The Dirac operator on generalized Taub-NUT spaces
aa r X i v : . [ m a t h . DG ] O c t THE DIRAC OPERATOR ON GENERALIZED TAUB-NUT SPACES
ANDREI MOROIANU AND SERGIU MOROIANU
Abstract.
We find sufficient conditions for the absence of harmonic L spinors on spinmanifolds constructed as cone bundles over a compact K¨ahler base. These conditions arefulfilled for certain perturbations of the Euclidean metric, and also for the generalizedTaub-NUT metrics of Iwai-Katayama, thus proving a conjecture of Vi¸sinescu and thesecond author. Introduction
The Taub-NUT metrics on R and their generalizations by Iwai-Katayama [9] providea fruitful framework for the study of classical and quantum anomalies in the presence ofconserved quantities, see e.g. [7]. To describe these metrics, consider the sphere S as theunit sphere inside the quaternions. There exist then three orthogonal unit vector fields I, J, K given by left translation with the unit quaternions i, j, k . The Berger metrics g λ on S are defined by setting the length of I, J to be 1, and that of K to be λ . TheIwai-Katayama metrics on R \ { } ≃ R + × S have the form(1.1) g IK = γ ( t )( dt + 4 t g λ ( t ) )where γ ( t ) = q a + btt , λ ( t ) = √ ct + dt for positive constants a, b, c, d . The apparent singularity at the origin is removable.We are interested here in the axial quantum anomaly already studied in [6, 16]. It wasfound in [6] that the axial anomaly, i.e., the difference between the number of null statesof positive and of negative chirality on a ball or annular domain, may become non-zero forsuitable choices of the parameters of the metric and of the domain when we impose theAtiyah-Patodi-Singer spectral condition at the boundary. Remarkably, when the radiusof the ball is sufficiently large the index was always 0. It was further proved in [16] thaton the whole space, although the Dirac operator is not Fredholm, it only has a finitenumber of null states. The method of proving the finiteness of the index in [16] relied on ageneral index formula due to Vaillant [19], and on a comparison between harmonic spinorsfor a pair of conformal metrics. On the standard Taub-NUT space, which is hyperk¨ahlerand therefore scalar-flat, it is easy to see that there are no harmonic L spinors usingthe Lichnerowicz identity and the infiniteness of the volume. It was somehow natural to Date : September 24, 2018.2000
Mathematics Subject Classification.
Key words and phrases.
Dirac operator, non-Fredholm L index, generalized Taub-NUT metrics. conjecture in [16] that the L index of the Dirac operator corresponding to the generalizedTaub-NUT metric is also zero. The motivation of the present work is to prove the aboveconjecture: Theorem 1.1.
There do not exist L harmonic spinors on R for the generalized Taub-NUT metrics. In particular, the L index of the Dirac operator vanishes. As we just mentioned, for the standard Taub-NUT metric this has been proved in [16].Our approach here is less analytic, and more geometric, than the previous attempt de-scribed above. We exploit the rich symmetries of the metric to decompose the spinors interms of frequencies along the fibers as in e.g. [18], and then further in terms of eigenvaluesof an associated spin c Dirac operator on S . We obtain a system of ordinary differentialequations which we show does not admit any L solutions. There are similarities with [11],[15] in the analysis of this system, but the essential difference is that large time behavioris not enough to rule out L harmonic spinors and we must use also the behavior nearthe origin. The method is more general and we can prove our results for a wider classof manifolds, constructed from a circle fibration over a compact Hodge base. Althoughthe one-point completion of such a manifold will not be in general a topological manifold,we consider it as a singular complete metric space, the appropriate condition on spinorsbeing boundedness in the L ∞ norm near the singular point. Our main result (Theorem5.1) applies both to the Iwai-Katayama metrics and to Euclidean metrics, and to certainperturbations thereof.The paper is organized as follows: In Section 2 we introduce the class of metrics studiedin the rest of the paper. In Section 3 we relate geometric objects – like the Levi-Civitaconnection and the Dirac operator – of a circle-fibered space to the corresponding objectson the base, and we introduce the announced splitting into frequencies along the fibers.Section 4 contains similar computations in the case of warped products, introducing anextra variable corresponding to the radial direction, and computing the corresponding spin c Dirac operator. The main analytic result is stated and proved in Section 5 by reducing theproblem to a linear system of ordinary differential equations on the positive real half-line,and a careful analysis both near infinity and 0 to exclude L solutions. Finally in Section6 we extend the result in a rather formal way to include the Iwai-Katayama generalizedTaub-NUT metrics. Acknowledgments.
The authors are indebted to Mihai Vi¸sinescu for his help concerninggeneralized Taub-NUT metrics and to the anonymous referee for valuable comments. Theauthors acknowledge the support of the Associated European Laboratory “MathMode”.A.M. was partially supported by ANR-10-BLAN 0105. S.M. was partially supported bygrant PN-II-ID-PCE 1187/2009.2.
Circle fibered warped products
Let (
B, g B , Ω) be a compact K¨ahler manifold of real dimension 2 m . Let h denote thewarped product metric dt + α ( t ) g B on N := R + × B and let p denote the projection N → B . HE DIRAC OPERATOR ON GENERALIZED TAUB-NUT SPACES 3
Assume that (
B, g B , Ω) is a Hodge manifold, i.e., [Ω] ∈ πH ( B, Z ). The classicalisomorphism of ˇCech cohomology groups H ( B, S ) ≃ H ( B, Z ) shows the existence of aHermitian line bundle L → B with first Chern class c ( L ) = − [Ω] / π . Let M denote thecircle bundle of L . The projection q : M → B can be viewed as a principal S -bundle.By Chern-Weil theory (cf. [14], Ch. 16 for instance) there exists an imaginary-valuedconnection 1-form iξ on M such that dξ = q ∗ Ω.We define L := R + × L and π := id × p the projection of L onto N . Then π : L → N isa Hermitian line bundle over N whose circle bundle, denoted by M , is just M := R + × M .We endow M with the Riemannian metric g := dt + α ( t )( p ◦ π ) ∗ g B + β ( t ) ξ ⊗ ξ forsome positive functions α and β defined on R + .The Riemannian manifold ( M, g ) obtained in this way will be referred to as the circle-fibered warped product (CFWP) over the Hodge manifold (
B, g B , Ω), with warping functions α and β . Notice that a CFWP can be viewed either as a generalized cylinder of a familyof metrics on the S -bundle M over B (cf. Proposition 2.3 below) or as a Riemanniansubmersion with 1-dimensional fibres over a warped product R + × α B . It is the latter pointof view which will be useful in order to relate spinors on M and B . Example 2.1.
The flat space C m +1 \ { } is the CFWP over the complex projective space( C P m , g F S , Ω F S ) endowed with the Fubini-Study metric, with warping functions α ( t ) = t √ , β ( t ) = t . The normalization g F S of the Fubini-Study metric used here is the one withscalar curvature equal to 2 m ( m + 1) or, equivalently, the one for which the projection S m +1 → ( C P m , g F S ) is a Riemannian submersion (cf. [14], Ch. 13).
Example 2.2.
The Taub-NUT metric on C is conformal to the one-point completion ofthe CFWP over the standard 2-sphere of radius 1 / √ α ( t ) = √ t , β ( t ) = t bt . More generally, the generalized Taub-NUT metrics of Iwai-Katayama on C are conformal to the one-point completion of the CFWP over ( C P , g F S ), i.e., the standard2-sphere of radius 1 / √
2, with warping functions α ( t ) = √ t , β ( t ) = t √ ct + dt for somepositive constants c and d (cf. [16], p. 6576):(2.1) g IK = a + btt (cid:0) dt + α ( t ) π ∗ g F S + β ( t ) ξ ⊗ ξ (cid:1) . By Remark 2.4 below, these are actually examples of CFWP’s.
Proposition 2.3.
Let ( M, g ) be the CFWP over a Hodge manifold ( B m , g B , Ω) with warp-ing functions α and β and assume that lim t → α ( t ) = lim t → β ( t ) = 0 . Let d denote the distanceon M induced by g . Then the metric completion ( ˆ M , d ) of ( M, d ) has exactly one extrapoint. If g extends to a smooth metric on ˆ M , then ( B, g B , Ω) is the complex projectivespace endowed with the Fubini-Study metric, and lim t → α ( t ) t − √ = lim t → β ( t ) t − . Proof.
The Riemannian manifold (
M, g ) will be viewed as a generalized cylinder (cf. [4])of the family of metrics g t := α ( t ) q ∗ g B + β ( t ) ξ ⊗ ξ on the S -bundle M over B (whichis a compact manifold). The first statement follows immediately from the fact that for ANDREI MOROIANU AND SERGIU MOROIANU every x ∈ M , the rays R + × { x } are geodesics parametrized by arc-length on M . Assumenow that g extends smoothly to ˆ M . The Gauss Lemma applied to a neighborhood of theorigin t = 0 in ˆ M shows that the distance spheres ( M , g t ) (renormalized by a factor 1 /t )tend to the standard sphere S m +1 in the Gromov-Hausdorff topology. In other words,there exist non-zero constants α , β such that lim t → α ( t ) /t = α , lim t → β ( t ) /t = β and α q ∗ g B + β ξ ⊗ ξ is the standard metric on S m +1 . On the other hand, this metric is bydefinition a Riemannian submersion over ( B, α g B ) with totally geodesic fibres of length2 πβ . Since the length of every closed geodesic on S m +1 is 2 π we get β = 1. The manifold( B, α g B ) is then the quotient of the sphere by an isometric S action, so B = C P m and α g B = g F S (cf. [14], Ch.13). The constant α is determined by the normalizationcondition c ( L ) = − [Ω B ] / π . Indeed, since L = R m +2 \ { } is clearly the tautologicalbundle Ω( −
1) of C P m , its first Chern class is equal to − [Ω F S ] / π , whence α = 1 / √ (cid:3) The converse holds under some extra smoothness assumption on α and β at t = 0 butwe will not need this in the sequel. Remark 2.4.
A metric conformal to a CFWP is itself a CFWP provided that the conformalfactor is a radial function (i.e., it only depends on t ). Indeed, if g = γ ( t ) ( dt + α ( t ) π ∗ g B + β ( t ) ξ ⊗ ξ ) , in the new coordinate s := s ( t ) defined by s = R t γ ( u ) du , g reads g = ds + α ( t ( s )) π ∗ g B + β ( t ( s )) ξ ⊗ ξ. The generalized Taub-NUT metrics from Example 2.2 are thus particular cases of CFWP.We will analyze these metrics in more detail in Section 6.Our main goal in this paper will be to study the L -index of the Dirac operator on aCFWP ( M, g ) when M is a spin manifold. As we will see below, this is automaticallythe case when B has a spin c structure whose auxiliary bundle is some tensor power of L ,i.e., if the second Stiefel-Whitney class of B satisfies w ( B ) = 0 or w ( B ) ≡ c ( L ) mod2. In the next two sections we will relate spinors on M to spin c spinors on N and thenfurther to spin c spinors on B . The results are quite general and can be viewed as a naturalextension of the theory of projectable spinors introduced in [12] to the case of submersionswith non-totally geodesic fibres.3. Spinors on circle fibrations
Let π : ( M, g ) → ( N, h ) be a Riemannian submersion with 1-dimensional fibres of length2 πβ for some function β : N → R + . The fibers of π are totally geodesic if and only if β isconstant, but we will mostly be interested in examples with non-constant β in the sequel.We can view M as a principal S -fibration over N . Indeed, the flow ϕ t of the verticalKilling vector field V on M of length π ∗ β closes up at time t = 2 π , i.e., ϕ π = id M ,thus it defines a free S -action on M whose orbit space is N . We denote by P U(1) N thisprincipal S -bundle with total space M . The Riemannian metric g can be written as g = π ∗ h + β ( t ) ξ ⊗ ξ , where ξ is the 1-form on M defined by ξ ( V ) = 1 and ker ξ = V ⊥ . HE DIRAC OPERATOR ON GENERALIZED TAUB-NUT SPACES 5
The 2-form dξ is basic, i.e., there exists some 2-form F on N such that dξ = π ∗ F . Thisfollows immediately from the Cartan formula and the fact that V is Killing, or alternatelysince iξ is a connection 1-form in the principal bundle P U(1) N (cf. Section 2).The following result holds without restriction on the dimension of N but we will stateit only for the case we will need in the sequel. Lemma 3.1.
Let P U(1) N → N be the principal S -bundle over the m + 1 -dimensionalmanifold N defined by the Riemannian submersion π : M → N . Let L → N be the complexline bundle associated to P U(1) N with respect to the canonical representation of S on C .Then every spin c structure P Spin c (2 m +1) N on N with auxiliary bundle L ⊗ k , k ∈ Z inducesa spin structure on M and all these spin structures are isomorphic.Proof. By enlargement of the structure groups, the two-fold covering θ : P Spin c (2 m +1) N → P SO(2 m +1) N × P U(1) N gives a two-fold covering θ : P Spin c (2 m +2) N → P SO(2 m +2) N × P U(1) N, which, by pull-back through π , gives rise to a Spin c structure on M : P Spin c (2 m +2) M π −−−→ P Spin c (2 m +2) N π ∗ θ y θ y P SO(2 m +2) M × P U(1) M π −−−→ P SO(2 m +2) N × P U(1) N y y M π −−−→ P. This construction actually yields a spin structure on M . Indeed, the pull back P U(1) M to M of P U(1) N is trivial since it carries a tautological global section σ ( u ) = ( u, u ) , ∀ u ∈ M = P U(1) N . Correspondingly, the pull-back to M of every associated bundle L ⊗ k is trivial. (cid:3) From now on we assume that N carries some spin c structure with auxiliary bundle L ⊗ k ,and we study M with the spin structure induced by the previous lemma. In particular, wewill consider the flat connection on the trivial bundle P U(1) M , rather than the pull-backconnection from P U(1) N , in order to define covariant derivatives of spinors on M . Thefollowing result, first proved in [13], relates an arbitrary connection on a principal bundle π : M = P U(1) N → N and the flat connection on π ∗ M = P U(1) M → M . π ∗ M = P U(1) M ≃ M × S π −−−→ M = P U(1) N π ∗ π y π y M π −−−→ N Lemma 3.2.
The connection form A of the flat connection on P U(1) M can be related toan arbitrary connection A on P U(1) N by A (( π ∗ s ) ∗ ( U )) = − A ( U ) , ANDREI MOROIANU AND SERGIU MOROIANU A (( π ∗ s ) ∗ ( X ∗ )) = A ( s ∗ X ) , where U is a vertical vector field on M , X ∗ is the horizontal lift (with respect to A ) of avector field X on N , and s is a local section of M → N .Proof. The identification M × U(1) ≃ π ∗ M is given by ( u, a ) ( u, ua ), for all ( u, a ) ∈ M × U(1). For some fixed u ∈ M , take a path u t in the fiber over x := π ( u ) such that u = u and ˙ u = U . We define a t ∈ U(1) by u t = s ( x ) a t , so via the above identification wehave ( π ∗ s )( u t ) = ( u t , s ( x )) = ( u t , ( a t ) − ) , and thus A (( π ∗ s ) ∗ ( U )) = − a − ˙ a = − A ( ˙ u ) = − A ( U ) . Similarly, for x ∈ N and X ∈ T x N , take a path x t in N such that x = x and ˙ x = X .Let u ∈ π − ( x ) and u t the horizontal lift of x t such that u = u . We define a t ∈ U(1) by s ( x t ) = u t a t , which by derivation gives s ∗ ( X ) = R a ˙ u + u ˙ a . Then( π ∗ s )( u t ) = ( u t , s ( x t )) = ( u t , a t ) , and thus, using the fact that ˙ u is horizontal, A (( π ∗ s ) ∗ ( X ∗ )) = a − ˙ a = A ( s ∗ ( X )) . (cid:3) Recall that the complex Clifford representation Σ m +2 = Σ +2 m +2 ⊕ Σ − m +2 can be identifiedwith Σ m +1 ⊕ Σ m +1 by defining in an orthonormal basis e j · ( ψ, φ ) = ( ( e j · φ, e j · ψ ) for j ≤ m + 1( − φ, ψ ) for j = 2 m + 2 . Accordingly, we obtain identifications, denoted by π ± , of the pull back π ∗ Σ N with Σ ± M .By a slight abuse of notation we will denote π ± and Σ ± M by π ε and Σ ε M for ε = ± X is a vector and Ψ is a spinor on N , then X ∗ · π ε Ψ = π − ε ( X · Ψ) , (3.1) β V · ( π ε Ψ) = επ − ε Ψ , (3.2)where β V is the unit vertical vector field defined at the beginning of this section, and X ∗ denotes the horizontal lift to M of a vector field X on N .We consider now a spin c structure P Spin c (2 m +1) N on ( N, h ) with auxiliary bundle L ⊗ k and denote by ∇ N the covariant derivative induced on Σ N by the connection form iξ of P U(1) N . By Lemma 3.1, the pull-back to M of P Spin c (2 m +1) N induces by enlargement aspin structure on ( M, g ), where we recall that g = π ∗ h + β ξ ⊗ ξ . Proposition 3.3.
Let ∇ M denote the covariant derivative on Σ ε M induced by the Levi-Civita connection on ( M, g ) and the flat connection on π ∗ P U(1) N . Let ∇ N denote the HE DIRAC OPERATOR ON GENERALIZED TAUB-NUT SPACES 7 spin c covariant derivative on Σ N induced by the Levi-Civita connection on ( N, h ) and theconnection form A = iξ on P U(1) N . Then ∇ M and ∇ N are related by ∇ MX ∗ ( π ε Ψ) = π ε ( ∇ NX Ψ − εβ T ( X ) · Ψ) , ∀ X ∈ T M, (3.3) ∇ MV ( π ε Ψ) = π ε (cid:16) β F · Ψ + ε dβ · Ψ − ki Ψ (cid:17) , (3.4) where T is the endomorphism of T N defined by dξ ( X ∗ , Y ∗ ) = F ( X, Y ) = h ( T X, Y ) .Proof. If V denotes as before the vertical vector field such that ξ ( V ) = 1, the Koszulformula and the fact that [ V, X ∗ ] = 0 for all vector fields X on N yield g ( ∇ MX ∗ Y ∗ , Z ∗ ) = h ( ∇ NX Y, Z )(3.5) g ( ∇ MV X ∗ , Y ∗ ) = g ( ∇ MX ∗ V, Y ∗ ) = − g ( V, [ X ∗ , Y ∗ ]) = − β ξ ([ X ∗ , Y ∗ ])= β dξ ( X ∗ , Y ∗ ) = β h ( T X, Y ) = β F ( X, Y ) , (3.6)and g ( ∇ MV X ∗ , V ) = g ( ∇ MX ∗ V, V ) = βX ( β ) , (3.7)for all vector fields X , Y and Z on N .Consider a spinor field on N locally expressed as Ψ = [ σ, ψ ], where ψ : U ⊂ N → Σ m +1 is a vector-valued function, and σ is a local section of P Spin c (2 m +1) N whose projectiononto P SO(2 m +1) N is a local orthonormal frame ( X , ..., X m +1 ) and whose projection onto P U(1) N is a local section s . Then π ε Ψ can be expressed as π ε Ψ = [ π ∗ σ, π ∗ ξ ]. Moreover,the projection of π ∗ σ onto P SO(2 m +2) M is the local orthonormal frame ( β V, X ∗ , ..., X ∗ m +1 )and its projection onto P U(1) M is just π ∗ s .Using the general formula for the covariant derivative on spinors, Lemma 3.2, and thefact that the bundle L ⊗ k is associated to P U(1) N via the representation ρ k ( z ) = z k of S on C , we obtain ∇ MX ∗ π ε Ψ = [ π ∗ σ, X ∗ ( π ∗ ψ )] + X j The Dirac operators on M and N are related by (3.8) D M ( π ε Ψ) = π − ε (cid:16) D N Ψ − εβ F · Ψ + dβ β · Ψ − εki β Ψ (cid:17) Proof. Simple computation using (3.1)–(3.4): D M ( π ε Ψ) = X j X ∗ j ·∇ MX ∗ j ( π ε Ψ) + β V ·∇ M β V ( π ε Ψ)= X j X ∗ j · π ε (cid:16) ∇ NX j Ψ − εβ T ( X j ) · Ψ (cid:17) + β V · β π ε (cid:16) β F · Ψ + ε dβ · Ψ − ki Ψ (cid:17) = π − ε (cid:16) D N Ψ − εβ X j · T ( X j ) · Ψ + εβ F · Ψ + dβ β · Ψ − εki β Ψ (cid:17) = π − ε (cid:16) D N Ψ − εβ F · Ψ + dβ β · Ψ − εki β Ψ (cid:17) . (cid:3) Note that Equation (3.4) is just the Bourguignon-Gauduchon formula [5] for the Liederivative of a spinor field with respect to a Killing vector field: ∇ V Φ = L V Φ + dV ♭ · Φ . Incidentally, this formula shows that if Φ = π ε Ψ is the pull-back of a spin c spinor on N corresponding to a spin c structure with auxiliary bundle L ⊗ k , then L V Φ = − ik Φ. Yetanother way to understand this fact is the following. A section s of the complex line bundle L → N can be identified with a complex function f s on M ⊂ L with “frequency” − s ( π ( x )) = xf s ( x ) (clearly f s ( ϕ t ( x )) = e − it f s ( x ) so L V ( f s ) = − if s ). Now, a spin c bundle on N with auxiliary bundle L ⊗ k is the tensor product between two locally defined bundles:the spin bundle of N and a square root of L ⊗ k . It is now clear that the pull-back to M ofits sections are spinors on M with “frequency” − k/ Lemma 3.5. Let H be the Hilbert space of L spinors on M and let H n be the Hilbert spaceof L sections of the spin c bundle on N with auxiliary bundle L ⊗ n . If the spin structure of M is induced as before by a spin c structure on N with auxiliary bundle L ⊗ k , H decomposesin a Hilbertian orthogonal direct sum H = M n ∈ Z ,ε = ± π ε ( H k +2 n ) . HE DIRAC OPERATOR ON GENERALIZED TAUB-NUT SPACES 9 If Φ ∈ H is smooth, the same holds for its components Φ n ∈ H k +2 n and the length of Φ n at any x ∈ M is bounded by the maximum of the lengths of Φ along the S -orbit of x .Proof. The space of L functions on M decomposes in a Hilbertian orthogonal direct sum L ( M ) = M n ∈ Z L n ( M ) , where L n ( M ) := { f ∈ L ( M ) | L V ( f ) = in · f } is the space of L functions on M offrequency n , identified as before with the space of L sections of ( L ∗ ) ⊗ n . The desireddecomposition now follows immediately from the fact that the tensor product between L ⊗ n and the spin c bundle on N with auxiliary bundle L ⊗ k is the spin c bundle on N withauxiliary bundle L ⊗ ( k +2 n ) .The last assertion follows from the fact thatΦ n ( x ) = 12 π Z π e − int (( ϕ t ) ∗ Φ)( x ) dt. (cid:3) A similar decomposition of the space of spinors on total space of S fibrations was usedby Ammann [1] and Ammann and B¨ar [3] in order to study the properties of the spectrumof the Dirac operator when the fibres collapse, and also by Nistor [18] in his study of the S -equivariant index.Note that Eqs. (3.3) and (3.4) already appeared (in a slightly different form becauseof different spinor identifications) as Lemma 3.2 and Eq. (2) respectively in Ammann [1].However, since the proofs of these formulas appear only in Ammann’s thesis [2], we havechosen for the reader’s convenience to include here the full details of the proofs.4. Spinors on warped products Let now ( B, g B ) be a Riemannian manifold and assume that ( N, h ) is the warped product( R + × B, dt + α ( t ) g B ) for some positive function α : R + → R + . We denote by p : N → B the standard projection. Let ∇ N and ∇ B denote the Levi-Civita covariant derivatives on N and B and let ∂ t denote the (unit) radial vector field on N . Every vector field X on B defines a “horizontal” vector field also denoted by X on N such that [ X, ∂ t ] = 0.The warped product formulae for the covariant derivatives ([17], p.206) are ∇ N∂ t ∂ t = 0 , (4.1) ∇ N∂ t X = ∇ NX ∂ t = α ′ α X, (4.2) ∇ NX Y = ∇ BX Y − αα ′ g B ( X, Y ) ∂ t . (4.3)Consider a spin c structure on B with auxiliary line bundle L ⊗ k and the induced pull-backspin c structure on N with auxiliary line bundle L ⊗ k , where L = p ∗ L is the pull-back of L . We continue to denote by ∇ B and ∇ N the spin c covariant derivatives induced by someconnection on L . Assume now that B has even dimension. The spinor bundle Σ N can be canonicallyidentified with π ∗ (Σ B ) such that the Clifford product satisfies α X · ( p ∗ Ψ) = p ∗ ( X · Ψ) , ∀ X ∈ T B, (4.4)and ∂ t · ( p ∗ Ψ) = ip ∗ ( ¯Ψ) , (4.5)where ¯Ψ := Ψ + − Ψ − is the “conjugate” of Ψ = Ψ + + Ψ − with respect to the chiraldecomposition Σ M = Σ + M ⊕ Σ − M . From (4.4) we easily obtain α q ( p ∗ ω ) · ( p ∗ Ψ) = p ∗ ( ω · Ψ)(4.6)for every q -form ω on B . Using the warped product formulae one can easily relate thespin c covariant derivatives ∇ N and ∇ B like before: ∇ NX ( p ∗ Ψ) = p ∗ ( ∇ BX Ψ − iα ′ X · ¯Ψ) , ∀ X ∈ T B, (4.7)and ∇ N∂ t ( p ∗ Ψ) = 0 . (4.8)In particular, the Dirac operators on N and B are related by D N ( p ∗ Ψ) = α p ∗ ( D B Ψ + imα ′ ¯Ψ) . (4.9)Indeed, if ( X , . . . , X m ) is a local orthonormal basis on B , then ( α X , . . . , α X m , ∂ t ) is alocal orthonormal basis on N , whence D N ( p ∗ Ψ) = X j α X j ·∇ N α X j ( p ∗ Ψ) + ∂ t ·∇ N∂ t ( p ∗ Ψ)= X j α X j · (cid:16) α p ∗ ( ∇ BX j Ψ − iα ′ X j · ¯Ψ) (cid:17) = α p ∗ ( D B Ψ + imα ′ ¯Ψ) . More generally, one can identify a spinor Ψ on N with a 1-parameter family Ψ t of spinorson B , and (4.9) becomes(4.10) D N Ψ = p ∗ (cid:16) α D B Ψ t + imα ′ α ¯Ψ t + i ˙¯Ψ t (cid:17) . Harmonic spinors on CFWP’s We now have all necessary ingredients in order to prove the main result of this paper: Theorem 5.1. Let ( M, g ) , g = dt + α ( t ) π ∗ g B + β ( t ) ξ ⊗ ξ be a circle-fibered warpedproduct (CFWP) over the Hodge manifold ( B m , g B , Ω) , endowed with the spin structuredefined by some spin c structure on B as before. Assume that the positive warping functions α and β satisfy the conditions HE DIRAC OPERATOR ON GENERALIZED TAUB-NUT SPACES 11 (a) Z ∞ x e − R tx √ α ( s ) ds dt = ∞ for all x > ; (b) lim t → α ( t ) = 0 ; (c) 2 α ( t ) ≥ β ( t ) > m − m α ( t ) for all t > .Then ( M, g ) carries no non-trivial harmonic L spinors which are bounded near the sin-gularity t = 0 .Proof. Assume that Φ is a non-zero harmonic L spinor, bounded near t = 0. By ellipticregularity, Φ is smooth. We can of course assume that Φ is chiral, i.e., it is a section of Σ ε M for some ε = ± 1. By Lemma 3.5, and using the fact that the Dirac operator commuteswith the Lie derivative L V , we can also assume that Φ = π ε Ψ is the pull-back of some L section Ψ of the spin c structure on N with auxiliary bundle L ⊗ k (the integer k is even orodd, depending on whether the spin structure on M projects onto a spin structure on N or not).Using (3.8) we infer(5.1) D N Ψ = β ( εβ F · Ψ − dβ · Ψ + ε ik Ψ) . We now view Ψ as a family Ψ t of spin c spinors on B . Recalling that F = p ∗ Ω and taking(5.1), (4.6) and (4.10) into account, we get(5.2) (cid:16) α D B Ψ t + imα ′ α ¯Ψ t + i ˙¯Ψ t (cid:17) = β (cid:16) ε β α Ω · Ψ t − iβ ′ ¯Ψ t + ε ik Ψ t (cid:17) The spin c bundle Σ B decomposes in a direct sum (cf. [10])Σ B = m M l =0 Σ l B of eigenspaces of the operator of Clifford multiplication by the K¨ahler form Ω, i.e.,Σ l B = { Ψ ∈ Σ B | Ω · Ψ = i (2 l − m )Ψ } . One has Σ l B ⊂ Σ + B if l is even and Σ l B ⊂ Σ − B if l is odd. Moreover D B maps sectionsof Σ l B to sections of Σ l − B ⊕ Σ l +1 B and each Σ l B is stable by ( D B ) . This easily showsthat every eigenspinor of D B is a finite sum of eigenspinors of D B in C ∞ (Σ l B ⊕ Σ l +1 B )for 0 ≤ l ≤ m − B is compact and D B is elliptic, the space of L spinors on B is the Hilbertiandirect sum of the eigenspaces of D B . By the above, there exits l ∈ { , . . . , m − } , λ ∈ R and a spinor Φ = Φ l + Φ l +1 ∈ C ∞ (Σ l B ⊕ Σ l +1 B ) with D B Φ = λ Φ such that the functions u ( t ) := Z B h Ψ t , Φ l i dv B , v ( t ) := Z B h Ψ t , Φ l +1 i dv B do not vanish identically.Taking the scalar product with Φ l and Φ l +1 in (5.2) and integrating over B yields λα v + ( − l imα ′ α u + ( − l iu ′ = ε i (2 l − m ) β βα u − ( − l iβ ′ β u + ε ik β u λα u − ( − l imα ′ α v − ( − l iv ′ = ε i (2( l +1) − m ) β βα v + ( − l iβ ′ β v + ε ik β v which can be written after setting w := iv : u ′ = (cid:16) ε ( − l (2 l − m ) β − α β ′ + ε ( − l α k − mαα ′ β βα (cid:17) u + ( − l λα ww ′ = ( − l λα u + (cid:16) ε ( − l ( m − l +1)) β − α β ′ − ε ( − l α k − mαα ′ β βα (cid:17) w. Denoting U := uβ α m and W := wβ α m this simplifies to( − l U ′ = ε (cid:16) (2 l − m ) β +2 α k βα (cid:17) U + λα W ( − l W ′ = λα U + ε (cid:16) ( m − l +1)) β − α k βα (cid:17) W. (5.3)We have shown that if ( M, g ) carries a non-trivial harmonic spinor, then (5.3) has anon-trivial solution ( U, W ) for some l ∈ { , . . . , m − } , k ∈ Z , λ ∈ Spec( D B ) ⊂ R , and ε ∈ {± } . Moreover, since the volume form on ( M, g ) is dv M = α m βdt ∧ ξ ∧ dv B and B is compact, Fubini’s theorem shows that the original spinor Φ is L on M if andonly if Z ∞ | α m β Ψ t ( x ) | dt < ∞ , ∀ x ∈ B. We thus get that U and W are L functions on R + and satisfy U, W ∈ O ( α m β ) at t = 0.From conditions (b), (c) and the definition of U, V we get(5.4) lim t → U ( t ) = lim t → W ( t ) = 0 . The system (5.3) reads(5.5) ( U ′ = ρU + σWW ′ = σU + τ W where ρ := ε ( − l (cid:16) (2 l − m ) β +2 α k βα (cid:17) , τ := ε ( − l (cid:16) ( m − l +1)) β − α k βα (cid:17) , σ := ( − l λα . Notice that the coefficients of the system (5.5) are real functions, thus we can assumethat U, W are real by considering separately their real and imaginary parts. Lemma 5.2. If a linear combination of the functions U and W is monotonous, it mustvanish identically.Proof. Using (5.4) we see that if aU + bW does not vanish identically, then | aU + bW | isbounded from below by a non-zero constant on [ x , ∞ ) for some x > 0, so it cannot be L . (cid:3) HE DIRAC OPERATOR ON GENERALIZED TAUB-NUT SPACES 13 The previous lemma together with (5.5) show that λ = 0: indeed, for λ = 0 the system(5.5) uncouples into two first-order linear ODE’s, whose nontrivial solutions never vanishby uniqueness, hence they have constant sign and so Lemma 5.2 applies. By changing U to − U if necessary, we can therefore assume that σ ( t ) > t > Lemma 5.3. If σ ( t ) > for all t > then we must have ( U W )( t ) ≤ for all t ∈ R + .Proof. Assume that U W > I . From (c) we easily infer(5.6) τ + ρ = − ε ( − l β α ≥ − √ α , so (5.5) yields(5.7) ( U W ) ′ = ( τ + ρ ) U W + σ ( U + W ) ≥ − √ α U W. Consider the maximal interval J := ( x , x ) containing I on which U W > 0. For every x < x ≤ t < x , (5.7) implies(5.8) ( U W )( t ) ≥ ( U W )( x ) e − R tx √ α ( s ) ds . If x < ∞ then by continuity ( U W )( x ) ≥ ( U W )( x ) e − R x x √ α ( s ) ds > 0, contradicting themaximality of J . Therefore x = ∞ , so U W ( t ) > t > x . By integration, (5.8)implies Z ∞ x ( U W )( t ) dt ≥ ( U W )( x ) Z ∞ x e − R tx √ α ( s ) ds dt. By hypothesis (a), the last integral is infinite, however U, W ∈ L ( R + , dt ) implies that R ∞ x ( U W )( t ) dt < ∞ , contradiction. (cid:3) Lemma 5.4. ( U W )( t ) < for all t > .Proof. Assume for instance that U ( x ) = 0. The Cauchy-Lipschitz theorem gives W ( x ) =0 and the first equation in (5.5) shows that U ( x ) has the same sign as W ( x ) for every x in some small interval ( x , x + δ ) , contradicting Lemma 5.3. The same argument workswhen W ( x ) = 0 by considering the second equation in (5.5). (cid:3) We proved so far that U and W have opposite signs and σ is positive. Condition (c)implies that τ and ρ have constant signs on R + since 0 ≤ l ≤ m − 1. If τ ≤ 0, it meansthat τ and σ have opposite signs, and since also U and W have opposite signs by Lemma5.4, it follows from the second equation in (5.5) that W ′ has constant sign. By Lemma 5.2we get a contradiction. This shows that τ > ρ > 0. By condition(c), this can only happen for k = 0, m = 2 l + 1 and ( − l = − ε .Assuming this to be the case, the system (5.5) reads U ′ = β α U + | λ | α WW ′ = | λ | α U + β α W. (5.9)The difference D := U − W is thus a non-vanishing function satisfying(5.10) D ′ = (cid:16) β α − | λ | α (cid:17) D, so for every t > D ( t ) = D ( t ) e R tt β ( s )4 α ( s )2 − | λ | α ( s ) ds . To conclude the proof of the theorem we distinguish two cases. If λ ≤ − / we get | D ( t ) | > | D ( t ) | e − R tt √ α ( s ) ds so D cannot be square-integrable because of hypothesis (a).If λ ≥ − / , (5.6) together with (5.10) show that D is decreasing, contradicting Lemma5.2. (cid:3) Axial anomaly for generalized Taub-NUT metrics on R Radial perturbations of the Euclidean metric on R m +2 . Recall from Example2.1 that the Euclidean space is the metric completion of the CFWP with α = t √ , β = t and with basis B = C P m endowed with the Fubini-Study metric. Note that by ellipticregularity, bounded spinors which are harmonic on a punctured ball B ( ǫ ) \{ } are actuallysmooth and harmonic on B ( ǫ ), while conversely harmonic spinors on R m +2 are clearlybounded near 0. Theorem 5.1 applies therefore to the Euclidean metric on R m +2 , for all m ≥ 1. It is of course well-known that there are no harmonic L spinors on the Euclideanspace. Our results generalize this to metrics which are radial perturbations of the standardEuclidean metric with any α, β satisfying the conditions of Theorem 5.1.6.2. Generalized Taub-NUT metrics. The main application of Theorem 5.1 that wehave in mind is the vanishing of the index for the generalized Tab-NUT metric of Iwai-Katayama. The difficulty of the problem resides in the non-Fredholmness of the Diracoperator as an unbounded operator in L . Nevertheless in [16] it was proved that the L kernel of D is finite-dimensional, and vanishes for the standard Taub-NUT metric.We cannot apply Theorem 5.1 directly because of the conformal factor γ ( t ) in (1.1). Asin Remark 2.4 we set ds = γ ( t ) dt . Notice that s = s ( t ), t = t ( s ) are diffeomorphisms of R + onto itself provided Z γ ( t ) dt < ∞ , Z ∞ γ ( t ) dt = ∞ . (6.1)This condition clearly holds for the conformal factor in (1.1), which is asymptoticallyconstant near infinity and of order t / near t = 0. Thus we obtain a CFWP metric γ ( t ) g where g is itself a CFWP metric. Lemma 6.1. Let g be a CFWP metric with coefficients α ( t ) , β ( t ) , and γ ( t ) a conformalfactor satisfying (6.1) . Then the CFWP metric γ g satisfies the hypotheses of Theorem5.1 if and only if (a’) Z ∞ x γ ( t ) e − R tx √ α ( u ) du dt = ∞ for all x > ; (b’) lim t → γ ( t ) α ( t ) = 0 ; (c’) 2 α ( t ) ≥ β ( t ) > m − m α ( t ) for all t > . HE DIRAC OPERATOR ON GENERALIZED TAUB-NUT SPACES 15 Proof. The coefficients of the CFWP metric γ g are˜ α ( s ) = γ ( t ( s )) α ( t ( s )) , ˜ β ( s ) = γ ( t ( s )) β ( t ( s ))so conditions ( b ) , ( c ) from Theorem 5.1 for ˜ α, ˜ β are clearly equivalent to conditions (b’),(c’). Now α ( t ) dt = α ( s ) ds and by definition γ ( t ) dt = ds so by two changes of variables,condition (a’) is equivalent to condition (a) for ˜ α . (cid:3) As a corollary to Theorem 5.1 we deduce that the Iwai-Katayama metrics on R do notadmit non-trivial L harmonic spinors. Proof of Theorem 1.1. It is straightforward to check that the conditions of Lemma 6.1 holdfor the coefficients of Example 2.2, namely m = 1 , α ( t ) = √ t, β ( t ) = t √ ct + dt , γ ( t ) = q a + btt . It follows from Theorem 5.1 that there do not exist non-trivial L harmonic spinors on( R \ { } , g IK ) bounded near the origin. Of course, the metric g IK is smooth at the origin,as can be seen by the change of variable r = t . In particular we have proved that theredo not exist L harmonic spinors on ( R , g IK ). (cid:3) We could have also used the conformal covariance of the Dirac operator (cf. [8], see also[18]) to related harmonic spinors for the metrics g and λ ( t ) g = g IK . We do not give detailssince this approach is essentially equivalent to the above proof. References 1. B. Ammann, The Dirac operator on collapsing S -bundles , S´emin. Th´eor. Spec. G´eom., Univ. Grenoble, (1998), 33–42.2. B. Ammann, Spin-Strukturen und das Spektrum des Dirac-Operators , PhD Thesis, Freiburg, 1998.3. B. Ammann, C. B¨ar, The Dirac Operator on Nilmanifolds and Collapsing Circle Bundles , Ann. GlobalAnal. Geom. (1998), 221–253.4. C. B¨ar, P. Gauduchon and A. 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Centre de Math´ematiques, ´Ecole Polytechnique, 91128 Palaiseau Cedex, France E-mail address : [email protected] Institutul de Matematic˘a al Academiei Romˆane, P.O. Box 1-764, RO-014700 Bucharest,RomaniaS¸coala Normal˘a Superioar˘a Bucharest, calea Grivit¸ei 21, Bucharest, Romania E-mail address ::