The Dirac Operator on Nilmanifolds and Collapsing Circle Bundles

by

Bernd Ammann and Christian Bär

November, 1997

Abstract

We compute the spectrum of the Dirac operator on 3-dimensional Heisenberg manifolds. The behavior under collapse to the 2-torus is studied. Depending on the spin structure either all eigenvalues tend to ±∞ or there are eigenvalues converging to those of the torus. This is shown to be true in general for collapsing circle bundles with totally geodesic fibers. Using the Hopf fibration we use this fact to compute the Dirac eigenvalues on complex projective space including the multiplicities.

Finally, we show that there are 1-parameter families of Riemannian nilmanifolds such that the Laplacian on functions and the Dirac operator for certain spin structures have constant spectrum while the Laplacian on 1-forms and the Dirac operator for the other spin structures have nonconstant spectrum. The marked length spectrum is also constant for these families.

Keywords: Dirac operator, nilmanifolds, Heisenberg manifolds, circle bundles, collapse, isospectral deformation

Mathematics Classification: 58G25, 58G30, 53C25, 53C30

0 Introduction

The Dirac operator on a Riemannian spin manifold is a natural elliptic first order differential operator. If the underlying manifold is closed, then the spectrum is discrete. In order to understand the relation of the Dirac eigenvalues and geometric data it is desirable to have examples for which one can explicitly compute the spectrum. These examples can be used to test conjectures. For example, they show that the spectrum does not determine the manifold, not even topologically. There are examples of Dirac isospectral pairs of spherical space forms with different fundamental groups [3].

In the following table we collect those closed Riemannian spin manifolds known to us for which the Dirac spectrum has been computed.

ℝ^{n}/Γ  flat tori  [15]
S^{n}  spheres of constant curvature  [33], [3], [34, 35], [11]
S^{n}/Γ  spherical space forms  [3]
S^{2m+1}  spheres with Berger metrics  [20] for m = 1

[4] for general m

S^{3}/ℤ_{k}  3-dim. lens spaces with Berger metric  [2]
G  simply connected compact Lie groups  [14]
ℂP^{2m−1}  complex projective spaces  [9, 10], [30]
ℍP^{m}  quaternionic projective spaces  [8] for m = 2

[22] for general m

Gr_{2}(ℝ^{2m}) real Grassmannians [31] for m = 3

[32] for general m

Gr_{2p}(ℝ^{2m})  real Grassmannians  [28, 29]
Gr_{2}(ℂ^{m+2})  complex Grassmannians  [23]
G_{2}/SO(4)  [28, 29]

We compute the Dirac spectrum for another class of Riemannian spin manifolds, for the 3-dimensional Heisenberg manifolds. They are of the form M = Γ\G where G is the 3-dimensional Heisenberg group and Γ is a cocompact lattice. The metric comes from a left invariant metric on G. We restrict to three dimensions mostly for simplicity. The methods also work in higher dimensions.

After collecting some facts from spin geometry and from the representation theory of nilpotent groups in the first two sections we perform the computation of the Dirac eigenvalues in Section 3 (Theorem 3.1). Our Heisenberg manifolds are circle bundles over the flat 2-torus. The explicit formulas for the Dirac eigenvalues allow us to study their behavior under collapse to the 2-torus, i.e. we let the length of the fibers tend to zero. It turns out that the behavior of the eigenvalues depends essentially on the spin structure. For some spin structures all eigenvalues tend to ±∞. For the other spin structures most of the eigenvalues also tend to ±∞ but some converge to exactly the eigenvalues of the torus (Corollary 3.3).

As it turns out in Section 4 this is a general fact for the Dirac operator on the total space of a collapsing circle bundle with totally geodesic fibers over any Riemannian manifold. There are two types of spin structures which we call projectable and nonprojectable. In the nonprojectable case all eigenvalues tend to ±∞ (Theorem 4.5) whereas in the projectable case there are also eigenvalues converging to exactly the eigenvalues of the base (Theorem 4.1).

As an example we apply this to the Hopf fibration over the complex projective space ℂP^{m}. Using explicit formulas for the Dirac eigenvalues on Berger spheres [4] we see that for m even (when ℂP^{m} is nonspin) all eigenvalues go to ±∞. If m is odd (when ℂP^{m} is spin) the limits of the eigenvalues give us a new computation of the Dirac spectrum of ℂP^{m} confirming (and simplifying) the results in [9, 10, 30]. We also obtain formulas for the multiplicities (Theorem 4.6) which have not been computed before.

In the last section we discuss the change of the spectrum of the Dirac operator on certain pairs or continuous families of Riemannian nilmanifolds. A Riemannian nilmanifold is a nilmanifold Γ\G endowed with a Riemannian metric whose lift to G is left invariant. Here we assume that G is a simply connected nilpotent Lie group and that Γ is a cocompact discrete subgroup. This generalizes Heisenberg manifolds. Besides the Dirac spectrum we look at several different spectral invariants which have already been studied by other authors: the spectrum of the Laplacian on functions, the spectrum of the Laplacian on forms, the length spectrum, i.e. the set of all lengths of closed geodesics, and the marked length spectrum, i.e. the set of all lengths of closed geodesics in each free homotopy class.

The questions behind the study of these examples are: “How far is the geometry on a Riemannian nilmanifold determined by these spectral invariants?” and “How are these invariants related to each other?” The class of nilmanifolds provides us with many examples for the discussion of these questions.

Milnor gave in [24] the first example of a pair of Riemannian manifolds (16dimensional tori) which are nonisometric but isospectral for the Laplace operator on functions.

Gordon and Wilson [18] constructed a continuous family of nonisometric Riemannian 2-step nilmanifolds with the same spectrum of the Laplacian on functions and forms. This is the first example of a nontrivial Laplace-isospectral deformation. This family and other deformations of the Gordon-Wilson type, i.e. via almost inner automorphisms, do not only have the same spectrum for the Laplacian on functions and on forms, but also the same marked length spectrum. In the 2-step case with a left invariant metric Eberlein [13, Theorem 5.20] showed that two 2-step nilmanifolds having the same marked length spectrum arise via almost inner automorphisms and therefore have the same spectrum of the Laplacian on functions and on forms.

This statement is false if we drop the condition “2-step”. Gornet [17] constructed two families of deformations of 3-step Riemannian nilmanifolds which are isospectral for the Laplacian on functions and for the marked length spectrum, but not for the spectrum of the Laplacian on 1-forms. We will show that these families are Dirac isospectral for the nonprojectable spin structures while they have nonconstant Dirac spectrum for the projectable spin structures.

1 Review of Spin Geometry

In this section we collect a few facts on spin manifolds and the Dirac operator which we will use later on. For details the reader may consult [6] or [21]. Let M be an oriented Riemannian manifold of dimension n. Let P_{SO}(M) be the set of all oriented orthonormal bases of all tangent spaces of M. Obviously, P_{SO}(M) is an SO(n)-principal bundle. A spin structure on M consists of a Spin(n)-principal bundle P_{Spin}(M) over M together with a twofold covering map φ : P_{Spin}(M) → P_{SO}(M) such that the diagram

P_{Spin}(M) × Spin(n) → P_{Spin}(M)
↘
↓ φ × Θ ↓ φ M
↗
P_{SO}(M) × SO(n) → P_{SO}(M)

commutes. Here Θ : Spin(n) → SO(n) is the standard twofold covering map and the horizontal arrows are given by the principal bundle structure.

A manifold which admits such a spin structure is called a spin manifold. Being spin is a global condition which is equivalent to the vanishing of the second Stiefel-Whitney class of the tangent bundle, w_{2}(M) = 0. The number of different spin structures, if any, is given by #H^{1}(M, ℤ_{2}). In particular, a simply connected manifold has at most one spin structure.

Let us from now on assume that M is spin and that a spin structure on M is fixed. There is a unitary representation Σ_{n} of Spin(n) of dimension 2^{[n/2]}, called the spinor representation. If n is odd, then Σ_{n} is irreducible. The associated hermitian vector bundle ΣM := P_{Spin}(M)×_{Spin(n)}Σ_{n} is called the spinor bundle.

The Levi-Civita connection on M induces a natural connection ∇^{Σ} on ΣM. It can be described as follows. Let e_{1}, …, e_{n} be a local orthonormal tangent frame defined over U ⊂ M. Let Γ_{ij}^{k} : U → ℝ be the corresponding Christoffel symbols,

∇_{e_{i}}e_{j} = ∑_{k=1}^{n} Γ_{ij}^{k}e_{k}. Then (e_{1}, …, e_{n}) is a local section of P_{SO}(M). Let q be a lift

to P_{Spin}(M), i.e. φ ∘ q = (e_{1}, …, e_{n}). Then q defines a trivialization of ΣM over U, ΣM|_{U} = U × Σ_{n}, with respect to which we have the following formula for ∇^{Σ}:

∇_{e_{i}}^{Σ}σ = ∂_{e_{i}}σ + \frac{1}{4} ∑_{j,k=1}^{n} Γ_{ij}^{k}γ(E_{j})γ(E_{k})σ  (1.1)

Since the spinor representation of Spin(n) extends to a representation of the Clifford algebra Cl(n) there is a well-defined Clifford multiplication T_{p}M⊗Σ_{p}M → Σ_{p}M, X ⊗ σ → γ(X)σ. It satisfies the relations γ(X)γ(Y) + γ(Y)γ(X) + 2⟨X, Y ⟩ id = 0.

For example in dimension n = 3, one can choose a basis for Σ_{3} with respect to which

γ(E_{1}) = ( \begin{matrix}0 & i \\ i & 0 \end{matrix} ) , γ(E_{2}) = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) , γ(E_{3}) = ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) ,

where E_{1}, E_{2}, E_{3} denotes the standard basis of ℝ^{3} ⊂ Cl(3).

The Dirac operator D acts on the sections of ΣM. It is defined by Dσ = ∑_{i=1}^{n} γ(e_{i})∇_{e_{i}}^{Σ}σ where e_{1}, …, e_{n} is any orthonormal basis of the tangent space. If one

has, in addition, a complex vector bundle E over M equipped with a connection one can form the twisted Dirac operator D^{E} acting on sections of ΣM ⊗E by using the tensor product connection and Clifford multiplication on the first factor.

The Dirac operator is a formally self adjoint elliptic differential operator of first order. If the manifold M is closed, then D has discrete real spectrum.

Let G be a simply connected Lie group with a left invariant metric. We regard the elements of the Lie algebra 𝔤 as left invariant vector fields. The choice of an oriented orthonormal basis of 𝔤 yields a trivialization of the frame bundle P_{SO}(G) = G × SO(n). The unique spin structure can be written as P_{Spin}(G) = G×Spin(n), φ = id×Θ. Spinor fields are then simply maps G → Σ_{n}.

Let Γ ⊂ G be a lattice. Spin structures of M = Γ\G correspond to homomorphisms ϵ : Γ → ℤ/2ℤ = {−1, 1}. The corresponding spin structure is given by P_{Spin,ϵ}(Γ\G) = G×_{Γ}Spin(n) where g_{0} ∈ Γ acts on G by left multiplication and on Spin(n) by multiplication with the central element ϵ(g_{0}). Spinor fields on M can then be identified with ϵ-equivariant maps σ : G → Σ_{n}, i.e. σ(g_{0}g) = ϵ(g_{0})σ(g) for all g ∈ G, g_{0} ∈ Γ. Denote the corresponding Hilbert space of square integrable spinors by L^{2}(Σ_{ε}M).

2 Review of Kirillov Theory

In this section we summarize some facts from the representation theory of nilpotent groups which will be of importance for our study of nilmanifolds. For details see [12]. Let G be a simply connected nilpotent Lie group with Lie algebra 𝔤. The exponential map exp : 𝔤 → G is a global diffeomorphism whose inverse we denote by log : G → 𝔤.

Example. Let

G = \{( \begin{matrix}1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{matrix} )| x, y, z ∈ ℝ\}

be the 3-dimensional Heisenberg group. Its Lie algebra is

𝔤 = \{( \begin{matrix}0 & x & z \\ 0 & 0 & y \\ 0 & 0 & 0 \end{matrix} )| x, y, z ∈ ℝ\} ,

the 3-dimensional Heisenberg algebra. Write

\begin{aligned}g(x, y, z) & := & ( \begin{matrix}1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{matrix} ) and \\ X(x, y, z) & := & ( \begin{matrix}0 & x & z \\ 0 & 0 & y \\ 0 & 0 & 0 \end{matrix} ) . \end{aligned}

Then the exponential map is given by

exp(X(x, y, z)) = g (x, y, z + \frac{1}{2}x · y) and its inverse by
log(g(x, y, z)) = X (x, y, z − \frac{1}{2}x · y) .

Define generators of 𝔤 by

\begin{aligned}X & := & X(1, 0, 0), \\ Y & := & X(0, 1, 0), \\ Z & := & X(0, 0, 1). \end{aligned}

Then [X, Y] = Z ist the only nontrivial commutator and Z spans the center of 𝔤.

Any Lie group acts on its Lie algebra via the adjoint representation

Ad : G → End(𝔤), Ad_{g}(Y) = \frac{d}{dt} (g · exp(tY) · g^{−1})|_{t=0}

and on the dual space 𝔤^{∗} via the coadjoint representation

Ad^{∗} : G → End(𝔤^{∗}), (Ad_{g}^{∗}l)(Y) = l(Ad_{g^{−1}}Y), g ∈ G, Y ∈ 𝔤, l ∈ 𝔤^{∗}.

The orbits for the coadjoint representation are called coadjoint orbits. Given any l ∈ 𝔤^{∗} there exists, in the nilpotent case, a maximal isotropic subspace 𝔪 ⊂ 𝔤 for the antisymmetric bilinear form (X, Y) → l([X, Y]) which is also a subalgebra. Such subalgebras are called polarizing subalgebras.

Example. Let G be the 3-dimensional Heisenberg group. Let l ∈ 𝔤^{∗}. We have to distinguish two cases depending on whether l vanishes on the center or not.

Case 1: l(Z) = 0.

(Ad_{g(x,y,z)}^{∗}l) (X(ξ, η, ζ)) = l (Ad_{g(−x,−y,−z+xy)}X(ξ, η, ζ))
= l (X(ξ, η, ζ + ξy − xη))
= l(ξ · X + η · Y + (ζ + ξy − xη) · Z)
= l(ξ · X + η · Y)
= l (X(ξ, η, ζ)) .

Hence Ad_{g}^{∗} l = l for all g ∈ G, i.e. the orbit of l consists of l only. The bilinear form (X_{1}, X_{2}) → l([X_{1}, X_{2}]) vanishes in this case. Hence 𝔪 = 𝔤 is the unique polarizing subalgebra.

Case 2: l(Z) = τ ≠ 0.

Write l = α · X^{∗} + β · Y^{∗} + τ · Z^{∗} where X^{∗}, Y^{∗}, Z^{∗} ∈ 𝔤^{∗} is the basis dual to X, Y, Z. The computation above shows

\begin{aligned}Ad_{g(x,y,z)}^{∗}l & = & (α + τy) · X^{∗} + (β − τx) · Y^{∗} + τ · Z^{∗} \\ & = & l + τy · X^{∗} − τx · Y^{∗}. \end{aligned}

Hence the coadjoint orbit through l is the affine hyperplane through l spanned by X^{∗} and Y^{∗}. As a polarizing subalgebra we can choose e.g. 𝔪 = ℝ · X ⊕ ℝ · Z or 𝔪 = ℝ · Y ⊕ ℝ · Z.

For a general simply connected nilpotent Lie group G pick an l ∈ 𝔤^{∗}. We will construct a continuous unitrary representation of G associated to l. Choose a polarizing subalgebra 𝔪 ⊂ 𝔤 for l. Let M := exp(𝔪) be the corresponding subgroup of G. By the definition

ρ̃_{l}(exp(X)) := e^{2πil(X)}

we obtain a well-defined 1-dimensional unitary representation ρ̃_{l} of M. Induction yields a continuous unitary representation ρ_{l} of G, ρ_{l} = Ind(M ↑ G, ρ̃_{l}). One checks that ρ_{l} does not depend (up to equivalence) upon the choice of 𝔪, that ρ_{l_{1}} is equivalent to ρ_{l_{2}} if and only if l_{1} and l_{2} are in the same coadjoint orbit and that every irreducible unitary representation of G is equivalent to ρ_{l} for some l. In other words, Kirillov theory sets up a bijection between the set of coadjoint orbits and Ĝ, the set of equivalence classes of irreducible unitary representations of G.

Example. Let G be the 3-dimensional Heisenberg group, let l = αX^{∗} + βY^{∗} + τZ^{∗} ∈ 𝔤^{∗}. Again, we have to distinguish the two cases τ = 0 and τ ≠ 0.

Case 1. τ = 0.

In this case M = G, hence ρ_{l} = ρ̃_{l} is a 1-dimensional irreducible representation of G. Since coadjoint orbits of those l with τ = 0 are just points the corresponding representations are parametrized by the two parameters α, β ∈ ℝ, ρ_{l} =: ρ_{α,β}, where

ρ_{α,β}(g(x, y, z)) = e^{2πi(αX^{∗}+βY^{∗}+τZ^{∗})(X(x,y,z− \frac{1}{2} xy))}  (2.1)
= e^{2πi(αx+βy)}

Case 2. τ ≠ 0.

Since the representation ρ_{l} does not change if we replace l by another linear form in its coadjoint orbit we may assume l = τ · Z^{∗}. Choose the polarizing subalgebra 𝔪 = ℝ · Y ⊕ ℝ · Z.

The representation space of the induced representation ρ_{l} is the space of L^{2}- sections over the homogeneous space G/M in the line bundle associated with ρ̃_{l}, the Hilbert space L^{2}(G/M, G ×ρ̃_{l} ℂ). Sections in G ×ρ̃_{l} ℂ can be identified with ρ̃_{l}-equivariant maps f : G → ℂ, i.e. f(gm^{−1}) = ρ̃_{l}(m)f(g) for all m ∈ M, g ∈ G. There is a diffeomorphism ℝ → G/M, t → [g(t, 0, 0)], where [·] denotes the equivalence class in G/M. Its inverse is simply given by [g(x, y, z)] → x. Using this diffeomorphism we can construct out of a complex valued L^{2}-function u ∈ L^{2}(ℝ, ℂ) a ρ̃_{l}-equivariant map f_{u} : G → ℂ by the formula

f_{u}(g(x, y, z)) = f_{u}(g(x, 0, 0) · g(0, y, z − xy)
= ρ̃_{l} (g(0, y, z − xy)^{−1}) · f_{u}(g(x, 0, 0))
= ρ̃_{l}(g(0, −y, −z + xy) · u(x)
= e^{2πil(X(0,−y,−z+xy))} · u(x)
= e^{−2πiτ(z−xy)} · u(x).

Using the isomorphism L^{2}(ℝ, ℂ) → L^{2}(G/M, G ×ρ̃_{l} ℂ), u → f_{u}, we can replace L^{2}(G/M, G ×ρ̃_{l} ℂ) by L^{2}(ℝ, ℂ) as the representation space for ρ_{l}.

How does a given element g(x, y, z) ∈ G act on u ∈ L^{2}(ℝ, ℂ) via ρ_{l}? On the corresponding f_{u} it acts by

(ρ_{l}(g(x, y, z))f_{u})(g(x̃, ỹ, z̃)) = f_{u}(g(x, y, z)^{−1} · g(x̃, ỹ, z̃))
= f_{u}(g(−x + x̃, −y + ỹ, −z + z̃ + xy − xỹ))
= e^{−2πiτ(−z+z̃−x̃(ỹ−y))} · u(x̃ − x).

Putting ỹ = z̃ = 0 yields the corresponding function in L^{2}(ℝ, ℂ)

(ρ_{l}(g(x, y, z))u)(x̃) = e^{−2πiτ(−z+x̃y)} · u(x̃ − x).

Denote the representation ρ_{l} considered as acting on L^{2}(ℝ, ℂ) by ρ_{τ}.

Summarizing we see that the 3-dimensional Heisenberg group has two families of irreducible unitary representations, the first one parametrized by α, β ∈ ℝ and acting on ℂ:

ρ_{α,β}(g(x, y, z)) = e^{2πi(αx+βy)}.

The second family is parametrized by τ ∈ ℝ − {0} and acts on L^{2}(ℝ, ℂ):

(ρ_{τ}(g(x, y, z))u) (t) = e^{2πiτ(z−ty)}u(t − x).  (2.2)

Note that ρ_{τ} is determined by the action of the center of G

ρ_{τ}(g(0, 0, z)) = e^{2πiτz} · Id.

We compute the action of the Heisenberg algebra 𝔤 on L^{2}(ℝ, ℂ):

(ρ_{τ})_{∗}(X)u(t) = \frac{d}{ds}ρ_{τ}(expsX)u(t)|_{s=0}
= \frac{d}{ds}ρ_{τ}(g(s, 0, 0))u(t)|_{s=0}
= \frac{d}{ds}e^{2πiτ·0}u(t − s)|_{s=0}
= −u′(t).

Similarly,

\begin{aligned}(ρ_{τ})_{∗}(Y)u(t) & = & −2πiτ t u(t) \\ (ρ_{τ})_{∗}(Z)u(t) & = & 2πiτ u(t). \end{aligned}

The Hilbert space L^{2}(ℝ, ℂ) has the well-known basis of Hermite functions

h_{k}(t) = e^{\frac{t^{2}}{2}} (\frac{d}{dt})^{k} e^{−t^{2}}, k ∈ ℕ_{0}.

They satisfy the relations

\begin{matrix}hk^{′}(t) = t h_{k}(t) + h_{k+1}(t), \\ h_{k+2}(t) + 2t h_{k+1}(t) + 2(k + 1)h_{k}(t) = 0. \end{matrix}

We look at the L^{2}-basis u_{k}(t) = h_{k}(\sqrt{2π|τ|}t). Then the relations translate into

2π|τ|t u_{k}(t) + \sqrt{2π|τ|} u_{k+1}(t) = u′_{k}(t),  (2.3)
u_{k+2}(t) + 2\sqrt{2π|τ|} t u_{k+1}(t) + 2(k + 1)u_{k}(t) = 0.  (2.4)

This basis will be used in the next section for the computation of the Dirac spectrum of 3-dimensional Heisenberg manifolds.

3 Heisenberg Manifolds

In this section let G denote the 3-dimensional Heisenberg group. Let r be a positive integer. Define

Γ_{r} := {g(r · x, y, z) ∈ G | x, y, z ∈ ℤ} .

Then Γ_{r} is a uniformly discrete cocompact lattice in G. For every lattice in G there is an automorphism of G mapping the lattice to some Γ_{r}, [19, §2]. We want to compute the Dirac spectrum of the Heisenberg manifold M = Γ_{r}\G.

For positive reals d, T > 0 we equip G and M with the left invariant metric for which

\begin{aligned}e_{1} & := & −d · X, \\ e_{2} & := & −d · Y, \\ e_{3} & := & T^{−1} · Z \end{aligned}

form an orthonormal frame. Here X, Y, Z denote the standard generators of 𝔤 (compare last section). Using the Koszul formula we compute the Christoffel symbols

Γ_{12}^{3} = Γ_{23}^{1} = Γ_{32}^{1} = −Γ_{21}^{3} = −Γ_{31}^{2} = −Γ_{13}^{2} = \frac{d^{2}T}{2} , Γ_{ij}^{k} = 0 otherwise.

To emphasize the dependence of M on the parameters r, d, and T we also write M(r, d, T) instead of M. Note that d and T are differential geometric parameters whereas r is topological. The fundamental groups π_{1}(M(r, d, T)) = Γ_{r} are nonisomorphic for different values of r. We regard spinor fields on the universal covering as maps G → Σ_{3}. Choose a basis of Σ_{3} ≅ ℂ^{2} such that Clifford multiplication of the standard basis E_{1}, E_{2}, E_{3} ∈ ℝ^{3} is given by

γ(E_{1}) = ( \begin{matrix}0 & i \\ i & 0 \end{matrix} ) , γ(E_{2}) = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) , γ(E_{3}) = ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) ,

compare first section. The spinor connection on G with respect to the left invariant metric described above is given by (1.1)

∇_{e_{1}}^{Σ}σ = ∂_{e_{1}}σ + \frac{1}{4} · \{\frac{d^{2}T}{2} γ(E_{2})γ(E_{3}) − \frac{d^{2}T}{2} γ(E_{3})γ(E_{2})\} · σ
= ∂_{e_{1}}σ + \frac{d^{2}T}{4} ( \begin{matrix}0 & i \\ i & 0 \end{matrix} ) · σ,
∇_{e_{2}}^{Σ}σ = ∂_{e_{2}}σ + \frac{d^{2}T}{4} ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) · σ,
∇_{e_{3}}^{Σ}σ = ∂_{e_{3}}σ + \frac{d^{2}T}{4} ( \begin{matrix}−i & 0 \\ 0 & i \end{matrix} ) · σ.

For the Dirac operator we obtain

D = ∑_{i=1}^{3} γ(e_{i})∇_{e_{i}}^{Σ} = ∑_{i=1}^{3} γ(e_{i})∂_{e_{i}} − \frac{d^{2}T}{4} .

Spin structures of M(r, d, T) are given by homomorphisms ϵ : Γ_{r} → ℤ/2ℤ = {−1, 1}. Since ℤ/2ℤ is abelian ϵ must factor through Γ_{r}/[Γ_{r}, Γ_{r}]. The commutator subgroup is [Γ_{r}, Γ_{r}] = {g(0, 0, r z)|z ∈ ℤ}. Hence Γ_{r}/[Γ_{r}, Γ_{r}] ≅ r ℤ ⊕ ℤ ⊕ (ℤ/r ℤ). The homomorphism ϵ is determined by its image on generators of Γ_{r}/[Γ_{r}, Γ_{r}], i.e. by

\begin{aligned}δ_{1} & := & ϵ(g(r, 0, 0)), \\ δ_{2} & := & ϵ(g(0, 1, 0)), \\ δ_{3} & := & ϵ(g(0, 0, 1)). \end{aligned}

Put δ := (δ_{1}, δ_{2}, δ_{3}). The first two components δ_{1} and δ_{2} can take the values 1 and −1. The third component δ_{3} can also take both values if r is even, but if r is odd δ_{3} is necessarily δ_{3} = 1. Summarizing, we see that spin structures on M(r, d, T) are given by triples δ = (δ_{1}, δ_{2}, δ_{3}) ∈ (ℤ/2ℤ)^{3} where δ_{3} must be +1 in case r is odd. The corresponding homomorphism is then

ϵ(g(rx, y, z)) = δ_{1}^{x} · δ_{2}^{y} · δ_{3}^{z} ∈ ℤ/2ℤ

and spinor fields are maps σ : G → Σ_{3} = ℂ^{2} such that

σ(g_{0}g) = ϵ(g_{0}) · σ(g), g_{0} ∈ Γ_{r}, g ∈ G.  (3.1)

The Heisenberg group G acts on the Hilbert space L^{2}(Σ_{ε}M) of L^{2}-spinor fields on M by the right regular representation R:

(R(g_{0})σ)(g) = σ(g g_{0}).

At the same time the Clifford algebra Cl(3) acts on this Hilbert space by pointwise Clifford multiplication γ. Note that this action is compatible with condition (3.1). Moreover, the actions of G and of Cl(3) commute. Modules with commuting Gand Cl(3)-operations will be called G-Cl(3)-bimodules.

Differentiation of a spinor field is given by the derived right regular action:

∂_{X}σ = \frac{d}{dt}|_{t=0} R(exp tX)σ = R_{∗}(X)σ, X ∈ 𝔤.

We will decompose the Hilbert space L^{2}(Σ_{ε}M) into closed subspaces invariant under the actions R and γ. These subspaces are then also left invariant by the Dirac operator. Expressed in terms of R and γ the Dirac operator is given by

D = ∑_{i=1}^{3} R_{∗}(e_{i}) ⊗ γ(E_{i}) − \frac{d^{2}T}{4} .  (3.2)

Let the spin structure corresponding to δ = (δ_{1}, δ_{2}, δ_{3}) ∈ (ℤ/2ℤ)^{3} be fixed. Let us start with the case δ_{3} = 1. Then for fixed g ∈ G and spinor field σ : G → Σ_{3} the map

φ_{g} : ℝ → Σ_{3}, φ_{g}(z) = (R(g(0, 0, z))σ)(g),

is 1-periodic, φ_{g}(z + 1) = φ_{g}(z). We expand φ_{g} into a Fourier series

φ_{g}(z) = ∑_{τ∈ℤ}φ_{τ}(g) · e^{2πiτz},

where the Fourier coefficients are given by φ_{τ}(g) = ∫_{0}^{1} R(g(0, 0, t))σ(g)e^{−2πiτt}dt.

Putting z = 0 we obtain

σ(g) = φ_{g}(0) = ∑_{τ∈ℤ}φ_{τ}(g).

We found a first decomposition of L^{2}(Σ_{ε}M) = ⨁_{τ∈ℤ}H_{τ} = H_{0} ⊕ ⨁_{τ∈ℤ−{0}}H_{τ}. On

H_{τ} central elements g(0, 0, z) act via R by multiplication by e^{2πiτz}. Hence H_{τ} decomposes into copies of the irreducible representations ρ_{τ}. The subspace H_{0} is the part on which the center of G acts trivially. Thus it decomposes under R into 1-dimensional representations of the form ρ_{α,β}. We now determine this decomposition.

For σ ∈ H_{0} the map ℝ^{2} ↦ Σ_{3}, (x, y) → σ(g(x, y, 0)), is periodic for the lattice 2(rℤ ⊕ ℤ). Hence we can expand it into the Fourier series

σ(g(x, y, z)) = ∑ ; \begin{matrix}α ∈ \frac{1}{2r}ℤ \\ β ∈ \frac{1}{2}ℤ \end{matrix} a_{α,β} · e^{2πi(αx+βy)}.

The equivariance property (3.1) imposes further restrictions on α and β, namely

σ(g(x, y, z)) = ∑ ; \begin{matrix}α ∈ \frac{1}{2r}ℤ, e^{2πirα} = δ_{1} \\ β ∈ \frac{1}{2}ℤ, e^{2πiβ} = δ_{2} \end{matrix} a_{α,β} · e^{2πi(αx+βy)}.

This yields the decomposition

H_{0} = ⨁ ; \begin{matrix}α ∈ \frac{1}{2r}ℤ, e^{2πirα} = δ_{1} \\ β ∈ \frac{1}{2}ℤ, e^{2πiβ} = δ_{2} \end{matrix} H^{α,β}

where each summand H^{α,β} is isomorphic to ℂ ⊗ Σ_{3} = Σ_{3} with G acting via R by ρ_{α,β} on the first factor (see 2.1 for the definition) and Cl(3) acting via γ on the second.

How does the Dirac operator D (3.2) act on H^{α,β}? We compute

D = −d R_{∗}(X) · γ(E_{1}) − d R_{∗}(Y) · γ(E_{2})

+T^{−1}R_{∗}(Z) · γ(E_{3}) − \frac{d^{2}T}{4}

= −d · 2πiα · ( \begin{matrix}0 & i \\ i & 0 \end{matrix} ) − d · 2πiβ · ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} )

+T^{−1} · 0 · ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) − \frac{d^{2}T}{4} · ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} )

= ( \begin{matrix}−\frac{d^{2}T}{4} & 2πid(β − iα) \\ −2πid(β + iα) & −\frac{d^{2}T}{4} \end{matrix} ) .

The eigenvalues are −\frac{d^{2}T}{4} ± 2πd\sqrt{α^{2} + β^{2}}.

Next we decompose H_{τ}, τ ≠ 0, under the actions of G via R and of Cl(3) via γ. Let σ ∈ H_{τ}. Then σ is determined by f_{σ} : ℝ^{2} → ℂ, for f_{σ}(x, y) = σ(g(x, y, 0)), because

σ(g(x, y, z)) = e^{2πiτz} · f_{σ}(x, y).

Fixing x ∈ ℝ we expand y → f_{σ}(x, y) into a Fourier series and using (3.1) we obtain

f_{σ}(x, y) = ∑_{β∈ \frac{1}{2} ℤ,e^{2πiβ}=δ_{2}} b_{β}^{σ}(x) · e^{2πiβy}.

Using (3.1) once more we get the periodicity property of b_{β}^{σ},

b_{β}^{σ}(x + ξ) = δ_{1}^{ξ/r} · b_{β+τξ}^{σ}(x), x ∈ ℝ, ξ ∈ r · ℤ.

Hence we have |τ| · r independent functions b_{1+ε}^{σ}, …, b_{|τ|·r+ε}^{σ}, with ε = 1/2 for δ_{2} = −1 and ε = 0 for δ_{2} = 1. The map

H_{τ} → ⨁_{j=1}^{|τ|r} L^{2}(ℝ, Σ_{3}) = ⨁_{j=1}^{|τ|r} L^{2}(ℝ, ℂ) ⊗ Σ_{3}
σ ↦ (b_{1+ε}^{σ}, …, b_{|τ|r+ε}^{σ})

is an isomorphism. This decomposition is left invariant by the actions R and γ. Each summand carries a G-Cl(3)-bimodule structure given by the action of R on the first factor and γ on the second. This G-Cl(3)-bimodule structure is equivalent to the G-Cl(3)-bimodule structure given by ρ_{τ} and γ (for the definition of ρ_{τ} see 2.2).

We compute the eigenvalues of the Dirac operator on L^{2}(ℝ, ℂ) ⊗ Σ_{3}.

D = −d(ρ_{τ})_{∗}(X) · γ(E_{1}) − d(ρ_{τ})_{∗}(Y) · γ(E_{2})

+T^{−1}(ρ_{τ})_{∗}(Z) · γ(E_{3}) − \frac{d^{2}T}{4}

= −d · ( \begin{matrix}0 & i \\ i & 0 \end{matrix} ) · (−\frac{d}{dt}) − d · ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) (−2πiτt)

+T^{−1} · ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) · 2πiτ − \frac{d^{2}T}{4} · ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} )

= ( \begin{matrix}−2πT^{−1}τ − \frac{d^{2}T}{4} & −2πidτt + id\frac{d}{dt} \\ 2πidτt + id\frac{d}{dt} & 2πT^{−1}τ − \frac{d^{2}T}{4} \end{matrix} ) .

We use the L^{2}-basis u_{0}, u_{1}, … for L^{2}(ℝ, ℂ) where u_{k}(t) = h_{k} (\sqrt{2π|τ|t} ), see Section 2. In case τ > 0 we apply D to \binom{u_{k}}{0} and obtain

D\binom{u_{k}}{0} =  ( \begin{matrix}(−2πT^{−1}τ − \frac{d^{2}T}{4} ) u_{k} \\ id(2πτ t u_{k} + u′_{k}) \end{matrix} )
\begin{matrix}(2.3) \\ = \end{matrix} ( \begin{matrix}(−2πT^{−1}τ − \frac{d^{2}T}{4} ) u_{k} \\ id(4πτ t u_{k} + \sqrt{2πτ}u_{k+1}) \end{matrix} )

\begin{matrix}(2.4) \\ = \end{matrix} ( \begin{matrix}(−2πT^{−1}τ − \frac{d^{2}T}{4} ) u_{k} \\ −id\sqrt{2πτ} · 2k u_{k−1} \end{matrix} ) .

Similarly, D\binom{0}{u_{k−1}} = ( \begin{matrix}id\sqrt{2πτ}u_{k} \\ (2πT^{−1}τ − \frac{d^{2}T}{4} ) u_{k−1} \end{matrix} ). Hence \binom{u_{0}}{0} is an eigenvector for

D with eigenvalue −2πT^{−1}τ − \frac{d^{2}T}{4} and for k ∈ ℕ the vectors \binom{u_{k}}{0} and \binom{0}{u_{k−1}} span a D-invariant subspace. With respect to this basis D is given by the 2×2-matrix

D ∼ ( \begin{matrix}−2πT^{−1}τ − \frac{d^{2}T}{4} & id\sqrt{2πτ} \\ −2k id\sqrt{2πτ} & 2πT^{−1}τ − \frac{d^{2}T}{4} \end{matrix} ) .

The eigenvalues are −\frac{d^{2}T}{4} ± 2\sqrt{π^{2}τ^{2}T^{−2} + kπd^{2}τ}. Analogous reasoning for the case τ < 0 yields an eigenvalue −2πT^{−1}|τ| − \frac{d^{2}T}{4} and for k ∈ ℕ the eigenvalues −\frac{d^{2}T}{4} ± 2\sqrt{π^{2}τ^{2}T^{−2} + kπd^{2}|τ|}.

It remains to consider the case δ_{3} = −1. In this case we get the decomposition

L^{2}(Σ_{ε}M) = ⨁_{τ∈ℤ+ \frac{1}{2}} H_{τ},

the representations ρ_{α,β} do not occur. The computation of the eigenvalues of D on H_{τ} is the same as above. We summarize

THEOREM 3.1 Spin structures on the 3-dimensional Heisenberg manifold M(r, d, T), d, T > 0, r ∈ ℕ, correspond to triples δ = (δ_{1}, δ_{2}, δ_{3}) ∈ (ℤ/2 ℤ)^{3} where the case δ_{3} = −1 occurs if and only if r is even. The eigenvalues of the Dirac operator are as follows

A. If δ_{3} = +1:

a) λ_{α,β}^{±} := −\frac{d^{2}T}{4} ± 2π d\sqrt{α^{2} + β^{2}} with multiplicity 1 for each α ∈ \frac{1}{2r} · ℤ with e^{2π i r α} = δ_{1} and each β ∈ \frac{1}{2} · ℤ with e^{2π i β} = δ_{2}.

b) λ_{τ} := −\frac{d^{2}T}{4} − 2π τ T^{−1} with multiplicity 2τr for each τ ∈ ℕ.

c) λ_{τ,k}^{±} := −\frac{d^{2}T}{4} ± 2\sqrt{π^{2}τ^{2}T^{−2} + k πd^{2}τ} with multiplicity 2τr for all τ, k ∈ ℕ.

B. If δ_{3} = −1:

a) λ_{τ} := −\frac{d^{2}T}{4} − 2π τ T^{−1} with multiplicity 2τr for each τ ∈ ℕ_{0} + \frac{1}{2}.

b) λ_{τ,k}^{±} := −\frac{d^{2}T}{4} ± 2\sqrt{π^{2}τ^{2}T^{−2} + k πd^{2}τ} with multiplicity 2τr for each τ ∈ ℕ_{0} + \frac{1}{2} and each k ∈ ℕ.

Note that the Dirac spectrum for the spin structure δ = (δ_{1}, δ_{2}, −1) is independent of δ_{1} and of δ_{2}.

The eigenvalue λ_{τ,k}^{+} is zero for suitable T > 0. Its multiplicity is 2τr hence unbounded if we let τ → ∞. Thus we have

COROLLARY 3.2 For each N ∈ ℕ the 3-dimensional Heisenberg manifold M = Γ_{r}\G with spin structure corresponding to δ possesses a left invariant metric such that the dimension of the space of harmonic spinors is at least N.

A similar observation for Berger metrics on S^{3} is due to Hitchin [20]. It is known that every closed 3-manifold with a fixed spin structure admits a Riemannian metric such that the corresponding Dirac operator has nontrivial kernel [4]. It is conjectured that it can be made arbitrarily large.

The manifold M(r, d, T) is a circle bundle over the torus T^{2} = ℝ^{2}/r ℤ ⊕ ℤ. If one shrinks the fiber length (T → 0) then M(r, d, T) is said to collapse to T^{2}.

It is interesting to examine the behavior of the eigenvalues under this collapse. We see that eigenvalues of the type λ_{τ} or λ_{τ,k}^{±} tend to ±∞. But the eigenvalues λ_{α,β}^{±} converge to ±2π d\sqrt{α^{2} + β^{2}} which are precisely the Dirac eigenvalues of the torus T^{2}.

COROLLARY 3.3 If the Heisenberg manifold M(r, d, T) collapses to the 2-torus, i.e. if T → 0, then the Dirac eigenvalues behave as follows:

A. If δ_{3} = +1: There are eigenvalues converging to those of the 2-torus. All other eigenvalues tend to ±∞.

B. If δ_{3} = −1: All eigenvalues tend to ±∞.

This phenomenon will be studied in larger generality in the next section.

To conclude this section we mention that a computation of the Dirac eigenvalues on Heisenberg manifolds can also be carried out in higher dimensions. It just becomes notationally more complicated, see [1] for more details. In all dimensions ≥ 5 it is possible to find Heisenberg manifolds with nonisomorphic fundamental groups having the same Dirac spectrum. See Section 5 for a general discussion.

4 S^{1}-Actions and Collapse

In this section we will study the asymptotic behavior of the spectrum of the Dirac operator on a larger class of collapsing manifolds. This will generalize the observations we made in the last section for 3-dimensional Heisenberg manifolds. Our discussion in this section is similar in spirit to the one in [5] for the Laplace operator on functions.

We suppose that S^{1} acts freely and isometrically on the compact connected Riemannian spin manifold (M, g̃) of dimension n + 1. We view M^{n+1} as the total space of an S^{1}-principal bundle over the base space N^{n} = M^{n+1}/S^{1}.

This base space shall carry the unique metric g such that the projection

π : (M, g̃)⟶(N, g)

is a Riemannian submersion.

This S^{1}-principal bundle has a unique connection 1-form iω : T M → iℝ such that ker ω|_{m} is orthogonal to the fibers for all m ∈ M with respect to g̃.

The S^{1}-action induces a Killing vector field K. To keep the discussion simple we will assume that the length |K| is constant on M. This is equivalent to saying that the fibers of π are totally geodesic. One can relax this assumption, compare Remark 4.2.

By rescaling the metric g̃ on M along the fibers while keeping it the same on ker ω we obtain a 1-parameter family of metrics g̃_{ℓ} on M for which π : M → N is a Riemannian submersion, ω(K) ≡ 1, and 2πℓ is the length of the fibers. The length of K with respect to g̃_{ℓ} is ℓ.

As the metric on M is completely characterized by the connection 1-form iω, the fiber length 2πℓ and the metric g on N, we can express the Dirac operator in terms of ω, ℓ, and g. This will allow us to analyze the behavior of the spectrum for collapsing S^{1}-fibers (ℓ → 0). It will turn out that the spin structure on M determines whether there are converging eigenvalues or not.

The S^{1}-action on M induces an S^{1}-action on P_{SO}(M). A spin structure φ̃ : P_{Spin}(M) → P_{SO}(M) will be called projectable if this S^{1}-action on P_{SO}(M) lifts to P_{Spin}(M). Otherwise it will be called nonprojectable.

Any projectable spin structure induces a spin structure on N as follows: Let φ̃ : P_{Spin}(M) → P_{SO}(M) denote the projectable spin structure on M and let P_{SO(n)}(M) consist of those frames over M having K/ℓ as the first vector. We can identify P_{SO}(N) with P_{SO(n)}(M)/S^{1}. Now φ̃^{−1}(P_{SO(n)}(M))/S^{1} is a Spin(n)- bundle over N and φ̃ induces a corresponding spin structure on N.

On the other hand, any spin structure on N canonically induces a projectable spin structure on M via pull-back: Let φ : P_{Spin}(N) → P_{SO}(N) be a spin structure of N. Then π^{∗}φ : π^{∗}P_{Spin}(N) → π^{∗}P_{SO}(N) =: P_{SO(n)}(M) is a Θ_{n} : Spin(n) → SO(n) equivariant map. Enlarging the structure group to Spin(n + 1) by

φ̃ := π^{∗}φ×_{Θ_{n}} Θ_{n+1} : π^{∗}P_{Spin}(N)×_{Spin(n)}Spin(n+1) → P_{SO(n)}(M)×_{SO(n)}SO(n+1)

yields a spin structure on M.

Projectable spin structures and projectable spinors have already been studied by Moroianu [25].

Throughout this section we use the following convention: When we say that an operator has the eigenvalues μ_{1}, μ_{2}, … we suppose that each eigenvalue be repeated according to its multiplicity.

THEOREM 4.1 Let M be a closed Riemannian spin manifold. Let S^{1} act isometrically on M. We assume that the orbits have constant length. Let N = M/S^{1} carry the induced Riemannian metric and let g̃_{ℓ} be the metric on M described above. Let E → N be a Hermitian vector bundle with a metric connection ∇^{E}.

We suppose that the spin structure on M is projectable and that N carries the induced spin structure. Let μ_{1}, μ_{2}, … be the eigenvalues of the twisted Dirac operator D^{E} on ΣN ⊗ E → N.

Then we can number the eigenvalues (λ_{j,k}(ℓ))_{j∈ℕ,k∈ℤ} of the twisted Dirac operator D̃^{ℓ} for g̃_{ℓ} on ΣM ⊗ π^{∗}E → M such that they depend continuously on ℓ and such that for ℓ → 0:

(1) For any j ∈ ℕ and k ∈ ℤ

ℓ · λ_{j,k}(ℓ) → k.

In particular, λ_{j,k}(ℓ) → ±∞ if k ≠ 0.

(2) If n = dim N is even, then

λ_{j,0}(ℓ) → μ_{j}.

On the other hand if n = dim N is odd, then

λ_{2j−1,0}(ℓ) → μ_{j}
λ_{2j,0}(ℓ) → −μ_{j}

In both cases, the convergence of the eigenvalues λ_{j,0}(ℓ) is uniform in j.

REMARK 4.2 The theorem also holds for a family of nonconstant length functions ℓ : N → ℝ^{+} provided ℓ and (1/ℓ) grad ℓ converge uniformly to 0. The proof is analogous to the one below except that some formulas contain an additional term depending on grad ℓ.

In the proof of the theorem we will assume for simplicity that E is a trivial line bundle. For nontrivial E every spinor bundle on N or M simply has to be twisted by E or π^{∗}E resp.

In order to prove Theorem 4.1 we write the Dirac operator D̃^{ℓ} as a sum of a vertical Dirac operator, a horizontal Dirac operator, and a zero order term.

For the definition of the vertical Dirac operator we need the Lie derivative of a spinor along the S^{1}-fibers. The action of S^{1} on P_{Spin}M induces an action of S^{1} on ΣM = P_{Spin}(M^{n+1}) ×_{Spin(n+1)} Σ_{n+1} which we denote by κ. A spinor with base point m will be mapped by κ(e^{it}) to a spinor with base point m · e^{it}. We define the Lie derivative of a smooth spinor Ψ in the direction of the Killing field K by

L_{K}(Ψ)(m) = \frac{d}{dt}|_{t=0} κ(e^{−it})(Ψ(m · e^{it})).

See [7] for a general discussion of Lie derivatives for spinors. Since L_{K} is the differential of a representation of the Lie group S^{1} on L^{2}(ΣM), we get the decomposition

L^{2}(ΣM) = ⨁_{k∈ℤ}V_{k}

into the eigenspaces V_{k} of L_{K} for the eigenvalue ik, k ∈ ℤ. The S^{1}-action commutes with the Dirac operator on M, hence this decomposition is preserved by the Dirac operator.

We now want to calculate the difference between the covariant derivative and the Lie derivative in the direction of K. We always use the convention that any r-form α acts on a spinor Ψ by

γ(α)Ψ := ∑_{i_{1}<...<i_{r}} α(e_{i_{1}}, …, e_{i_{r}}) γ(e_{i_{1}}) ⋯ γ(e_{i_{r}})Ψ

where the e_{i} form an orthonormal basis of the tangent space.

LEMMA 4.3 If Ψ̃ is a smooth section of ΣM, then

∇_{K} Ψ̃ − L_{K} Ψ̃ = \frac{ℓ^{2}}{4} γ(dω)Ψ̃

Proof First we fix some notations. In the rest of this section, we denote the horizontal lift of X ∈ T N by X̃ ∈ T M. Let (f_{1}, …, f_{n}) be a local orthonormal frame of N on the open subset U and let A be a lift to the spin structure. Then (e_{0} := K/ℓ, e_{1} := f̃_{1} ∘ π, …, e_{n} := f̃_{n} ∘ π) is a local orthonormal frame on π^{−1}(U) with lift π^{∗}A. Using the Koszul formula we can express the Christoffel symbols Γ̃_{ij}^{k} for (e_{0}, …, e_{n}) in terms of the Christoffel symbols Γ_{ij}^{k} of (f_{1}, …, f_{n}), the S^{1}-bundle curvature form dω and the length ℓ. For i, j, k ∈ {1, …, n} we get

Γ̃_{ij}^{k} = Γ_{ij}^{k}  − Γ̃_{ij}^{0} = Γ̃_{i0}^{j} = Γ̃_{0i}^{j} = \frac{ℓ}{2} dω(e_{i}, e_{j})  (4.1)
Γ̃_{i0}^{0} = Γ̃_{0i}^{0} = Γ̃_{00}^{i} = Γ̃_{00}^{0} = 0

With respect to these local frames on M and N we apply (1.1) for M and we obtain

\frac{1}{ℓ}∇_{K} Ψ̃ = \frac{1}{ℓ}∂_{K} Ψ̃ + \frac{1}{4} ∑_{j,k=0}^{n} Γ̃_{0j}^{k}γ(e_{j})γ(e_{k})Ψ̃
= \frac{1}{ℓ}∂_{K} Ψ̃ + \frac{ℓ}{8} ∑_{j,k=1}^{n} dω(e_{j}, e_{k})γ(e_{j})γ(e_{k})Ψ̃
= \frac{1}{ℓ}∂_{K} Ψ̃ + \frac{ℓ}{4} γ(dω)Ψ̃.

On the other hand the choice of the frames implies that

L_{K} Ψ̃ = ∂_{K} Ψ̃,

thus the lemma follows. □

Now we associate to the S^{1}-bundle M → N the complex line bundle L := M ×_{S^{1}} ℂ with the natural connection given by iω.

LEMMA 4.4 Let n be even. There is a homothety of Hilbert spaces

Q_{k} : L^{2}(ΣN ⊗ L^{−k}) → V_{k}

such that the horizontal covariant derivative is given by

∇X̃Q_{k}(Ψ) = Q_{k}(∇_{X}Ψ) + \frac{ℓ}{4} γ(K/ℓ)γ(Ṽ_{X})Q_{k}(Ψ)

where V_{X} is the vector field on N satisfying dω(X̃, ·) = ⟨Ṽ_{X}, ·⟩. Clifford multiplication is preserved, i.e.

Q_{k}(γ(X)Ψ) = γ(X̃)Q_{k}(Ψ).

Proof Since n is even, we have Σ_{n} = Σ_{n+1}. We define a vector bundle map

Π_{k} : ΣM → ΣN ⊗ L^{−k}
(m, [π^{∗}A, σ]) ↦ (π(m), [A, σ] ⊗ [m, 1]^{−k}),

where [π^{∗}A, σ] (resp. [A, σ], [m, 1]) denotes the equivalence class of (π^{∗}A, σ) (resp. (A, σ), (m, 1)) in π^{∗}P_{Spin}(N) ×_{Spin(n)} Σ_{n} = ΣM (resp. P_{Spin}(N) ×_{Spin(n)} Σ_{n} = ΣN, resp. M ×_{S^{1}} ℂ = L). This bundle map is fiberwise a vector space isomorphism preserving Clifford multiplication. Therefore for any spinor Ψ : N → ΣN ⊗ L^{−k} there is a unique Q_{k}(Ψ) such that

\begin{matrix}M & ΣM \\ N & ΣN ⊗ L^{−k} \end{matrix} Q_{k}(Ψ) π Π_{k} Ψ

commutes. Hence Q_{k} is a well defined homomorphism of Hilbert spaces and injectivity of Q_{k} follows from the surjectivity of π.

For any section Ψ̃ of ΣM we get the equivalences

Ψ̃ ∈ image(Q_{k})
⟺ Π_{k} ∘ Ψ̃(m) = Π_{k} ∘ Ψ̃(me^{it}) ∀m ∈ M, e^{it} ∈ S^{1}
⟺ Ψ̃(m) = e^{−ikt}κ(e^{−it})Ψ̃(me^{it}) ∀m ∈ M, e^{it} ∈ S^{1}
⟺  ik Ψ̃ = L_{K} Ψ̃.

Thus the image of Q_{k} is precisely V_{k}. The map Q_{k} is an isometry up to the factor ℓ.

Finally it remains to show the formula for the covariant derivative in the horizontal directions. For this it suffices to prove it locally and we can restrict to the case X = f_{i} with the notations as in the previous lemma. As before we use (1.1) and (4.1).

∇f̃_{i}Q_{k}(Ψ) = Q_{k}(∇_{f_{i}}Ψ) + \frac{1}{4} ∑_{j=1}^{n} Γ̃_{i0}^{j}γ(K/ℓ)γ(e_{j})Q_{k}(Ψ)

+\frac{1}{4} ∑_{j=1}^{n} Γ̃_{ij}^{0}γ(e_{j})γ(K/ℓ)Q_{k}(Ψ)

= Q_{k}(∇_{f_{i}}Ψ) + \frac{ℓ}{4} γ(K/ℓ)γ(Ṽ_{f_{i}})Q_{k}(Ψ)
□

Proof of Theorem 4.1 for n even. We define the horizontal Dirac operator as the unique closed linear operator D_{h} : L^{2}(ΣM) → L^{2}(ΣM) on each V_{k} given by

D_{h} := Q_{k} ∘ D ∘ Q_{k}^{−1}

where D is the (twisted) Dirac operator on ΣN ⊗ L^{−k}.

We define the vertical Dirac operator

D_{v} := γ(K/ℓ) L_{K}

and the zero order term

Z := −(1/4) γ(K/ℓ) γ(dω).

Using Lemma 4.3 and Lemma 4.4 we can express the Dirac operator D̃^{ℓ} as a sum

D̃^{ℓ} = ∑_{i=0}^{n} γ(e_{i})∇_{e_{i}}
= \frac{1}{ℓ}D_{v} + D_{h} + ℓZ.

Since D_{h}, γ(K/ℓ) and Z commute with the S^{1}-action they also commute with L_{K}. On the other hand, we have γ(K/ℓ)Q_{k}(Ψ) = Q_{k}(cγ(dvol_{n})Ψ) with c ∈ {1, i, −1, −i} depending on n and the representation of Cl_{n+1}. Since γ(dvol_{n}) anticommutes with any twisted Dirac operator on N, we know that γ(K/ℓ) anticommutes with D_{h}. Let Ψ be a common eigenspinor for L_{K} and D_{h} for the eigenvalues ik and μ resp.

On U := span{Ψ, γ(K/ℓ)Ψ} the operator A_{ℓ} := (1/ℓ)D_{v} + D_{h} is represented by the matrix

\frac{1}{ℓ} ( \begin{matrix}0 & −ik \\ ik & 0 \end{matrix} ) + ( \begin{matrix}μ & 0 \\ 0 & −μ \end{matrix} ) = ( \begin{matrix}μ & −ik/ℓ \\ ik/ℓ & −μ \end{matrix} ) .

Thus for k = 0 the restriction A_{ℓ}|_{U} has eigenvalues ±μ. For k ≠ 0 the eigenvalues of A_{ℓ}|_{U} are the square roots of (k/ℓ)^{2} + μ^{2}. Therefore the eigenvalues (λ_{j,k}^{0}(ℓ))_{j∈ℕ,k∈ℤ} of A_{ℓ} can be numbered such that they are continuous in ℓ and satisfy properties (1) and (2) of Theorem 4.1. The additional term ℓZ does not change this behavior because Z : L^{2}(ΣM) → L^{2}(ΣM) is bounded. □

Proof of Theorem 4.1 for n odd. In the case “n odd” the proof is similar but we have to do some modifications as the vector space equality Σ_{n+1} = Σ_{n} is no longer valid.

For the standard basis E_{1}, …, E_{n} ∈ ℝ^{n} and E_{0}, …, E_{n} ∈ ℝ^{n+1} we define the volume element by

ν_{n} := i^{\frac{n+1}{2}} γ(E_{1}) ⋯ γ(E_{n}) ∈ Cl(n + 1).

Its square is the identity and hence has the eigenvalues ±1. Let

Σ_{n+1} = Σ_{n}^{(+)} ⊕ Σ_{n}^{(−)}

be the splitting into the eigenspaces. These Σ_{n}^{(±)} are the two irreducible representations of Cl(n). The Clifford multiplication γ(E_{0}) anticommutes with ν_{n} and therefore restricts to an isomorphism of Σ_{n}^{(+)} → Σ_{n}^{(−)}. This isomorphism changes the sign of γ(E_{i}) for i > 0 as γ(E_{0})γ(E_{i}) = −γ(E_{i})γ(E_{0}).

Denote the corresponding spinor bundles by Σ^{(±)}N. Lemma 4.4 is now valid with

Q_{k} : L^{2}((Σ^{(+)}N ⊕ Σ^{(−)}N) ⊗ L^{−k}) → V_{k}.

The same arguments as in the even case can be used to prove that the eigenvalues of D̃^{ℓ}|_{V_{k}} converge to ±∞ for k ≠ 0 and to the eigenvalues of the Dirac operator on Σ^{(+)}N ⊕Σ^{(−)}N for k = 0. The Dirac operator on Σ^{(+)}N has the same eigenvalues as the Dirac operator on Σ^{(−)}N but with opposite signs. This yields the statement of Theorem 4.1. □

In the case of a nonprojectable spin structure we get a similar result. Here the variable k takes no longer integer values but

k ∈ ℤ + \frac{1}{2} .

THEOREM 4.5 Let M be a closed Riemannian spin manifold. Let S^{1} act isometrically on M. We assume that the orbits have constant length. Let N = M/S^{1} carry the induced Riemannian metric and let g̃_{ℓ} be the metric on M described above. Let E → N be a Hermitian vector bundle with a metric connection ∇^{E}.

We suppose that the spin structure on M is nonprojectable.

Then we can number the eigenvalues (λ_{j,k}(ℓ))_{j∈ℕ,k∈ℤ+(1/2)} of the twisted Dirac operator D̃^{ℓ} for g̃_{ℓ} on ΣM ⊗ π^{∗}E → M such that they depend continuously on ℓ and such that for ℓ → 0

ℓ · λ_{j,k}(ℓ) → k ∀j = 1, 2, ….

In particular, λ_{j,k}(ℓ) → ±∞.

Proof The proof is a variation of the proof of Theorem 4.1. We will restrict to the case n even and E trivial.

Let φ̃ : P_{Spin}(M) → P_{SO}(M) be a nonprojectable spin structure on M. In this case N may or may not be spin. But as before the principal SO(n + 1)-bundle P_{SO}(M) can be restricted to a principal SO(n)-bundle P_{SO(n)}(M). Moreover P := ϕ̃^{−1}(P_{SO(n)}(M)) is a principal Spin(n)-bundle over M.

The action of S^{1} ≅ (ℝ/2πℤ) does not lift to P, but the double covering of S^{1}, i.e. S^{1} ≅ (ℝ/4πℤ), does act on P. We define Spin^{ℂ}(n) to be Spin(n) ×_{ℤ_{2}} S^{1} where −1 ∈ ℤ_{2} identifies (−A, c) with (A, −c). The Lie group Spin^{ℂ}(n) naturally acts on Σ_{n}. The actions of Spin(n) and S^{1} on P induce a free action of Spin^{ℂ}(n) on P and we can view P as a principal Spin^{ℂ}(n)-bundle over N. Then we can form the bundle P ×_{Spin^{ℂ}(n)} Σ_{n}.

If N is spin, this bundle is just ΣN ⊗ L^{\frac{1}{2}} . If N is not spin, then neither ΣN nor L^{\frac{1}{2}} exist but ΣN ⊗ L^{\frac{1}{2}} does exist.

Again we get a splitting

L^{2}(ΣM) = ⨁_{k∈ℤ+ \frac{1}{2}} V_{k}

into eigenspaces V_{k} for L_{K} to the eigenvalue ik. The rest of the proof of Theorem 4.5 is the same as the one for the case k ≠ 0 in Theorem 4.1. □

Example We apply Theorems 4.1 and 4.5 to the Hopf fibration S^{2m+1} → ℂP^{m}. The Dirac eigenvalues for the metric g̃_{ℓ} on S^{2m+1} have been computed by the second author in [4, Theorem 3.1]:

The Dirac operator of S^{2m+1} with Berger metric g̃_{ℓ} has the following eigenvalues:

(i) λ(ℓ) = \frac{1}{ℓ} (a + \frac{m+1}{2}) + \frac{ℓm}{2}, a ∈ ℕ_{0}, with multiplicity \binom{m+a}{a}.

(ii) λ(ℓ) = (−1)^{m+1} · \{\frac{1}{ℓ} (a + \frac{m+1}{2}) + \frac{ℓm}{2}\}, a ∈ ℕ_{0}, with multiplicity \binom{m+a}{a}.

(iii) λ_{1,2}(ℓ) = (−1)^{j} \frac{ℓ}{2} ± \{ [\frac{ℓ}{2}(m − 2j − 1) + \frac{1}{ℓ} (a_{1} − a_{2} + \frac{m − 1}{2} − j)]^{2}

+ 4(m − j + a_{1})(j + 1 + a_{2})}^{1/2}

a_{1}, a_{2} ∈ ℕ_{0}, j = 0, …, m − 1, with multiplicity

\frac{(a_{1} + m)! (a_{2} + m)! (a_{1} + a_{2} + m + 1)}{a_{1}! a_{2}! (a_{1} + m − j) (a_{2} + j + 1) m! j! (m − j − 1)!}.

We observe for the eigenvalues of type (i) and (ii)

lim_{ℓ→0} ℓ · λ(ℓ) = ± (a + \frac{m + 1}{2}) .

We see that the spin structure on S^{2m+1} is projectable if and only if m is odd. In fact, ℂP^{m} is spin if and only if m is odd.

To obtain a finite limit of λ(ℓ) itself we must necessarily have an eigenvalue of type (iii) and a_{1} − a_{2} + \frac{m−1}{2} = j. In particular, m must be odd again. Plugging this expression for j into the formula for the eigenvalue of type (iii) and passing to the limit ℓ → 0 yields the eigenvalues of the complex projective space:

THEOREM 4.6 The Dirac operator on ℂP^{m}, m odd, has the eigenvalues

±2\sqrt{(a_{1} + \frac{m + 1}{2}) (a_{2} + \frac{m + 1}{2})}

where a_{1}, a_{2} ∈ ℕ_{0}, |a_{1} − a_{2}| ≤ \frac{m−1}{2} with multiplicity

\frac{(a_{1} + m)! (a_{2} + m)! (a_{1} + a_{2} + m + 1)}{a_{1}! a_{2}! (a_{1} + \frac{m+1}{2}) (a_{2} + \frac{m+1}{2}) m! (a_{1} − a_{2} + \frac{m−1}{2})! (a_{2} − a_{1} + \frac{m−1}{2})!}.

The formula for the eigenvalues coincides with the results in [9, 10, 30] after the right parameter substitutions. Our expression however is simpler than the previous ones. The formula for the multiplicities is new.

5 Isospectral Deformations

We now study the isospectrality question for the Dirac operator on Riemannian nilmanifolds Γ\G. The spectrum of the Dirac operator does not only depend on the Riemannian structure but also on the spin structure, i.e. on the homomorphism ε : Γ → {−1, 1}. For ε ≡ 1 it will turn out in the examples under consideration that the spectrum of the Dirac operator behaves much like the spectrum of the Laplacian on 1-forms. For other ε the behavior can be different.

At first, let L_{ε}^{2}(Γ\G) be the space of all complex-valued locally squareintegrable functions on G, satisfying the ε-equivariance condition

f(g_{0}g) = ε(g_{0})f(g),

for all g_{0} ∈ Γ and g ∈ G. The nilpotent group G acts on this space via the right regular representation

R : G → End(L_{ε}^{2}(Γ\G))
g_{0} ↦ (R(g_{0}) : f ↦ f( · g_{0})).

Let Γ′ be another cocompact discrete subgroup of G, ε′ : Γ′ → {−1, 1} a homomorphism. We say that (Γ, ε) and (Γ′, ε′) are representation equivalent if the right regular representations on L_{ε}^{2}(Γ\G) and on L_{ε′}^{2}(Γ′\G) are isomorphic as G-representations.

As already mentioned at the end of Section 1, sections of the spinor bundle Σ_{ε}(Γ\G) may be identified with Γ-equivariant maps G → Σ_{n}. Elements of Σ_{n} will also be interpreted as constant maps G → Σ_{n} and therefore as sections of ΣG.

Hence f ⊗s ↦ f ·s defines an isomorphism of Hilbert spaces L_{ε}^{2}(Γ\G)⊗_{ℂ}Σ_{n} → L^{2}(Σ_{ε}(Γ\G)). In the rest of this section we will identify via this isomorphism. Clifford multiplication is performed on the second factor

γ(X)(f ⊗ s) = f ⊗ (γ(X)s)

for all X ∈ 𝔤, f ∈ L_{ε}^{2}(Γ\G), s ∈ Σ_{n}.

THEOREM 5.1 If (Γ, ε) and (Γ′, ε′) are representation equivalent, then the corresponding Dirac operators have the same spectrum for any left-invariant Riemannian metric on G.

Examples:

• Milnor’s 16-dimensional tori [24] satisfy the condition of the theorem for ε ≡ ε′ ≡ 1.

• The Gordon-Wilson deformations [18] preserve the spectrum of the Dirac operator. In fact, if Ψ_{t} is a family of almost inner automorphisms of G, Γ a cocompact discrete subroup and ε : Γ → {−1, 1} a homomorphism, then (Ψ_{t}(Γ), ε ∘ Ψ_{t}^{−1}) are pairwise representation equivalent. This deformation preserves the length spectrum, the spectrum of the Laplacian on functions and on forms, and the Dirac spectrum for all spin structures.

Since the spectrum of the Dirac operator determines the volume, we get

COROLLARY 5.2 If (Γ, ε) and (Γ′, ε′) are representation equivalent, then Γ\G and Γ′\G have the same volume for any left invariant volume form on G.

Proof of Theorem 5.1 Since L_{ε}^{2}(Γ\G) and L_{ε′}^{2}(Γ′\G) are isomorphic as representations of the Lie group G they are isomorphic as representations of its Lie algebra 𝔤. Let

ϑ : L_{ε}^{2}(Γ\G) → L_{ε′}^{2}(Γ′\G)

be such an isomorphism, i.e. ϑ ∘ R = R′ ∘ ϑ and ϑ ∘ R_{∗} = R′_{∗} ∘ ϑ. In order to prove the theorem we show that ϑ ⊗ id maps eigenspinors of the Dirac operator in L^{2}(Σ_{ε}(Γ\G)) to eigenspinors of the Dirac operator in L^{2}(Σ_{ε′}(Γ′\G)). It is important to keep in mind that these two Dirac operators are just restrictions of the Dirac operator on ΣG to spinors equivariant under ε (resp. ε′).

Let s = ∑_{i} f_{i}s_{i} be an arbitrary eigenspinor in L^{2}(Σ_{ε}(Γ\G)) for the eigenvalue λ with f_{i} ∈ L_{ε}^{2}(Γ\G) and s_{i} ∈ Σ_{n}. Let E_{1}, …, E_{n} an orthonormal basis of 𝔤. Using the Leibniz rule

∇_{E_{j}}(f_{i}s_{i}) = (R_{∗}(E_{j})f_{i})s_{i} + f_{i}∇_{E_{j}}s_{i}  (5.1)

we get

λ(ϑ ⊗ id)(s) = (ϑ ⊗ id)(Ds)
= ∑_{ij}(ϑ ⊗ id) ((id ⊗γ(E_{j}))∇_{E_{j}}(f_{i}s_{i}))
= ∑_{ij} (ϑ(R_{∗}(E_{j})f_{i})γ(E_{j})s_{i} + ϑ(f_{i})γ(E_{j})∇_{E_{j}}s_{i})
= ∑_{ij} (R′_{∗}(E_{j})(ϑ(f_{i}))γ(E_{j})s_{i} + ϑ(f_{i})γ(E_{j})∇_{E_{j}}s_{i})
= ∑_{ij}(id ⊗γ(E_{j}))∇_{E_{j}}(ϑ(f_{i})s_{i})
= D((ϑ ⊗ id)(s)).

Thus ϑ ⊗ id is an isomorphism of Hilbert spaces that preserves eigenspinors and eigenvalues and therefore the Dirac spectra coincide. □

To get further applications of Theorem 5.1 we need criteria telling us under which conditions (Γ, ε) and (Γ′, ε′) are representation equivalent. In the examples of Theorem 5.1 we already mentioned that almost inner automorphisms give us representation equivalent families. This criterion is sufficient but not necessary. However, in the special case when G is stricly nonsingular, we will get a necessary and sufficient criterion. A nilpotent Lie group G with center Z(G) will be called stricly nonsingular if for every z ∈ Z(G) and x ∈ G − Z(G) there exists an a ∈ G such that

xax^{−1}a^{−1} = z.

Note that the Heisenberg group is strictly nonsingular.

Let Γ be a cocompact discrete subgroup of G. The quotient G/Z(G) will be denoted by \overline{G} and the image of Γ under the canonical projetion π : G → \overline{G} by \overline{Γ}. According to [27, Proposition 2.17] and [12, Lemma 5.1.4] \overline{Γ} is a cocompact discrete subgroup of \overline{G}. If ε|_{Γ∩Z(G)} ≡ 1 then there is an \overline{ε} such that

Γ  \{−1, 1\} ; \overline{Γ} ε\overline{ε}

commutes. Such an ε will be called projectable. For dim Z(G) = 1 this definition agrees with our definition of projectable spin structures in Section 4.

THEOREM 5.3 Let G be a simply connected, strictly nonsingular nilpotent Lie group with left invariant metric. Let Γ_{1}, Γ_{2} be cocompact, discrete subgroups of G such that

Γ_{1} ∩ Z(G) = Γ_{2} ∩ Z(G) (=: Γ_{Z})

and let ε_{i} : Γ_{i} → {−1, 1} be homomorphisms. We define the volume quotient

v := \frac{vol(\overline{Γ}_{1}\\overline{G})}{vol(\overline{Γ}_{2}\\overline{G})} = \frac{vol(Γ_{1}\G)}{vol(Γ_{2}\G)} ∈ ℚ^{+}.

Then (Γ_{1}, ε_{1}) and (Γ_{2}, ε_{2}) are representation equivalent if and only if one of the following conditions is satisfied:

(1) ε_{1}|_{Γ_{Z}} ≡ ε_{2}|_{Γ_{Z}} ≡ 1 and (\overline{Γ}_{1}, \overline{ε}_{1}) and (\overline{Γ}_{2}, \overline{ε}_{2}) are representation equivalent for \overline{G}.

(2) ε_{1}|_{Γ_{Z}} ≡ ε_{2}|_{Γ_{Z}} ≢ 1 and v = 1.

REMARK 5.4 If under the assumptions of the above theorem ε_{1}|_{Γ_{Z}} ≡ ε_{2}|_{Γ_{Z}} ≢ 1 but v ≠ 1, then (Γ_{1}, ε_{1}) and (Γ_{2}, ε_{2}) are “representation equivalent up to multiplicity v”, i.e. L_{ε_{1}}^{2}(Γ_{1}\G) and L_{ε_{2}}^{2}(Γ_{2}\G) have the same irreducible components and for every irreducible component H with multiplicities m_{1}, m_{2} we get v = m_{1}/m_{2}.

Let us compare this result to Theorem 3.1. Using the notation of Section 3 for the 3-dimensional Heisenberg groups write M(r_{i}, d, T) = Γ_{i}\G. Then v = r_{1}/r_{2}. The remark explains why the eigenvalues of the Dirac operator in Theorem 3.1 B do not depend on r and why the multiplicity is proportional to r.

Proof of Theorem 5.3 and Remark 5.4 If (Γ_{1}, ε_{1}) and (Γ_{2}, ε_{2}) are representation equivalent, then L_{ε_{1}}^{2}(Γ_{1}\G) and L_{ε_{2}}^{2}(Γ_{2}\G) are also equivalent as Γ_{Z}-modules. Since the action of g_{0} ∈ Γ_{Z} on L_{ε_{i}}^{2}(Γ_{i}\G) is just multiplication by ε_{i}(g_{0}) we get

ε_{1}|_{Γ_{Z}} ≡ ε_{2}|_{Γ_{Z}}.

Let T be the set of coadjoint orbits of 𝔤^{∗}. Via Kirillov theory T parametrizes the set of irreducible unitary representations of G. We write elements of T in the form [τ] with τ ∈ 𝔤^{∗}. The irreducible G-module corresponding to [τ] ∈ T will be denoted by H_{[τ]}. We write L_{ε_{i}}^{2}(Γ_{i}\G) as the discrete sum of irreducible representations

L_{ε_{i}}^{2}(Γ_{i}\G) = ⨁_{[τ]∈T} m_{i}([τ])H_{[τ]}

where m_{i}([τ]) ∈ ℕ_{0} denotes the multiplicity of H_{[τ]}. Let 𝔷 be the Lie algebra of the center Z(G). The action of Z(G) on H_{[τ]} is trivial if and only if τ|𝔷 ≡ 0. Therefore

H_{i}^{proj} := ⨁_{{[τ]∈T \atop τ|𝔷≡0}} m_{i}([τ])H_{[τ]} i = 1, 2

is the space of L^{2}-functions acted trivially upon by Z(G). The orthogonal complement is

H_{i}^{⊥} := ⨁_{{[τ]∈T \atop τ|𝔷≢0}} m_{i}([τ])H_{[τ]} i = 1, 2 .

The G-modules L_{ε_{i}}^{2}(Γ_{i}\G) for i = 1, 2 are isomorphic if and only if H_{1}^{proj} is isomorphic to H_{2}^{proj} and H_{1}^{⊥} is isomorphic to H_{2}^{⊥}. In a first step we investigate under which conditions H_{1}^{proj} and H_{2}^{proj} are isomorphic and then we treat H_{1}^{⊥} and H_{2}^{⊥}.

In the case ε_{i}|_{Γ_{Z}} ≢ 1 we have H_{i}^{proj} = {0}. Therefore H_{1}^{proj} and H_{2}^{proj} are trivially isomorphic G-modules.

In the case ε_{i}|_{Γ_{Z}} ≡ 1 elements in H_{i}^{proj} are exactly those functions in L^{2}(Γ_{i}\G) which are pullbacks of functions in L_{\overline{ε}_{i}}^{2}(\overline{Γ}_{i}\\overline{G}). Thus we get a module-isomorphism (for both groups G and \overline{G})

H_{i}^{proj} ⟶^{∼} L_{\overline{ε}_{i}}^{2}(\overline{Γ}_{i}\\overline{G}).

Therefore in this case H_{1}^{proj} and H_{2}^{proj} are isomorphic G-modules if and only if (\overline{Γ}_{1}, \overline{ε}_{1}) and (\overline{Γ}_{2}, \overline{ε}_{2}) are representation equivalent for \overline{G}.

It remains to study the H_{i}^{⊥}. In Lemma 5.5 below we will show that m_{i}([τ]) = c_{G,τ} · vol(\overline{Γ}_{i}\\overline{G}) for any [τ] ∈ T with τ|𝔷 ≢ 0 where c_{G,τ} only depends on G, on τ, and on the volume form on \overline{G} but not on the lattice Γ_{i}. This will complete the proof of the theorem: Suppose that condition (1) or (2) in the theorem is fulfilled. Because of Corollary 5.2 we then know that the volume quotient v is 1, and therefore H_{1}^{⊥} and H_{2}^{⊥} are isomorphic G-modules. Using the statement for v ≠ 1 we get Remark 5.4. □

To complete the proof of Theorem 5.3 and Remark 5.4 it remains to show Lemma 5.5. The lemma is a consequence of [26]. In order to simplify notation we drop the index i in the formulation from now on.

For any ρ ∈ 𝔷^{∗} we define a bilinear form on \overline{𝔤} by

b_{ρ}(\overline{X}, \overline{Y}) := ρ([X, Y]),

where X and Y are vectors in 𝔤 and \overline{X} and \overline{Y} their images in \overline{𝔤}. As 𝔤 is strictly nonsingular, b_{ρ} is nondegenerate for all ρ ≠ 0 [26, 16]. Therefore dim \overline{𝔤} is even and we suppose dim \overline{𝔤} = 2d. We define the affine lattice L_{Γ,ε} by

L_{Γ,ε} := {ρ ∈ 𝔷^{∗} | e^{2πiρ(z)} = ε(z) ∀z ∈ log(Γ_{Z})}

LEMMA 5.5 If ρ := τ|𝔷 ∈ L_{Γ,ε} − {0} then

m([τ]) = | ∫_{\overline{Γ}\\overline{G}} \underbrace{b_{ρ} ∧ … ∧ b_{ρ}}_{d-times} | = vol(\overline{Γ}\\overline{G}) · c_{G,τ}

where c_{G,τ} only depends on G, on τ and on the volume form on \overline{G}.

If ρ := τ|𝔷 ∈ 𝔷^{∗} − (L_{Γ,ε} ∪ {0}) then m([τ]) = 0.

The lemma does not make any statement for ρ = 0. The reader may notice that for strictly nonsingular Lie groups the orbit of τ|𝔷 ≠ 0 is the set of all τ′ ∈ 𝔤^{∗} with τ|𝔷 ≡ τ′|𝔷.

Proof of Lemma 5.5 In the case ε ≡ 1 the lemma is just a consequence of [26, Theorem 7]. In the case ε ≢ 1 we set Γ′ := ker(ε) and we decompose

L_{1}^{2}(Γ′\G) = L_{1}^{2}(Γ\G) ⊕ L_{ε}^{2}(Γ\G)

as an orthogonal sum of G-modules. We already now the lemma for (Γ, 1) and (Γ′, 1). Using the fact that every factor occurs only finitely many times, we can calculate the multiplicity in L_{ε}^{2}(Γ\G) as the difference of the multiplicities of L_{1}^{2}(Γ′\G) and L_{1}^{2}(Γ\G). If ε is projectable, then L_{Γ,ε} = L_{Γ,1} = L_{Γ′,1}, but vol(\overline{Γ}′\\overline{G}) = 2 vol(\overline{Γ}\\overline{G}) and if ε is nonprojectable then L_{Γ′,1} = L_{Γ,1} ˙∪L_{Γ,ε} but vol(\overline{Γ}\\overline{G}) = vol(\overline{Γ}′\\overline{G}). This proves the lemma for general ε. □

THEOREM 5.6 There is a family of nilmanifolds having

1. constant spectrum of the Laplacian on functions,

2. nonconstant spectrum of the Laplacian on 1-forms

3. constant marked length spectrum

4. nonconstant spectrum of the Dirac operator for projectable spin structures,

5. constant spectrum of the Dirac operator for nonprojectable spin structures.

Proof Example I and II of [17] will have all these properties for suitably chosen lattices. Properties 1–3 have been shown in [17]. We will prove 4 and 5 for Example I. The arguments for Example II are analogous. To describe Example I let 𝔤 be the 7-dimensional Lie algebra with orthonormal basis {X_{1}, X_{2}, X_{3}, X_{4}, Z_{1}, Z_{2}, Z} and Lie bracket

[X_{1}, X_{2}] = [X_{3}, X_{4}] = Z_{1} + Z
[X_{1}, X_{3}] = [X_{4}, X_{2}] = Z_{2}
[X_{2}, X_{3}] = [X_{1}, Z_{1}] = [Z_{2}, X_{4}] = Z.

All other Lie brackets of basis vectors shall be zero. This Lie algebra is strongly nonsingular and Z generates the center of 𝔤. The matrix

cos s  0  0  sin s  0  0  0
0  cos 2s − sin 2s 0  0  0  0
0  sin 2s cos 2s  0  0  0  0
− sin s 0  0  cos s  0  0  0
0  0  0  0  cos s  − sin s 0
0  0  0  0  sin s  cos s 0
0  0  0  0 −1 + cos s − sin s 1

defines a family of Lie algebra automorphisms φ_{s} of 𝔤 and induces a family of Lie group automorphisms Φ_{s} of G, i.e. Φ_{s}_{∗} = φ_{s}.

The center Z(G) is fixed under the automorphism Φ_{s}. Therefore Φ_{s} induces an automorphism \overline{Φ}_{s} of \overline{G} = G/Z(G) with commuting diagram

\begin{matrix}G & G \\ \overline{G} & \overline{G}. \end{matrix}Φ_{s} π π_{\overline{Φ}_{s}}

These \overline{Φ}_{s} also act isometrically.

Let Γ be an arbitrary discrete cocompact subgroup of G. We write Γ_{s} := Φ_{s}(Γ), \overline{Γ} = π(Γ) and \overline{Γ}_{s} := \overline{Φ}_{s}(\overline{Γ}).

We now prove that the family M_{s} := Γ_{s}\G, s ∈ ℝ, satisfies the theorem. Gornet showed in [17] that this family is isospectral for the Laplacian on functions, but not for the Laplacian on 1-forms. Furthermore she showed that Φ_{s} marks an isomorphism of the length spectrum from M_{0} to M_{s}.

We have to show 4 and 5 and we start with 5. Let ε : Γ → {−1, 1} describe a nonprojectable spin structure on M_{0}. As we want the spin structure to be chosen continuously in s, the spin structure on M_{s} is given by ε∘Φ_{s}^{−1}, and therefore also nonprojectable. Since \overline{Φ}_{s} is a family of isometric automorphisms, the volume of \overline{Γ}_{s}\\overline{G} is independent of s. Hence the second condition in Theorem 5.3 is satisfied and therefore (Φ_{s} ∘ Γ, ε ∘ Φ_{s}^{−1}) is a pairwise representation equivalent family. Theorem 5.1 implies that the spectrum of the Dirac operator does not depend on s.

To show 4 let M_{0} carry a projectable spin structure described by ε : Γ → \{−1, 1\}. Let \overline{ε} : \overline{Γ} → \{−1, 1\} describe the induced spin structure on \overline{Γ}\\overline{G}. By continuity as above ε_{s} := ε ∘ Φ_{s}^{−1} and \overline{ε}_{s} := \overline{ε} ∘ \overline{Φ}_{s}^{−1} describe the spin structures on M_{s} and \overline{M}_{s} := \overline{Γ}_{s}\\overline{G} resp.

Just as in the proof of Theorem 5.1 we write the space of L^{2}-spinors on M_{s} as

L^{2}(Σ_{ε_{s}}(Γ_{s}\G)) = L_{ε_{s}}^{2} (Γ_{s}\G) ⊗_{ℂ} Σ_{7}.

We can assume for our calculations that

ω := γ(E_{1}) · γ(E_{2}) · . . . · γ(E_{7})

acts as − Id on Σ_{7}. As in the proof of Theorem 5.3 we write

L_{ε_{s}}^{2} (Γ_{s}\G) = ⨁_{[τ]∈T} m_{s}([τ])H_{[τ]}.

The Leibniz rule 5.1 shows that H_{[τ]} ⊗ Σ_{7} is a subspace invariant under ∇_{E_{j}} : L^{2} (Σ_{ε_{s}}(Γ_{s}\G)) → L^{2} (Σ_{ε_{s}}(Γ_{s}\G)) and therefore it is also invariant under the Dirac operator D = ∑_{j=1}^{7}(id ⊗γ(E_{j})) · ∇_{E_{j}}. Hence

spec D^{M_{s}} = ⋃_{[τ]∈T} spec D|_{H_{[τ]}⊗Σ_{7}}

and the multiplicity of each eigenvalue λ of D^{M_{s}} is the sum of m_{s}([τ]) times the multiplicity of the eigenvalue λ in D|_{H_{[τ]}⊗Σ_{7}}.

To prove that spec D^{M_{s}} is not constant in s it is sufficient to show that there is one eigenvalue λ_{s} in spec D^{M_{s}} which is continuous and nonconstant in s.

Therefore we will show that there is a τ ∈ 𝔤^{∗} vanishing on [𝔤, 𝔤] such that τ_{s} := τ ∘ Φ_{s}^{−1} satisfies m_{s}([τ_{s}]) > 0 and D|_{H_{[τs]}⊗Σ_{7}} has at least one nonconstant eigenvalue depending continuously on s.

Because of τ_{s}|_{[𝔤,𝔤]} ≡ 0 we know that H_{[τ_{s}]} is one-dimensional and exp X ∈ G acts on H_{[τ_{s}]} by multiplication with e^{2πiτ_{s}(X)}. Elements f of H_{[τ_{s}]} satisfy f(exp X) = c · e^{2πiτ_{s}(X)} for all X ∈ 𝔤 with c ∈ ℂ. The multiplicity m_{s}([τ_{s}]) is 1 if e^{2πiτ_{s}(X)} = ε_{s}(exp X) for any exp X ∈ Γ_{s} and otherwise the multiplicity is 0. This condition is independent of s.

We will compute the determinant of D|_{H_{[τs]}⊗Σ_{7}}. Writing E_{1} := X_{1}, …, E_{4} := X_{4}, E_{5} := Z_{1}, E_{6} := Z_{2} and E_{7} := Z, we apply (1.1) and get for S ∈ Σ_{7} and f ∈ H_{[τ_{s}]}:

D^{M_{s}}(fS) = ∑_{i} (R_{∗}(E_{i})f) γ(E_{i})S + \frac{1}{4}f ∑_{i,j,k} Γ_{ij}^{k}γ(E_{i})γ(E_{j})γ(E_{k})S
= ∑_{i=1}^{4} 2πiτ_{s}(X_{i})fγ(E_{i})S + fA(S)

where A := (1/4) ∑_{i,j,k} Γ_{ij}^{k}γ(E_{i})γ(E_{j})γ(E_{k}) ∈ End(Σ_{7}) is constant in s. Using the Koszul formula one can express A in a suitable basis by the matrix

A = \frac{1}{4} ( \begin{matrix}−2 & 0 & 0 & 1 & 2 & i & i & −2 \\ 0 & 0 & 1 & 0 & −i & 0 & 0 & −i \\ 0 & 1 & 0 & 0 & i & 0 & 0 & i \\ 1 & 0 & 0 & 2 & 2 & −i & −i & −2 \\ 2 & i & −i & 2 & 2 & 0 & 0 & −1 \\ −i & 0 & 0 & i & 0 & 0 & −1 & 0 \\ −i & 0 & 0 & i & 0 & −1 & 0 & 0 \\ −2 & i & −i & −2 & −1 & 0 & 0 & −2 \end{matrix} )

Then

det D|_{H_{[τs]}⊗Σ_{7}} = det [A + 2πi ∑_{i=1}^{4} τ_{s}(X_{i})γ(E_{i})]

and

δ_{τ} := \frac{d}{ds}|_{s=0} det D|_{H_{[τs]}⊗Σ_{7}}
= − \frac{1}{64}π^{2} (−τ(X_{1}) τ(X_{2}) + τ(X_{3}) τ(X_{4}))
(4096 π^{4} τ(X_{1})^{4} + 4096 π^{4} τ(X_{2})^{4} + 4096 π^{4} τ(X_{3})^{4} + 4096 π^{4} τ(X_{4})^{4}
+ 8192 π^{4} τ(X_{1})^{2} τ(X_{2})^{2} + 8192 π^{4} τ(X_{1})^{2} τ(X_{3})^{2} + 8192 π^{4} τ(X_{2})^{2} τ(X_{3})^{2}
+ 8192 π^{4} τ(X_{1})^{2} τ(X_{4})^{2} + 8192 π^{4} τ(X_{2})^{2} τ(X_{4})^{2} + 8192 π^{4} τ(X_{3})^{2} τ(X_{4})^{2}
+ 512 π^{2} τ(X_{3}) τ(X_{1}) + 128 π^{2} τ(X_{4})^{2} + 512 π^{2} τ(X_{4}) τ(X_{2}) + 128 π^{2} τ(X_{2})^{2}
+ 128 π^{2} τ(X_{1})^{2} + 128 π^{2} τ(X_{3})^{2} + 1).

The set

{(τ(X_{1}), τ(X_{2}), τ(X_{3}), τ(X_{4})) | τ ∈ 𝔤^{∗}, τ|_{[𝔤,𝔤]} ≡ 0, m([τ]) > 0}

is an affine lattice in ℝ^{4} (i.e. up to translation a discrete cocompact subgroup). But the only polynomial vanishing on an affine lattice of ℝ^{4} is the zero polynomial. Therefore there exist τ ∈ 𝔤^{∗} with τ|_{[𝔤,𝔤]} ≡ 0, m([τ]) > 0 and δ_{τ} ≠ 0. □

References

[1] C. Bär, Das Spektrum von Dirac-Operatoren, Bonner Math. Schr. 217 (1991)

[2] C. Bär, The Dirac operator on homogeneous spaces and its spectrum on 3dimensional lens spaces, Arch. Math. 59, 65–79 (1992)

[3] C. Bär, The Dirac operator on space forms of positive curvature, J. Math. Soc. Japan 48, 69–83 (1994)

[4] C. Bär, Metrics with harmonic spinors, GAFA 6, 899–942 (1996)

[5] L. Bérard Bergery, J.-P. Bourguignon, Riemannian submersions with totally geodesic fibers, Illinois J. Math. 26, 181–200 (1982)

[6] N. Berline, E. Getzler, M. Vergne, Heat kernels and Dirac operators, Springer-Verlag 1991

[7] J.-P. Bourguignon, P. Gauduchon, Spineurs, opérateurs de Dirac et variations de métriques, Commun. Math. Phys. 144, 581–599 (1992)

[8] U. Bunke, Upper bounds of small eigenvalues of the Dirac operator and isometric immersions, Ann. Glob. Anal. Geom. 9, 109–116 (1991)

[9] M. Cahen, A. Franc, S. Gutt, Spectrum of the Dirac operator on complex projective space P_{2q−1}(ℂ), Lett. Math. Phys. 18, 165–176 (1989)

[10] M. Cahen, A. Franc, S. Gutt, Erratum to ’Spectrum of the Dirac operator on complex projective space P_{2q−1}(ℂ)’, Lett. Math. Phys. 32, 365–368 (1994)

[11] R. Camporesi, A. Higuchi, On the eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces, J. Geom. Phys. 20, no. 1, 1–18 (1996)

[12] L. Corwin, F.P. Greenleaf, Representations of nilpotent Lie groups and their applications, Part1: Basic theory and examples, Cambridge University Press 1990

[13] P. Eberlein, Geometry of two-step nilpotent groups with a left invariant metric, Ann. Scien. de l’Ecole Norm. Sup. 27, 611–660 (1994)

[14] H. Fegan, The spectrum of the Dirac operator on a simply connected compact Lie group, Simon Stevin 61, 97–108 (1987)

[15] T. Friedrich, Zur Abhängigkeit des Dirac-Operators von der Spin-Struktur, Colloq. Math. 48, 57–62 (1984)

[16] R. Gornet, A New Construction of Isospectral Riemannian Nilmanifolds with Examples, Michigan Math. J. 43, 159–188 (1996)

[17] R. Gornet, Continuous families of Riemannian manifolds isospectral on functions but not on 1-forms, to appear in J. Geom. Anal.

[18] C. Gordon, E. Wilson, Isospectral deformations of compact solvmanifolds, J. Diff. Geom. 19, 241–256 (1984)

[19] C. Gordon, E. Wilson, The spectrum of the Laplacian on Riemannaian Heisenberg manifolds, Mich. Math. J. 33, 253–271 (1986)

[20] N. Hitchin, Harmonic spinors, Adv. Math. 14, 1-55 (1974)

[21] H.-B. Lawson, M.-L. Michelsohn, Spin Geometry, Princeton University Press, Princeton 1989

[22] J.-L. Milhorat, Spectre de l’opérateur de Dirac sur les espaces projectifs quaternioniens, C. R. Acad. Sci., Paris 1, 69–72 (1992)

[23] J.-L. Milhorat, Spectrum of the Dirac operator on quaternion-Kähler spin manifolds: The case of the symmetric space Gr_{2}(ℂ^{m+2}), J. Math. Phys. 39 No. 1, 594–609 (1998)

[24] J. Milnor, Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci. 51, 542 (1964)

[25] A. Moroianu, Opérateur de Dirac et submersions riemanniennes, Thesis, École Polytechnique, Palaiseau 1996

[26] C.G. Moore, J.A. Wolf, Square integrable representations of nilpotent groups, Trans. AMS 185, 445–462 (1973)

[27] M.S. Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag 1972

[28] L. Seeger, Der Dirac-Operator auf kompakten symmetrischen Räumen, Diplomarbeit, Universität Bonn 1997

[29] L. Seeger, The Dirac operator on oriented Grassmann manifolds and G_{2}/SO(4), Preprint 1997

[30] S. Seifarth, U. Semmelmann, The Spectrum of the Dirac Operator on the Odd Dimensional Complex Projective Space P^{2m−1}(ℂ), SFB 288 Preprint 95 (1993)

[31] H. Strese, Über den Dirac-Operator auf Graßmann-Mannigfaltigkeiten, Math. Nachr. 98, 53–59 (1980)

[32] H. Strese, Spektren symmetrischer Räume, Math. Nachr. 98, 75–82 (1980)

[33] S. Sulanke, Die Berechnung des Spektrums des Quadrates des Dirac-Operators auf der Sphäre, Doktorarbeit, Humboldt-Universität Berlin, 1979

[34] A. Trautman, Spin structures on hypersurfaces and the spectrum of the Dirac operator on spheres, in Oziewicz, Zbigniew (ed.), Spinors, Twistors, Clifford Algebras and Quantum Deformations, Kluwer Academic Publishers 1993, 25–29 [35] A. Trautman, The Dirac operator on hypersurfaces, Acta Phys. Polon. B 26,

1283–1310 (1995)

Authors’ address:

Mathematisches Institut Universität Freiburg Eckerstr. 1 79104 Freiburg Germany

e-mail:

ammann@mathematik.uni-freiburg.de
baer@mathematik.uni-freiburg.de