Adiabatic Paths and Pseudoholomorphic Curves

Transcription

1 ESI The Erwin Schrödinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria Adiabatic Paths and Pseudoholomorphic Curves A.G. Sergeev Vienna, Preprint ESI 1676 (2005) August 15, 2005 Supported by the Austrian Federal Ministry of Education, Science and Culture Available via

2 ADIABATIC PATHS AND PSEUDOHOLOMORPHIC CURVES A.G.Sergeev Abstract. We consider the Taubes correspondence between solutions of the Seiberg Witten equations on a compact 4-dimensional symplectic manifold and pseudoholomorphic curves on this manifold. We compare it with its 3-dimensional analogue in which the roles of Seiberg Witten equations and pseudoholomorphic curves are played respectively by the Ginzburg Landau equations and adiabatic paths in the moduli space of static Ginzburg Landau solutions. C.H.Taubes in his papers [7,8] has established a correspondence between solutions of the Seiberg Witten equations on a compact 4-dimensional symplectic manifold and pseudoholomorphic curves. The Taubes correspondence involves a certain limiting procedure, called the scaling limit. It turns out that there exists a non-trivial 3-dimensional analogue of this procedure. In the 3-dimensional (or, better to say, in the (2+1)-dimensional) setting the role of Seiberg Witten equations is played by the hyperbolic Ginzburg Landau equations and the scaling limit is replaced by the adiabatic limit, involving the introduction of the slow time. The adiabatic limit construction establishes a correspondence between solutions of Ginzburg Landau equations and certain adiabatic paths in the moduli space of static solutions. The Taubes correspondence can be considered from this point of view as a complex (or, better to say, (2+2)-dimensional) analogue of the adiabatic limit construction in which pseudoholomorphic curves should be considered as complex adiabatic paths. In this paper I present an overview of the results, related to the Taubes correspondence and its 3-dimensional analogue. The work on this subject started when I first visited the Erwin Schrödinger Institute in 1999 and I would like to thank ESI for the hospitality. I. ABELIAN (2+1)-DIMENSIONAL HIGGS MODEL 1.1. Ginzburg Landau Equations We consider the (2+1)-dimensional Abelian Higgs model, governed by the Ginzburg Landau action functional S(A, Φ) = {T(A, Φ) U(A, Φ)}dt (1.1) 1991 Mathematics Subject Classification. Primary 59E15. Key words and phrases. adiabatic paths, Taubes correspondence, Seiberg Witten equations, Ginzburg Landau equations. 1 Typeset by AMS-TEX

3 2 A.G.SERGEEV on the space R 3 R 1+2, provided with coordinates (x 0, x 1, x 2 ) (t, x 1, x 2 ). In this formula the potential energy U(A, Φ) is defined by the formula U(A, Φ) = 1 {2 F d A,1 Φ 2 + d A,2 Φ } 2 (1 Φ 2 ) 2 dx 1 dx 2, (1.2) and the kinetic energy T(A, Φ) is given by T(A, Φ) = 1 2 {2 F F d A 0Φ 2} dx 1 dx 2. (1.3) The ingredients in these formulae have the following meaning. We denote by A a U(1)-connection on R 1+2, given by a 1-form A = A 0 dt + A 1 dx 1 + A 2 dx 2 with smooth pure imaginary coefficients A µ = A µ (t, x 1, x 2 ), µ = 0, 1, 2. We shall denote by A (resp. A 0 ) the space (resp. the time) component of A so that A = A 1 dx 1 + A 2 dx 2, A 0 = A 0 dt. The curvature F A = da of the connection A is represented by a smooth 2-form with F A = 2 µ,ν=0 F µν dx µ dx ν F µν = µ A ν ν A µ, µ := / x µ, µ, ν = 0, 1, 2. Accordingly, F A denotes the curvature of the space component A of A, given by F A = 2 F ij dx i dx j = 2F 12 dx 1 dx 2. i,j=1 The exterior covariant derivative d A = d + A is given by the formula d A Φ = 2 ( µ + A µ )Φdx µ µ=0 with its space and time components, having the form d A Φ = d A,1 Φ+d A,2 Φ = ( 1 +A 1 )Φdx 1 +( 2 +A 2 )Φdx 2, d A 0Φ = ( 0 +A 0 )Φdt. The Higgs field Φ = Φ(t, x 1, x 2 ) is a smooth complex-valued function on R 1+2. The Ginzburg Landau equations are the Euler Lagrange equations for the Ginzburg Landau action S(A, Φ), given by the formula (1.1). They have the following form 2 0 F 0j + ǫ jk k F 12 = iim( Φ A,j Φ), j = 1, 2 k=1 ( 2 A,0 2 A,1 2 A,2)Φ = 1 (1.4) 2 Φ(1 Φ 2 ) 1 F F 02 = iim( Φ A,0 Φ)

4 ADIABATIC PATHS AND PSEUDOHOLOMORPHIC CURVES 3 where A,µ = µ + A µ, µ = 0, 1, 2 ; ǫ 12 = ǫ 21 = 1, ǫ 11 = ǫ 22 = 0. The Ginzburg Landau equations (1.4), as well as the action (1.1), are invariant under gauge transformations, given by A µ A µ + i µ χ, Φ e iχ Φ (1.5) where χ is a smooth real-valued function on R 1+2. By a suitable choice of the gauge function χ, we can always satisfy the condition A 0 = 0. (1.6) Such a choice of χ is called the temporal gauge. Note that after fixing the condition (1.6) we still have the gauge freedom up to gauge transformations of the form (1.5) with a function χ, depending only on x 1,x 2. We call the latter gauge transformations (with χ, depending on x 1,x 2 but not on t) the static gauge transformations to distingush them from general dynamic gauge transfromations. In the temporal gauge we can rewrite the formula (1.3) for the kinetic energy in a more convenient form T(A, Φ) = 1 ) ( Ȧ 2 L 2 + Φ 2 2 L (1.7) 2 where we denote by dot the time derivative 0 = / t, and by = L 2 the L 2 -norm on R 2. Solutions of the Ginzburg Landau equations (1.4) will be called briefly the dynamic solutions. We are interested in the description of the moduli space of dynamic solutions, i.e. the space of dynamic solutions modulo gauge transformations. We start from the case of static solutions, given by (A, Φ) which do not depend on time. In this case there is an explicit description of the corresponding moduli space Moduli Space of Static Solutions It is proved in [1] that any static solution of the Ginzburg Landau equations (1.4) with a finite potential energy U(A, Φ) < is a minimum of U(A, Φ). More precisely, the finite energy condition implies that Φ 1 for x = x x2 2. So we can define a topological invariant of the problem, called the vortex number d, by the following formula d = the winding number of Φ at infinity. (1.8) More precisely, d is the winding number of the map, given by Φ, of a circle S 1 R of a sufficiently large radius R onto the topological circle S 1 = { Φ 1}.

5 4 A.G.SERGEEV From now on, we shall fix d > 0 and call by d-vortices the minima of U(A, Φ) < in a given topological class, determined by the vortex number d. To describe d-vortices, we introduce the complex coordinate z := x 1 + ix 2 on R 2 (x, thus identifying it with C 1,x 2) z. Then, according to [1], the d-vortices are solutions with finite energy of the following vortex equations A Φ = 0 if A = 1 2 (1 (1.9) Φ 2 ) where is the Hodge star-operator with respect to the Euclidean metric on R 2 (x 1,x 2) and A := z +A 0,1. The second vortex equation may be rewritten also as: if 12 = 1 2 (1 Φ 2 ). The equations (1.9) for the minima of U(A, Φ) follow from the Bogomolnyi identity: U(A, Φ) = 1 {2 A Φ 2 + if } 2 (1 Φ 2 ) 2 dvol + i F A (1.10) 2 where dvol = i 2dz d z. If the vortex number of (A, Φ) is equal to d then i F A = πd 2 by the Gauss Bonnet formula and (1.10) implies that U(A, Φ) πd. The equality here is achieved only on solutions of equations (1.9), i.e. for the d-vortices (A, Φ). (Note that for d < 0 there exists an analogous Bogomolnyi identity, which implies that the d-vortices with d < 0 are solutions of the antivortex equations. The latter equations are obtained from (1.9) by replacing A with A in the first equation of (1.9) and changing the sign in the right hand side of the second equation). The moduli space of d-vortices, defined by M d = has the following description, due to [1]. {d-vortices (A, Φ)} {gauge transforms}, (1.11) Theorem (Taubes). For any effective divisor k k=1 d jz j on C with integers d j > 0 and degree d := d j there exists a unique (up to gauge) d-vortex (A, Φ) such that the zero-divisor of Φ is equal to d j z j. This theorem implies immediately that M d = Sym d C = C d. (1.12) According to Taubes theorem, we have a 1-1 correspondence between d-vortices and effective divisors of degree d on the complex plane. This correspondence can be considered as a (2+0)-dimensional analogue of the (4 = 2 + 2)-dimensional Taubes correspondence, mentioned in the introduction.

6 ADIABATIC PATHS AND PSEUDOHOLOMORPHIC CURVES The Tangent Structure of the d-vortex Moduli Spaces In order to study the tangent structure of the moduli space M d, we introduce a Sobolev version of M d. Denote by V s = Vd s for s 1 the Sobolev space of d-vortices (A, Φ) on C, i.e. the space of solutions (A, Φ) of the vortex equations (1.9) with vortex number d where A is a 1-form on C with coefficients in the Sobolev space H s (C, ir), i.e. A Ω 1 (C) H s (C, ir) = Ω 1 s(c, ir) and 1 Φ H s (C, C), denoting this briefly by: (A, Φ) Ω 1 s H s. The Sobolev version of the moduli space M d is defined by M s d = Vs d {Sobolev d-vortices (A, Φ) Ω 1 s H s } = G s+1 {Sobolev gauge transforms, given by χ H s+1 (C, R)} where G s+1 is the group of Sobolev gauge transformations A A + idχ, Φ e iχ Φ, given by functions χ H s+1 (C, R). It can be proved that for s 1 the moduli space M s d coincides with Symd C and so does not depend on s. By varying the vortex equations (1.9) with respect to (A, Φ) Ω 1 s H s at some fixed solution (A, Φ), we obtain the following linearized vortex equations: { A ϕ + a 0,1 Φ = 0 ida + Re(ϕ Φ) = 0 (1.13) where (a, ϕ) Ω 1 s H s. The left hand side of these equations defines a linearized vortex operator sending D (A,Φ) : Ω 1 s H s Ω 0,1 s 1 (C, C) H s 1(C, R), (1.14) (a, ϕ) ( A ϕ + a 0,1 Φ, ida + Re(ϕ Φ)). The tangent space of Vd s at a given d-vortex (A, Φ) coincides with the kernel of the operator D (A,Φ) : T (A,Φ) V s d = Ker D (A,Φ) = {(a, ϕ) Ω 1 s H s : D (A,Φ) (a, ϕ) = 0}. (1.15) The tangent space T (A,Φ) Vd s is invariant under infinitesimal gauge transformations, given by a a + idχ, ϕ ϕ iφχ (1.16) where χ H s+1 (C, R). We introduce a tangential gauge operator δ (A,Φ) : χ (idχ, iφχ), H s+1 (C, R) Ω 1 s H s

7 6 A.G.SERGEEV and its adjoint operator δ (A,Φ) : Ω1 s H s H s (C, R), δ (A,Φ) : (a, ϕ) id a + iim( Φϕ). Since the whole Sobolev space Ω 1 s H s can be decomposed into the orthogonal direct sum Ω 1 s H s = T (A,Φ) (G s+1 (A, Φ)) Ker δ (A,Φ), we can use δ(a,φ) to fix the infinitesimal gauge by the gauge fixing condition δ(a,φ) (a, ϕ) = 0. (1.17) In these terms the tangent space of M s d at (A, Φ) can be described as: T (A,Φ) M s d = {(a, ϕ) Ω 1 s H s : D (A,Φ) (a, ϕ) = δ(a,φ) (a, ϕ) = 0} Adiabatic Paths We consider again our original problem of studying the moduli space of dynamic solutions. The Taubes theorem, giving a description of the static moduli space M d, is based on the fact that the static Ginzburg Landau equations can be reduced to the 1st-order vortex equations, due to the Bogomolnyi formula. In the general dynamic situation such a reduction does not exist and we cannot expect to have an explicit description of the moduli space of dynamic solutions, similar to the static case. However, one can try to find approximate methods of solving the dynamic equations. We consider here one of such methods, called the adiabatic limit construction, which allows to describe slowly moving dynamic solutions in terms of the moduli space of static solutions. In the framework of the gauge field theory this method was proposed heuristically by [2] and on a more rigorous basis by [6]. Consider the dynamic Euler Lagrange equations (1.4) and introduce the temporal gauge (1.6). Then solutions of dynamic equations (1.4) in this gauge may be treated as paths in the configuration space of d-vortices, consisting of all pairs (A, Φ) with a finite potential energy and the vortex number d. More precisely, a dynamic solution is represented by a path γ : t [A(t), Φ(t)], where [A, Φ] denotes the gauge class of (A, Φ) (with respect to static gauge transformations). The map γ takes values in the configuration space of d-vortices: N d = {(A, Φ) with U(A, Φ) < and vortex number d} {(static) gauge transformations}. Note that N d contains, in particular, the moduli space of d-vortices M d. Let γ ǫ be a family of dynamic solutions, depending on a parameter ǫ > 0, represented by paths γ ǫ : t [A ǫ (t), Φ ǫ (t)] with the kinetic energy T(γ ǫ ) := T(γ ǫ (t))dt ǫ.

8 ADIABATIC PATHS AND PSEUDOHOLOMORPHIC CURVES 7 Then for ǫ 0 the path γ ǫ approaches M d and in the limit collapses to a point of M d, i.e. to a static solution. But, if we introduce a slow-time variable τ on γ ǫ by setting: τ = ǫt, then in the limit ǫ 0 we get some path γ, lying in M d. We call this slow-time limit adiabatic and the limiting path γ the adiabatic path so that γ = the adiabatic limit of γ ǫ for ǫ 0. (1.18) Note that every point of γ is a static solution but γ itself cannot be a dynamic solution. However, in any neighborhood of γ there exists a dynamic solution with a small kinetic energy so adiabatic paths γ describe approximately slowly moving dynamic solutions. Hence, if we would be able to find an intrinsic description of adiabatic paths in terms of M d, it will give us a method of approximate solution of our dynamic equations, more precisely, a method of construction of slowly moving dynamic solutions. Suppose we have a path γ in M d. We consider it as a path γ 0 : t (A(t), Φ(t)) in the Sobolev space V d = Vd s of d-vortex solutions, satisfying the gauge fixing condition δ(a,φ) (Ȧ, Φ) = 0 (1.19) for any t. We want to find conditions on γ which guarantee that it is adiabatic, i.e. γ coincides with the adiabatic limit of some dynamic solutions γ ǫ for ǫ 0. We consider these γ ǫ as small perturbations of γ, represented by paths with γ ǫ : t (A ǫ (t), Φ ǫ (t)) (A ǫ (t), Φ ǫ (t)) = (A(t), Φ(t)) + ǫ 2 (a(t), ϕ(t)), in the Sobolev configuration space N d = Nd s, satisfying the gauge fixing condition (1.19). We shall also impose on γ ǫ the following orthogonality condition by requiring that (a(t), ϕ(t)) T (A(t),Φ(t)) M d for any t. (1.20) It means, in other words, that we exclude perturbations, tangent to M d. Under the gauge fixing condition (1.19) for γ ǫ, we can rewrite the orthogonality condition (a, ϕ) T (A,Φ) M d in the form If Ker D (A,Φ) has an H 2 -base {n µ }: (a, ϕ) Ker D (A,Φ). D (A,Φ) n µ = 0 for any t and µ = 1,..., 2d, then the orthogonality condition (1.20) may be rewritten in the form (a, ϕ), n µ = 0 for any t and µ = 1,..., 2d. (1.21)

9 8 A.G.SERGEEV Introduce now the slow-time parameter τ = ǫt and plug (A ǫ, Φ ǫ ) into the dynamic equations (1.4). Then we shall get (cf. [5]) the following adiabatic condition for γ 2 τ(a, Φ) Ker D (A,Φ) (1.22) under the gauge fixing condition (1.19). In terms of the base {n µ } the latter condition (1.22) may be written in the form of the adiabatic equation: 2 τ (A, Φ), n µ = 0 for µ = 1,..., 2d. (1.23) The adiabatic equation (1.23) has a transparent geometric interpretation, namely it coincides with the Euler geodesic equation on M d, provided with the Riemannian T-metric, generated by the kinetic energy functional, more precisely, the following theorem is true (cf. [5,6]). Theorem. Adiabatic paths γ on M d are the extremals of the kinetic energy functional T(γ) = T(γ(τ))dτ, restricted to paths γ(τ) = [A(τ), Φ(τ)], lying in M d. II. SEIBERG WITTEN EQUATIONS 2.1. Seiberg Witten Equations on Symplectic 4-Manifolds Let (X, ω) be a compact symplectic 4-manifold, provided with an almost complex structure J, compatible with ω. (Recall that an almost complex structure on X is compatible with ω if ω(jξ, Jη) = ω(ξ, η) for any ξ, η TX and g(ξ, η) := ω(ξ, Jη) determines a Riemannian metric on X, i.e. g is positive definite). Let E X be a complex Hermitian line bundle over X, provided with a Hermitian connection B. The Seiberg Witten equations on (X, ω, J), associated with these data, have the following form (cf. [3,4]): B α + Bβ = 0 λ 2 F 0,2 B = ᾱβ 4i λ F B ω = 1 + β 2 α 2 (2.1) Here, B is the (0, 1)-component of the exterior covariant derivative d B : Ω 0 (X, E) Ω 1 (X, E), generated by the connection B, and B is the L2 -adjoint of B. We denote by F 0,2 B (resp. F B ω ) the (0, 2)-component (resp. the component, parallel to ω) of the curvature F B, and by λ > 0 a scaling parameter. A solution of these equations, called briefly the SW λ -equations, is given by the connection B and the spinor field Φ = (α, β) where α is a section of E X, α Ω 0 (X, E), and β is a (0, 2)-form with values in E X, β Ω 0,2 (X, E). Note that the 1st equation (2.1) coincides with the covariant Dirac equation D B Φ = 0

10 ADIABATIC PATHS AND PSEUDOHOLOMORPHIC CURVES 9 for the section Φ of the semi-spinor bundle W + E := (Λ0 E) (Λ 0,2 E) where the covariant Dirac operator D B : Ω 0 (X, W + E ) Ω0 (X, W E ) sends sections of the positive semi-spinor bundle W + E into sections of the negative semi-spinor bundle W E := Λ0,1 E. The 2nd and 3rd equations (2.1) are respectively the (0, 2)- component and ω-component of the equation F + B = Φ Φ 1 2 Φ 2 id where F + B is the self-dual component of F B, acting on sections of W + E by the Clifford multiplication. We shall assume further on that c 1 (E) [ω] > 0 (this condition is similar to the inequality d > 0 in Sec.1.2) and the equations (2.1) have a solution (B λ, Φ λ ) = (B λ, (α λ, β λ )) for all λ λ 0 (this condition is satisfied if the Seiberg Witten invariant of X w.r. to the Spin c -structure, associated with E X, does not vanish, cf. [3,4]) The Scale Limit We shall study the scale limit of these equations, i.e. the behavior of solutions (B λ, Φ λ ) for λ +. This limit is analogous to the adiabatic limit of Sec.1, the role of the small parameter ǫ from Sec.1.4 being played now by ǫ := 1/ λ. Taubes in [7] has proved that SW λ -solutions (B λ, (α λ, β λ )) have the following behavior for λ + : α λ 1 almost everywhere on X ; Bλ α λ 2 0 ; β λ 0 everywhere on X where we denote by the L 2 -norm on X. Note that in the first assertion α λ cannot converge to 1 everywhere on X since α λ necessarily has zeros on X (cf. below). The second assertion means that α λ tends to become a Bλ -holomorphic section of E in the limit λ +. Denote by C λ := α 1 λ (0) the zero set of the section α λ. Taubes proved in [7] that for a subsequence λ n there exists a weak limit of C λn which coincides with a pseudoholomorphic divisor C of degree c 1 (E), i.e. k C = d j C j j=1 is a sum of compact connected (maybe, non smooth) non-intersecting pseudoholomorphic curves C j with multiplicities, given by positive integers d j, such that the homology class k j=1 d j[c j ] is the Poincarè-dual of c 1 (E). The weak limit of C λn here is understood in the sense of currents. The scaling limit λ n in the SW λ -equations may be considered as a complex analogue of the adiabatic limit from Sec.1.4. Namely, the SW λn -solutions (B λn, (α λn, β λn )) converge for n to a family (B, (α, 0)) of vortices, defined

11 10 A.G.SERGEEV in the normal planes N x of C, x C, with the zero divisor of α equal to C. So the limit of (B λn, Φ λn ) may be considered as a complex path, parametrized by the pseudoholomorphic divisor C, in the moduli space of vortices in the complex plane. We are going to derive the adiabatic equation for the limiting vortex families of SW λn -solutions along the same lines, as in Sec.1.4. Namely, we start with an arbitrary pseudoholomorphic path γ in the moduli space of vortices, parametrized by C, and obtain, by studying perturbations of γ in the configuration space, a condition which is necessary for γ to be the limit of SW λn -solutions. In order to simplify our argument, we restrict to the case when the limiting pseudoholomorphic divisor C consists of only one smooth compact connected pseudoholomorphic curve C 0 in X (so k = 1 in this case), having a multiplicity d > 0. We start with some comments on the complex geometry in a neighborhood of the pseudoholomorphic curve C 0. Let π : N C 0 be a normal bundle of C 0 in X which is identified with the orthogonal complement of TC 0 in the tangent bundle TX, restricted to C 0. Since the almost complex structure operator J preserves TC 0, it also preserves N, thus converting it into a complex line bundle. We introduce a fibre-constant almost complex structure J 0 on π : N C 0 by setting J 0 := π (J TC 0 ), and a J 0 -holomorphic fibre coordinate s on N so that the curve C 0, identified with the zero-section of π : N C 0, is given by the equation s = 0. We provide N with the Riemannian metric, induced by the Riemannian metric of X. Then a neighborhood U of the zero section C 0 in N C 0, formed by discs of some small radius in fibers of N C 0, can be identified by the exponential map exp : U X with a tubular neighborhood u of C 0 in X. Using this identification, we can transfer the almost complex structure J from u X to U by taking the inverse image of J u under the exponential map. We denote the obtained almost complex structure on U by the same letter J. By definition, J coincides with J 0 on the zero-section C 0 of π : N C 0 but is not fibre-constant on U. The J -operator of this structure in the neighborhood U has the following expansion on U J = J0 + νs + µ s + O( s 2 ) (2.2) where µ, ν are (0, 1)-forms (w.r. to J 0 ) on C. We define a linearization D J of the J -operator in fibre directions of π : N C 0 by D J := J0 + δ J = J0 + νs + µ s. (2.3) We introduce next a d-vortex bundle M d C 0, associated with N C 0. The fibre of this bundle is given by the moduli space M d of d-vortices on C and sections of M d are identified with families γ : x C 0 γ x = [A x, Φ x ] of d-vortices on N x = C. We can identify γ with a section of N d C 0 by associating with γ the zero set Φ 1 (0). In particular, by choosing the zero-section in N d C 0, we get a constant radial section of M d. We also consider a configuration bundle N d C 0, associated with N C 0. The fibre of this bundle coincides with the configuration space N d of d-vortices on C, introduced in Sec.1.4.

12 ADIABATIC PATHS AND PSEUDOHOLOMORPHIC CURVES Adiabatic Equation We are going to derive the adiabatic equation for d-vortex sections γ of M d along the same lines, as in Sec.1.4. Following [8], we can associate with any d-vortex section γ : C [A, Φ] the Seiberg Witten data (E, (B, α)) for the SW λ -equations on X, consisting of a Hermitian line bundle E X, a Hermitian connection B on it and its section α. These data are obtained by a smooth extension of, respectively, the line bundle π N d N, the connection A and the section Φ from U (identified with the tubular neighborhood u of C in X) to the whole of X. The Chern class c 1 (E) of the line bundle E will be Poincarè-dual to d[c 0 ]. The corresponding Seiberg Witten equations for these data (E, (B, α)) in a small neighborhood U of the zero-section C 0 in N C 0 may be written in the following form: J,A Φ = 0 d + J A+ i 4 (1 (2.4) Φ 2 )ω = 0 where the operator d + J is given by the composition of the exterior derivative d with the projection to the self-dual 2-forms (this projection depends on the Riemannian metric g and so on the almost complex structure J). As in the (2+1)-dimensional case, we consider perturbations γ ǫ = [A ǫ, Φ ǫ ] of the original d-vortex section γ = [A, Φ] in the configuration bundle N d C with A ǫ = A + ǫa, Φ ǫ = Φ + ǫϕ where (a, ϕ) satisfy the orthogonality condition (a, ϕ) T (A,Φ) M d. In terms of an H 2 -base {n µ } of solutions of the linearized vortex equations D (A,Φ) n µ = 0, this orthogonality condition is written the form: (a, ϕ), n µ = 0 for µ = 1,..., 2d. We assume, as in the (2 + 1)-dimensional case, that both n µ and (a, ϕ) satisfy the gauge fixing condition: δ(a,φ) n µ = 0, δ(a,φ) (a, ϕ) = 0. Plugging γ ǫ into the Seiberg Witten equations (2.4), we obtain: J,A Φ + ǫ J,A ϕ + ǫ(p 0,1 J a)φ 0 d + J A + i 4 (1 Φ 2 )ω + ǫd + J a iǫ 2 Re(ϕ Φ)ω 0 (2.5) where the sign denotes an equality up to terms of order, greater than 1, in ǫ and P 0,1 J is the projection operator to the subspace of (0, 1)-forms with respect to the almost complex structure J. Now comes a new feature, compared to Sec.1.4. We decompose all differential operators d and their - and covariant counterparts, involved in the equations

13 12 A.G.SERGEEV (2.5), into vertical (i.e. normal to TC 0 ) and horizontal (i.e. tangential to TC 0 ) components, e.g. d = d N + d C and so on. The vertical and horizontal components of (2.5) are treated in a different way. For d N -derivations we can suppose that the gauge class [A x, Φ x ] for x C 0 is fixed (since the change of the gauge class corresponds to the change inside M d, i.e. in the base of M d C 0 ). At the same time, we recall that the almost complex structure J x is not constant in the normal direction (since it depends on the fibre parameter s in N x ). For d C -derivations we can suppose, on the contrary, that the almost complex structure J x coincides with the (fibre-constant) almost complex structure J 0,x but the gauge class [A x, Φ x ] may change in the horizontal direction. By introducing a slow complex time variable ξ = ǫx, where x is a parameter along C 0, we get from (2.5) the following equation (cf. [5]) ( A C Φ, dc + A) + d ds ( δ J,A N Φ, δ+,n J A) + D(A,Φ) N (a, ϕ) 0 (2.6) s=0 where all d C -derivatives are taken with respect to ξ. Using the orthogonality condition to eliminate the terms in (2.6), containing (a, ϕ), we arrive at the following adiabatic equation ( AΦ, C d C +A), n µ + d ds ( δ J,AΦ, N δ +,N J A), n µ = 0. (2.7) s=0 It coincides with the equation, obtained in [8] from another considerations. The adiabatic equation (2.7) has the form of a non-linear -equation on sections γ = [A, Φ] of the d-vortex bundle M d C 0. We call d-vortex sections γ, satisfying (2.7), adiabatic. The equation (2.7) takes a simpler form for constant sections, i.e. for sections γ : x C 0 [A x, Φ x ] such that the gauge class [A x, Φ x ] does not depend on x (for the radial section, in particular). For such sections the first term in (2.7) vanishes and we obtain an equation d ds ( δ J,A N Φ, δ+,n J A), n µ = 0, µ = 1,..., 2d. s=0 Another simple case of (2.7) is d = 1 when it simplifies to (in notations of (2.3)) σ γ + νσ γ + µ σ γ = 0 where σ γ is the section of N C 0, given by Φ 1 (0) for γ = [A, Φ]. In particular, if σ γ coincides with the zero-section of N C 0 then the latter -equation reduces to the pseudoholomorphicity condition of C 0. REFERENCES [1] Jaffe A., Taubes C.H. Vortices and monopoles. Boston: Birkhäuser, [2] Manton N.S. A remark on the scattering of BPS monopoles.// Phys. Lett., 1982, vol. 110B, [3] Salamon D. Spin geometry and Seiberg Witten invariants. // Warwick: Warwick Univ. preprint, 1996.

14 ADIABATIC PATHS AND PSEUDOHOLOMORPHIC CURVES 13 [4] Sergeev A.G. Vortices and Seiberg Witten equations. // Nagoya: Nagoya Math. Lectures, vol. 5, [5] Sergeev A.G. Adiabatic limit in the Seiberg Witten equations. // Providence, R.I.: AMS Transl. Ser. 2, vol. 212, 2004; [6] Stuart D. Dynamics of Abelian Higgs vortices in the near Bogomolny regime // Commun. Math. Phys., 1994, vol. 159, [7] Taubes C.H. SW Gr: From the Seiberg Witten equations to pseudo-holomorphic curves // J. Amer. Math. Soc., 1996, vol. 9, [8] Taubes C.H. Gr SW: From pseudo-holomorphic curves to Seiberg Witten solutions // J. Diff. Geom., 1999, vol. 51, Steklov Mathematical Institute Moscow