 (cit. on p. 7).[Frø10] K.A. Frøyshov. Monopole Floer homology for rational homology 3-spheres . Duke Math-ematical Journal doi : . arXiv: . MR: . Zbl: (cit. on pp. 2, 18).[Gar93] O. García-Prada. Invariant connections and vortices . Communications in MathematicalPhysics doi : . MR: . Zbl: (cit. on p. 39).[Gra69] A. Gray. Vector cross products on manifolds . Trans. Amer. Math. Soc.
141 (1969), pp. 465–504. MR: . Zbl: (cit. on p. 5).[Hay08] A. Haydys.
Nonlinear Dirac operator and quaternionic analysis . Communications inMathematical Physics doi : .arXiv: . MR: MR2403610 . Zbl: (cit. on p. 48).[Hay12] A. Haydys.
Gauge theory, calibrated geometry and harmonic spinors . Journal of theLondon Mathematical Society doi : .arXiv: . MR: . Zbl: (cit. on p. 52).[Hay17] A. Haydys. G instantons and the Seiberg–Witten monopoles . 2017. arXiv: (cit. on p. 1).[Hil74] H. M. Hilden. Every closed orientable -manifold is a –fold branched covering spaceof S . Bulletin of the American Mathematical Society
80 (1974), pp. 1243–1244. doi : . MR: . Zbl: (cit. onp. 24).[Hil76] H. M. Hilden. Three-fold branched coverings of S . American Journal of Mathematics doi : . MR: . Zbl: (cit. on p. 24). 69Hit01] N. J. Hitchin. Stable forms and special metrics . Global differential geometry: the math-ematical legacy of Alfred Gray (Bilbao, 2000) . Vol. 288. Contemp. Math. Providence,RI, 2001, pp. 70–89. arXiv: math/0107101 . MR: . Zbl: (cit. onp. 5).[HKLR87] N. J. Hitchin, A. Karlhede, U. Lindström, and M. Roček.
Hyper-Kähler metrics andsupersymmetry . Communications in Mathematical Physics doi : . MR: . Zbl: (cit. on pp. 52, 54).[HL82] R. Harvey and H. B. Lawson Jr. Calibrated geometries . Acta Math.
148 (1982), pp. 47–157. doi : . MR: MR666108 . Zbl: (cit. on pp. 4, 6).[HW15] A. Haydys and T. Walpuski.
A compactness theorem for the Seiberg–Witten equa-tion with multiple spinors in dimension three . Geometric and Functional Analysis doi : . arXiv: . MR: . Zbl: (cit. on pp. 1, 35, 52).[JK17] D.D. Joyce and S. Karigiannis. A new construction of compact G –manifolds by gluingfamilies of Eguchi–Hanson spaces . 2017. arXiv: (cit. on p. 6).[Joy00] D. D. Joyce. Compact manifolds with special holonomy . Oxford Mathematical Mono-graphs. Oxford, 2000, pp. xii+436. MR: . Zbl: (cit. on pp. 5,6).[Joy02] D. D. Joyce.
On counting special Lagrangian homology –spheres . Topology and ge-ometry: commemorating SISTAG . Vol. 314. Contemp. Math. Providence, RI, 2002,pp. 125–151. doi :
10 . 1090 / conm / 314 / 05427 . MR: . Zbl: (cit. on p. 2).[Joy17] D. D. Joyce.
Conjectures on counting associative –folds in G –manifolds . 2017. arXiv: (cit. on pp. 2, 3, 9–14).[Joy96a] D. D. Joyce. Compact Riemannian –manifolds with holonomy G . I . Journal of Dif-ferential Geometry doi : . MR: MR1424428 . Zbl: (cit. on p. 5).[Joy96b] D. D. Joyce.
Compact Riemannian –manifolds with holonomy G . II . Journal ofDifferential Geometry doi : . MR: MR1424428 . Zbl: (cit. on pp. 5, 7).[KL11] A. Kovalev and N.-H. Lee. K surfaces with non-symplectic involution and compactirreducible G –manifolds . Mathematical Proceedings of the Cambridge PhilosophicalSociety doi : . MR: .Zbl: (cit. on p. 6).[KM07] P. B. Kronheimer and T. S. Mrowka. Monopoles and three-manifolds . Vol. 10. NewMathematical Monographs. Cambridge, 2007, pp. xii+796. doi : .MR: . Zbl: (cit. on pp. 2, 17–20, 37).70KMOS07] P. Kronheimer, T. Mrowka, P. Ozsváth, and Z. Szabó. Monopoles and lens space surg-eries . Ann. of Math. (2) doi : .arXiv: math/0310164 . MR: . Zbl: (cit. on pp. 2, 19).[Kov03] A. Kovalev. Twisted connected sums and special Riemannian holonomy . Journal für dieReine und Angewandte Mathematik
565 (2003), pp. 125–160. doi : .MR: MR2024648 . Zbl: (cit. on p. 6).[Lew98] C. Lewis.
Spin ( ) instantons . 1998 (cit. on p. 38).[Lim00] Y. Lim. Seiberg–Witten invariants for –manifolds in the case b = or . PacificJournal of Mathematics doi : .MR: . Zbl: (cit. on p. 2).[Lin15] F. Lin. Pin ( ) –monopole Floer homology, higher compositions and connected sums . 2015.arXiv: (cit. on pp. 2, 19, 20).[LM89] H. B. Lawson Jr. and M.-L. Michelsohn. Spin geometry . Vol. 38. Princeton Mathemat-ical Series. Princeton, NJ, 1989, pp. xii+427. MR: . Zbl: (cit. onp. 5).[Man03] C. Manolescu.
Seiberg–Witten–Floer stable homotopy type of three-manifolds with b = . Geometry and Topology doi : .MR: . Zbl: (cit. on pp. 2, 18).[McL98] R. C. McLean. Deformations of calibrated submanifolds . Communications in Analysisand Geometry . Zbl: (cit. on p. 9).[Mon74] J. M. Montesinos.
A representation of closed orientable –manifolds as –fold branchedcoverings of S . Bulletin of the American Mathematical Society
80 (1974), pp. 845–846. doi : . MR: . Zbl: (cit. onp. 24).[Mor96] J. W. Morgan. The Seiberg–Witten equations and applications to the topology of smoothfour-manifolds . Vol. 44. Mathematical Notes. 1996, pp. viii+128. MR: . Zbl: (cit. on p. 42).[MOY97] T. Mrowka, P. Ozsváth, and B. Yu.
Seiberg–Witten monopoles on Seifert fibered spaces . Communications in Analysis and Geometry doi : .arXiv: math/9612221 . MR: . Zbl: (cit. on p. 39).[MS12] D. McDuff and D. Salamon. J –holomorphic curves and symplectic topology . Second.Vol. 52. American Mathematical Society Colloquium Publications. 2012, pp. xiv+726.MR: . Zbl: (cit. on pp. 10, 48).[MST96] J. W. Morgan, Z. Szabó, and C. H. Taubes. A product formula for the Seiberg–Witteninvariants and the generalized Thom conjecture . Journal of Differential Geometry doi : . MR: . Zbl: (cit. on p. 39). 71MT96] G. Meng and C.H. Taubes. SW = Milnor torsion . Math. Res. Lett. doi : . MR: . Zbl: (cit. onpp. 2, 16, 17, 40).[MW01] M. Marcolli and B.-L. Wang. Equivariant Seiberg–Witten Floer homology . Communi-cations in Analysis and Geometry doi : .MR: . Zbl: (cit. on pp. 2, 18).[MW05] V. Muñoz and B.-L. Wang. Seiberg–Witten–Floer homology of a surface times a circlefor non-torsion spin c structures . Mathematische Nachrichten doi : . arXiv: math/9905050 . MR: . Zbl: (cit. on p. 39).[Nak16] H. Nakajima. Towards a mathematical definition of Coulomb branches of –dimensional N = gauge theories, I . Advances in Theoretical and Mathematical Physics doi :
10 . 4310 / ATMP . 2016 . v20 . n3 . a4 . arXiv: . MR: . Zbl: (cit. on p. 48).[Nak94] H. Nakajima.
Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras . Duke Mathematical Journal doi : .MR: . Zbl: (cit. on p. 53).[Nak99] H. Nakajima. Lectures on H ilbert schemes of points on surfaces . Vol. 18. UniversityLecture Series. Providence, RI, 1999, pp. xii+132. MR: . Zbl: (cit. on pp. 61, 62, 65).[Nog87] M. Noguchi.
Yang–Mills–Higgs theory on a compact Riemann surface . Journal ofMathematical Physics doi :
10 . 1063 / 1 . 527769 . MR: . Zbl: (cit. on p. 39).[Nor13] J. Nordström.
Desingularising intersecting associatives . in preparation. 2013 (cit. onpp. 2, 13).[Pid04] V. Ya. Pidstrigach.
Hyper-Kähler manifolds and Seiberg–Witten equations . English.
Algebraic geometry. Methods, relations, and applications. Collected papers. Dedicatedto the memory of Andrei Nikolaevich Tyurin. . Zbl: (cit. on p. 48).[PT09] R. Pandharipande and R. P. Thomas.
Curve counting via stable pairs in the derived cate-gory . Inventiones Mathematicae doi : .arXiv: . MR: . Zbl: (cit. on pp. 3, 41).[PT10] R. Pandharipande and R. P. Thomas. Stable pairs and BPS invariants . Journal of theAmerican Mathematical Society doi : .arXiv: . MR: . Zbl: (cit. on pp. 41, 42).[PT14] R. Pandharipande and R. P. Thomas. . Moduli spaces .Vol. 411. London Math. Soc. Lecture Note Ser. 2014, pp. 282–333. doi : .arXiv: . MR: . Zbl: (cit. on pp. 38, 41, 45).72Sal13] D.A. Salamon. The three-dimensional Fueter equation and divergence-free frames . Ab-handlungen aus dem Mathematischen Seminar der Universität Hamburg doi : . MR: . Zbl: (cit. on p. 48).[SL91] R. Sjamaar and E. Lerman. Stratified symplectic spaces and reduction . Ann. of Math.(2) doi : . MR: . Zbl: (cit. on p. 53).[Smi11] A. Smith. A theory of multiholomorphic maps . 2011. arXiv: (cit. on p. 24).[SW17] D. A Salamon and T. Walpuski.
Notes on the octonions . Proceedings of the 23rd GökovaGeometry–Topology Conference . 2017, pp. 1–85. arXiv: . MR: .Zbl: (cit. on pp. 4, 6, 20).[Tau13a] C. H. Taubes. P SL ( C ) connections on 3–manifolds with L bounds on curvature. Cam-bridge Journal of Mathematics doi : .arXiv: . MR: . Zbl: (cit. on pp. 35, 52).[Tau13b] C. H. Taubes. Compactness theorems for SL ( C ) generalizations of the 4–dimensionalanti-self dual equations . 2013. arXiv: (cit. on p. 35).[Tau16] C. H. Taubes. On the behavior of sequences of solutions to U ( ) Seiberg–Witten systemsin dimension . 2016. arXiv: (cit. on p. 35).[Tau17] C. H. Taubes. The behavior of sequences of solutions to the Vafa–Witten equations .2017. arXiv: (cit. on p. 35).[Tau99] C. H. Taubes.
Nonlinear generalizations of a –manifold’s Dirac operator . Trends inmathematical physics (Knoxville, TN, 1998) . Vol. 13. AMS/IP Studies in AdvancedMathematics. Providence, RI, 1999, pp. 475–486. MR: . Zbl: (cit. on p. 48).[Tia00] G. Tian.
Gauge theory and calibrated geometry. I . Annals of Mathematics doi : . arXiv: math/0010015 . MR: MR1745014 . Zbl: (cit. on p. 1).[TT04] T. Tao and G. Tian.
A singularity removal theorem for Yang–Mills fields in higherdimensions . Journal of the American Mathematical Society doi : . arXiv: math/0209352 . MR: . Zbl: (cit. on p. 1).[Wal13] T. Walpuski. Gauge theory on G –manifolds . Imperial College London, 2013. url : https://spiral.imperial.ac.uk/bitstream/10044/1/14365/1/Walpuski-T-2013-PhD-Thesis.pdf (cit. on p. 2).[Wal17] T. Walpuski. G –instantons, associative submanifolds, and Fueter sections . Communi-cations in Analysis and Geometry doi : .arXiv: . MR: . Zbl: (cit. on p. 1).73Wei94] C. A. Weibel. An introduction to homological algebra . Vol. 38. Cambridge Studies inAdvanced Mathematics. 1994, pp. xiv+450. doi : . MR: (cit. on p. 19).[Wir36] W. Wirtinger. Eine Determinantenidentität und ihre Anwendung auf analytische Gebildein euklidischer und Hermitescher Maßbestimmung . Monatshefte für Mathematik undPhysik doi :
10 . 1007 / BF01699328 . MR: . Zbl:0015.07602