THE SYMMETRY OF spin C DIRAC SPECTRUMS ON RIEMANNIAN PRODUCT MANIFOLDS

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1 J. Korean Math. Soc. 52 (2015), No. 5, pp 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 k of dimension 2k for k N is symmetric. In this article, we prove that over an odd-dimensional Riemannian product p 1 q+1 2 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 q+1 2 is symmetric or (e 1 2 c 1(L 1 ) Â( ))[ ] = 0, where L 1 is the associated line bundle for the given spin C structure of. 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 classand 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 multiplication γ D A = γ A : Γ(Σ(M,L)) Γ(T M Σ(M,L)) Γ(Σ(M,L)). ThenD A isaself-adjointellipticoperatoroffirst-order. Thereforethespectrum 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, Mathematics Subject Classification. 53C27, 58C40. Key words and phrases. Dirac operator, spin C manifold, spectrum, eta invariant c 2015 Korean Mathematical Society

2 1038 KYUSIK HONG AND CHANYOUNG SUNG (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 eigenspaceis 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) := λ 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 theoremforcompact manifoldswith 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 := p 1 M2q+1 2,h := g 1 +g 2 ) be a Riemannian product of two closed Riemannian spin C manifolds (p 1,g 1), p 1, and (q+1 2,g 2 ), q 0. Let π 1 : Q n p 1 and π 2 : Q n q+1 2 be the natural projections. Suppose that (resp. ) 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 (p 1 ) mod 2 (resp. c 1(L 2 ) w 2 (q+1 2 ) mod 2), and is equipped with the product spin C structure given by the complex line bundle L = π1(l 1 ) π2(l 2 ). Let A 1 (resp. A 2 ) be a U(1)-connection on L 1 (resp. L 2 ), and A = π1 (A 1)+ π 2(A 2 ) be a connection of L. Let D A1,D A2, and D A be the associated spin C Dirac operators of (p 1,g 1), (q+1 2,g 2 ), and (Q n,h), respectively. Then we have the following: Spec(D A ) is symmetric if and only if either Spec(D A2 ) is symmetric or (e 1 2 c1(l1) Â( ))[ ] = 0, where Â() denotes the Â-class of T. 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.

3 THE SYMMETRY OF spin C DIRAC SPECTRUMS 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 tensorproduct 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 ) π SO(k 1 +k 2 ) S 1 Spin C (k 1 ) S 1 Spin C (k 2 ) (±a 1 Z2 ±e iθ 1 2 ) S 1 (±a 2 Z2 ±e iθ 2 2 ) SO(k 1 ) SO(k 2 ) S 1 (a 1,a 2,e i(θ1+θ2) ), where π is a 2-fold covering map. Let P M1,P M2, 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 k1 1,g 1), (M k2 2,g 2), and (Q n := M k1 1 Mk2 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 k1 1 ) mod 2 (resp. c 1(L 2 ) w 2 (M k2 2 ) mod 2), and let L = π1l 1 π2l 2, where π 1 : Q and π 2 : Q are the natural projections. To denote the double cover of a principal bundle, we will put. Then the above pointwise diagram globalizes over the whole manifold to give the following commutative diagram: P Q L π 1 ( P M1 L 1 ) S 1 π 2 ( P M2 L 2 ) P Q (π 1 L 1 π 2 L 2) π (π 1 P π 2 P ) (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 π1( P M1 L 1 ) π2( P M2 L 2 ) in a π-equivariant way. Let (E 1,...,E k1 ) and (F 1,...,F k2 ) be local orthonormal frames on (M k1 1, g 1 ) and (M k2 2,g 2), respectively. We identify (E 1,...,E k1 ) and (F 1,...,F k2 ) with their lifts to (Q n,h). We may then regard (E 1,...,E k1,f 1,...,F k2 ) as a local orthonormal frame on (Q n,h). Let µ 1 = E 1 E k1, E t := g 1 (E t, ), and µ 2 = F 1 F k2, F l := g 2 (F l, ), be the volume forms of (M k1 1,g 1) and (M k2 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 1Spin C (k 2 )-action on k1 k2 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,

4 1040 KYUSIK HONG AND CHANYOUNG SUNG where ϕ 1 k1 and ϕ 2 k2. 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 π1(σ(m 1,L 1 )) π2(σ(m 2,L 2 )) for the principal bundle π1( P M1 L 1 ) S 1π2( P M2 L 2 ) is also an associated vector bundle for P Q L. Recall that there is a unique spinor bundle Σ(,L) on, if k 1 + k 2 is even, and there exist two of them, if k 1 + k 2 is odd. Suppose that dim = 2p and dim = 2q + 1 for p 1 and q 0. Then we have two non-isomorphic spinor bundles Σ(,L 1 ) Σ (,L 2 ) and Σ(,L 1 ) Σ (,L 2 ) on, whereσ(,l 1 ) isauniquespinorbundleon, andσ (,L 2 ), Σ (,L 2 ) denote two non-isomorphic spinor bundles corresponding to Σ(, L 2 ) on (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(,L 1 ) Σ (,L 2 ) and Σ(,L 1 ) Σ (,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, respectively. Since the tensor product bundle π 1 (Σ(,L 1 )) π 2 (Σ(,L 2 )) and the spinor bundle Σ(,L = π 1L 1 π 2L 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(,L = π 1 L 1 π 2 L 2) = π 1 (Σ(,L 1 )) π 2 (Σ(,L 2 )). If both k 1 and k 2 are odd, then Σ(,L = π 1L 1 π 2L 2 ) = (π 1 (Σ (,L 1 )) π 1 (Σ (,L 1 ))) π 2 (Σ(,L 2 )). The proofof 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 M1 L 1 ) S 1 π 2( P M2 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 A1, A2, and A be the spinor derivatives of Σ(,L 1 ), Σ(,L 2 ), and Σ(,L) respectively, we have (2.4) A X(ϕ 1 ϕ 2 ) = ( A1 π 1 (X) ϕ 1) ϕ 2 +ϕ 1 ( A2 π 2 (X) ϕ 2) for ϕ i Γ(π i (Σ(M i,l i ))) and X T( ).

5 THE SYMMETRY OF spin C DIRAC SPECTRUMS 1041 Now we prove some analysis lemmas on general vector bundles. Lemma 2.5. Let E and F be hermitian vector bundles over,, respectively, and π 1E π 2F be the induced bundle over, where each π l : M l for l = 1,2 is the natural projection. Let L 2 (π 1 E π 2 F), L2 (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 (L2 (E)) π 2 (L2 (F)), where the over-line denotes the L 2 -completion. Proof. Let {ϕ α (x)} and {ψ β (y)} be orthonormal bases for π1 (L2 (E)) and π2(l 2 (F)) respectively. Then {ϕ α (x) ψ β (y)} forms an orthonormal set in L 2 (π1 E π 2F), and hence L 2 (π 1E π 2F) π 1 (L2 (E)) π 2 (L2 (F)). To provethe reversedirection, we will provethat {ϕ α (x) ψ β (y)} is actually a maximal orthonormal set, i.e., basis. Let rank(e) = m and f C 0 (π 1E π 2 F). Take U, where U is a small open neighborhood of a point in. Since f x {M2} for each x U is continuous and hence in L 2 (F), the section f on U is expressed as f(x,y) = (f 1 (x,y),...,f m (x,y)) = β a β (x)ψ β (y) = ( β a β,1 (x)ψ β (y),..., β a β,m (x)ψ β (y)). Since f is continuous, we have the continuity of a β,k (x) = f k (x,y),ψ β (y) L 2 (F) = f k (x,y),ψ β (y) F dy, where, F is the hermitian inner product on F. Applying the Hölder s inequality, we obtain a β,k (x) 2 dx ( f k (x,y) F ψ β (y) F dy) 2 dx U U M 2 ( f k (x,y) 2 Fdy ψ β (y) 2 Fdy) dx U M 2 = ( f k (x,y) 2 Fdy) dx <, U where the finiteness is due to the fact that f L 2 (π1 E π 2F) and hence f k L 2 (π1 E U(p) π2 F). This implies that a β,k and hence a β are locally in L 2. Moreover, since f Γ(π1 E π 2 F) and ψ β Γ(π2 F), we have a β (x) = f(x,y),ψ β (y) F dy Γ(π1 E).

6 1042 KYUSIK HONG AND CHANYOUNG SUNG Therefore, we can write a β (x) = α c αβ ϕ α (x) for c αβ C, and hence f can be expressed as f(x,y) = c αβ ϕ α (x) ψ β (y). β α 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 and, respectively. Corollary 2.7. Let D M1 : E E (resp. D M2 : F F) denote a linear selfadjoint elliptic differential operator on a complex vector bundle over a closed manifold (resp. ) and let Γ ρ (D Mj ), for j {1,2}, denote the space of all eigenvectors of D Mj with eigenvalue ρ R. If D M1 Id + Id D M2 : π1 E π 2 F π 1 E π 2 F as an operator on is elliptic, then we have Γ γ (D M1 Id+Id D M2 ) = (π1(γ χ (D M1 )) π2(γ ν (D M2 ))). γ=χ+ν Proof. part is obvious, and we will show the other direction. Note that D M1 Id+Id D M2 is also self-adjoint. Since the unit norm eigenvectors of D M1, D M2, and D M1 Id+Id D M2 form orthonormalbases of L 2 (E), L 2 (F), and L 2 (π1 E π 2F), 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 standard 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 j. Note that the dimension 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 algebra M(2 m ;C) M(2 m ;C) for n = 2m+1. For an explicit isomorphism map

7 THE SYMMETRY OF spin C DIRAC SPECTRUMS 1043 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 ( Then 1 ǫ 1 ), ǫ = ±1. (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 2m, 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) m α=1 αǫα u( ǫ 1,..., ǫ m ), j 1 u(ǫ 1,...,ǫ m ) = ( 1) m α=1 (m α+1)ǫα 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 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) j 0 (ψ),j 0 (ϕ) = j 1 (ψ),j 1 (ϕ) = ϕ,ψ,ϕ,ψ n, where, is the standard hermitian inner product on n. Thus, j 0 (for n 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 α1α 2 : U α1 U α2 Spin C (n) = Spin(n) Z2 S 1 }. Define g α1α 2 : U α1 U α2 Spin C (n) by x f(x) Z2 h(x), where f(x) Z2 h(x) = g α1α 2 (x) for x U α1 U α2,f(x) Spin(n) and h(x) S 1. Then g α1α 2 is the transition function of a local trivialization for Σ(M, L). By the property (A), j 0 (f(x) Z2 h(x)) = (f(x) Z2 h(x)) j 0 for n 1 mod 4. Also, by the property (B), j 1 (f(x) Z2 h(x)) = (f(x) Z2 h(x)) j 1 for n 2.

8 1044 KYUSIK HONG AND CHANYOUNG SUNG We then have the following commutative diagram: U α1 n g α1 α 2 U α2 n j r j r U α1 n g α1 α 2 U α2 n, where r = 0 with n 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 awarethatthemappingj 0 iswell-definedforn 1mod4,andj 1 iswell-defined for all n 2. Lemma 3.4. and D A j 0 = j 0 D A for n 1 mod 4, 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 Xs) = j k (X(s)+ 1 2 ( 1A(X)+ i<j ω j,i (X)e i e j ) s) = X(j k (s))+ 1 2 ( 1A(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(TM), 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). Asacorollary,wehaveprovedthatifmisodd(resp. even),thenspec(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 A1 ϕ 1 ) ϕ 2 +( 1) p (µ 1 ϕ 1 ) (D A2 ϕ 2 ), (3.7) (D A ) 2 (ϕ 1 ϕ 2 ) = ((D A1 ) 2 ϕ 1 ) ϕ 2 +ϕ 1 ((D A2 ) 2 ϕ 2 ) for ϕ i Γ(π i (Σ(M i,l i ))).

9 THE SYMMETRY OF spin C DIRAC SPECTRUMS 1045 Consider the partial spin C Dirac operators D A1 +,D A2 acting on sections ψ Γ(Σ(Q,L)) of the spinor bundle over (Q n,h): D A1 + ψ = 2p k=1 E k A1 E k ψ, D A2 2q+1 ψ = l=1 F l A2 F l ψ. Define the twist D A of the spin C Dirac operator D A = D A1 + +DA2 by By (2.1) and (2.2), for i = 1,2 and D A = D+ A1 DA2. D A1 + µ i = µ i D A1 +, D A2 µ i = µ i D A2, (3.8) D A µ i = µ i D A, DA µ i = µ i D A. Since the Spin C (2p+2q +1)-principle bundle P Q L over (Q n := p 1 M2q+1 2,h) reduces to the Spin C (2p) S 1 Spin C (2q + 1)-principal bundle π1 ( P M1 L 1 ) π2 ( P M2 L 2 ), the complex-antilinear mapping j : 2p+2q+1 2p+2q+1 defined 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 and hence A j = j A, (3.9) D+ A1 j = j D+ A1, D A2 j = ( 1) p+q+1 j D A2. Similarly we also define complex-antilinear mappings ĵ, j 0, j 1 : Σ(Q,L) Σ(Q, L) as ĵ 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 0u(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 ĵ 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 (3.10) D+ A1 ĵ = ( 1) p+1 ĵ D+ A1, D A2 ĵ = ( 1) p ĵ D A2,

10 1046 KYUSIK HONG AND CHANYOUNG SUNG (3.11) (3.12) D+ A1 j0 = j0 D+ A1, D A2 j0 = ( 1) p j0 D A2, D+ A1 j1 = ( 1) p+1 j1 D+ A1, D A2 j1 = ( 1) p+q+1 j1 D A2. Putting these together, we can produce various operators which anti-commute with spin C Dirac operators: Proposition 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 A1 + ) = (µ i D A1 + ) DA. (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 ĵ ) = (µ i ĵ ) 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 DA. Proof. Since the Riemann curvatures R(E k,f l,, ) = 0 vanish and the Clifford multiplication anti-commutes, we have (3.14) D A2 DA1 + +DA1 + DA2 = 0. Using (3.8) and (3.14), we obtain D A (µ i D A1 + ) = µ i D A D A1 + = µ i ((D A1 + )2 D A2 DA1 + ) = µ i ((D A1 + )2 +D A1 + DA2 ) = µ i D A1 + DA. 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 D A ) j = µ i (D A1 + D A2 = µ i j (D A1 + +DA2 ) = (µ i j ) D A. ) j 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 ĵ ) = (µ i D A ) ĵ = (µ i ĵ ) D A and D A (µ i j 0 ) = (µ i D A ) j 0 = (µ i j 0 ) DA. 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 A1 ) 2 and one of (D A2 ) 2.

11 THE SYMMETRY OF spin C DIRAC SPECTRUMS 1047 From now on, we omit the projections and rewrite the formula of Corollary 2.7 as (4.1) Γ γ ((D A ) 2 ) = (Γ χ ((D A1 ) 2 ) Γ ν ((D A2 ) 2 )). Using the decomposition where γ=χ+ν Σ(M,L) = Σ + (M,L) Σ (M,L), Σ ± (,L 1 ) := {ϕ Σ(,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 ± (,L 1 ) Σ(,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 (DA1 ) be the space of all positive (resp. negative) harmonic spinors of D A1. For any λ 0 Spec(D A ), define a complex vector space H λ := {ψ Γ λ (D A ) D+ A1 ψ = 0, D A2 ψ = λψ}. Then, in the notation (4.1), we have H λ = {Γ + 0 (DA1 ) Γ ( 1) p λ(d A2 )} {Γ 0 (DA1 ) Γ ( 1) p λ(d A2 )}. Proof. By (3.6), it is enough to show that H λ {Γ + 0 (DA1 ) Γ ( 1) p λ(d A2 )} {Γ 0 (DA1 ) Γ ( 1) p λ(d A2 )}. Suppose that ψ H λ. Then ψ Γ λ 2((D A ) 2 ) and (4.1) implies that (4.3) ψ = k,l c k,l ϕ 0,k ϕ λ 2,l, c k,l 0 C, is a finite linear combination of tensor products of some ϕ 0,k Γ 0 ((D A1 ) 2 ) and some ϕ λ 2,l Γ λ 2((D A2 ) 2 ). By the decomposition of Σ(Q,L), we can rewrite (4.3) as ψ = k,l c k,l ϕ + 0,k ϕ λ 2,l + k,l c k,l ϕ 0,k ϕ λ 2,l, where ϕ ± 0,k Γ± 0 ((DA1 ) 2 ). By Lemma 2.8, the eigenvalue λ 2 of (D A2 ) 2 comes from the eigenvalue λ or λ of D A2. Since D A2 ψ = λψ, one can easily see from (3.6) ψ {Γ + 0 (DA1 ) Γ ( 1) p λ(d A2 )} {Γ 0 (DA1 ) Γ ( 1) p λ(d A2 )}.

12 1048 KYUSIK HONG AND CHANYOUNG SUNG 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 µ 1 D A1 + 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 A1 + ) 1 µ 1. By Lemma 4.2, (4.4) H λ = {Γ + 0 (DA1 ) Γ ( 1)p λ(d A2 )} {Γ 0 (DA1 ) Γ ( 1)p λ(d A2 )}. The symmetry of Spec(D A ) holds if and only if dim C H λ = dim C H λ for any λ 0 Spec(D A ). Letting dim C (Γ + 0 (DA1 )) = a 1, dim C (Γ 0 (DA1 )) = a 2, dim C (Γ ( 1)p λ(d A2 )) = b 1,λ, and dim C (Γ ( 1)p λ(d A2 )) = 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 A1 Σ+ (,L 1)))(b 1,λ b 2,λ ) = (e 1 2 c1(l1) Â( ))[ ](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 sign(n) n s = 0 for Re(s) 1, n Z\{0} η D 0(s) = 0 for all s C. In [7], it is shown that the Dirac spectrum of (p 1 S1, g 1 +g 2 ) for a spin manifold p 1 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 2Z. In this case, by Theorem 1.2, the spin C Dirac spectrum of (p 1 S 1, g 1 + g 2 ) for any product spin C structure is symmetric, and hence the corresponding eta invariant vanishes.

13 THE SYMMETRY OF spin C DIRAC SPECTRUMS 1049 The eta invariant of a spin C Dirac operator on a product spin C manifold 1 S1 appearswhencomputing thedimensionofthe moduli spaceofseiberg- Witten equations on a cylindrical-ended 4-manifold with asymptotic boundary equal to 1 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 ). The second-named author was supported by the National Research Foundation of Korea grant funded by the Korea government. (NRF-2014R1A1A4A ). References [1] M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry I, Math. Proc. Cambridge Philos. Soc. 77 (1975), [2], Spectral asymmetry and Riemannian geometry II, Math. Proc. Cambidge Philos. Soc. 78 (1975), [3] H. Baum, A remark on the spectrum of the Dirac operator on pseudo-riemannian spin manifolds, SFB 288, Preprint 136, [4] Th. Friedrich, Dirac Operators in Riemannian Geometry, Graduate Studies in Mathematics, vol. 25, AMS, Providence, RI, [5] Th. Friedrich and E. C. Kim, The Einstein-Dirac equation on Riemannian spin manifolds, J. Geom. Phys. 33 (2000), no. 1-2, [6] E. C. Kim, Die Einstein-Dirac-Gleichung über Riemannschen Spin-Mannigfaltigkeiten, Thesis, Humboldt-Universität zu Berlin, [7], The Â-genus and symmetry of the Dirac spectrum on Riemannian product manifolds, Differential Geom. Appl. 25 (2007), [8] H.-B. Lawson and M.-L. Michelsohn, Spin Geometry, Princeton University Press, Princeton, NJ, [9] A. Lichnerowicz, Champs spinoriels et propagateurs en relativité général, Bull. Sci. Math. France 92 (1964), [10] L. I. Nicolaescu, Notes on Seiberg-Witten Theory, Graduate Studies in Mathematics, 28. American Mathematical Society, Providence, RI, Kyusik Hong School of Mathematics Korea Institute for Advanced Study Seoul , Korea address: kszoo@kias.re.kr Chanyoung Sung Department of Mathematics Education Korea National University of Education Chungbuk , Korea address: cysung@knue.ac.kr