Instability of Reducible Critical Points of the Seiberg-Witten Functional

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1 Instability of Reducible Critical Points of the Seiberg-Witten Functional Celso. Doria UFSC - Depto. de atemática February 7, Introduction Let (, g) be a closed riemannian manifold with scalar curvature k g. There exist smooth 4-manifolds admitting a spin c structure c such that the Seiberg- Witten invariant SW (c) 0. These spin c structures are named basic classes and they are in the realm of the 4-dim differential topology. The space Spin c () of spin c structures on is {c = α c + β c H 2 (X, Z) H 1 (X, Z 2 ) w 2 (X) = α(mod 2)}. (1) From the analytical point of view, a basic classes carries a SW c -monopole, which is a special solution of a partial differential equation, as it will be defined next. The motivation for this research was to use variational techniques to measure the instability of reducible solutions in the jacobian torus J = H1 (,R). Based on the fact that the Seiberg-Witten invariants are H 1 (,Z) also defined as expectation values of a N = 4 supersymmetric twisted gauge theory [8], one might believe that either there exists a monopole or J achieves the minimum. Because Spin c 4 = (SU 2 SU 2 U 1 )/ Z2 and U 2 = (SU 2 U 1 )/Z 2 = Spin c 3 we get two representations ρ ± : Spin 4 U 2 = (SU 2 U 1 )/Z 2 GL(2, C). In practice, a spin c -structure on is given by a pair of rank 2 complex vector bundles S c ±, which fibers are Spin c 4 -modules corresponding to ρ ±, respec., and isomorphisms det(s c + ) = det(sc ) = L c, where det(s c ± ) are the research partially supported by FAPESC 2568/

2 determinant line bundle such that c 1 (L c ) = α c H 2 (, Z). Let Ω 0 (S c + ) the space of sections on S c + and A c be the space of U 1 -connections 1-forms. Each A A c induces a covariant derivative A : Ω 0 (L c ) Ω 1 (L c ) on L c. E.Witten introduced in [11] the SW -monopole equations, the coupled system of 1 st -order PDE, D + Aφ = 0, (2.1) F + A = σ(φ), (2.2), (2) where φ Ω 0 (S + c ), D + A is the positive component of the Dirac operator, F + A is the self-dual component of the curvature F A and σ : Ω 0 (S + c ) Ω 2 +(ir) is the self-dual 2-form σ(v)(x, Y ) =< X.Y.v, v > < X, Y > v 2. performing the coupling between a self-dual 2-form F A and a positive spinor field v; σ(v) 2 = 1 4 v 4. The configuration space is C c = A c Ω 0 (S + c ). Definition An element (A, φ) is a SW c -monopole if it verifies the SW-equations (2). There are two kinds of SW c -monopoles (i) irreducible if φ 0 and (ii) reducible if φ = 0. The irreducibles exist only for a finite number of classes in Spin c (). The monopole eqs. (2) fits in a variational formulation whose Euler-Lagrange eqs. are the 2 nd -order SW-equations d F A + 4iIm(< A φ, φ >= 0, A φ + φ 2 + k g (3) φ = 0. 4 Thus, J = {(A, 0) C c d F A = 0} is a solution set of (3), corresponding to those connections whose curvature is harmonic, and whose existence is guarantee by Hodge theory. Later, it will be shown that monopoles are the ground state of the theory and satisfies eqs (3). In order to measure the instability, we introduced { } λ c () = sup sup λ c g(a).[vol(, g)] 1/2 (4) A J g where is the space of riemannian metrics on. Thus, J is defined to be unstable if λ c () < 0. 2

3 Theorem Assume k g is not non-negative. If there exists an irreducible solution (A, φ) of eqs (3), then J is unstable. Theorem If c Spin c () admits a parallel spinor and the Yamabe invariant satisfies Y () < 0, then J is unstable. In particular, k g can t be non-negative. Theorem If c Spin c () is a basic class admitting a parallel spinor, and α 2 c > 0, then λ c () < π α 2 c. A class c Spin c () admitting a parallel spinor imposes strong restriction on ([2], [4]). Assuming π 1 () = 0 and being irreducible as cartesian product, it turn out that either is Kähler or is spin Ricciflat. The former case is characterized by the surjectivity of the Ricci tensor and the existence of a a integrable complex structure J on such that α c = c 1 (J) or c 1 (J). In the last, the author is not aware of any sort of classification theorem but its importance for physicists. Because the gauge group action is not free, the use of variational techniques on C c are inappropriate to answer the main question about the existence of basic classes on a smooth manifold, or a simpler question about the existence of irredutible solutions to eqs. (3). 2 Background Consider π : E a vector bundle with structural group G and denote F (E) the G-principal bundle of frames on E. 2.1 Gauge Group Consider G a Lie group with Lie algebra g. The Gauge group G of a principle G-bundle P G is the set of G-equivariant automorphism Φ : P G P G such that π Φ = π. This can be set by noting the existence of a map s : P G G such that Φ(p) = p.s(p), s(p.g) = g 1.s(p).g. Taking the adjoint action Ad g : G G, Ad g (x) = g 1.x.g, and defining the bundle Ad(G) = P G Ad G, the gauge group G is the space of sections of Ad(P ). The representation ad : G End(g), defined by ad(g) = g 1 vg, induces the associated vector bundle ad(g) = P G ad g. If G is abelian, then Ad(P ) = ap(, G); e.g.: G = U 1, G = ap(, U 1 ) and ad(u 1 ) = ir. 3

4 2.2 Spin and Spin c Structures on Whenever admits a spin structure or a spin c structure it carries a Dirac operator useful to study geometric and topological properties on by analytical methods. In order to define such structures we consider the Lie groups Spin 4 = SU 2 SU 2, recalling that Ad : Spin 4 SO 4 is the universal covering map, and Spin c 4 = Spin 4 Z2 U 1. Let π : F () be the frame bundle of. A spin structure on is a principal Spin 4 -bundle P s such that the projection π : P s lifts to a map ζ : P s F () satisfying the following conditions (i) ζ(p.g) = ζ(p).ad(g), for all p P s (E) and g Spin 4, (ii) π ζ = π : P s It turns out that admits a spin structure if and only if w 2 () = 0; in this case the space of spin-structures on is Spin() = H 1 (, Z 2 ). All spin structure on a smooth 4-manifold carries an irreducible representation ρ s : Spin 4 GL 2 (H) and the spin vector bundle S = P s ρs H 2 over, whose fibers are a Cl 4 -module (Cl 4 is the real Clifford Algebra isomorphic to 2 (H)). For all smooth 4-manifold Spin c (X), indeed a spin c -structure on corresponds to define an almost complex structure on \{pt}. When is a spin manifold, the cohomology classes α c H 2 (, 2Z), defined in (1), satisfy α c : H 2 (, Z) 2Z. In this case, S c = S (L c ) 1/2 where (L c ) 1/2 is the square root bundle of L c. Neverthless, if w 2 () 0, then the identity local S c = S (L c ) 1/2 is only locally true because neither S nor (L c ) 1/2 are globally defined on. 3 Geometric Structures A brief introduction of covariant derivatives and curvature is given in order to fix the concepts and the notations needed in the text. 3.1 Covariant Derivatives and Connections 1-forms Let s consider the general case of a smooth vector bundle E over. A connection, or equivalently a covariant derivative, on a vector bundle E over is a R-linear operator : Ω 0 (E) Ω 1 (E) satisfying the Leibnitz rule: for all f : R and V Ω 0 (E), (fv ) = df V + f V. Using the exterior derivative d : Ω p () Ω p+1, it can be extended to a linear operator d : Ω p (E) Ω p+1 (E), by 4

5 d (V ω) = V ω + V dω Fixing an origin at 0 A E, any connection is written as A = 0 + A, A Ω 1 (ad(g)), so the space of connection A E on E is an afim space, which vector space structure is isomorphic to Ω 1 (ad(g)). The group G acts on A E by g. = g 1 g inducing on Ω 1 (ad(g)) the G-action g.a = g 1 Ag + g 1 dg. Let : Ω 0 (T ) Ω 1 (T ) be the riemannian connection on and β = {e i 1 i 4} be a local orthonormal frames of T defined on local chart U with the following properties: for all i, j ( i = ei ) (i) [e i, e j ] = i e j j e i = 0, (ii) i e k = l Γl ik e l, The covariant derivative operator is locally given by = i ( i)dx i = d+γ, where i = i + Γ i and Γ = i Γ idx i Ω 0 (T ) is the connection 1-form. The set of linear maps e k e l : R 4 R 4, given by (e k e l )(v) =< v, e l > e k < v, e k > e l, (so 4 Λ 2 (R 4 )), defines a so 4 basis on which Γ i = k,l (Γ i) kl (e k e l ). The riemannian connection on (, g) induces a connection on S. Let Cl(, g) be the Clifford Algebra Bundle and c : T Cl(, g) be the Clifford map perfoming the inclusion. A Ω 0 (Cl(, g))-module structure is defined on Ω 0 (S) by the pointwise product (γ.φ)(x) = γ(x).φ(x), for all γ Ω 0 (Cl(, g)) and φ Ω 0 (S). In order to describe locally a connection on S we consider γ i = c(e i ). The whole procedure to induce the connection on S relies on the lie algebra isomorphism Θ : so n spin n, Θ(e k e l ) = 1 2 γ k.γ l ([9], prop 6.1). Through Θ, the Christoffel symbols of induces on S the operator Γ s i : Ω0 (S) Ω 0 (S), Γ s i = 1 2 l,k Γk il (γ k.γ l ). The spin connection on S is the 1-form Γ s = i Γs i dxi defining the covariant derivative s : Ω 0 (S) Ω 1 (S), s = d + Γ s. The bundle S c being locally equal to S Lc 1/2, a covariant derivative operator A : Ω 0 (S c ) Ω 1 (S c ) is defined by taking the spin connection s on S and a U 1 -connection A on L 1/2 c, as follows: let ψ Ω 0 (S), λ Ω 0 (L 1/2 c ) and ψ λ Ω 0 (S c ), A (ψ λ) = s ψ λ + ψ A λ. (5) 5

6 3.2 Curvature The curvature of a covariant derivative on E is the C -linear operator F : Ω 0 (E) Ω 2 (E), F = d d, whose expression F (V )(X, Y ) = ( X Y Y X [Y,X] ) V, for all V Ω 0 (E) and X, Y Ω 0 (T ), in a orthonormal frame β E = {f α 1 α r} on E is ( Aj F ij (V ) = F (e i, e j )(V ) = A ) i + [A i, A j ] (V ), x i x j where ei f α = β Aβ iα f β and F ij End(E). Besides, A i being a skew symmetric operator implies F ij End(E x ) is also skew-symmetric. Let A Ω 1 (E) be the connection 1-form and = d + A, the curvature 2- forms F A Ω 2 (E) is F A = i,j F ijdx i dx j. The gauge group acts on Ω 2 (E) by g.ω = g 1.ω.g, motivated by the fact that curvature of g.a is g.f A = g 1.F A.g. When E = T, the curvature 2-form R : Ω 0 (T ) Ω 2 (T ) of the riemannian metric is locally written, using the frame β, as R = i,j R ijdx i dx j. The components R ij (e k ) = l Rl ijk e l, (R ij ) lk = Rijk l satisfy the following identities; (i) R l ijk + Rl jki + Rl kij = 0, (ii)r l ijk = Rl jik Using the so 4 basis {e k e l } we have (iii) R l ijk = Rk ijl (iv) R l ijk = Rj kli (6) R ij = k,l R k ijl e k e l In this way, the curvature 2-form induced on S by the riemannian connection on T is R s = 1 Rijl k 2 dxi dx j γ k.γ l Ω 2 (S) i,j k,l Definition The Ricci curvature of the riemannian manifold (X, g) is the bilinear form Ric : Ω 0 (T X) Ω 0 (T X) C (X) 6

7 Ric(u, v) = trace g [w R(u, w)v]. Using the frame β = {e j } on, the Ricci curvature is given by Ric(u, v) = g( k R(e k, v)e k, u), u, v T. Using the symmetry Ric ij = Ric ji, we define the linear self-adjoint Ricci operator Ric : Ω 0 (T X) Ω 0 (T X), Ric(u, v) = g(u, Ric(v)), locally given by Ric(v) = k R(e k, v)e k It induces on S the operator Ric s (v) = k R(e k, v)γ k So far, it has been showed how the riemannian connection induces on S a connection. Applying equation (5), it can also induces a connection on S c. In the latter case, the curvature 2-form F A : Ω 0 (S c ) Ω 2 (S c ) locally decomposes into F A = R s + if A, f A Ω 2 () (7) 4 Variational Formulation and 2 nd Variation By fixing an origin at 0 A c a connection on L c is written as A = 0 + A, where A Ω 1 (, ir) is a u 1 -valued 1-forms. A topology on the configuration space C c = A c Ω 0 (S c + ) is defined by considering the Sobolev spaces A c = L 1,2 (Ω 1 (, ir)) and Γ (S c + ) = L 1,2 (Ω 0 (X, S c + )); the gauge group is taken to be G = L 2,2 (ap(x, U 1 )). The G action on C c is not free, the isotropy group are G (A,φ) = {I}, when φ 0, and G (A,0) U 1 for all A A c. An element (A, φ) C c is named irreducible if φ 0, otherwise is reducible. The subspace of irreducibles Cc = {(A, φ) C c φ 0} is a universal principal G-bundle over the moduli space Bc = Cc /G. The homotopy type of Bc is isomorphic to the subspace CP J. oreover, the free action of U 1 = {g G g constant} on Cc 7

8 defines a principal U 1 -bundle over Bc whose first Chern class c 1 (Cc ) = SW (c) is the generator for H 2 (Bc ; Z) corresponding to the CP factor. The riemannian structure on the tangent bundle T C c = C c (Ω 1 (ir) Ω 0 (S c + )) is the product defined on each component as follows; (i) on A c, for all η, θ Ω p (, ir), < η, θ >= (η θ)dv g ; recalling that the Hodge operator is minus the usual star operator because the forms take values in ir instead of R. (ii) on Ω 0 (S c + ), for any sections V, W Ω 0 (S c + ), (z C, Re(z) = real part) < V, W >= Re(< V, W >)dv g. Thus, the inner product <, >: T (A,φ) C c T (A,φ) C c R is X < η + V, θ + W >=< η, θ > + < V, W >. The Seiberg-Witten equations fit into a variational set up by defining the functional SW : C c R, SW c (A, φ) = X { 1 4 F A 2 + A φ ( φ 2 +k g ) k2 g}dv g + 2π 2 N c, where k g is the scalar curvature of (X, g) and N c = c 2 1(c) = c 1 (c) c 1 (c) = 1 4π 2 [ F + A 2 F A 2 ]dv g. Due to the gauge invariance, the functional is well defined as SW c : B c R. The gradient at (A, φ) is X (8) grad(sw c )(A, φ) = (d F A + 4iIm(< A φ, φ >, A φ + φ 2 + k g φ). (9) 4 The solution sets of the 2 nd -order SW -equations grad(sw c ) = 0 are invariant by the G-action because grad(sw c )(g.(a, φ)) = g 1 grad(sw c )(A, φ). The SW -monopoles also satisfy grad(sw c )(A, φ) = 0, as it can be seen from the identities 8

9 d (F A ) = 2d F + A = d [σ(φ)] = 4iIm ( < D + φ, X.φ > + < X φ, φ > ) D + φ = 0 0 = D D + φ = A + k g 4 φ + F + A 2.φ = Aφ + k g 4 φ + φ 2 φ J = H1 (,R) H 1 (,Z). Due to identity SW c (A, φ) = ( F + A σ(φ) 2 + D + A φ 2 )dv g, the stable critical points are exactly the SW-monopoles. In [5] they proved that the functional SW c : B c R satisfies the Palais-Smale condition, so the critical sets are compact and the minimum is always achieved (SW c 0). There are irreducible and reducible solution of grad(sw) = 0 to be considered, moreover, the scalar curvature plays an important role. Assuming k g 0, the minimum is achieved at reducible points because the monopole equations reduces to F + A = 0 and the SW c-equations reduces to d F A = 0. Independently of the sign of k g, the anti-self-dual solutions can be ruled out ([3]) by assuming b + 2 () 2. If d F A = 0, then F A is a harmonic 2-form, so the existence is guarantee by Hodge theory. Indeed, the space J = {(A, 0) C c d F A = 0} G is diffeomorphic to the jacobian torus A local slice of B c at (A, φ) is given ([7]) by the kernel of the operator Tφ : Ω1 (X, ir) Ω 0 (S c + ) Ω 0 (X, ir), Tφ (θ, V ) = d θ < V, φ >, namely ker(tφ ) = ker(d ) φ. Because (d ) 2 = 0, it can be further decomposed into ker(d ) = d (Ω 2 (, ir)) H 1, where H 1 = {θ Ω 1 (, ir) dθ = d θ = 0}, the subspace of harmonic 1- forms, is the tangent space to the Jacobian torus J at (A, 0). The instability of J is established by performing the analysis of the 2nd variation δ2 SW δαδβ of the SW-functional. The tangent space of C c at (A, φ) is T (A,φ) C c = Ω 1 (X; ir) Ω 0 (S c + ), so δ2 SW δαδβ defines a symmetrical bilinear form H(A,φ) SW ((θ 1, V 1 ), (θ 2, V 2 )) =< (θ 1, V 1 ), H(θ 2, V 2 ) >, where the operator ( ) h11 h H = 12 has entries given by h 21 h 22 δ 2 SW c δλδθ (A,φ).(θ, Λ) =< θ, (d dλ + 4 < Λ(φ), φ >) =< θ, h 11 (Λ) >, δ 2 SW c δw δθ (A,φ).(θ, W ) = 2 ( < A φ, θ(w ) > + < A W, θ(φ) > ) = =< θ, h 12 (W ) >, (h 21 = h 12 ) δ 2 SW c δw δv (A,φ).(V, W ) =< V, A W + k g+ φ 2 W + 1 < φ, W > φ >= 4 4 =< V, h 22 (W ) >. 9

10 The restriction of the 2 nd -variation to the slice of B c at (A, φ) is an elliptic operator H : ker(tφ ) ker(t φ ) whose leading terms d d = and A = ( A ) A are laplacians. The spectrum σ(h) is a discrete set such that each eigenvalue has finite multiplicity and no accumulation points, besides, there are but a finite number of eigenvalues ( below any given d number. At (A, 0), the hessian operator becomes H = ) d 0, where 0 L A L A : Ω 0 (S c + ) Ω 0 (S c + ) is the elliptic self-adjoint operator L A (V ) = A V + k g V. (10) 4 The spectrum σ(l A ) is bounded below. For each λ σ(l A ), the corresponding eingenspace V λ T (A,0) B c has finite dimension. In this way, ker(h) = T (A,0) J X V 0.The lower eigenvalue of L A is given by Rayleigh s quotient λ c g(a) = inf V S + c { A V 2 + kg 4 V 2 }dv g V 2 dv g (11) 4.1 Parallel Spinor A spinor ψ Ω 0 (S c + ) is parallel with respect to a connection when ψ = 0. In general, it is difficult to compute the spectrum of L A. Using Kato s inequality, σ(l A ) can be compare with the spectrum of L = g + kg 4 defined on functions f : R ( g =Lapl-Belt). Consider on a smooth atlas A() = {(U λ, ξ λ ) λ λ} such that, for each λ Λ, (i) U λ is convex, (ii) the local coordinates are {(x 1, x 2, x 3, x 4 ) U λ x i R} and (iii) attached to U λ there exists a local orthonormal frame β λ = {e i e i = i, 1 i 4} trivializing locally the tangent bundle T. Let F () be the principal G-bundle of frames on, so β λ F (). Proposition Let A A c and V Ω 0 (S + c ). Then, V 2 A V 2 (12) The equality holds if, and only if, there exists a 1-form ω Ω 1 () such that A V = ωv. Proof. Taking the orthonormal frame β = {e i 1 i 4}, locally we get V 2 = i i V 2 and A V 2 = i A i V 2. From the identities 10

11 i V 2 = 2 V. i V and i V 2 = i < V, V >= 2 < A V, V >, we have V. i V =< A i V, V >. Assuming V 0 and applying Cauchy-Scwartz inequality it follows the inequality i V A i V. Hence, ineq. (12) is verified. The equality is attained whenever there exists functions α i : C such that A i V = α iv, that is, A V = i [ ] A i V dx i = α i dx i V = ωv i If V is a harmonic spinor (D A V = 0) and A V = ωv, then A V = 0. It is rather restrictive to assume V as a harmonic spinor, but under an extra assumption on the functions α i : C the existence of a parallel spinor can be achieved. The 1-form ω = df f = d(ln(f)) is exact. The reverse claim is also true; Proposition There exists a parallel spinor V Ω 0 (S c + ) if, and only if, there exists a spinor V 0 Ω 0 (S c + ) and a class ω HdR 1 () such that A V 0 = ωv 0. Proof. Suppose V Ω 0 (S c + ) is parallel, A V = 0. So, V has constant length. Let V = fv 0, where f : C, so f(x) 0 and V 0 (x) 0, x. Furthermore, A V = df V 0 + f A V 0 = 0 A V 0 = df f V 0. (13) The 1-form ω = df f = d(ln(f)) is exact. Now, let s prove the reverse assuming that V 0 = ωv 0. The equation A (fv 0 ) = 0 is equivalent to df fω = 0; in this case ω = d(ln(f)). Taking a local chart (U λ, φ λ ) from the atlas defined at the beginning of this section, say a chart (U λ, φ λ ) with frame β λ = {e i 1 i 4}, and defining α i = w(e i ), we get ω = i α idx i and df fω = 0 becomes locally described by the system i f α i f = 0, 1 i 4. The closedness of ω is equivalent to the conditions j α i = i α j, for all i, j. A necessary condition to the existence of f is j i f = i j f, but this is a consequence of ω being close; j i f = ( j α i )f α i α j f = i j f. The identity j α i = i α j allow us to integrate and write 11

12 x1 α i (x 1, x 2, x 3, x 4 ) = i α 1 (t, x 2, x 3, x 4 )dt, 2 i 4. 0 Therefore, the function x1 f(x 1, x 2, x 3, x 4 ) = e 0 α 1 (t,x 2,x 3,x 4 )dt, satisfies i f α i f = 0, for 1 i 4, and is C. The function f is globally defined because it depends only on the 1-form ω. As before, consider β = {e α ; 1 α 4} an orthornormal frame on and γ α = c(e α ). The Ricci operator induces the operator c(ric(.)) : T X Cl(X), c(ric(x)) = α Rs (e α, X)γ α, such that [c(ric(x))] 2 = α R s (e α, X) 2 = Ric(X) 2. Definition Let A A c be a connection 1-form with curvature if A Ω 2 (, ir); 1. Let H A : T X Ω 0 (S c ) Ω 0 (S c ) be the linear operator defined by H A (X, ψ) = 1 γ α.f c 2 A(e α, X)(ψ), 2. Let I A : Ω 0 (T ) Ω 0 (T ) be the skew-symmetric operator α I A (X) = α [(X f A )(e α )]e α (14) (I A ) αβ = i(e β f A )(e α ) = 2if βα. The existence of a parallel spinor implies H A (X, ψ) = 0, for all X T, upon which deep global implications are draw, as we shall see next. Proposition Let A A c and assume ψ Ω 0 (S c ) is a parallel spinor ( A ψ = 0). So, for all X T, Ric 2 = f A 2. The proof of proposition relies strongly on the identity ([4], chap 3) [c(ric(x)) ic(i A (X))].ψ = 0, X T for all X T. It is straightforward from the proposition that f A = 0 if, and only if, Ric = 0. oreover, (i) c(ric(x)) = c(i A (X)) and (ii) 12

13 < c(ric(x)), ic(i A (X)) >= 0. The identity c(ric(x))(ψ) = ic(i A )(X))(ψ) is a key point. Defining R = {Ric(X) X Ω 0 (T X)}, the self-adjointness of Ricci operator results the decomposition T X = R ker(i A ) (ker(i A ) = R ). Let s consider the operators Ψ : T X Ω 0 (S) X c(x).ψ Ψ ι : T X Ω 0 (S) X ic(x).ψ and the vector space E = Ψ 1 (Imag(Ψ) Imag(Ψ ι )). The existence of a parallel spinor ψ means that R E and so E ker(i A ). These spaces define the distributions E = {E x x } and E = {E x x } Proposition If ψ is a parallel spinor, then the distributions E and E are integrable. Proof. For all X E there exist an unique Y T X such that X.ψ = iy.ψ. The space E is closed under the action of the covariant derivative because X.ψ = iy.ψ implies ( X).ψ = i( Y ).ψ. (i) E is integrable. Note that for all X E we get X f A = 0, in particular f A annihilates X. Since df A = 0, the distribution E is integrable. (ii) E is integrable. Taking X, Y E, it follows from the -invariance of E that the commutator [X, Y ] = X Y Y X E. Corollary There exist submanifolds 1, 2 such that 1 is Kähler and 2 is spin. Proof. Let 1 be the submanifold whose tangent space T x 1 = E x. Since for each X E there exists only one Y such that X.ψ = iy.ψ, define the automorphism J : T X T X, X.ψ = ij(x).ψ. Thus, for each x, J : T x 1 T x 1 defines a complex structure since ij 2 (X).ψ = ij(j(x)).ψ = J(X).ψ = ix.ψ J 2 = I. oreover, J = 0 because J( X) = Y and, for all X, Y Ω 0 (T ), ( X).ψ = [( J)X + J( X)].ψ = ij( X).ψ J = 0. Furthermore, from the identity 13

14 [J(X).J(Y )+J(Y ).J(X)].ψ = (XY +Y X).ψ+2i[g(J(X), Y )+g(j(y ), X)].ψ we get g(j(x), J(Y )).ψ = {g(x, Y ) + i[g(j(x), Y ) + g(j(y ), X)}.ψ, and so, g(j(x), J(Y )) = g(x, Y ) and g(j(x), Y ) = g(j(y ), X). Therefore, 1 is Kähler because J = 0, and 2 is spin Ricci-flat because f A 2 = 0 and so Ric 2 = f A 2 = 0 on 2 ; hence c 2 = 0 and w 2 ( 2 ) = 0. In the context of the arguments above, if Ricc : T T is onto, then is Kähler. Taking the restriction L 1 = L c 1, the canonical class of ( 1, J) is κ J = L 1. Assuming π 1 () = 0, orianu [2] proved κ J and κ J to be the only spin c classes on carrying parallel spinors, which are known to be the only basic class in ([7]). Using de Rham s decomposition theorem, oroianu concluded that a simply connected manifold carries a parallel spinor if, and only if, it is isometric to the riemannian product 1 2 where 1 is Kähler and 2 is spin Ricci-flat. Of course, if we assume is irreducible as cartesian product, then either is Kähler or is spin. To the best of author s knowledgment, it is not known the classification of spin Ricci-flat 4-manifolds beyond the one quoted in [6]. 5 Theorems proofs In this section, Kato s inequality (12) is used to compare the lower eigenvalue λ c g(a) of operator L A in (10) with the the lower eigenvalue λ g of operator L = g + kg 4 acting on functions f : R. By the Rayleigh s formula, each lower eigenvalue is given by λ g = inf f Ω 0 () λ c g(a) = inf V S c Definition λ() = sup g λ g [vol(, g)] 1/2 { f 2 + kg 4 f 2 }dv g f 2 dv g (15) { A V 2 + kg 4 V 2 }dv g V 2 dv g (16) Let be the space of riemannian metrics on and [g] = {ζ.g ζ : (0, )} the conformal class of g. The Yamabe constant of [g] is defined by Y [g] = inf k ĝdvĝ. (17) ĝ [g] [vol(, ĝ)] 1/2 14

15 The condition Y [g] 0 implies the existence of unique metric realizing the Yamabe constant ([10]). The smooth Yamabe invariant is defined as Y () = sup [g] Y [g]. Assuming Y () 0, it is proved in [1] the relation Y () = λ(). By analogy, associated to the operator L A we define λ c (A) = sup λ c g(a).[vol(, g)] 1/2. g Thus, if Y () < 0 and c Spin c (X) is a class such that there exists a parallel spinor ψ Ω 0 (S c + ), then λ c (A) < 0 and J is unstable. If a irreducible solution (A, φ) exists, it follows from the SW-equations that [ A φ 2 + k ] g 4 φ 2 dv g = 1 φ 4 dv g. 4 X Assuming k g isn t non-negative, let (A, φ) be an irreducible solution of the sw-eqs. and apply eq. (16) to get λ c (g, A). φ 2 dv g 1 φ 4 dv g λ c (g, A) < 0. 4 Applying the Cauchy-Schwartz inequality we get and so [ ] 1/2 φ 2 dv g [vol(, g)] 1/2. φ 4 dv g, [ ] 1/2 λ c g(a).[vol(, g)] 1/2. φ 4 dv g λ c g(a) φ 2 dv g 1 φ 4 dv g. 4 X Hence, λ c (A) 0, proving theorem If Y () < 0 and there exists a spin c class c carrying a parallel spinor, then λ c () = λ() < 0, concluding theorem Whenever there exists a SW c -monopole (A, φ), then 1 φ 4 dv g = F + A 4 2 dv g = 4π 2 c 2 1(J)[] + F A 2 dv g X 1 4 [ X φ 4 dv g ] 1/2 = 1 2 [ F + A 2 dv g ] 1/2 π α 2 c []. 15

16 Let s consider the case α 2 c [] > 0, otherwise φ = 0. So, defining λ c () = sup A J λc (A), we get the upper bound in theorem λ c () π α 2 c []. Therefore, if admits a SW c -monopole, then λ c () < 0. It can be conjectured that λ c () is a smooth invariant, for each c Spin c (X). References [1] AKUTAGAWA,K., ISHIDA,. and Le BRUN - Perelman s Invariant, Ricci Flow, the Yamabe Invariants of Smooth anifolds, Arch. ath. 48, 2007, [2] Andrei oroianu - Parallel and Killing Spinors on Spin c -manifolds, Comm. ath. Phys 187, (1997). [3] DONALDSON, S.K. and KRONHEIER, P. - The Geometry of 4- anifold, Oxford University Press, [4] Thomas Friedrich - Dirac Operators in Riemannian Geometry, Graduate Studies in ath 25, AS (2000). [5] JOST, J., PENG, X. and WANG, G. - Variational Aspects of the Seiberg-Witten Functional, Calculus of Variation 4 (1996), [6] itsuhiro Itoh - Conformal Geometry of Ricci Flat 4-anifolds, Kodai ath. J., 17 (1994), [7] ORGAN, J. - The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-anifolds, ath. Notes 44, Princeton Press. [8] J..E Labastida 1, Carlos Lozano - athai-quillen formulation of twisted N - 4 supersymmetric gauge theories in four dimensions, Nuclear Physics B 502 [P] (1997) [9] H. Blaine Lawson and arie-louise ichelson - Spin Geometry, Princeton Unv. Press, [10] Richard Schoen - Conformal Deformation of a Riemannian etric to Constant Scalar Curvature, J. Diff. Geom. 20, (1984). [11] WITTEN, E. - onopoles on Four anifolds, ath.res.lett. 1, n o 6 (1994),

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