The symmetry of $text{spin}^{mathbb{C}}$ Dirac spectrums on Riemannian product manifolds

http://dx.doi.org/10.4134/JKMS.2015.52.5.1037

THE SYMMETRY OF spin ^C DIRAC SPECTRUMS ON

RIEMANNIAN PRODUCT MANIFOLDS

Kyusik Hong and Chanyoung Sung

Abstract. It is well-known that the spectrum of a spin ^C Dirac operator on a closed Riemannian spin ^C manifold M ^2k of dimension 2k for k ∈ N is symmetric. In this article, we prove that over an odd-dimensional Riemannian product M ₁ ^2p × M ₂ ^2q+1 with a product spin ^C structure for p ≥ 1, q ≥ 0, the spectrum of a spin ^C Dirac operator given by a product connection is symmetric if and only if either the spin ^C Dirac spectrum of M ₂ ^2q+1 is symmetric or (e ^{1 2 c 1 (L 1 )} A(M ˆ 1 ))[M 1 ] = 0, where L 1 is the associated line bundle for the given spin ^C structure of M 1 .

1. Introduction

This article is a generalization of the paper [7] to a spin ^C Dirac operator on a spin ^C manifold. Let (M ⁿ , g) be an n-dimensional closed Riemannian manifold with a spin ^C structure given by an associated complex line bundle L with c 1 (L) ≡ w ² (M ⁿ ) mod 2. Here c 1 is the first Chern class and w 2 is the 2-nd Stiefel-Whitney class. Let A be a U (1)-connection on the line bundle L. This combined with the Levi-Civita connection of g induces a covariant derivative

∇ ^A : Γ(Σ(M, L)) → Γ(T ^∗ M ⊗ Σ(M, L))

in the associated spinor bundle Σ(M, L). The associated spin ^C Dirac operator D ^A is the composition of the covariant derivative ∇ ^A and Clifford multiplica- tion γ

D ^A = γ ◦ ∇ ^A : Γ(Σ(M, L)) → Γ(T ^∗ M ⊗ Σ(M, L)) → Γ(Σ(M, L)).

Then D ^A is a self-adjoint elliptic operator of first-order. Therefore the spectrum Spec(D ^A ) of D ^A is discrete and real. The behavior of Spec(D ^A ) generally depends on the U (1)-connection A, the metric, and the spin ^C structure. For general properties of Dirac operators we refer to [4, 8].

Definition 1.1. The Spec(D ^A ) is called symmetric, if the following conditions hold:

Received December 7, 2014; Revised February 12, 2015.

2010 Mathematics Subject Classification. 53C27, 58C40.

Key words and phrases. Dirac operator, spin ^C manifold, spectrum, eta invariant.

c

2015 Korean Mathematical Society

1037

(1) There exits −λ ∈ Spec(D ^A ) whenever λ ∈ Spec(D ^A ).

(2) The multiplicity of −λ is equal to that of λ.

If n is even, then the volume form µ of (M ⁿ , g) anti-commutes with the spin ^C Dirac operator D ^A

D ^A ◦ µ = −µ ◦ D ^A . Thus, in this case Spec(D ^A ) is symmetric.

Since D ^A is elliptic, each eigenspace is finite-dimensional. The asymmetry of Spec(D ^A ) on an odd-dimensional manifold was investigated by Atiyah, Patodi and Singer [1] via the eta funtion defined as

η _D ^A (s) := X

λ6=0

sign(λ)

|λ| ^s , s ∈ C,

where λ runs through the eigenvalues according to their multiplicities. The series η D ^A (s) converges for sufficiently large Re(s) and has the meromorphic continuation to the whole C with η D ^A (0) finite. They showed that the value η D ^A (0), called the eta invariant of D ^A , appears as a global correction term for the index theorem for compact manifolds with boundary. Note that η D ^A (s) ≡ 0 is a necessary condition for the symmetry of Spec(D ^A ).

In this paper, we prove a necessary and sufficient condition for the symmetry of Spec(D ^A ) on odd-dimensional Riemannian spin ^C product manifolds using the ideas mainly adapted from that of E. C. Kim [7]. We will take a convention that a superscript on a manifold denotes its dimension.

Theorem 1.2. Let (Q ⁿ := M ₁ ^2p ×M 2 ^2q+1 , h := g 1 +g 2 ) be a Riemannian product of two closed Riemannian spin ^C manifolds (M ₁ ^2p , g 1 ), p ≥ 1, and (M 2 ^2q+1 , g 2 ), q ≥ 0. Let π 1 : Q ⁿ → M 1 ^2p and π 2 : Q ⁿ → M 2 ^2q+1 be the natural projections.

Suppose that M 1 (resp. M 2 ) is equipped with a spin ^C structure given by a complex line bundle L 1 (resp. L 2 ) with c 1 (L 1 ) ≡ w 2 (M ₁ ^2p ) mod 2 (resp. c 1 (L 2 )

≡ w ² (M ₂ ^2q+1 ) mod 2), and M 1 ×M ² is equipped with the product spin ^C structure given by the complex line bundle L = π ^∗ ₁ (L 1 ) ⊗ π 2 ^∗ (L 2 ).

Let A 1 (resp. A 2 ) be a U (1)-connection on L 1 (resp. L 2 ), and A = π ^∗ ₁ (A 1 )+

π ^∗ ₂ (A 2 ) be a connection of L. Let D _M ^{A 1} ₁ , D _M ^{A 2} ₂ , and D ^A be the associated spin ^C Dirac operators of (M ₁ ^2p , g 1 ), (M ₂ ^2q+1 , g 2 ), and (Q ⁿ , h), respectively. Then we have the following:

Spec(D ^A ) is symmetric if and only if either Spec(D _M ^{A 2} ₂ ) is symmetric or (e ^{1 2 c 1 (L 1 )} A(M ˆ 1 ))[M 1 ] = 0, where ˆ A(M 1 ) denotes the ˆ A-class of T M 1 .

As a result auxiliary, we generalize real and quaternionic structures of spinor

bundles on spin manifolds to complex anti-linear mappings between Σ(M, L)

and Σ(M, −L) on spin ^C manifolds, and study several variations on product

spin ^C manifolds. These may be used as tools for studying the spectrum of a

spin ^C Dirac operator on a product spin ^C manifold.

2. Preliminaries

This section is divided into two parts. In the first part, we explain that the spinor bundle of a Riemannian spin ^C product manifold has a natural tensor- product splitting. In the second part, we prove the decomposition property of L ² -sections of a vector bundle given by the tensor product of two vector bundles.

Consider the following commutative diagram:

Spin ^C (k 1 + k 2 ) oo ? _

π



Spin ^C (k 1 ) ⊗ ^S

Spin ^C (k 2 ) ∋ (±a ¹ ⊗ ^Z

±e

) ⊗ ^S

(±a ² ⊗ ^Z

±e

)



SO(k 1 + k 2 ) ⊕ S ¹ oo ? _

SO(k 1 ) ⊕ SO(k 2 ) ⊕ S ¹ ∋ (a 1 , a 2 , e ^i(θ

^+θ

⁾ ),

where π is a 2-fold covering map.

Let P M 1 , P M 2 , and P Q be the SO(k 1 )-, SO(k 2 )-, and SO(k 1 + k 2 )-principal bundles of positively oriented orthonormal frames of (M ₁ ^{k 1} , g 1 ), (M ₂ ^{k 2} , g 2 ), and (Q ⁿ := M ₁ ^{k 1} × M 2 ^{k 2} , h := g 1 + g 2 ), respectively. Let L 1 (resp. L 2 ) be a complex line bundle with c 1 (L 1 ) ≡ w 2 (M ₁ ^{k 1} ) mod 2 (resp. c 1 (L 2 ) ≡ w 2 (M ₂ ^{k 2} ) mod 2), and let L = π ^∗ ₁ L 1 ⊗π 2 ^∗ L 2 , where π 1 : Q → M 1 and π 2 : Q → M 2 are the natural projections. To denote the double cover of a principal bundle, we will put e.

Then the above pointwise diagram globalizes over the whole manifold to give the following commutative diagram:

P ^ Q ⊕ L oo ? _

π



π ₁ ^∗ ( ^ P M 1 ⊕ L ¹ ) ⊗ S ¹ π ₂ ^∗ ( ^ P M 2 ⊕ L ² )



P Q ⊕ (π ^∗ 1 L 1 ⊗ π ^∗ 2 L 2 ) oo ? _

(π ₁ ^∗ P M 1 ⊕ π ^∗ 2 P M 2 ) ⊕ (π 1 ^∗ L 1 ⊗ π 2 ^∗ L 2 ).

Thus, the Spin ^C (k 1 + k 2 )-principle bundle ^ P Q ⊕ L over (Q ⁿ , h) reduces to the Spin ^C (k 1 ) ⊗ S ¹ Spin ^C (k 2 )-principal bundle π ^∗ ₁ ( ^ P M 1 ⊕ L 1 ) ⊗ π 2 ^∗ ( ^ P M 2 ⊕ L 2 ) in a π-equivariant way.

Let (E 1 , . . . , E k 1 ) and (F 1 , . . . , F k 2 ) be local orthonormal frames on (M ₁ ^{k 1} , g 1 ) and (M ₂ ^{k 2} , g 2 ), respectively. We identify (E 1 , . . . , E k 1 ) and (F 1 , . . . , F k 2 ) with their lifts to (Q ⁿ , h). We may then regard (E 1 , . . . , E k 1 , F 1 , . . . , F k 2 ) as a local orthonormal frame on (Q ⁿ , h). Let µ 1 = E ¹ ∧ · · · ∧ E ^{k 1} , E ^t := g 1 (E t , ·), and µ 2 = F ¹ ∧ · · · ∧ F ^{k 2} , F ^l := g 2 (F l , ·), be the volume forms of (M 1 ^{k 1} , g 1 ) and (M ₂ ^{k 2} , g ₂ ), respectively, as well as their lifts to (Q ⁿ , h).

Now let’s assume that at least one of k 1 and k 2 is even. Say, k 1 = 2p for p ∈ N. Using the following Clifford action on the tensor product vector space [7, Section 2], one can extend the Spin ^C (k 1 ) ⊗ S ¹ Spin ^C (k 2 )-action on △ k 1 ⊗ △ k 2

to the Spin ^C (k 1 + k 2 )-action:

(2.1) E t · (ϕ 1 ⊗ ϕ 2 ) = (E t · ϕ 1 ) ⊗ ϕ 2 , t = 1, . . . , 2p, (2.2) F l · (ϕ 1 ⊗ ϕ 2 ) = ( √

−1) ^p (µ 1 · ϕ 1 ) ⊗ (F l · ϕ 2 ), l = 1, . . . , k 2 ,

where ϕ 1 ∈△ ^{k 1} and ϕ 2 ∈△ ^{k 2} . If k 2 = 2p, the Clifford actions are defined as:

E t · (ϕ 1 ⊗ ϕ 2 ) = (E t · ϕ 1 ) ⊗ ( √

−1) ^p (µ 2 · ϕ 2 ), t = 1, . . . , k 1 , F l · (ϕ ¹ ⊗ ϕ ² ) = ϕ 1 ⊗ (F ^l · ϕ ² ), l = 1, . . . , 2p.

Hence we can conclude that the associated vector bundle π ₁ ^∗ (Σ(M 1 , L 1 )) ⊗ π ^∗ ₂ (Σ(M 2 , L 2 )) for the principal bundle π ₁ ^∗ ( ^ P M 1 ⊕ L ¹ ) ⊗ S ¹ π ₂ ^∗ ( ^ P M 2 ⊕ L ² ) is also an associated vector bundle for ^ P Q ⊕ L.

Recall that there is a unique spinor bundle Σ(M 1 × M ² , L) on M 1 × M ² , if k 1 + k 2 is even, and there exist two of them, if k 1 + k 2 is odd. Suppose that dim M 1 = 2p and dim M 2 = 2q + 1 for p ≥ 1 and q ≥ 0. Then we have two non-isomorphic spinor bundles

Σ(M 1 , L 1 ) ⊗ Σ ^′ (M 2 , L 2 ) and Σ(M 1 , L 1 ) ⊗ Σ ^′′ (M 2 , L 2 )

on M 1 ×M ² , where Σ(M 1 , L 1 ) is a unique spinor bundle on M 1 , and Σ ^′ (M 2 , L 2 ), Σ ^′′ (M 2 , L 2 ) denote two non-isomorphic spinor bundles corresponding to Σ(M 2 , L 2 ) on M 2 (more precisely, they are the eigenbundles of +( √

−1) ^q+1 and

−( √

−1) ^q+1 under the action of µ 2 , respectively). One can easily check that Σ(M 1 , L 1 ) ⊗ Σ ^′ (M 2 , L 2 ) and Σ(M 1 , L 1 ) ⊗ Σ ^′′ (M 2 , L 2 ) are the eigenbundles of +( √

−1) ^p+q+1 and −( √

−1) ^p+q+1 under the action of the volume form µ 1 ∧ µ ² on M 1 × M ² , respectively.

Since the tensor product bundle π ^∗ ₁ (Σ(M 1 , L 1 )) ⊗ π 2 ^∗ (Σ(M 2 , L 2 )) and the spinor bundle Σ(M 1 × M ² , L = π ₁ ^∗ L 1 ⊗ π ^∗ 2 L 2 ) have the same dimension, we have proved the first part of the following:

Lemma 2.3. If at least one of k 1 and k 2 is even, then

Σ(M 1 × M ² , L = π ₁ ^∗ L 1 ⊗ π 2 ^∗ L 2 ) = π ^∗ ₁ (Σ(M 1 , L 1 )) ⊗ π 2 ^∗ (Σ(M 2 , L 2 )).

If both k 1 and k 2 are odd, then

Σ(M 1 × M 2 , L = π ^∗ ₁ L 1 ⊗ π 2 ^∗ L 2 )

= (π ₁ ^∗ (Σ ^′ (M 1 , L 1 )) ⊕ π 1 ^∗ (Σ ^′′ (M 1 , L 1 ))) ⊗ π ^∗ 2 (Σ(M 2 , L 2 )).

The proof of the second part can be done in the same as the spin case, whose details can be found in [7, Section 4].

Moreover the connections on the LHS of the previous diagram are induced ones from the RHS, which are subbundles. A spin ^C connection on

π ₁ ^∗ ( ^ P M 1 ⊕ L ¹ ) ⊗ S ¹ π ₂ ^∗ ( ^ P M 2 ⊕ L ² ),

which is lifted from downstairs is given by the tensor-product. Therefore, letting A i for i = 1, 2 and A = π ₁ ^∗ (A 1 ) + π ₂ ^∗ (A 2 ) be connections on L i and L respectively, and ∇ ^{A 1} , ∇ ^{A 2} , and ∇ ^A be the spinor derivatives of Σ(M 1 , L 1 ), Σ(M 2 , L 2 ), and Σ(M 1 × M ² , L) respectively, we have

(2.4) ∇ ^A X (ϕ 1 ⊗ ϕ ² ) = (∇ ^A π _1∗ ¹ (X) ϕ 1 ) ⊗ ϕ ² + ϕ 1 ⊗ (∇ ^A π _2∗ ² (X) ϕ 2 )

for ϕ i ∈ Γ(π ^∗ i (Σ(M i , L i ))) and X ∈ T (M 1 × M 2 ).

Now we prove some analysis lemmas on general vector bundles.

Lemma 2.5. Let E and F be hermitian vector bundles over M 1 , M 2 , respec- tively, and π ₁ ^∗ E ⊗ π 2 ^∗ F be the induced bundle over M 1 × M 2 , where each π l : M 1 ×M 2 → M l for l = 1, 2 is the natural projection. Let L ² (π ₁ ^∗ E ⊗π ^∗ 2 F ), L ² (E), L ² (F ) be the completion, with respect to the L ² -norm, of C ⁰ (π ^∗ ₁ E ⊗ π 2 ^∗ F), C ⁰ (E), C ⁰ (F ), respectively. Then we have

L ² (π ^∗ ₁ E ⊗ π ^∗ 2 F ) = π ₁ ^∗ (L ² (E)) ⊗ π 2 ^∗ (L ² (F )), where the over-line denotes the L ² -completion.

Proof. Let {ϕ ^α (x)} and {ψ ^β (y)} be orthonormal bases for π 1 ^∗ (L ² (E)) and π ^∗ ₂ (L ² (F )) respectively. Then {ϕ ^α (x) ⊗ ψ ^β (y)} forms an orthonormal set in L ² (π ^∗ ₁ E ⊗ π ^∗ 2 F ), and hence

L ² (π ^∗ ₁ E ⊗ π ^∗ 2 F ) ⊇ π 1 ^∗ (L ² (E)) ⊗ π 2 ^∗ (L ² (F )).

To prove the reverse direction, we will prove that {ϕ ^α (x)⊗ψ ^β (y)} is actually a maximal orthonormal set, i.e., basis. Let rank(E) = m and f ∈ C ⁰ (π ^∗ ₁ E ⊗ π ^∗ ₂ F ). Take U × M ² ⊂ M ¹ × M ² , where U is a small open neighborhood of a point in M 1 . Since f | x×{M 2 } for each x ∈ U is continuous and hence in L ² (F ), the section f on U × M ² is expressed as

f (x, y) = (f 1 (x, y), . . . , f m (x, y)) = X

β

a β (x)ψ β (y)

= ( X

β

a β,1 (x)ψ β (y), . . . , X

β

a β,m (x)ψ β (y)).

Since f is continuous, we have the continuity of a β,k (x) = hf ^k (x, y), ψ β (y)i L ² (F ) =

Z

M 2

hf ^k (x, y), ψ β (y)i ^F dy,

where h·, ·i F is the hermitian inner product on F . Applying the H¨ older’s in- equality, we obtain

Z

U |a ^β,k (x)| ² dx ≤ Z

U

( Z

M 2

|f ^k (x, y)| ^F |ψ ^β (y)| ^F dy) ² dx

≤ Z

U

( Z

M 2

|f k (x, y)| ² F dy Z

M 2

|ψ β (y)| ² F dy) dx

= Z

U

( Z

M 2

|f ^k (x, y)| ² F dy ) dx < ∞,

where the finiteness is due to the fact that f ∈ L ² (π ₁ ^∗ E ⊗ π ^∗ 2 F ) and hence f k ∈ L ² (π ^∗ ₁ E | U(p) ⊗ π 2 ^∗ F). This implies that a β,k and hence a β are locally in L ² . Moreover, since f ∈ Γ(π 1 ^∗ E ⊗ π 2 ^∗ F ) and ψ β ∈ Γ(π 2 ^∗ F), we have

a β (x) = Z

M 2

hf(x, y), ψ β (y)i F dy ∈ Γ(π ^∗ 1 E).

Therefore, we can write

a β (x) = X

α

c αβ ϕ α (x) for c αβ ∈ C, and hence f can be expressed as

f (x, y) = X

β

X

α

c αβ ϕ α (x) ⊗ ψ ^β (y).

Because the subset of continuous sections is dense in the space of L ² -sections,

the proof is completed. 

Remark 2.6. Lemma 2.5 remains valid when E and F are real vector bundles with Riemannian metrics over M 1 and M 2 , respectively.

Corollary 2.7. Let D M 1 : E → E (resp. D M 2 : F → F ) denote a linear self- adjoint elliptic differential operator on a complex vector bundle over a closed manifold M 1 (resp. M 2 ) and let Γ ρ (D M j ), for j ∈ {1, 2}, denote the space of all eigenvectors of D M j with eigenvalue ρ ∈ R. If D ^M 1 ⊗ Id + Id ⊗ D ^M 2 : π ^∗ ₁ E ⊗ π ^∗ 2 F → π ^∗ 1 E ⊗ π 2 ^∗ F as an operator on M 1 × M ² is elliptic, then we have

Γ γ (D M 1 ⊗ Id + Id ⊗ D ^{M 2} ) = M

γ=χ+ν

(π ₁ ^∗ (Γ χ (D M 1 )) ⊗ π ^∗ 2 (Γ ν (D M 2 ))).

Proof. “ ⊇ ” part is obvious, and we will show the other direction. Note that D M 1 ⊗ Id + Id ⊗ D ^M 2 is also self-adjoint. Since the unit norm eigenvectors of D M 1 , D M 2 , and D M 1 ⊗Id+Id⊗D ^M 2 form orthonormal bases of L ² (E), L ² (F ), and L ² (π ^∗ ₁ E ⊗ π ^∗ 2 F ), respectively, the proof follows from Lemma 2.5.  Lemma 2.8. Let D : E → E be a linear self-adjoint elliptic operator, where E is a complex vector bundle over a closed manifold M . Then all the eigenvectors of D ² come from the eigenvectors of D, and the squares of the eigenvalues of D are exactly the eigenvalues of D ² .

Proof. Obviously, the eigenvector of D is the eigenvector of D ² . Since the unit norm eigenvectors of D form an orthonormal basis of L ² (E), and the same is true for the unit norm eigenvectors of D ² , the conclusion follows. 

3. Real and quaternionic structures

Let’s first review some basic properties of real or quaternionic structures j 0

and j 1 of the spinor representation. For more details, the readers are referred to [3, 4, 6, 7, 9].

The real Clifford algebra Cl(R ⁿ ) is multiplicatively generated by the stan-

dard basis {e 1 , . . . , e n } of the Euclidean space R ⁿ subject to the relations

e ² _i = −1 for all i ≤ n and e i e j = −e j e i for all i 6= j. Note that the di-

mension of Cl(R ⁿ ) is 2 ⁿ . The complexification Cl(R ⁿ ; C) := Cl(R ⁿ ) ⊗ ^R C is

isomorphic to the matrix algebra M (2 ^m ; C) for n = 2m and to the matrix al-

gebra M (2 ^m ; C) ⊕ M(2 ^m ; C) for n = 2m + 1. For an explicit isomorphism map

for n ≥ 2, we refer to [5, Section 1] (or [7, Section 2]). Following them, let us denote by u(ǫ) ∈ C ² the vector

(3.1) u(ǫ) := 1

√ 2

 1

−ǫ √

−1



, ǫ = ±1.

Then

(3.2) u(ǫ 1 , . . . , ǫ m ) := u(ǫ 1 ) ⊗ . . . , ⊗u(ǫ m ), m = [ n 2 ] ≥ 1,

form an orthonormal basis for the spinor space △ n := C ^{2 m} , m = [ ⁿ ₂ ] ≥ 1, with respect to the standard hermitian inner product.

Definition 3.3. The complex-antilinear mappings j 0 , j 1 :△ n →△ ⁿ defined, in the notations of (3.1) and (3.2), by

j 0 u(ǫ 1 , . . . , ǫ m ) = ( √

−1) ^{P m α=1 αǫ α} u (−ǫ 1 , . . . , −ǫ m ), j 1 u(ǫ 1 , . . . , ǫ m ) = ( √

−1) ^{P m α=1 (m−α+1)ǫ α} u (−ǫ 1 , . . . , −ǫ m ), m = [ n 2 ], are called the j 0 -structure and j 1 -structure, respectively.

The following facts are well-known [3, 7]. Fix m = [ ⁿ ₂ ].

(A) j 0 ◦ e k = e k ◦ j 0 for all k = 1, . . . , 2m and j 0 ◦ e 2m+1 = (−1) ^m+1 e 2m+1 ◦ j 0 . Thus, the mapping j 0 :△ n →△ n is Spin(n)-equivariant for n, n 6≡

1 mod 4.

(B) j 1 ◦ e l = (−1) ^m+1 e l ◦ j 1 for all l = 1, . . . , n. Thus, the mapping j 1 :△ n →△ n is Spin(n)-equivariant for all n ≥ 2.

(C) j 0 ◦ j ⁰ = j 1 ◦ j ¹ = (−1) ^m(m+1)/2 and j 0 ◦ j ¹ = j 1 ◦ j ⁰ .

(D) hj ⁰ (ψ), j 0 (ϕ)i = hj ¹ (ψ), j 1 (ϕ)i = hϕ, ψi, ϕ, ψ ∈△ ⁿ , where h·, ·i is the standard hermitian inner product on △ n .

Thus, j 0 (for n 6≡ 1 mod 4) and j 1 give real (resp. quaternionic) structures on

△ _n as Spin(n)-representations, if m ≡ 0, 3 mod 4 (resp. m ≡ 1, 2 mod 4).

Let us now fix a local trivialization of Σ(M, L) on a spin ^C manifold M ⁿ . Namely, let ∪ α U α be an open covering of M for which there exits a system of transition functions {g α 1 α 2 : U α 1 ∩ U α 2 → Spin ^C (n) = Spin(n) ⊗ ^Z 2 S ¹ }. Define g α 1 α 2 : U α 1 ∩ U α 2 → Spin ^C (n) by

x 7→ f(x) ⊗ ^Z 2 h(x),

where f (x) ⊗ ^Z 2 h(x) = g α 1 α 2 (x) for x ∈ U α 1 ∩ U α 2 , f (x) ∈ Spin(n) and h(x) ∈ S ¹ . Then g α 1 α 2 is the transition function of a local trivialization for Σ(M, −L).

By the property (A), j 0 ◦(f(x)⊗ ^Z 2 h (x)) = (f (x)⊗ ^Z 2 h (x))◦j ⁰ for n 6≡ 1 mod 4.

Also, by the property (B), j 1 ◦ (f(x) ⊗ ^Z 2 h (x)) = (f (x) ⊗ ^Z 2 h (x)) ◦ j 1 for n ≥ 2.

We then have the following commutative diagram:

U α 1 × △ n g _α1α2

//

j r



U α 2 × △ n



j r



U α 1 × △ ^{n g α1α2} // U α 2 × △ ⁿ ,

where r = 0 with n 6≡ 1 mod 4, or r = 1 with n ≥ 2. Thus, the mapping j r is compatible with the transition functions of Σ(M, L) and Σ(M, −L) so that the j 0 - and j 1 -structure can be globalized to mappings j 0 , j 1 : Σ(M, L) → Σ(M, −L), and we can carry all the properties (A)–(D) over to Σ(M, L). Be aware that the mapping j 0 is well-defined for n 6≡ 1 mod 4, and j ¹ is well-defined for all n ≥ 2.

Lemma 3.4.

D ^−A ◦ j ⁰ = j 0 ◦ D ^A for n 6≡ 1 mod 4, and

D ^−A ◦ j ¹ = (−1) ^m+1 j 1 ◦ D ^A .

Proof. Let (ω i,j ) be the so(n)-valued 1-form on U α coming from the Levi- Civita connection of (M ⁿ , g ). The spinor derivative ∇ ^A with respect to a U (1)-connection A in the bundle L is locally expressed as

j k (∇ ^A X s) = j k (X(s) + 1 2 ( √

−1A(X) + X

i<j

ω j,i (X)e i · e j ) · s)

= X(j k (s)) + 1 2 (− √

−1A(X) + X

i<j

ω j,i (X)e i · e j ) · j k (s)

= ∇ ^−A X j k (s),

where s ∈ Γ(Σ(M, L)), X ∈ Γ(T M), and k = 1, 2. Thus, we have (3.5) ∇ ^−A ◦ j ⁰ = j 0 ◦ ∇ ^A and ∇ ^−A ◦ j ¹ = j 1 ◦ ∇ ^A .

Now the conclusion immediately follows from the properties (A) and (B).  As a corollary, we have proved that if m is odd (resp. even), then Spec(D ^A ) = Spec(D ^−A ) (resp. −Spec(D ^−A )) with the same multiplicities.

Now we take up the case of Theorem 1.2. Using (2.1), (2.2), and (2.4), one can easily verify the following formulas:

(3.6) D ^A (ϕ 1 ⊗ ϕ ² ) = (D ^A _M ¹ ₁ ϕ 1 ) ⊗ ϕ ² + ( √

−1) ^p (µ 1 · ϕ ¹ ) ⊗ (D M ^{A 2} 2 ϕ 2 ), (3.7) (D ^A ) ² (ϕ 1 ⊗ ϕ 2 ) = ((D ^A _M ¹ ₁ ) ² ϕ 1 ) ⊗ ϕ 2 + ϕ 1 ⊗ ((D ^A M ² 2 ) ² ϕ 2 )

for ϕ i ∈ Γ(π ^∗ i (Σ(M i , L i ))).

Consider the partial spin ^C Dirac operators D ^A ₊ ¹ , D ^A ₋ ² acting on sections ψ ∈ Γ(Σ(Q, L)) of the spinor bundle over (Q ⁿ , h):

D ₊ ^{A 1} ψ = X 2p k=1

E k · ∇ ^A E ¹ k ψ, D ₋ ^{A 2} ψ =

2q+1 X

l=1

F l · ∇ ^A F l ² ψ.

Define the twist e D ^A of the spin ^C Dirac operator D ^A = D ₊ ^{A 1} + D ^A ₋ ² by D e ^A = D ^A ₊ ¹ − D − ^{A 2} .

By (2.1) and (2.2), for i = 1, 2

D ^A ₊ ¹ ◦ µ i = −µ i ◦ D + ^{A 1} , D ^A ₋ ² ◦ µ i = µ i ◦ D − ^{A 2} , and

(3.8) D ^A ◦ µ ⁱ = −µ ⁱ ◦ e D ^A , D e ^A ◦ µ ⁱ = −µ ⁱ ◦ D ^A . Since the Spin ^C (2p + 2q + 1)-principle bundle ^ P Q ⊕ L over

(Q ⁿ := M ₁ ^2p × M 2 ^2q+1 , h)

reduces to the Spin ^C (2p) ⊗ S ¹ Spin ^C (2q + 1)-principal bundle π ^∗ ₁ ( ^ P M 1 ⊕ L ¹ ) ⊗ π ^∗ ₂ ( ^ P M 2 ⊕ L ² ), the complex-antilinear mapping j ^∗ :△ 2p+2q+1 →△ ^2p+2q+1 de- fined by

j ^∗ u(ǫ 1 , . . . , ǫ p , ǫ p+1 , . . . , ǫ p+q ) = {j ⁰ u(ǫ 1 , . . . , ǫ p )} ⊗ {j ¹ u(ǫ p+1 , . . . , ǫ p+q )}, combining the j 0 - and j 1 -structure in Definition 3.3, globalizes to mapping j ^∗ : Σ(Q, L) → Σ(Q, −L). Then the mapping j ^∗ is well-defined for all p ≥ 1, q ≥ 1.

Using the properties (A) and (B) below Definition 3.3, the formulas (2.1) and (2.2), we have the following:

j ^∗ ◦ E ^k = E k ◦ j ^∗ , k = 1, . . . , 2p, j ^∗ ◦ F ^l = (−1) ^p+q+1 F l ◦ j ^∗ , l = 1, . . . , 2q + 1.

From the formulas (2.4) and (3.5), it follows that

∇ ^−A ◦ j ^∗ = j ^∗ ◦ ∇ ^A , and hence

(3.9) D ^−A ₊ ¹ ◦ j ^∗ = j ^∗ ◦ D + ^{A 1} , D ₋ ^{−A 2} ◦ j ^∗ = (−1) ^p+q+1 j ^∗ ◦ D ^A − ² .

Similarly we also define complex-antilinear mappings b j ^∗ , j ₀ ^∗ , j ₁ ^∗ : Σ(Q, L) → Σ(Q, −L) as

bj ^∗ u(ǫ 1 , . . . , ǫ p , ǫ p+1 , · · · , ǫ p+q ) = {j 1 u(ǫ 1 , . . . , ǫ p )} ⊗ {j 0 u(ǫ p+1 , . . . , ǫ p+q )}, j ₀ ^∗ u(ǫ 1 , . . . , ǫ p , ǫ p+1 , . . . , ǫ p+q ) = {j 0 u(ǫ 1 , . . . , ǫ p )} ⊗ {j 0 u(ǫ p+1 , . . . , ǫ p+q )}, j ₁ ^∗ u(ǫ 1 , . . . , ǫ p , ǫ p+1 , . . . , ǫ p+q ) = {j ¹ u(ǫ 1 , . . . , ǫ p )} ⊗ {j ¹ u(ǫ p+1 , . . . , ǫ p+q )}.

The mappings b j ^∗ and j ₀ ^∗ are well-defined when q is odd, and j ₁ ^∗ is well-defined for all p ≥ 1, q ≥ 1. One can easily check that

D ₊ ^{−A 1} ◦ bj ^∗ = (−1) ^p+1 bj ^∗ ◦ D ^A + ¹ , D ^−A ₋ ² ◦ bj ^∗ = (−1) ^p bj ^∗ ◦ D ^A − ² ,

(3.10)

D ₊ ^{−A 1} ◦ j 0 ^∗ = j ₀ ^∗ ◦ D ^A + ¹ , D ^−A ₋ ² ◦ j 0 ^∗ = (−1) ^p j ₀ ^∗ ◦ D ^A − ² , (3.11)

D ₊ ^{−A 1} ◦ j 1 ^∗ = (−1) ^p+1 j ^∗ ₁ ◦ D ^A + ¹ , D ^−A ₋ ² ◦ j 1 ^∗ = (−1) ^p+q+1 j ₁ ^∗ ◦ D − ^{A 2} . (3.12)

Putting these together, we can produce various operators which anti-commute with spin ^C Dirac operators:

Proposition 3.13. Under the assumptions of Theorem 1.2 and with the above notations, for i = 1, 2:

(1) For p ≥ 1 and q ≥ 0, D ^A ◦ (µ i ◦ D ^A + ¹ ) = −(µ i ◦ D ^A + ¹ ) ◦ D ^A .

(2) Let p ≥ 1 and q ≥ 1. If p and q are either both even or both odd, then D ^−A ◦ (µ i ◦ j ^∗ ) = −(µ i ◦ j ^∗ ) ◦ D ^A .

(3) Let p ≥ 1 and q ≥ 1. If p and q are odd, then D ^−A ◦ (µ ⁱ ◦ bj ^∗ ) =

−(µ i ◦ bj ^∗ ) ◦ D ^A and D ^−A ◦ (µ i ◦ j 0 ^∗ ) = −(µ i ◦ j 0 ^∗ ) ◦ D ^A .

(4) Let p ≥ 1 and q ≥ 1. If p and q are even, then D ^−A ◦ j 1 ^∗ = −j 1 ^∗ ◦ D ^A . Proof. Since the Riemann curvatures R(E k , F l , ·, ·) = 0 vanish and the Clifford multiplication anti-commutes, we have

(3.14) D ₋ ^{A 2} D ^A ₊ ¹ + D ₊ ^{A 1} D ^A ₋ ² = 0.

Using (3.8) and (3.14), we obtain

D ^A ◦ (µ ⁱ ◦ D ^A + ¹ ) = −µ ⁱ ◦ e D ^A ◦ D ^A + ¹

= −µ ⁱ ◦ ((D ^A + ¹ ) ² − D − ^{A 2} ◦ D ^A + ¹ )

= −µ ⁱ ◦ ((D ^A + ¹ ) ² + D ₊ ^{A 1} ◦ D ^A − ² )

= −µ ⁱ ◦ D + ^{A 1} ◦ D ^A .

Suppose that p and q are either both even or both odd. By (3.8) and (3.9), we have

D ^−A ◦ (µ i ◦ j ^∗ ) = −(µ i ◦ e D ^−A ) ◦ j ^∗ = −µ i ◦ (D + ^{−A 1} − D − ^{−A 2} ) ◦ j ^∗

= −µ i ◦ j ^∗ ◦ (D + ^{A 1} + D ^A ₋ ² ) = −(µ i ◦ j ^∗ ) ◦ D ^A .

For the statement (3), we assume that p and q are odd. By (3.8), (3.10) and (3.11), we see that

D ^−A ◦ (µ ⁱ ◦ bj ^∗ ) = −(µ ⁱ ◦ e D ^−A ) ◦ bj ^∗ = −(µ ⁱ ◦ bj ^∗ ) ◦ D ^A and D ^−A ◦ (µ i ◦ j 0 ^∗ ) = −(µ i ◦ e D ^−A ) ◦ j 0 ^∗ = −(µ i ◦ j ^∗ 0 ) ◦ D ^A .

The statement (4) follows from (3.12). 

4. Proof of Theorem 1.2

By Corollary 2.7 and (3.7), we see that the eigenvalues of (D ^A ) ² are all

possible sums of one eigenvalue of (D _M ^{A 1} ₁ ) ² and one of (D _M ^{A 2} ₂ ) ² .

From now on, we omit the projections and rewrite the formula of Corollary 2.7 as

(4.1) Γ γ ((D ^A ) ² ) = M

γ=χ+ν

(Γ χ ((D ^A _M ¹

1 ) ² ) ⊗ Γ ν ((D _M ^{A 2}

2 ) ² )).

Using the decomposition

Σ(M, L) = Σ ⁺ (M, L) ⊕ Σ ⁻ (M, L), where

Σ ^± (M 1 , L 1 ) := {ϕ ∈ Σ(M 1 , L 1 ) | µ 1 · ϕ = ±( √

−1) ^p ϕ },

we have a decomposition Σ(Q, L) = Σ ⁺ (Q, L) ⊕ Σ ⁻ (Q, L) due to the action of the volume form µ 1 = E ¹ ∧ · · · ∧ E ^2p ,

Σ ^± (Q, L) := Σ ^± (M 1 , L 1 ) ⊗ Σ(M 2 , L 2 ).

The positive part ψ ⁺ (resp. negative part ψ ⁻ ) of ψ ∈ Γ(Σ(Q, L)) is in fact equal to

ψ ^± = 1 2 ψ ± 1

2 (− √

−1) ^p µ 1 · ψ.

Lemma 4.2. Let Γ ^± ₀ (D ^A _M ¹ ₁ ) be the space of all positive (resp. negative) har- monic spinors of D _M ^{A 1}

1 . For any λ 6= 0 ∈ Spec(D ^A ), define a complex vector space

H λ := {ψ ∈ Γ ^λ (D ^A ) | D ^A + ¹ ψ = 0, D ^A ₋ ² ψ = λψ}.

Then, in the notation (4.1), we have

H λ = {Γ ⁺ 0 (D _M ^{A 1} ₁ ) ⊗ Γ (−1) ^p λ (D ^A _M ² ₂ )} ⊕ {Γ ⁻ 0 (D ^A _M ¹ ₁ ) ⊗ Γ −(−1) ^p λ (D ^A _M ² ₂ )}.

Proof. By (3.6), it is enough to show that

H λ ⊂ {Γ ⁺ 0 (D _M ^{A 1} ₁ ) ⊗ Γ (−1) ^p λ (D ^A _M ² ₂ )} ⊕ {Γ ⁻ 0 (D ^A _M ¹ ₁ ) ⊗ Γ −(−1) ^p λ (D ^A _M ² ₂ )}.

Suppose that ψ ∈ H ^λ . Then ψ ∈ Γ λ ² ((D ^A ) ² ) and (4.1) implies that

(4.3) ψ = X

k,l

c k,l ϕ 0,k ⊗ ϕ λ ² ,l , c k,l 6= 0 ∈ C,

is a finite linear combination of tensor products of some ϕ 0,k ∈ Γ ⁰ ((D _M ^{A 1} ₁ ) ² ) and some ϕ λ ² ,l ∈ Γ λ ² ((D ^A _M ² ₂ ) ² ). By the decomposition of Σ(Q, L), we can rewrite (4.3) as

ψ = X

k,l

c k,l ϕ ⁺ _0,k ⊗ ϕ λ ² ,l + X

k,l

c k,l ϕ ⁻ _0,k ⊗ ϕ λ ² ,l ,

where ϕ ^± _0,k ∈ Γ ^± 0 ((D ^A _M ¹ ₁ ) ² ). By Lemma 2.8, the eigenvalue λ ² of (D _M ^{A 2} ₂ ) ² comes from the eigenvalue λ or −λ of D ^A M ² 2 . Since D ^A ₋ ² ψ = λψ, one can easily see from (3.6)

ψ ∈ {Γ ⁺ 0 (D _M ^{A 1} ₁ ) ⊗ Γ (−1) ^p λ (D ^A _M ² ₂ )} ⊕ {Γ ⁻ 0 (D _M ^{A 1} ₁ ) ⊗ Γ −(−1) ^p λ (D ^A _M ² ₂ )}. 

Proof of Theorem 1.2. At first, using Proposition 3.13(1), one can define a map:

f : Γ λ (D ^A ) ∩ H λ ^⊥ −→ Γ ^−λ (D ^A ) ∩ H −λ ^⊥ defined by ψ 7→ µ ¹ · D + ^{A 1} ψ, where H _λ ^⊥ is the orthogonal complement of H λ . Note that f is bijective via the inverse map

f ⁻¹ := (−1) ^p (D ^A ₊ ¹ ) ⁻¹ · µ 1 . By Lemma 4.2,

(4.4) H −λ = {Γ ⁺ 0 (D _M ^{A 1} ₁ ) ⊗ Γ −(−1) ^p λ (D ^A _M ² ₂ )} ⊕ {Γ ⁻ 0 (D _M ^{A 1} ₁ ) ⊗ Γ (−1) ^p λ (D ^A _M ² ₂ )}.

The symmetry of Spec(D ^A ) holds if and only if dim C H λ = dim C H −λ for any λ 6= 0 ∈ Spec(D ^A ). Letting dim C (Γ ⁺ ₀ (D _M ^{A 1} ₁ )) = a 1 , dim C (Γ ⁻ ₀ (D _M ^{A 1} ₁ )) = a 2 , dim C (Γ (−1) ^p λ (D ^A _M ²

2 )) = b 1,λ , and dim C (Γ −(−1) ^p λ (D ^A _M ²

2 )) = b 2,λ , Lemma 4.2 and (4.4) give

dim C H λ − dim ^C H −λ = (a 1 − a 2 )(b 1,λ − b 2,λ )

= (ind (D ^A _M ¹ ₁ | Σ ⁺ (M 1 ,L 1 ) ))(b 1,λ − b 2,λ )

= (e ^{1 2 c 1 (L 1 )} A(M ˆ 1 ))[M 1 ](b 1,λ − b 2,λ ),

where the last equality is due to the Atiyah-Singer index theorem [8]. Now the

desired conclusion follows immediately. 

5. Examples

In this section, applying Theorem 1.2, we present an important example.

Let D ⁰ := i∂ θ be the Dirac operator of (S ¹ , g 2 = dθ ² ). Then the eigenvectors of unit L ² -norm are

e ^inθ

√ 2π for n ∈ Z

with multiplicity 1, and hence Spec(D ⁰ ) = Z. Thus Spec(D ⁰ ) is symmetric, and since

X

n∈Z\{0}

sign(n)

|n| ^s = 0 for Re(s) ≫ 1,

η _D ⁰ (s) = 0 for all s ∈ C. In [7], it is shown that the Dirac spectrum of (M ₁ ^2p × S ¹ , g 1 + g 2 ) for a spin manifold M ₁ ^2p is symmetric. We can generalize this to the spin ^C case:

Example 5.1. Note that D ^−iadθ = D ⁰ + a for a ∈ R is a spin ^C Dirac operator of (S ¹ , g 2 = dθ ² ). Thus

Spec(D ^−iadθ ) = {n + a | n ∈ Z},

and hence Spec(D ^−iadθ ) is symmetric if and only if a ∈ Z ∪ ¹ 2 Z. In this

case, by Theorem 1.2, the spin ^C Dirac spectrum of (M ₁ ^2p × S ¹ , g 1 + g 2 ) for

any product spin ^C structure is symmetric, and hence the corresponding eta

invariant vanishes.

The eta invariant of a spin ^C Dirac operator on a product spin ^C manifold M ₁ ² ×S ¹ appears when computing the dimension of the moduli space of Seiberg- Witten equations on a cylindrical-ended 4-manifold with asymptotic boundary equal to M ₁ ² × S ¹ . For details, the readers are referred to [10].

For general a, it is known that the eta invariant of D ^−iadθ is 1 − 2a. (See [10, Example 4.1.7] or [2].)

Acknowledgments. The first author was supported by National Researcher Program of the National Research Foundation (NRF) funded by the Ministry of Science, ICT and Future Planning (No.2014028806). The second-named author was supported by the National Research Foundation of Korea grant funded by the Korea government. (NRF-2014R1A1A4A01008987).

References

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[2] , Spectral asymmetry and Riemannian geometry II, Math. Proc. Cambidge Phi- los. Soc. 78 (1975), 405–432.

[3] H. Baum, A remark on the spectrum of the Dirac operator on pseudo-Riemannian spin manifolds, SFB 288, Preprint 136, 1994.

[4] Th. Friedrich, Dirac Operators in Riemannian Geometry, Graduate Studies in Mathe- matics, vol. 25, AMS, Providence, RI, 2000.

[5] Th. Friedrich and E. C. Kim, The Einstein-Dirac equation on Riemannian spin mani- folds, J. Geom. Phys. 33 (2000), no. 1-2, 128–172.

[6] E. C. Kim, Die Einstein-Dirac-Gleichung ¨ uber Riemannschen Spin-Mannigfaltigkeiten, Thesis, Humboldt-Universit¨ at zu Berlin, 1999.

[7] , The ˆ A-genus and symmetry of the Dirac spectrum on Riemannian product manifolds, Differential Geom. Appl. 25 (2007), 309–321.

[8] H.-B. Lawson and M.-L. Michelsohn, Spin Geometry, Princeton University Press, Prince- ton, NJ, 1989.

[9] A. Lichnerowicz, Champs spinoriels et propagateurs en relativit´ e g´ en´ eral, Bull. Sci.

Math. France 92 (1964), 11–100.

[10] L. I. Nicolaescu, Notes on Seiberg-Witten Theory, Graduate Studies in Mathematics, 28. American Mathematical Society, Providence, RI, 2000.

Kyusik Hong

School of Mathematics

Korea Institute for Advanced Study Seoul 130-722, Korea

E-mail address: [email protected] Chanyoung Sung

Department of Mathematics Education Korea National University of Education Chungbuk 363-791, Korea

E-mail address: [email protected]