THE MORSE INDEX OF REDUTIBLE SOLUTIONS OF THE SW-EQUATIONS CELSOM.DORIA 7 UFSC-DEPTO.DEMATEMA´TICA 0 0 2 n a Abstract. The2nd variationformulaoftheSeiberg-Wittenfunctionalisob- J tainedinordertoestimatetheMorseindexofredutiblesolutions(A,0). Itis 1 shownthattheirMorseindexisgivenbythedimensionofthelargestnegative 3 eigenspace oftheoperator△A+ k4g,henceitisfinite. 1 v 1. Introduction 0 Let (X,g) be a closed riemannian 4-manifold and 8 0 1 Spinc(X)={β+γ ∈H2(X,Z)⊕H1(X,Z )|w (X)=α(mod 2)}. 0 2 2 7 0 the space of Spinc structures on X. / Originally,the SW -monopole equations in 2 were not obtained through a vari- h α ationalprinciple,but they canbe describedas the stable solutonsofthe functional p - in3,namedSWα-functional. Since thisfunctionalsatisfiesthe Palais-Smalecondi- h tion it is allowedto relatedits criticalpoints with the topology of its configuration t a spaceA × Γ(S+). Asshownin[3],byconsideringtheembeddingoftheJacobian m α Gα α Torus : v i H1(X,R) X i:Tb1(X) = H1(X,Z) ֒→Aα×Gα Γ(Sα+), b1(X)=dimRH1(X,R) r a into the configuration space , the variational formulation of the SW -equations α gives us an interpretation for the topology of A × Γ(S+); α Gα α Theorem 1.0.1. Let (X,g) be a closed riemannian 4-manifold with scalar curvature k . Thus, g (1) ifk ≥0,thenthegradientflowoftheSW -functionaldefinesanhomotopy g α equivalence among A × Γ(S+) and i(Tb1(X)). α Gα α (2) if k <0, then A × Γ(S+) has the same homotopy type of Tb1(X). g α Gα α Therefore, the existence of SW -monopoles are not a consequence of the ho- α motopy type of A × Γ(S+). It is also known that they may exist for at α Gα α most a finite number of classes α and, whenever X has simple type, they sat- isfy α2 =2χ(X)+3σ(X)1 corresponding,eachone,to aalmostcomplex structure on X. 1χ(X)-eulercharacteristicofX,σ(X)=signature ofX. 1 2 CELSOM.DORIA UFSC-DEPTO.DEMATEMA´TICA An application of theorem 1.0.1 is that the space A × Γ(S+)−{(A,0)|A∈ α Gα α A } has the homotopy type of Tb1(X)×CP∞. α I am grateful to Professor Clifford Taubes for sharing his knowledgment about the spectral properties of the operator △ + kg. A 4 2. Background In order to define the SW functional, we fix a Spinc structure on X and also α describe the Sobolev Space structure defined on the configuration space. For each α ∈ Spinc(X), there is a representation ρ : SO → Cl inducing a pair of vector α 4 4 bundles (S +,L ) over X, where α α (i): S =P × V =S +⊕S −. ThebundleS ± arethepositiveandthe α SO4 ρα α α α negativecomplexspinorsbundle(fibersareSpinc−modulesisomorphicto 4 C2). (ii): L =P × C. It is calledthe determinant line bundle associated α SO4 det(α) to α. (c (L )=α) 1 α Thus, given α∈Spinc(X) we associate a pair of bundles α∈Spinc(X) (L ,S +) α α From now on, we considered on X a Riemannian metric g and on S an hermitian α structure h. Let P be the U -principal bundle over X obtained as the frame bundle of L α 1 α (c (P )=α). Also, we consider the associated bundles 1 α Ad(U )=P × U ad(u )=P × u , 1 U1 Ad 1 1 U1 ad 1 where Ad(U ) is a fiber bundle with fiber U , and ad(u ) is a vector bundle with 1 1 1 fiber isomorphic to the Lie Algebra u . 1 Let A be (formally) the space of connections (covariant derivative) on L , α α Γ(S+) the space of sections of S+ and G =Γ(Ad(U )) the gauge group acting on α α α 1 A ×Γ(S+) as follows: α α (1) g.(A,φ)=(A+g−1dg,g−1φ). So, the action is free whenever φ 6= 0, otherwise the pair (A,0) has isotropy subgroup isomorphic to U . A is an affine space which vector space structure, after 1 α fixing an origin, is isomorphic to the space Ω1(ad(u )) of ad(u )-valued 1-forms. 1 1 Once a connection ▽0 ∈A is fixed, a bijection A ↔Ω1(ad(u )) is explicited by α α 1 ▽A ↔A,where▽A =▽0+A. G =Map(X,U ),sinceAd(U )≃X×U . Thecur- α 1 1 1 vatureofa1-connectionformA∈Ω1(ad(u ))isthe2-formF =dA∈Ω2(ad(u )). 1 A 1 The configuration space is C . A topology is fixed on C by considering the α α following Sobolev structures: • A =L1,2(Ω1(ad(u )))); α 1 • Γ(S +) = L1,2(Ω0(X,S +)); α α • C =A ×Γ(S +); α α α • G =L2,2(X,U )=L2,2(Map(X,U )). α 1 1 (G is an ∞-dimensional Lie Group which Lie algebra is g=L1,2(X,u )). α 1 THE MORSE INDEX OF REDUTIBLE SOLUTIONS OF THE SW-EQUATIONS 3 2.1. Seiberg-Witten Monopole Equation. In dimension 4, the vector bundle Ω2(ad(u )) splits as 1 Ω2(ad(u ))⊕Ω2(ad(u )), + 1 − 1 where (+) is the seld-dual component and (-) the anti-self-dual. Fixed α∈Spinc(X), the 1st-orderSW -monopoleequations, defined overC = α α A ×Γ(S +), are α α D+(φ)=0, (2) A (FA+ =σ(φ), where (i): D+ is the Spincc-Dirac operator defined on Γ(S+), A α (ii): The quadratic form σ :Γ(S+)→End0(S+) is defined as α α |φ|2 σ(φ)=φ⊗φ∗− .I 2 performs the coupling of the ASD-equation with the Diracc operator. A solution of equations 2 is named as SW -monopole. It can be described as the α space F−1(0), where F :C →Ω2(X)⊕Γ(S −) is the map α α + α F (A,φ)=(F+−σ(φ),D+(φ)). α A A The SW -equations are G -invariant and the map F is a Fredholm map up to α α gauge equivalence. Definition 2.1.1. (A,φ) is named a SW -monopole if it satisfies the equation 2 α and φ6=0. 2.2. SW Lagrangean. α Definition 2.2.1. For each α ∈ Spinc(X), the Seiberg-Witten Functional is the functional SW :C →R given by α α 1 1 k (3) SW(A,φ)= { |F |2 +|▽Aφ|2 + |φ|4 + g |φ|2}dv +π2α2, A g 4 8 4 ZX where k = scalar curvature of (X,g) and α2 = Q (α,α) (Q is the intersection g X X form of X) Remark 2.2.2. . (i): The functional is well defined on C , since in dimension n=4 there is the α Sobolev embeddings L4(Ω1(ad(u )))⊂L1,2(Ω1(ad(u ))) and 1 1 L4(Ω0(X,S +)⊂L1,2(Ω0(X,S +)). α α (ii): the functional SW :C →R is gauge invariant. α α (iii): Whenever k ≥ 0, SW (A,φ) ≥ | F |2 dx+π2α2. Therefore, the g α X A stable solutions are (A,0). R (iv): (A,0) is a SW -monopole if and only if A is anti-self-dual connection. α Whenever b+ ≥ 3, it is known [?] that such solutions do not exist for a 2 dense set of metrics on X. 4 CELSOM.DORIA UFSC-DEPTO.DEMATEMA´TICA The tangent bundle TC is trivial because C is contractible. The fiber over α α (A,φ) is T C =Ω1(ad(u ))⊕Γ(S+). (A,φ) α 1 α A riemannian structure on TC is defined by the inner product α <,>:T C ×T C →R (A,φ) α (A,φ) α ((Θ+V),(Λ+W))→<Θ+V,Λ+W >=<Θ,Λ>+. where <Θ,Λ>= <Θ(x),Λ(x)>dx, ZX = {dx. ZX The derivative of SW-functional defines a 1-form dSW ∈ Ω1(C ). For each point α (A,φ), the functional dSW :Ω1(ad(u ))⊕Γ(S+)→R is decomposed in com- (A,φ) 1 α ponents as dSW .(Θ+Λ)=d SW .Θ+d SW .Λ, (A,φ) 1 (A,φ) 2 (A,φ) where (i): d 1 SW(A+tΘ,φ)−SW(A,φ) d SW .Θ= lim = 1 (A,φ) t→0 t 1 = Re{+4<▽A(φ),Θ(φ)>}dx= A 2 ZX 1 = Re dx= A 2 (cid:18)ZX (cid:19) 1 = Re . A 2 (cid:0) (cid:1) where Φ:Ω1(u )→Ω1(S +) is the linear operator Φ(Θ)=Θ(φ), and 1 α 1 Φ∗(▽A(φ))= d(|φ|2). 2 (ii): d 2 SW(A,φ+tV)−SW(A,φ) d SW .V = lim = 2 (A,φ) t→0 t |φ|2 +k =2. Re{<▽Aφ,▽AV >+< gφ,V >}dx= 4 ZX |φ|2 +k =2. Re{<△ φ+ gφ,V >}dx= A 4 ZX |φ|2 +k =2.Re <△ φ+ gφ,V > . A 4 (cid:18) (cid:19) THE MORSE INDEX OF REDUTIBLE SOLUTIONS OF THE SW-EQUATIONS 5 where △ =(▽A)∗▽A. A Thus, the Euler-Lagrangeequations of SW -functional are α (4) d∗F +4Φ∗(▽Aφ)=0, A |φ|2 k (5) ∆ φ+ φ+ gφ=0,. A 4 4 2.3. Hodge Solutions. It follows from the identity SW (A,φ)= {|D φ|2 +|F+−σ(φ)|2}dx, α A A ZX (seein[4])thattheSW monopolesarethestablesolutions. From[11],itisknown α that such SW-monopoles exist only for a finite number of classes α ∈ Spinc(X). Thus,itarisethequestionaboutasufficientconditiontoguaranteetheirexistence. The easiest solutions of the Euler-Lagrange equations above are the ones of type (A,0), where the curvature F is a harmonic 2-form. Hodge theory guaranteesthe A existence of such solution. The gauge isotropy subgroup of (A,0) is isomorphic to U , justifying their nickname redutible. Furthermore, 1 Proposition 2.3.1. The space of solutions of d∗F =0, module the G -action, is A α diffeomorphic to the Jacobian Torus H1(X,R) Tb1(X) = H1(X,Z), b1(X)=dimRH1(X,Z). Proof. The equation d∗F = 0 implies that F is an harmonic 2-form and, by A A Hodgetheory,itisthe onlyone. LetAandB besolutionsandconsiderB =A+b; so, d∗F +d∗F +d∗db=0 ⇒db=0. B A By fixing the origin at A, we associate B b ∈ H1(X,R); note that F = F . B A Let B be a solution gauge equivalent to B , so there exists g ∈G such that B = 1 2 α A+g−1dg and F =F . However,the 1-formg−1dg ∈H1(X,Z). Consequently, B1 B2 if B =A+b and B =A+b , then [b ]=[b ] in H1(X,R). (cid:3) 1 1 2 2 2 1 H1(X,Z) 3. 2nd - Variation Formula The 2nd - variation formula is the symmetric bilinear form HSW :T C ×T C →R. (A,φ) (A,φ) α (A,φ) α obtained by computing the hessian of SW. A neat exposition may achieved by consideringtheoperatorHSW asalinearfunctionalHSW :T C ⊗T C → (A,φ) (A,φ) (A,φ) α (A,φ) α R. By taking U =Ω1(ad(u )) and V =Γ(S+), we have the functional 1 α HSW :(U ⊗U) (U ⊗V) (V ⊗U) (V ⊗V)→R (A,φ) M M M decomposing into 6 CELSOM.DORIA UFSC-DEPTO.DEMATEMA´TICA HSW ((Θ,V),(Λ,W))=HSW (Θ⊗Λ+Θ⊗W +V ⊗Λ+V ⊗W)= (A,φ) (A,φ) =d (Θ⊗Λ)+d (Θ⊗W)+d (V ⊗Λ)+d (V ⊗W). 11 12 21 22 Thus, HSW =d +d +d +d , where (A,φ) 11 12 21 22 (1) d :Ω1(ad(u ))⊗Ω1(ad(u ))→R 11 1 1 1 d SW (Θ,Λ)= lim {d SW .Θ−d SW .Θ}= 11 (A,φ) t→0 t 1 (A+tΛ,φ) 1 (A,φ) 1 = Re{+4<Θ(φ),Λ(φ)>}dx= 2 ZX =Re{+4<Θ(φ),Λ(φ)>}= 1 = Re{<Θ,(d∗d+4Φ∗Φ)(Λ)>}. 2 (2) d :Ω1(ad(u ))⊗Γ(S+)→R 12 1 α 1 d SW (Θ,W)= lim {d SW .Θ−d SW .Θ}= 12 (A,φ) t→0 t 1 (A,φ+tW) 1 (A,φ) =2 Re{<▽Aφ,Θ(W)>+<▽AW,Θ(φ)>}dx ZX =2.Re{<▽Aφ,Θ(W)>+<▽AW,Φ(Θ)>}= =2.Re{<▽Aφ,Θ(W)>+<Θ,Φ∗▽AW >} Introducing the operator W :Γ(S+)→Ω1(ad(u )), as W(Θ)=Θ(W), α 1 d SW (Θ,W)=2.Re{<Θ,P(W)>} 12 (A,φ) where P :Γ(S+)→Ω1(ad(u )) is P(W)=W∗(▽Aφ)+Φ∗(▽AW). α 1 (3) d :Γ(S+)⊗Ω1(ad(u ))→R 21 α 1 1 d SW (V,Λ)= lim {d SW .V −d SW .V}= 21 (A,φ) t→0 t 2 (A+tΛ,φ) 2 (A,φ) =2. Re{<▽Aφ,Λ(V)>+<Λ(φ),▽AV >}dx ZX =2.Re{<▽Aφ,Λ(V)>+<Φ(Λ),▽AV >} =2.Re{<▽Aφ,Λ(V)>+} Analogously, by considering the operator Q:Ω1(ad(u ))→Γ(S+), 1 α Q(Λ)=Λ∗▽Aφ+(▽A)∗Φ(Λ), d SW (V,Λ)=2.Re{}. 21 (A,φ) It is straight forward that Q=P∗. THE MORSE INDEX OF REDUTIBLE SOLUTIONS OF THE SW-EQUATIONS 7 (4) d :Γ(S+)⊗Γ(S+)→R 22 α α 1 d SW (V,W)= lim {d SW .V −d SW .V}= 22 (A,φ) t→0 t 2 (A,φ+tW) 2 (A,φ) |φ|2 =2 Re{<▽AW,▽AV >+ + 4 ZX 1 k + Re(<φ,W >)<φ,V >+ g }dx= 2 4 k +|φ|2 1 =2.Re{φ>} 4 2 From the computation above, there exists a linear operator H :T C → (A,φ) (A,φ) α T C , H=H , given by (A,φ) α (A,φ) HSW ((Θ,V),(Λ,W))=<(Θ,V),H(Λ,W)>, (A,φ) which matrix representation is d∗d+4Φ∗Φ P (6) H= Q (▽A)∗▽A+ 1 <φ,.>φ+ kg+|φ|2I!. 2 4 Therefore, the induced quadratic form HSW :Ω1(ad(u ))⊕Γ(S+)→R is (A,φ) 1 α (7) b HSW (Θ,V)=|dΘ|2 + |Φ(Θ)|2 + 2.<▽Aφ,Θ(V)>+ 2<Φ(Θ),▽AV >+ (A,φ) k +|φ|2 1 (8) +|▽AV |2 + g .|V |2 + <φ,V >2 . 4 2 Hence, HSW :T C ⊗T C →R is a bounded operator. (A,φ) (A,φ) α (A,φ) α 3.1. Morse Index of Redutible Solutions. From Hodge theory, we have the complex (9) Ω0(ad(u ))−d→Ω1(ad(u ))−d→Ω2(ad(u ))−d→...Ω4(ad(u )). 1 1 1 1 Since Ω1(ad(u )) = d(Ω0)⊕ker(d∗(Ω1)), and d(Ω0) is the tangent space to the 1 orbits,the tangentspace ofA × Γ(S+)at(A,φ) canbe decomposedintoW ⊕ α Gα α 1 W , where 2 W ={Θ∈Ω1(ad(u )); d∗Θ=0}, 1 1 W ={W ∈Γ(S+); =0, ∀A∈Ω0(ad(u ))} 2 α 1 Furthermore,W =d∗(Ω2(ad(u )))⊕H ,whereH isspaceofharmonicformsand 1 1 1 1 is also the tangent space to the Jacobian torus Tb1(X) at (A,0). The quadratic form in 7, when evaluated at a redutible solution (A,0), is given by 8 CELSOM.DORIA UFSC-DEPTO.DEMATEMA´TICA k (10) H(Θ,V)=||dΘ||2 +||▽AV ||2 + g |V |2 dx. L2 L2 4 ZX Indeed, at (A,0), it follows from expression 7 d∗d 0 (11) H= , HSW (Θ,V)=<(Θ,V),H(Θ,V)>. 0 △ + kg (A,φ) (cid:18) A 4 (cid:19) TheMorseindexof(A,0)isequaltothedimensionofthelargestnegativeeigenspace of the operator L = △ + kg : Γ(S ) → Γ(S ). Let V ⊂ T A × Γ(S+) A A 4 α α λ (A,0) α Gα α be the eigenspace associated to the eigenvalue λ. Since L is an elliptic operator, A V has finite dimension; 0 Proposition 3.1.1. Let (A,0) be a redutible solution. So, (1) If k > 0, then V = T Tb1(X) and so SW : A × Γ(S+) → R is a g 0 (A,0) α α Gα α Morse-Bott function at (A,0). (2) If k =0, then V =T Tb1(X)⊕{V ∈Γ(S+)|▽AV =0}. g 0 (A,0) α Proof. It is straightforwardfrom equation 10. (cid:3) Proposition 3.1.2. The Morse index of a redutible solution (A,0) is finite. Proof. It follows from the spectral theory applied to the elliptic, self-adjoint operator L ( [1], [12]) that the spectrum of L A A (i): is discrete, (ii): each eigenvalue has finite multiplicity, (iii): there are no points of accumulation, (iv): there are but a finite number of eigenvalues below any given number. Therefore, the spectrum of L is bounded from below. A (cid:3) References [1] CHENG,SHIUandLI,PETER-Heat Kernelestimatesand LowerBounds of Eigenvalues, CommentariiMath.Helvetia,56,1981, 3,327-338, MR0639355(83b:58076) [2] DONALDSON, S.K. - The Seiberg-Witten Equations and 4-Manifold Topology, Bull.Am.Math.Soc.,NewSer.33,no1(1996), 45-70. [3] DORIA,CELSO M-The Homotopy Type of the Seiberg-WittenConfiguration Space, Bull. Soc.ParanaensedeMat.22,no 2,49-62,2004. (http://www.spm.uem.br/spmatematica/index.htm) [4] DORIA, CELSO M - Variational Principle for the Seiberg-Witten Equations, Progress in Nonlinear Diffrential Equations and Their Applications, 66, pp 247-261, 2005, Birkhaüser Verlag. [5] GILBARG, D. and TRUDINGER, N.S. - Elliptic Partial Differential Equations of Second Order,2nd-edition,SCSM224,Springer-Verlag,1983. [6] JOST,J.,PENG,X.andWANG,G.-VariationalAspectsoftheSeiberg-WittenFunctional, CalculusofVariation4(1996),205-218. [7] KATO, T.- Perturbation Theory for Linear Operators, 2nd-edition, SCSM 132, Springer- Verlag,1984 [8] MORGAN, J. - The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds,Math.Notes44,PrincetonPress. [9] TAUBES, C. - The Seiberg-Witten invariants and Sympletic Forms, Math.Res.Lett. 1, no6 (1994), 809-822. THE MORSE INDEX OF REDUTIBLE SOLUTIONS OF THE SW-EQUATIONS 9 [10] TAUBES,C.-Asymptotic Spectral Flow for Dirac Operators,arXiv.org,math.DG/0612126 -http://xxx.if.usp.br/pdf/math.DG/0612126. [11] WITTEN,E.-Monopoles on Four Manifolds,Math.Res.Lett.1,no6(1994), 769-796. [12] VAFA,CUMRUNandWITTEN,EDWARD-Eigenvalue inequalitiesforfermionsingauge theories,Comm.Math.Phys.95(1984), no.3,257–276. 81E20(58G25 81E13), MR0765268 (86c:81075). Universidade Federal de Santa Catarina Campus Universitario , Trindade Florianopolis - SC , Brasil CEP: 88.040-900

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