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Revista de la Unión Matemática Argentina
versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.46 n.1 Bahía Blanca ene./jun. 2005
Eta series and eta invariants of Z4-manifolds
Ricardo A. Podestá,
Universidad Nacional de Córdoba, Argentina.
Supported by Conicet and Secyt-UNC
Abstract
In this paper, for each , we construct a family of compact flat spin -manifolds with holonomy group isomorphic to such that the spectrum of the Dirac operator is asymmetric. For these manifolds we will obtain explicit expressions for the eta series, , in terms of Hurwitz zeta functions, and for the eta-invariant, , associated to . The explicit expressions will show the meromorphic continuation of to is in fact everywhere holomorphic.
Keywords and phrases: Dirac operator; Eta series; Eta invariant; Flat manifolds
2000 Mathematics Subject Classification. Primary 58J53; Secondary 58C22, 20H15.
Introduction
If is a positive self-adjoint elliptic differential operator on a compact -manifold , then it has a discrete spectrum, denoted by , consisting of positive eigenvalues with finite multiplicity . This spectrum can be properly studied by the zeta function , where the sum is taken over the non zero eigenvalues of and , with the order of .
If is no longer positive, then the eigenvalues can now be positive or negative. In this case, the spectrum is said to be asymmetric if for some we have . To study this phenomenon, Atiyah, Patodi and Singer introduced in [APS] the "signed" version of the zeta function, namely, the so called eta series defined by
This series converges for and defines a holomorphic function which has a meromorphic continuation to . It is a non trivial fact that this function is really finite at (See [APS2] for odd, and [Gi], [Gi2], for even). The number is a spectral invariant, called the eta invariant, which gives a measure of the spectral asymmetry of .
In this paper, we will take to be the Dirac operator . It is a first order elliptic essentially self-adjoint operator defined on spin manifolds, that is, manifolds admitting a spin structure. Consider the eta function associated to , denoted simply by . The determination of and is in general a difficult task and explicit computations of these objects are not easy to find in the literature.
For compact flat spin manifolds (see Preliminaries) we have the following picture. Pfäffle computed the -invariants in the 3-dimensional case ([Pf]). A general expression for for an arbitrary -manifold with a spin structure is given in [MP2]. There, in the particular case of manifolds with holonomy group , , the authors obtained a very explicit expression for , in terms of differences of Hurwitz zeta functions , for and . This allowed to compute the -invariant simply by evaluation at . Also in [MP2], similar results were obtained for a family of compact flat spin -manifolds with holonomy group , with prime of the form .
These results led us to expect that the eta series of any compact flat spin manifold with abelian holonomy group should be expressible in terms of differences of Hurwitz zeta functions for .
The goal of the present paper is to deal with the simplest case not covered in [MP2], that is, when . More precisely, we consider a rather large family of compact flat spin manifolds with holonomy group having asymmetric Dirac spectrum and we compute the corresponding eta series and eta invariant for every manifold in the family (in the case of symmetric spectrum one has that , see (4.1)). The general expression for is either of the form
or
where are explicit constants depending on , the spin structure of , and the dimension of .
Acknowledgements. I would like to thank Professor Roberto Miatello for several useful comments on a preliminary version that have contributed to improve the exposition.
1. Preliminaries
Flat manifolds. We refer to [Ch]. A Bieberbach group is a discrete, cocompact, torsion-free subgroup of the isometry group of . Such acts properly discontinuously on , thus is a compact flat Riemannian manifold with fundamental group . By the Killing-Hopf theorem any such manifold arises in this way. Any element decomposes uniquely as , with , and denotes translation by . The translations in form a normal maximal abelian subgroup of finite index, with a lattice in which is -stable for each . As usual, one identifies with . The restriction to of the canonical projection given by is a group homomorphism with kernel and is a finite subgroup of . The group is called the holonomy group of and is isomorphic to the linear holonomy group of the Riemannian manifold . The action of on by conjugation is usually called the integral holonomy representation of . By an -manifold we understand a Riemannian manifold with holonomy group . In this paper we shall consider -manifolds which, by the Cartan-Ambrose-Singer theorem, are necessarily flat.
Spin groups. For standard results on spin geometry we refer to [LM] or [Fr]. Let denote the Clifford algebra of with respect to the standard inner product on and let be its complexification. If is the canonical basis of then a basis for is given by the set . One has that holds for all , thus and for . Inside the group of units of we have the compact Lie group
which is connected if and simply connected if . There is a Lie group epimorphism with kernel given by .
If is a matrix for , we will denote by the block matrix having the block in the diagonal position . For let and, for, define | (1.1) |
| (1.2) |
| (1.3) |
Spin representations. We consider an irreducible complex representation of the Clifford algebra , restricted to . The complex vector space has dimension with . If is odd, then is irreducible for and is called the spin representation. If is even, then where each is irreducible of dimension . If denote the restricted action of on then are called the half-spin representations of . We shall write and for and when we wish to specify the dimension.
We will make repeatedly use of the following result (see [MP2]) which gives the values of the characters and of the spin and half spin representations on . If , then
| (1.4) |
Furthermore, for or .
Spin structures. If is an orientable Riemannian manifold, let be the bundle of oriented frames on and the canonical projection. is a principal -bundle over . A spin structure on is an equivariant 2-fold cover where is a principal -bundle and . Such endowed with a spin structure is called a spin manifold.
On compact flat manifolds , a Bieberbach group, the spin structures are in a one to one correspondence with group homomorphisms commuting the diagram
| (1.5) |
that is, satisfying where if (see [Fr], [LM]). We shall denote by the spin manifold endowed with the spin structure as in (1.5).
The spectrum of the Dirac operator. If is the spin representation, the vector bundle with action , where , , is called the spinor bundle of . The space of smooth sections of the spinor bundle can be identified with the set . One can consider the Dirac operator acting on smooth sections of by , where for . One has that is an elliptic first-order differential operator, which is symmetric and essentially self-adjoint. Furthermore, over compact manifolds, has a discrete spectrum consisting of real eigenvalues , , of finite multiplicity . If , is called a harmonic spinor.
In [MP2] explicit expressions for the multiplicities for any compact flat spin manifold with translation lattice and holonomy group were obtained. We now recall the ingredients for these expressions.
Let where . Put , with the dual lattice of , and | (1.6) |
| (1.7) |
Furthermore, for , let be a fixed, though arbitrary, element in the maximal torus of , conjugate in to . Finally, define a sign , depending on and on the conjugacy class of in , in the following way. If and , let such that . Hence, . Take if is conjugate to in and otherwise. As a consequence, and for every (see Definition 2.3, Remark 2.4 and Lemma 6.2 in [MP2] for details).
For odd, the multiplicity of the eigenvalue , for , is given by | (1.8) |
while for even, it is given by the first term in (1.8) (i.e., the sum over ) with replaced by . For , with even or odd, we have that , if , and , otherwise.
2. A family of spin -manifolds
In this section, for each , we shall construct a family of -dimensional pairwise non-homeomorphic spin -manifolds, having asymmetric Dirac spectrum. In this way, we will obtain non trivial eta series and eta invariants, to be computed in the next sections. Put. For each and , we set | (2.1) |
where .
Then , and if and only if is even. Let be the canonical lattice of and for as before define the groups | (2.2) |
| (2.3) |
of compact flat manifolds with holonomy group . It is easy to see that the cardinality of is given by .
Lemma 2.1. For we have | (2.4) |
Hence the manifolds in are non-homeomorphic to each other.
Proof. We compute . For , we have Using this, and the fact that , it is easy to see that Thus, if and are homeomorphic then and . Since we have that and . Therefore, the manifolds in are non-homeomorphic to each other.
We now study the existence of spin structures on -dimensional -manifolds following [MP], where the existence of spin structures on -manifolds was considered. Let be a Bieberbach group with holonomy group and translation lattice . Then with where , , , and .
Assume there is a spin structure defined on , that is, a group homomorphism such that . Then, necessarily , for . Thus, if is a -basis of and we set , for every with , we have .
For any we will fix a distinguished (though arbitrary) element in , denoted by . Thus, where depends on and on the choice of .
The morphism is determined by its action on the generators of . Hence, we will identify this morphism with the -tuple | (2.5) |
where and is defined by the equation .
Now, since is a morphism and , for any we have | (2.6) |
| (2.7) |
| (2.8) |
The next result says that these necessary conditions for the existence of spin manifolds are also sufficient in the case of manifolds with cyclic holonomy groups. We adapt the proof of Theorem 2.1 in [MP] to our case.
Proposition 2.2. If is a Bieberbach group with holonomy group and is as in (2.5), then the map defines a bijective correspondence between the spin structures on and the set . Hence, the number of spin structures on is either or for some .
Proof. It suffices to prove that, given , we can extend it into a group homomorphism from to , also called , satisfying (1.5).
Let . Since is normal in and , we see that any can be written uniquely as , with , . For any choice of , we set , with , and, for a general element in , we define
for . Thus, we get a well defined map such that , and we claim it is a group homomorphism.
In fact, note that if then where . We have that and also . Hence, by using these relations and condition we get
for any .
It is clear that the number of spin structures is either , for some , or 0, in case the equations given by conditions () and () are incompatible ones. Since each of these equations divides by 2 the number of spin structures and the covering torus has exactly such structures we have that . This completes the proof.
Since spin manifolds are orientable, we need to restrict ourselves to the manifolds with even. We have the following result
Corollary 2.3. Every orientable -manifold , , has spin structures parametrized by the -tuples as in (2.5) satisfying: | (2.9) |
where .
Proof. We first note that in the notation of (1.1). Hence, , by (1.3) and we take .
Let . Since and for every , condition gives
where we have used (1.2) in the third equality. Moreover, condition gives for , hence (2.9) holds. Since there are no more relations imposed on the 's the result follows.
Remark 2.4. It would be natural to consider the larger family with matrices in (2.1) replaced by | (2.10) |
where , and . That is, consider where if and if .
However, this larger family of -manifolds has trivial eta series unless and , i.e. the case previously considered. In fact, we have for every with and, by Corollary 2.6 in [MP2], the spectrum of is symmetric. The remaining case ( and ) is more involved, but one checks it by proceeding similarly as in the next section.
3. The spectrum of the Dirac operator
Since we are looking for spectral asymmetry of , by Corollary 2.6 in [MP2], we will restrict ourselves to orientable odd dimensional manifolds in with (recall that). Hence, we fix and take | (3.1) |
| (3.2) |
| (3.3) |
By using expression (1.8), we will explicitly compute the multiplicity of the eigenvalues of the Dirac operator of the spin manifolds .
We shall need the following auxiliary function. Let , i.e. | (3.4) |
Theorem 3.1. Let . The -manifolds with spin structures as in (3.2) have asymmetric Dirac spectrum and, in the notation of (1.6), the multiplicity of the non zero eigenvalue of is given by
| (3.5) |
for , and by
for , where in both cases and is as in (3.4).
Furthermore, for every there are no non-trivial harmonic spinors, that is, .
Proof. For fixed , let be the generator of . For , let be defined by and we put | (3.6) |
| (3.7) |
where is Kronecker's delta function.
Now, since , the maximal torus in , we can take , and since
and for (see (1.2)). Furthermore, since and for we can take , then for each . Moreover, and , because (see Preliminaries). On the other hand .
Since is the canonical lattice then and we can write where . Furthermore, since for and (see (3.2)), we have that if is even and if is odd. Thus, | (3.8) |
Clearly, if is such that for every , only the identity of can give a non-zero contribution to (1.8) and the multiplicity formula now reads . Thus, from now on, we will assume that satisfies for some . Then, we have with for even and for odd and hence we have
| (3.9) |
| (3.10) |
By (1.4), the characters have the expression
| (3.11) |
Case 1: . Let or . Suppose that is even, then . If is even, because . Let with . Since , by (3.10) we have
Replacing these values in (3.7) we get
where we have used that and that is odd, because is even and .
Now, if is odd, with . One has that | (3.13) |
Then, by using that is even because is odd, we get that
Taking , with , from and we finally obtain expression (3.5).
Case 2: . Then is odd and with . We have that | (3.15) |
by (3.10); and, for , by (3.11) we obtain
| (3.16) |
By introducing the values of (3.15) and (3.16) in (3.7), we get to the expressions in the statement of the proposition.
Finally, the claim concerning to the multiplicity of the 0-eigenvalue follows directly from the expressions just after (1.4) and (1.8), respectively.
Remark 3.2. From the multiplicity formulae obtained in Proposition 3.1 we see that, generically, there are no Dirac isospectrality between manifolds in .
On the other hand, if , that is, if , then the spectrum of is symmetric with multiplicities given by for every pair . This follows by and in the proof of Theorem 3.1, in the case , and by (3.7) for , since both (3.15) and (3.16) vanish in this case. Hence, for a fixed , , is a set of mutually -isospectral -manifolds, and similarly for .
4. Eta series and eta invariants of -manifolds
In general, for a differential operator having positive and negative eigenvalues we can decompose the spectrum where and are the symmetric and the asymmetric components of the spectrum, respectively. That is, if , if and only if . We say that is symmetric if .
As a measure of the spectral asymmetry of a differential operator on a manifold , Atiyah, Patodi and Singer introduced the eta series | (4.1) |
generalizing the zeta functions for the Laplacian. It is known that this series converges absolutely for , where is the order of , and defines a holomorphic function in this region, having a meromorphic continuation to that is holomorphic at ([APS2], [Gi2]). The eta invariant is defined by . It is known that if then for every Riemannian manifold (see [Fr]).
Now, we let , the Dirac operator. By using the results obtained in the previous section we shall compute the expression for the eta series and the values of the -invariants for the spin -manifolds considered. We have the following result
Theorem 4.1. Let . The eta series for the -manifolds with spin structures as in (3.2) are given for by | (4.2) |
| (4.3) |
where is the Hurwitz zeta function for , , and is as defined in (3.4).
Furthermore, the meromorphic continuation of to is everywhere holomorphic for all manifolds .
Note. Observe that, for , all the eta functions are mutually proportional and the same happens with .
Proof. By Proposition 3.1 we have that | (4.4) |
| (4.5) |
where .
If is even, the series in (4.5) equals
| (4.6) |
For odd, the series in (4.5) now equals
where we have separated the contributions of , with , and hence
| (4.7) |
By putting together expressions (4.6) and (4.7) we get formula (4.2).
On the other hand, for , by (4.1) and (4.4), we have
The series above equals
| (4.8) |
From here it is clear that (4.3) holds.
The last assertion clearly follows from the explicit expressions for eta series obtained in (4.6), (4.7) and (4.8) since the Hurwitz zeta function has a simple pole at with residue 1 (see [Ap]). □
Corollary 4.2. The eta invariants of the spin -manifolds with spin structures are given by
| (4.9) |
where is as in . In particular .
Proof. This is a consequence of the expressions given in Theorem 4.1 and the fact that for every . □
We now illustrate the results in the lowest dimensions considered, that is and .
Example 4.3. For there is only one manifold in , namely where . Since and , by (4.3) we have
and by (4.9)
This is in agreement with the values obtained in [Pf].
Example 4.4. For there are 3 manifolds in . They are , and where , , and . Now, by (4.2) and (4.3) we get
and, again by (4.9), also
Remark 4.5. To conclude, we conjecture that for a compact flat manifold of dimension , with a "nice" integral holonomy representation, the eta series can be put in terms of differences of Riemann-Hurwitz zeta functions , where , and that the meromorphic continuation to is holomorphic everywhere. Hence, from this expression, the -invariant is easily computed simply by evaluation at . More precisely, we claim that the eta series has the expression
where , is a constant depending on and on the spin structure and each is an entire function (trigonometric or constant). The results in this paper, together with those in [MP2] bring support to this conjecture. We plan to get deeper into this question in the future.
References
[Ap] Apostol T., Introduction to analytic number theory, Springer Verlag, NY, 1976. [ Links ]
[APS] Atiyah, M.F., Patodi V.K., Singer, I.M., Spectral asymmetry and Riemannian geometry, Bull. Lond. Math. Soc. 5, (229–234) 1973. [ Links ]
[APS2] Atiyah M.F., Patodi V.K., Singer I.M., Spectral asymmetry and Riemannian geometry I, II, III, Math. Proc. Cambridge Philos. Soc. 77, (43–69) 1975, 78 (405–432) 1975, 79, (71–99) 1976. [ Links ]
[Ch] Charlap L., Bieberbach groups and flat manifolds, Springer Verlag, Universitext, 1988. [ Links ]
[Fr] Friedrich T., Dirac operator in Riemannian geometry, Amer. Math. Soc. GSM 25, 1997. [ Links ]
[Gi] Gilkey P., The Residue of the Local Eta Function at the Origin, Math. Ann. 240, (183–189) 1979. [ Links ]
[Gi2] Gilkey P., The Residue of the Global Function at the Origin, Adv. in Math. 40, (290–307) 1981. [ Links ]
[LM] Lawson H.B., Michelsohn M.L., Spin geometry, Princeton University Press, NJ, 1989. [ Links ]
[MP] Miatello R.J., Podestá R.A., Spin structures and spectra of Z2k-manifolds, Math. Zeitschrift 247, (319–335) 2004. arXiv:math.DG/0311354. [ Links ]
[MP2] Miatello R.J., Podestá R.A., The spectrum of twisted Dirac operators on compact flat manifolds, TAMS, to appear. arXiv:math.DG/0312004. [ Links ]
[MR] Miatello R., Rossetti J.P., Flat manifolds isospectral on -forms, Jour. Geom. Anal. 11, (647–665) 2001. [ Links ]
[Pf] Pfäffle F., The Dirac spectrum of Bieberbach manifolds, J. Geom. Phys. 35, (367–385) 2000. [ Links ]
Ricardo A. Podestá
FaMAF–CIEM
Universidad Nacional de Córdoba
Córdoba, Argentina.
podesta@mate.uncor.edu
Recibido: 10 de agosto de 2004
Aceptado: 17 de junio de 2005