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ISSN 1364-0380 (on line) 1465-3060 (printed) 965 eometry & opology GVolume 7 (200T3) 965–999 GGGGTTTGTTT Published: 21 December 2003 G T G T G T G T GGGTGGGTTTT A non-abelian Seiberg–Witten invariant for integral homology 3–spheres Yuhan Lim Dept of Mathematics, Univ of California Santa Cruz, CA 95064,USA Email: [email protected] Abstract Anewdiffeomorphisminvariant of integral homology 3–spheres is definedusing a non-abelian “quaternionic” version of the Seiberg–Witten equations. AMS Classification numbers Primary: 57R57 Secondary: 57M27 Keywords Seiberg–Witten, 3–manifolds Proposed: Ronald Stern Received: 9 January 2003 Seconded: Ronald Fintushel, John Morgan Revised: 10 December 2003 c eometry & opology ublications (cid:13) G T P 966 Yuhan Lim 1 Introduction The Seiberg–Witten equations when applied to the study of oriented integral homology 3–spheres yield an invariant which was shown in [9] to coincide with Casson’s invariant. In [3], Boden and Herald introduced a generalization of Casson’s invariant from SU(2) to the higher structure group SU(3) based on the gauge theory approach of Taubes [12]. This SU(3)–Casson invariant uti- lizes valuesoftheChern–Simonsfunctionwhichmakesitarealvaluedinvariant rather than an integral one. In the present article we define a non-Abelian version of the Seiberg–Witten equations which we call quaternionic and construct a topological invariant of integral homology 3–spheres in a manner parallel to the SU(3)–Casson invariant. This new invariant has the property that it is independent of orientation of the 3–manifold and a linear combination with the SU(3)–Casson invariant gives a Z mod 4Z invariant forunoriented integral homology 3–spheres. The contents of this article are as follows. In section 2 we introduce the generalization of the SW–equations we use. The technical issue of admissible perturbations is also discussed. We use the novel approach of non-gradient perturbations. Section 3 gives the main results which are Theorems 3.7 and 3.8. The remaining sections take up the proofs. We assume the reader has some familiarity with [3], [9] and [12]. 2 Quaternionic gauge theory in 3–dimensions Standing Convention Throughout this article Y will denote an oriented closed integral homology 3–sphere (ZHS). Y will also be assumed to have a fixed Riemannian metric g. The aim is to introduce a quaternionic setting in which the Seiberg–Witten equations will make sense. Since Y is a ZHS it has a unique spin structure, up to equivalence. With respect to g this is given by a principal spin(3) = SU(2) ∼ bundle P Y . In the (real) Clifford bundle CL(T∗Y) = CL(Y) the volume → ∼ form ω has the property that ω2 = 1. The action of ω on CL(Y) induces a Y Y Y splitting into 1 eigenbundles CL+ CL−. Both CL+ and CL− are bundles ± ⊕ of algebras over Y with each fibreisomorphic, as an algebra, to thequaternions H. Let S Y be the complex spinor bundle on which CL+ acts non-trivially. → This is a rank 2 complex Hermitian vector bundle. Since the fibres of CL+ eometry & opology, Volume 7 (2003) G T A non-abelian Seiberg–Witten invariant 967 are quaternionic vector spaces, S possesses an additional action by H which commutes with the Clifford action (see [8]); we may take this to be a right action S H S. × → Suppose now that E Y is a given fixed rank one metric quaternionic vector → bundle—weassumetheactionby H isaleftaction. E alsohasadescriptionas a complex Hermitian rank 2 vector with trivial determinant, i.e. with structure group SU(2). We can twist the spinor bundle S by tensoring with E over the quaternions to form the bundle S H E. This is a real rank 4 Riemannian ⊗ vector bundleand does not naturally inherit a complex structure from S or E. Given an SU(2)–connection A on E (henceforth any connection on E mentioned will be assumed to be such type) we may construct using the canonical Riemiannian connection on S, a metric (i.e. SO(4)) connection on S H E. ⊗ This then defines in the usual way a Dirac operator 3 D = e A. A i·∇ei Xi=1 Here the ei are an orthonormal frame and A is the connection on S H ∇ ⊗ E mentioned above. We emphasize that D is in general only a real linear A operator on S HE. ⊗ Lemma 2.1 Thecomplexification of S HE isnaturallyisomorphicasacom- ⊗ plexCliffordmodulewith S CE. Underthisisomorphismthecomplexification ⊗ C D C corresponds to the complex Dirac operator D . A⊗ A Proof Introduce the notation to denote the tensor product of elements in ⊗ S HE and thecomplex tensor productin S CE. Definethevector bundle ⊗ ⊗ ⊗ map h from S CE to (S HE) C by ⊗ ⊗ ⊗ h(e f)= e f √ 1(ei f). ⊗ ⊗ − − ⊗ One checks directly that this map is a complex isomorphism and commutes with Clifford multiplication. Since the real two forms Λ2 naturally include in CL(Y) we have by Clifford mutiplication the action of Λ2 on S. This representation of Λ2 on S is well- known to be injective and with image the adjoint bundle adS, the bundle of skew-Hermitian transformations of S. The bundle adE acts on E from the left. Define an action of Λ2 adE on S HE by the rule ⊗ ⊗ (ω l) (φ e) := (ω φ) l(e). ⊗ · ⊗ · ⊗ This is well-defined since the actions of Λ2 and adE commute with the quaternionic structures. eometry & opology, Volume 7 (2003) G T 968 Yuhan Lim Remark 2.2 The Clifford action of β Λ2 is the same as the action of ∈ β Λ1 on S since the volume form ω acts by the identity. Thus we may Y −∗ ∈ equivalently work (up to multiplication by 1) with the action of Λ1 adE on − ⊗ S HE. ⊗ Lemma 2.3 Therepresentation Λ2 adE EndR(S HE) aboveisinjective ⊗ → ⊗ and has image the subbundle Sym0R(S H E) of trace zero real symmetric ⊗ transformations of S HE. ⊗ Proof That the representation of the lemma is injective is easily verified. We may rewrite the action of Λ2 adE as (ω l) (φ e) = (iω φ) il(e). Since ⊗ ⊗ · ⊗ − · ⊗ iadS is exactly the trace zero Hermitian symmetric bundle endomorphisms of S, and similiarly for iadE, the image of the representation clearly lies in the trace zero real symmetric endomorphisms of S H E. That it is onto follows ⊗ by a dimension count giving both Λ2 adE and Sym0R(S H E) real vector ⊗ ⊗ bundles of rank 9. The above lemma shows that we may regard the bundle Sym0R(S H E) as ⊗ identical to Λ2(Y) adE. Thus whenever convenient we can think of a trace ⊗ zero real symmetric endomorphism of S HE as a twisted 2–form with values ⊗ in adE. Lemma 2.4 Thereisauniquefibrewisesymmetricbilinearform 0 on S H {·} ⊗ E with values in Λ2 adE determined by the rule that ⊗ ω, φ ψ = ω ψ,φ = ω φ,ψ 0 h { · } i h · i h · i holds for all sections ω of Λ2 adE. As a section of Sym0R(S HE), φ ψ 0 ⊗ ⊗ { · } is given by the expression 1 1 ∗ ∗ φ ψ = φ ψ +ψ φ φ,ψ I . 0 { · } 2(cid:18) ⊗ ⊗ − 2h i (cid:19) Here φ ψ∗(ν)= φ ν,ψ and similiarly for ψ φ∗. ⊗ h i ⊗ Proof Let φi , ωj be a local orthonormal frames for S H E, Λ2 adE { } { } ⊗ ⊗ respectively. Let φ φ = ck ω . Then we see that ck = ω ψ ,ψ = { i · j}0 i,j k i,j h k · i ji hωk ·ψj,ψii = ckj,i determines {·}0. Identify Λ2 ⊗adE with Sym0R(S ⊗H E). In a local trivialization we may regard sections of Sym0R(S HE) as functions ⊗ with values in Sym0 (R4), the 4 4 real symmetric matrices, and sections of R × S H E as R4–valued functions. As such the inner product in Sym0R(R4) is ⊗ eometry & opology, Volume 7 (2003) G T A non-abelian Seiberg–Witten invariant 969 given by M,N = Tr(MN). The right side of the defining equation for 0 h i {·} can be expressed locally as 1 1 Tr(WΦΨT)+Tr(WΨΦT) = Tr(W(ΦΨT +ΨΦT)) 2 2 (cid:16) (cid:17) 1 = W, (ΦΨT +ΨΦT) . h 2 i The claimed expression for is exactly the trace-free component of the 0 {·} symmetric expression 1(ΦΨT +ΨΦT). 2 The configuration space is the space of all pairs (A,Φ) consisting of an C SU(2)–connection A on E and a section Φ of S H E. As usual we should ⊗ work within the framework of a certain functional space; for us choose A and Φ to be of class L2 (for A this means A A is L2 where A is a fixed 2 − 0 2 0 C∞–connection). is an affine space modelled on the Hilbert space C L22(Λ1⊗adE)×L22(S ⊗HE). The gauge automorphism group in this case will consist of the L2–bundle G 3 automorphisms which preserve the quaternionic structure of E, or equivalently the L2–sections of AdE. Since L2 C0 in dimension 3, and consists of 3 2 ⊂ C G continuous objects. acts on by g (A,Φ) = (g(A),g−1Φ). This action is G C · differentiable and the quotient we denote by . Our convention is that g(A) is B the pull-back of A by g. We have the following observation: the stabilizer 1 if Φ = 0 stab(A,Φ) =  { } 6  stab(A) if Φ = 0. Thepossiblechoicesfor stab(A) are 1 , U(1) or SU(2). Notethatinthelast {± } possibility A is necessarily a trivial connection. The pair (A,Φ) is irreducible if Φ = 0 and reducible otherwise. Thus acts freely on ∗, the irreducible 6 G C portion of and the quotient ∗ by is denoted ∗. C C G B is a Hilbert Lie group with tangent space at the identity T = L2(adE). G eG 3 Let , g (g(A),g−1Φ) be the map which is the orbit of (A,Φ) under G → C 7→ the action of . The derivative at the identity is the map G δA0,Φ: L23(adE) → L22(Λ1⊗adE)⊕L22(S ⊗HE), (2.1) δ0 (γ) = (d γ, γ(Φ)). A,Φ A − A slice for the action of on at (A,Φ) is given by (A,Φ)+X where X A,Φ A,Φ G C is the slice space which is the L2–orthogonal complement in L2(Λ1 adE) 2 ⊗ ⊕ eometry & opology, Volume 7 (2003) G T 970 Yuhan Lim sLp22a(cSe⊗toH E∗)aotfatnheirrimedaugceibolef oδAr0b,Φit. [AW,eΦm].ay also regard XA,Φ as the tangent B Define a bilinear product B: (S H E) (S H E) adE by the rule that ⊗ ⊗ ⊗ → γ(φ),ψ = γ,B(ψ,φ) holds for all γ adE. Then X has the description A,Φ h i h i ∈ as the subspace of L22(Λ1⊗adE)⊕L22(S ⊗HE) defined by the equation δ0∗ (a,ψ) = 0 d∗a B(Φ,ψ) = 0. (2.2) A,Φ ⇐⇒ A − A reducible we will often simply denote by A instead of (A,0). Corresponding reducible subspaces of and are denoted and A. At a reducible A the C B A B slice XA splits into a product XAr ×L22(S⊗HE) where XAr is the slice for the action of on . Then the normal space to A in near [A] is modelled on G A B B L2(S E)/stab(A). 2 ⊗ For instance if A is irreducible as a connection then this normal space is a cone on the quotient of the unit sphere in a separable Hilbert space by the antipodal map v v. 7→ − On we have the Chern–Simons–Dirac function csd: R (with respect to C C → a choice of trivial connection Θ say) given by 1 2 csd(A,Φ) = Tr a d a+ a a a D Φ,Φ , a =A Θ. 8π2 ZY (cid:16) ∧ Θ 3 ∧ ∧ (cid:17)−ZYh A i − A direct computation gives dcsd (a,φ) = Tr(F a)+ a Φ,Φ D Φ,φ A,Φ A A Z ∧ Z h∗ · i−Z h i Y Y Y = F + Φ Φ ,a D Φ,φ . A 0 A Z h−∗ ∗{ · } i−Z h i Y Y Thus the negative of the L2–gradient of csd is the ‘L2–vector field’ on 1 C (A,Φ) d=ef ( F Φ Φ ,D Φ) L2. (2.3) X ∗ A −∗{ · }0 A ∈ 1 By this we mean that is a section of the L2–version of the tangent bundle X 1 to . The Quaternionic Seiberg–Witten equation is the equation for the zeros C of , i.e. the critical points of csd. X Definition 2.5 The Quaternionic Seiberg–Witten equation is the equation defined for a pair (A,Φ) consisting of a connection on E and a section Φ (‘spinor’) of S HE. The equation reads: ⊗ F Φ Φ = 0  A−{ · }0 (2.4)  D Φ = 0 A  eometry & opology, Volume 7 (2003) G T A non-abelian Seiberg–Witten invariant 971 where F is the curvature of A, and since A is an SU(2)–connection, a section A of Λ2 adE. DA is the Dirac operator on S H E and 0 denotes the ⊗ ⊗ { } quadratic form of Lemma 2.4. If g is gauge transformation then csd(g(A),g−1Φ) = csd(g,Φ) deg(g), so ± csd descends to an R/Z–valued function on . This implies that (A,Φ) B X ∈ X L2 and the portion of over ∗ descends to a ‘L2–vector field’ over A,Φ∩ 1 X C 1 X ∗. B b Definition 2.6 The moduli space of solutions to (2.4) we denote by def = (A,Φ) solving (2.4) / . M { } G ⊂ B ∗ will denote irreducible and r will denote the reducible portion of M M M respectively. Thus ∗ is the zeros of and following Taubes, will be the basis for defining M X a Poincare–Hopf index for ∗. bB Remark 2.7 In our Quaternionic SW–theory the reducible portion r of M is just the moduli space of flat SU(2)–connections on Y . This is the space M dealt with by Taubes [12] in the gauge theory approach to Casson’s invariant. We need to now address the issue of an admissible class of perturbations which will make a finite number of non-degenerate points (made precise below) to M apply the idea of a Poincare–Hopf index. Unlike the holonomy perturbations used by Taubes and Boden–Herald which are gradient perturbations we elect to perturb directly rather than csd; i.e. at the level of vector fields, for X this avoids a number of technical problems which the author has presently no satisfactory solution. This approach will be adequate for defining a Poincare– Hopf index butnot a Floer type homology theory wheregradient perturbations are required. Definition 2.8 An admissible perturbation π consists of a differentiable – G equivariant map of the form (∗k,l): C → L22(Λ1⊗adE)×L22(S ⊗HE) where (i) π = ( k ,l ) X A,Φ A,Φ A,Φ A,Φ ∗ ∈ (ii) the linearization of ( k,l) at (A,Φ) is a bounded linear operator ∗ (Lπ)A,Φ: L22(Λ1⊗adE)⊕L22(S ⊗HE) → L22(Λ1⊗adE)⊕L22(S ⊗HE) eometry & opology, Volume 7 (2003) G T 972 Yuhan Lim (iii) there is a uniform bound 2 π d=ef ( A)iπ C. k A,ΦkL22,A Xi=0k ∇ A,ΦkL2 ≤ Remark 2.9 In the unperturbed case, can be easily shown to be com- M pact. The preceding uniform L2–type bound requirement on the perturbation 2 is crucial to retain compactness of the moduli space for the perturbed equation below. This is a gauge invariant bound. Definition 2.10 The perturbed Quaternionic Seiberg–Witten equations are the equations F Φ Φ +k = 0  A −{ · }0 A,Φ  D Φ+l = 0. A A,Φ The corresponding moduli space is denoted , the irreducible portion ∗ Mπ Mπ and the reducible portion r where π is the restriction to or equivalently Mπ A the k–component of π. Note that when Φ = 0, stab(A)–invariance forces l = 0 and the only effective portion of π on is the k–component. A,0 A Let = +π, the perturbation of . The linearization at a zero (A,Φ) is π X X X a map (LXπ)A,Φ: L22(Λ1⊗adE)⊕L22(S ⊗HE) → XA,Φ∩L21 (2.5) (L ) (a,φ) = ( d a φ Φ ,D φ+a Φ)+(Lπ) (a,φ). π A,Φ A 0 A A,Φ X ∗ −∗{ · } · Definition 2.11 Call (A,Φ) or [A,Φ] non-degenerate if L is surjective at π X (A,Φ). is non-degenerate if it consists entirely of non-degenerate points. π M In this instance we also call π non-degenerate. The standard Kuranishi local model argument shows that a non-degenerate point is isolated in . (This B includes reducible points.) Fix a connection 0 and let L2 denote the Sobolev norm with respect to 0. ∇ 2 ∇ A metric on is defined by the rule B d([A,Φ],[A′,Φ′]) = inf (A g(A′),Φ g−1Φ′) . (2.6) g∈Gnk − − kL22o Proposition2.12 Foranyadmissibleperturbation isacompactsubspace π M of . Furthermore there is an ε > 0 such that for any 0 < ε < ε , if 0 0 B π < ε uniformly then given any [A,Φ] there is a [A′,Φ′] k A,ΦkL22,A ∈ Mπ ∈ M such that d([A,Φ],[A′,Φ′]) < ε. eometry & opology, Volume 7 (2003) G T A non-abelian Seiberg–Witten invariant 973 Proposition 2.13 There exists non-degenerate admissible perturbations. Furthermore such a perturbation π may be choosen so that π is arbi- k A,ΦkL22,A trarily small (uniformly) and π vanishes on any given closed subset of which C is disjoint from the subspace of unperturbed SW–solutions. The proofs are in sections 4 and 5. 3 Spectral flow and definition of the invariant Fix a perturbation π (not necessarily non-degenerate). Regard the image of Xπ as lying in the larger space L2(Λ1⊗adE)⊕L2(S⊗HE) since XA,Φ∩L21 is a subspace of the former. The analog of the operator used by Taubes to define relative signs between non-degenerate zeros of is the unbounded operator π X on L2(adE)⊕L2(Λ1 ⊗adE)⊕L2(S ⊗HE) givecn in block matrix form: 0 δ0∗ Lπ =  A,Φ . A,Φ δ0 (L )  A,Φ Xπ A,Φ    Here δ0∗ is the formal L2–adjoint of δ0 and the splitting used above is the A,Φ A,Φ (first) (2nd and 3rd factors). Lπ has dense domain the subspace of L2– ⊕ A,Φ 2 sections. The ellipticity of Lπ implies that it is closed and unbounded as A,Φ an operator on L2. In general Lπ will not have a real spectrum, due to the A,Φ non-gradient perturbations we are using. Remark 3.1 If it were the case that Lπ is formally self-adjoint (i.e. on A,Φ smooth sections) then it is well-known that Lπ has only a discrete real spec- A,Φ trum which is unbounded in both directions in R and is without any accumulation points. It can be shown in general that since Lπ is a L –compact A,Φ A,Φ perturbationof L on L2 thespectrumcontinues to bediscrete, the real part A,Φ of the spectrum is also unbounded in both directions in R and is without any accumulation points, see [7]. Let us consider the behaviour of Lπ along the reducible stratum . A,Φ A ⊂ C Since Φ = 0 we abbreviate the operator to Lπ. This has a natural splitting A Lπ = Kπ Dπ (3.1) A A⊕ A corresponding to the splitting (Λ0+1 adE) (S H E). We call Kπ the ⊗ ⊕ ⊗ A tangential operator (the dependenceon only the restriction π of π will be clear eometry & opology, Volume 7 (2003) G T 974 Yuhan Lim below) and Dπ the normal operator. Explicitly A 0 d∗ Kπ =  A  and Dπ = D +(Lνπ) . A A A A d d +(Lπ)  A A A  ∗   Here (Lνπ)A denotes (Lπ)A,0 restricted to the normal space L22(S ⊗H E) fol- lowed byprojection ontothenormalspaceagain. Thenormalspaceisnaturally acted on by stab(A) and Dπ commutes with this action. If A is irreducible A as a connection then stab(A) = 1 and Dπ remains real linear but when {± } A A = Θ a trivial connection, stab(A) = SU(2) and it is quaternionic linear. ∼ (There is a stab(A) = U(1) case but this will not play a role so we will omit ∼ discussing it.) A fact established in section 5 is: Lemma 3.2 The operator (Lνπ) is multiplication by a real function f A A ∈ L2. Thus Dπ = D + f extends to an unbounded self-adjoint operator on 2 A A A L2(S HE) and spectral flow is defined for this operator. ⊗ To define relative signs between non-degenerate zeros of one usually uses π X the mod2 spectral flow of Lπ when this operator is self-adjoint. In the gen- A,Φ c eral case we use the determinant line detindLπ regarding Lπ as a family A,Φ parameterized by (A,Φ) . This is equivalent to the spectral flow definition ∈ C in the self-adjoint case. detindLπ descends to a line bundle over which we B also denote by the same notation. However we note: Lemma 3.3 detindLπ is non-orientable over , i.e. there exists closed loops B γ: S1 such that γ∗(detindLπ) is a non-trivial line bundle over S1. → B Proof It suffices to consider the determinant index detindL of the unperturbed family over A . Then LA = KA DA where KA is essentially B ⊂ B ⊕ the boundary of the (twisted) Self-dual operator in dimension 4. Spectral flow around closed loops for K is equivalent to the index of the (twisted) Self-dual A operator over Y S1. The latter index is well-known to be 0 mod 8. By × ≡ Lemma 2.1 the spectral flow for D around closed loops is equivalent to the A index of the twisted complex Dirac operator over Y S1, where the twisting × bundle E is rank 2 complex. According to the Atiyah–Singer Index Theorem this index is the negative of 2nd Chern class of E evaluated over the funda- mental class of Y S1. We may choose any closed loop so that this is 1. (See × ± the proof of Lemma 3.5 for more details on this part of the calculation.) eometry & opology, Volume 7 (2003) G T