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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
![∑ ηA(s) = sign(λ) ∣λ ∣- s. 0⁄= λ∈SpecA](/img/revistas/ruma/v46n1/1a0426x.png)
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
![CM,ɛ( 1 3 ) η(s) = (8π)s ζ(s, 4) - ζ(s, 4)](/img/revistas/ruma/v46n1/1a0465x.png)
or
![C ( ( ) ( )) η(s) = (8Mπ,ɛ)s an ζ(s, 18) - ζ(s, 78) + bn ζ(s, 38) - ζ(s, 58)](/img/revistas/ruma/v46n1/1a0466x.png)
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
![Spin(n) = {v1 ⋅⋅⋅v2k : ∥vj∥ = 1, 1 ≤ j ≤ 2k},](/img/revistas/ruma/v46n1/1a04124x.png)
which is connected if and simply connected if
. There is a Lie group epimorphism
with kernel
given by
.
![Bj](/img/revistas/ruma/v46n1/1a04130x.png)
![1 ≤ j ≤ m](/img/revistas/ruma/v46n1/1a04131x.png)
![diag(B1, ...,Bm)](/img/revistas/ruma/v46n1/1a04132x.png)
![B j](/img/revistas/ruma/v46n1/1a04133x.png)
![j](/img/revistas/ruma/v46n1/1a04134x.png)
![t ∈ ℝ](/img/revistas/ruma/v46n1/1a04135x.png)
![B(t) = [cost - sint] sint cost](/img/revistas/ruma/v46n1/1a04136x.png)
![t1,...,tm ∈ ℝ](/img/revistas/ruma/v46n1/1a04137x.png)
![]() | (1.1) |
![k ∈ ℤ](/img/revistas/ruma/v46n1/1a04139x.png)
![x(t1,...,tm)](/img/revistas/ruma/v46n1/1a04140x.png)
![]() | (1.2) |
![Spin(n)](/img/revistas/ruma/v46n1/1a04142x.png)
![SO(n)](/img/revistas/ruma/v46n1/1a04143x.png)
![T = {x(t1,...,tm) : tj ∈ ℝ, 1 ≤ j ≤ m}](/img/revistas/ruma/v46n1/1a04145x.png)
![T0 = {x0(t1,...,tm) :](/img/revistas/ruma/v46n1/1a04146x.png)
![tj ∈ ℝ,j = 1, ...,m}](/img/revistas/ruma/v46n1/1a04147x.png)
![μ : T → T0](/img/revistas/ruma/v46n1/1a04148x.png)
![]() | (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.
![F1 = {B ∈ F = r(Γ ) : nB = 1}](/img/revistas/ruma/v46n1/1a04228x.png)
![nB := dim ker(B - Id)](/img/revistas/ruma/v46n1/1a04229x.png)
![* * 2πiλ⋅u Λ ɛ = {u ∈ Λ : ɛ(λ) = e ,λ ∈ Λ}](/img/revistas/ruma/v46n1/1a04230x.png)
![* Λ](/img/revistas/ruma/v46n1/1a04231x.png)
![Λ](/img/revistas/ruma/v46n1/1a04232x.png)
![]() | (1.6) |
![γ = BLb ∈ Γ](/img/revistas/ruma/v46n1/1a04234x.png)
![(Λ*ɛ,μ)B](/img/revistas/ruma/v46n1/1a04235x.png)
![B](/img/revistas/ruma/v46n1/1a04236x.png)
![Λ *ɛμ](/img/revistas/ruma/v46n1/1a04237x.png)
![]() | (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).
![n](/img/revistas/ruma/v46n1/1a04261x.png)
![± 2πμ](/img/revistas/ruma/v46n1/1a04262x.png)
![μ > 0](/img/revistas/ruma/v46n1/1a04263x.png)
![]() | (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
![n ≥ 3](/img/revistas/ruma/v46n1/1a04275x.png)
![n](/img/revistas/ruma/v46n1/1a04276x.png)
![ℤ4](/img/revistas/ruma/v46n1/1a04277x.png)
![J˜:= [01 -01]](/img/revistas/ruma/v46n1/1a04278x.png)
![j,l ≥ 1](/img/revistas/ruma/v46n1/1a04279x.png)
![k ≥ 0](/img/revistas/ruma/v46n1/1a04280x.png)
![]() | (2.1) |
where .
![B ∈ O(n) j,k](/img/revistas/ruma/v46n1/1a04283x.png)
![B 4 = Id j,k](/img/revistas/ruma/v46n1/1a04284x.png)
![B ∈ SO(n) j,k](/img/revistas/ruma/v46n1/1a04285x.png)
![k](/img/revistas/ruma/v46n1/1a04286x.png)
![Λ = ℤe1 ⊕ ⋅⋅⋅ ⊕ ℤen](/img/revistas/ruma/v46n1/1a04287x.png)
![n ℝ](/img/revistas/ruma/v46n1/1a04288x.png)
![j,k,l](/img/revistas/ruma/v46n1/1a04289x.png)
![]() | (2.2) |
![Λ](/img/revistas/ruma/v46n1/1a04291x.png)
![B j,k](/img/revistas/ruma/v46n1/1a04292x.png)
![(Bm - Id) me = 0 ∈ Λ j,k 4 n](/img/revistas/ruma/v46n1/1a04293x.png)
![0 ≤ m ≤ 3](/img/revistas/ruma/v46n1/1a04294x.png)
![∑3 m e ∑3 m ( m=0 Bj,k)-n4 = en ∈ Λ \( m=0 B j,k)Λ](/img/revistas/ruma/v46n1/1a04295x.png)
![Γ j,k](/img/revistas/ruma/v46n1/1a04296x.png)
![]() | (2.3) |
of compact flat manifolds with holonomy group . It is easy to see that the cardinality of
is given by
.
![Mj,k ∈ Fn](/img/revistas/ruma/v46n1/1a04301x.png)
![]() | (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
.
![ɛ](/img/revistas/ruma/v46n1/1a04349x.png)
![Γ](/img/revistas/ruma/v46n1/1a04350x.png)
![(n + 1)](/img/revistas/ruma/v46n1/1a04351x.png)
![]() | (2.5) |
where and
is defined by the equation
.
![ɛ](/img/revistas/ruma/v46n1/1a04356x.png)
![γ = BLb ∈ Γ](/img/revistas/ruma/v46n1/1a04357x.png)
![λ ∈ Λ](/img/revistas/ruma/v46n1/1a04358x.png)
![]() | (2.6) |
![ɛ](/img/revistas/ruma/v46n1/1a04360x.png)
![M Γ](/img/revistas/ruma/v46n1/1a04361x.png)
![γd ∈ L Λ](/img/revistas/ruma/v46n1/1a04362x.png)
![ɛ∣Λ : Λ → {±1}](/img/revistas/ruma/v46n1/1a04363x.png)
![γ = BLb ∈ Γ](/img/revistas/ruma/v46n1/1a04364x.png)
![]() | (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
![k k ɛ(γ Lλ) = ɛ(γ) ɛ(L λ)](/img/revistas/ruma/v46n1/1a04393x.png)
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
![ɛ(γkL λγlLλ′) = ɛ(BkLb(k)+ λBlLb(l)+λ′) = ɛ(Bk+lLB -l(b(k)+λ)+b(l)+λ′) k+l k+l ′ k l = ɛ(γ LB -lλ+λ′) = ɛ(γ) ɛ(L λ)ɛ(L λ) = ɛ(γ L λ)ɛ(γ L λ′)](/img/revistas/ruma/v46n1/1a04403x.png)
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
![ℤ4](/img/revistas/ruma/v46n1/1a04413x.png)
![Mj,k](/img/revistas/ruma/v46n1/1a04414x.png)
![k = 2k0](/img/revistas/ruma/v46n1/1a04415x.png)
![n- j 2](/img/revistas/ruma/v46n1/1a04416x.png)
![ɛ](/img/revistas/ruma/v46n1/1a04417x.png)
![(n + 1)](/img/revistas/ruma/v46n1/1a04418x.png)
![(δ1,...,δn,σuBj,k)](/img/revistas/ruma/v46n1/1a04419x.png)
![]() | (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
![δ = ɛ(γ )4 = u4 = x(π,...,π ,2π,...,2π ,0,...,0) = (- 1)j, n j,k Bj,k ◟--◝◜--◞ ◟---◝◜---◞ j k0](/img/revistas/ruma/v46n1/1a04430x.png)
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.
![n F](/img/revistas/ruma/v46n1/1a04435x.png)
![]() | (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![D](/img/revistas/ruma/v46n1/1a04454x.png)
![Fn](/img/revistas/ruma/v46n1/1a04455x.png)
![F ⁄= ∅ 1](/img/revistas/ruma/v46n1/1a04456x.png)
![n B F1 = {B ∈ F : dim( ℝ ) = 1}](/img/revistas/ruma/v46n1/1a04457x.png)
![n = 2m + 1](/img/revistas/ruma/v46n1/1a04458x.png)
![]() | (3.1) |
![n Mj,k ∈ F 1](/img/revistas/ruma/v46n1/1a04460x.png)
![]() | (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
.
![j ω(j) := 3-(-21)-](/img/revistas/ruma/v46n1/1a04465x.png)
![]() | (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
![{ ( t) 4r-1∣Λɛσm,0,μ∣ ± (- 1)r 2r-1 (- 1)t2r + σ (- 1)[2] μ = 2t+21 d±μ(ɛσm,0) = r-1 4 ∣Λɛσm,0,μ∣ otherwise](/img/revistas/ruma/v46n1/1a04475x.png)
for , where
in both cases and
is as in (3.4).
Furthermore, for every there are no non-trivial harmonic spinors, that is,
.
![j,k](/img/revistas/ruma/v46n1/1a04481x.png)
![γ = Bj,kL en4-](/img/revistas/ruma/v46n1/1a04482x.png)
![Γ j,k](/img/revistas/ruma/v46n1/1a04483x.png)
![1 ≤ h ≤ 3](/img/revistas/ruma/v46n1/1a04484x.png)
![b ∈ ℝn h](/img/revistas/ruma/v46n1/1a04485x.png)
![γh = Bh L j,k bh](/img/revistas/ruma/v46n1/1a04486x.png)
![]() | (3.6) |
![F1(Γ j,k) ⁄= ∅](/img/revistas/ruma/v46n1/1a04488x.png)
![nB = nB3 = 1](/img/revistas/ruma/v46n1/1a04489x.png)
![nB2 = 1](/img/revistas/ruma/v46n1/1a04490x.png)
![k = 0](/img/revistas/ruma/v46n1/1a04491x.png)
![]() | (3.7) |
where is Kronecker's delta function.
Now, since , the maximal torus in
, we can take
,
and
since
![xγh = ɛ(γh) = ɛ(γ)h = (σx γ)h](/img/revistas/ruma/v46n1/1a04499x.png)
and for
(see (1.2)). Furthermore, since
and for
we can take
, then
for each
. Moreover,
and
, because
(see Preliminaries). On the other hand
.
![Λ](/img/revistas/ruma/v46n1/1a04511x.png)
![n ℤ](/img/revistas/ruma/v46n1/1a04512x.png)
![* Λ = Λ](/img/revistas/ruma/v46n1/1a04513x.png)
![* Λɛ = Λ + u ɛ](/img/revistas/ruma/v46n1/1a04514x.png)
![∑ uɛ = {i:δi= -1}ei](/img/revistas/ruma/v46n1/1a04515x.png)
![δi = 1](/img/revistas/ruma/v46n1/1a04516x.png)
![1 ≤ i ≤ n - 1](/img/revistas/ruma/v46n1/1a04517x.png)
![δn = (- 1)j](/img/revistas/ruma/v46n1/1a04518x.png)
![Λ *ɛσ = ℤe1 ⊕ ⋅⋅⋅ ⊕ ℤen j,k](/img/revistas/ruma/v46n1/1a04519x.png)
![j](/img/revistas/ruma/v46n1/1a04520x.png)
![* 1 Λ ɛσj,k = ℤe1 ⊕ ⋅⋅⋅ ⊕ ℤen -1 ⊕ (ℤ + 2)en](/img/revistas/ruma/v46n1/1a04521x.png)
![j](/img/revistas/ruma/v46n1/1a04522x.png)
![]() | (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) |
![χL -n- 1(x γh) = - χL+n-1(x γh)](/img/revistas/ruma/v46n1/1a04538x.png)
![]() | (3.10) |
By (1.4), the characters have the expression
![h m-1( hπ j hπ k0 m hπ j hπ k0) χL±n-1(xγh) = σ 2 (cos(-4 )) (cos(-2 )) ± i (sin( 4-))(sin(-2 )) .](/img/revistas/ruma/v46n1/1a04541x.png)
![1 ≤ h ≤ 3](/img/revistas/ruma/v46n1/1a04542x.png)
![]() | (3.11) |
Case 1: . Let
or
. Suppose that
is even, then
. If
is even,
because
. Let
with
. Since
, by (3.10) we have
![± μ-21+[h2] S h (μ) = - 2i (- 1) χL ±n-1(x γh).](/img/revistas/ruma/v46n1/1a04555x.png)
Replacing these values in (3.7) we get
![( ) d± (ɛσ ) = 1 2m- 1∣Λ σ ∣ - 2 i(- 1)t(χ ± (x ) - χ ± (x 3)) μ j,k 4( ɛj,k,μ Ln-1 γ Ln-1 γ ) 1 m- 1 σ t+1 m m+1 √2- j( k0) = 4 2 ∣Λ ɛj,k,μ∣ ± σ (- 1) 2 i ( 2 ) 1 - (- 1) (3.12) r-1 t+r m-1-[j] = 4 ∣Λɛσj,k,μ ∣ ± σ (- 1) 2 2](/img/revistas/ruma/v46n1/1a04556x.png)
where we have used that and that
is odd, because
is even and
.
![j](/img/revistas/ruma/v46n1/1a04561x.png)
![2t+1 μ = 2](/img/revistas/ruma/v46n1/1a04562x.png)
![t ∈ ℕ0](/img/revistas/ruma/v46n1/1a04563x.png)
![]() | (3.13) |
Then, by using that is even because
is odd, we get that
![( √- ) d± (ɛσ ) = 1 2m- 1∣Λ σ ∣ - 2 i(- 1)[t2](-2)(χ ± (x ) + χ ± (x 3)) μ j,k 4 ( ɛj,k,μ 2 Ln-1 γ Ln-1 γ ) 1 m- 1 [t2]+1 m m+1 √2-j+1( k0) = 4 2 ∣Λ ɛσj,k,μ∣ ± σ (- 1) 2 i ( 2 ) 1 + (- 1) (3.14) r-1 r+[t] m -2-[j] = 4 ∣Λɛσj,k,μ∣ ± σ (- 1) 2 2 2.](/img/revistas/ruma/v46n1/1a04567x.png)
Taking , with
, from
and
we finally obtain expression (3.5).
![k = 0](/img/revistas/ruma/v46n1/1a04572x.png)
![j](/img/revistas/ruma/v46n1/1a04573x.png)
![2t+1- μ = 2](/img/revistas/ruma/v46n1/1a04574x.png)
![t ∈ ℕ0](/img/revistas/ruma/v46n1/1a04575x.png)
![]() | (3.15) |
by (3.10); and, for , by (3.11) we obtain
![√- ( π π ) S± (μ) = σ 2m -1(-2)m e--2ihμ((- 1)[h2] ± im)+ e2 ihμ((- 1)[h2] ∓ im) h 2 ( ) m -1 √2-m [h2] πhμ m πhμ = σ 2 (2 ) (- 1) 2cos( 2 ) ± i (- 2i)sin( 2 ) .](/img/revistas/ruma/v46n1/1a04578x.png)
![( ) ( ) cos π(2t+1) - cos 3π(2t+1)- = 0 4 4](/img/revistas/ruma/v46n1/1a04579x.png)
![]() | (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
.
![A](/img/revistas/ruma/v46n1/1a04607x.png)
![M](/img/revistas/ruma/v46n1/1a04608x.png)
![]() | (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
![n = 2m + 1 = 4r + 3](/img/revistas/ruma/v46n1/1a04623x.png)
![ℤ 4](/img/revistas/ruma/v46n1/1a04624x.png)
![M ∈ Fn j,k 1](/img/revistas/ruma/v46n1/1a04625x.png)
![σ ɛj,k](/img/revistas/ruma/v46n1/1a04626x.png)
![k > 0](/img/revistas/ruma/v46n1/1a04627x.png)
![]() | (4.2) |
![j Cj,σ = σ (- 1)r 2m+1-ω(j)- [2]](/img/revistas/ruma/v46n1/1a04629x.png)
![]() | (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
.
![]() | (4.4) |
![k > 0](/img/revistas/ruma/v46n1/1a04642x.png)
![]() | (4.5) |
where .
If is even, the series in (4.5) equals
![∞ ( ∞ ∞ ) ∑ --(--1)t-- 1- ∑ ---1---- ∑ ---1---- 1-( 1 3) (2t + 1)s = 4s (t + 1)s - (t + 3)s = 4s ζ(s, 4) - ζ(s, 4) . t=0 t=0 4 t=0 4](/img/revistas/ruma/v46n1/1a04646x.png)
![2t](/img/revistas/ruma/v46n1/1a04647x.png)
![2t + 1](/img/revistas/ruma/v46n1/1a04648x.png)
![]() | (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
![r r ∑∞ [t2] t r ηɛσ (s) = (--1)-2- σ-(--1)--+-(--1)-2-. m,0 (2 π)s t=0 (t + 12)s](/img/revistas/ruma/v46n1/1a04656x.png)
The series above equals
![∑∞ (σ + 2r) (σ - 2r) (σ - 2r) (σ + 2r) ------1-s + ------3s-- -----5-s-- ------7-s t=0 (4t + 2) (4t + 2) (4t + 2) (4t + 2)](/img/revistas/ruma/v46n1/1a04657x.png)
![4t + h](/img/revistas/ruma/v46n1/1a04658x.png)
![0 ≤ h ≤ 3](/img/revistas/ruma/v46n1/1a04659x.png)
![]() | (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
![-2--( 1 7 ) --2-( 3 5) ηɛ+1,0(s) = (8π)s ζ(s,8) - ζ(s,8) , ηɛ-1,0(s) = (8π)s ζ(s, 8) - ζ(s, 8)](/img/revistas/ruma/v46n1/1a04680x.png)
and by (4.9)
![ηɛ+ (0) = 32, ηɛ- (0) = - 12. 1,0 1,0](/img/revistas/ruma/v46n1/1a04681x.png)
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
![ηɛσ1,4(0) = - 4σ, ηɛσ2,2(0) = - 2σ, ηɛ+ (0) = - 4, ηɛ- (0) = 3. 3,0 3,0](/img/revistas/ruma/v46n1/1a04692x.png)
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
![∑N ( ) ηΓ ,ɛ(s) = C-Γ ,ɛ- fj,Γ ,ɛ(s) ζ(s,αj) - ζ(s,1 - αj) (2 π)s j=1](/img/revistas/ruma/v46n1/1a04700x.png)
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