Rigidity and Vanishing Theorems for Almost Even-Clifford Hermitian Manifolds

We prove the rigidity and vanishing of several indices of"geometrically natural"twisted Dirac operators on almost even-Clifford Hermitian manifolds admitting circle actions by automorphisms.


Introduction
There are two classical vanishing theorems for the A-genus (the index of the Dirac operator) on Spin manifolds: the Lichnerowicz vanishing [17] which assumes a metric of positive scalar curvature, and the Atiyah-Hirzebruch vanishing [3] which assumes smooth circle action. These vanishings can be seen and have been used frequently as obstructions to the existence of such metrics or actions. More vanishing theorems for the indices of Spin c Dirac operators were explored by Hattori [11] on almost complex manifolds and Spin c manifolds with compatible circle actions, which have parallels on complex manifolds with ample line bundles (a positivity condition for certain curvature) as in the case of the Kodaira vanishing theorem. Vanishing theorems have also been proven for indices of twisted Dirac operators on compact quaternion-Kähler manifolds with positive scalar curvature [16], and for almost quaternion-Hermitian manifolds with isometric circle actions that preserve the almost quaternion-Hermitian structure [12].
The vanishings of such indices on manifolds with isometric circle actions are instances of the rigidity of elliptic operators under such actions, an important property in the context of elliptic genera [6,8,13,15,18,22,23,24]. In this paper, we prove the rigidity and vanishing of the indices of several "geometrically natural" twisted Dirac operators on almost even-Clifford manifolds admitting circle actions by automorphisms, resembling those studied on almost quaternionic-Hermitian manifolds.
The note is organized as follows. In Section 2, we recall some material on Clifford algebras, Spin groups and representations, maximal tori of classical Lie groups, almost even-Clifford Hermitian manifolds and their structure groups. In Section 3, we examine the weights of the Spin representation in terms of the weights of the aforementioned structure groups and explore which representations to use in the twisted Dirac operators. In Section 4, we prove the vanishing Theorems 4.7, 4.8 and 4.9, using the Atiyah-Singer fixed point theorem.

Preliminaries
The material presented in this section can be consulted in [1,7,9]. arXiv:1609.01509v3 [math.DG] 23 Apr 2017 2.1 Clif ford algebra, spin group and representation Let Cl n denote the 2 n -dimensional real Clifford algebra generated by the orthonormal vectors e 1 , e 2 , . . . , e n ∈ R n subject to the relations e i e j + e j e i = −2δ ij , and Cl n = Cl n ⊗ R C its complexification. The even Clifford subalgebra Cl 0 r is defined as the invariant (+1)-subspace of the involution of Cl r induced by the map −Id R r .
The Spin group Spin(n) ⊂ Cl n is the subset endowed with the product of the Clifford algebra. It is a Lie group and its Lie algebra is The restriction of κ to Spin(n) defines the Lie group representation which is, in fact, special unitary. We have the corresponding Lie algebra representation Recall that the Spin group Spin(n) is the universal double cover of SO(n), n ≥ 3. For n = 2 we consider Spin(2) to be the connected double cover of SO (2). The covering map will be denoted by Its differential is given by λ n * (e i e j ) = 2E ij , where E ij = e * i ⊗ e j − e * j ⊗ e i is the standard basis of the skew-symmetric matrices, and e * denotes the metric dual of the vector e. Furthermore, we will abuse the notation and also denote by λ n the induced representation on the exterior algebra * R n . By means of κ, we have the Clifford multiplication The Clifford multiplication µ n is skew-symmetric with respect to the Hermitian product is Spin(n)-equivariant and can be extended to a Spin(n)-equivariant map When n is even, we define the following involution where vol n = e 1 · · · e n . The ±1 eigenspace of this involution is denoted ∆ ± n . These spaces have equal dimension and are irreducible representations of Spin(n). Note that our definition differs from the one given in [9] by a (−1) n 2 . The reason for this difference is that we want the spinor u 1,...,1 to be always positive. In this case, we will denote the two representations by For future use, let us recall the effect of vol n on ∆ n = ∆ + n ⊕ ∆ − n when n is even: Furthermore, for n ≡ 0 (mod 4), n = 4, For r even, let and for r ≡ 0 (mod 4) let Note that we will always denote by 1 and Id r the identity elements of Spin(r) and SO(r) respectively.

SO(n)
Recall that a maximal torus of SO(n) is given by if n is odd.
By using the explicit description (2.2) of the isomorphisms (2.1), we can check that i.e., the basis vectors u ε 1 ,...,ε k are weight vectors of the spin representation with weight which in coordinate vectors with respect to the basis {ϕ j } give the well known expressions Indeed, in terms of the (appropriately ordered) basis B, the matrix associated to an element . . . . . .
Note that, when n is even, ∆ + n is generated by the basis vectors u ε 1 ,...,ε n 2 with an even number of ε j equal to −1, and ∆ − n is generated by the basis vectors u ε 1 ,...,ε n 2 with an odd number of ε j equal to −1. Therefore, after reordering the basis, the matrix above can be rearranged to have two diagonal blocks of equal size: one block in which the exponents contain an even number of negative signs and another block in which the exponents contain an odd number of negative signs

U(m)
The standard maximal torus of U(m) is

Sp(m)
The standard maximal torus of Sp(m) is

Almost even-Clif ford Hermitian structures
Definition 2.1. Let N ∈ N and (e 1 , . . . , e r ) an orthonormal frame of R r .
• A linear even-Clifford structure of rank r on R N is an algebra representation Φ : Cl 0 r −→ End R N .
• A linear even-Clifford Hermitian structure of rank r on R N (endowed with a positive definite inner product) is a linear even-Clifford structure of rank r such that each bivector e i e j , 1 ≤ i < j ≤ r, is mapped to a skew-symmetric endomorphism Φ(e i e j ) = J ij .
• Note that J 2 ij = −Id R N . • Given a linear even-Clifford structure of rank r on R N , we can average the standard inner product , on R N as follows where (e 1 , . . . , e r ) is an orthonormal frame of R r , so that the linear even-Clifford structure is Hermitian with respect to the averaged inner product.
• Given a linear even-Clifford Hermitian structure of rank r, the subalgebra spin(r) is mapped injectively into the skew-symmetric endomorphisms End − (R N ).
• A rank r almost even-Clifford structure on a smooth manifold M is a smoothly varying choice of a rank r linear even-Clifford structure on each tangent space of M .
• A smooth manifold carrying an almost even-Clifford structure will be called an almost even-Clifford manifold.
• A rank r almost even-Clifford Hermitian structure on a Riemannian manifold M is a smoothly varying choice of a linear even-Clifford Hermitian structure on each tangent space of M .
• A Riemannian manifold carrying a rank r almost even-Clifford Hermitian structure will be called a rank r almost even-Clifford Hermitian manifold, or an almost-Cl 0 r -Hermitian manifold for short.
Remark 2.4. Our definition of almost even-Clifford Hermitian structure does not require the existence of a Riemannian vector bundle of rank r. Therefore, it includes both the notions of even Clifford structure and projective even Clifford structure introduced in [19, Definition 2.2 and Remark 2.5].

Structure groups of almost even-Clif ford manifolds
Thanks to [2], we know that the complexification of the tangent space of an almost-Cl 0 r -Hermitian manifold decomposes as follows where the different C p denote the corresponding standard complex representations of the classical Lie groups SO(p), U(p) or Sp(p). Note that the dimension of an almost even-Clifford Hermitian manifold depends of two or three parameters: the rank r of the even-Clifford structure and the multiplicity m or multiplicities m 1 , m 2 . The structure groups of the aforementioned manifolds, for r ≥ 3, are given as follows (see [1]): • For r ≡ 0 (mod 4) Note that for r = 2, the structure group is actually U(m). Since all of these groups are quotients of products G × Spin(r), where G is a (product of) classical Lie group(s), it will be useful to know if they can be mapped to either Spin(r), or SO(r) or PSO(r). It is easy to see that they map as follows • For r ≡ 4 (mod 8) This can be summarized roughly as follows: the structure group of an almost-Cl 0 r -Hermitian manifold of rank r maps to SO(r) if r is odd, and maps to PSO(r) if r is even.
For future use, we will establish the notation for the decomposition of the complexified tangent bundle of an almost-Cl 0 r -Hermitian manifold: where E, E 1 , E 2 are locally defined vector bundles with fibre C p which correspond to the standard complex representation of the different Lie groups mentioned in (2.3).

A useful lemma
3 Twisted spinor bundles on almost-Cl 0 r -Hermitian manifolds In this subsection, we present some calculations relevant to the global definiton of twisted spinor bundles. When the structure group of an oriented N -dimensional Riemannian manifold reduces to a proper subgroup G ⊂ SO(N ), one can associate vector bundles to the corresponding Gprincipal bundle P G by means of the representations of G. If the manifold is Spin, one can ask if there exists a lifting mapĩ making the following diagram commute Spin(N ) in which case, the Spin representation ∆ N may decompose according to G.
Even when such mapĩ does not exist (necessarily π 1 (G) = {1}), there may be a finite covering space G of G = G /Γ for which it does, and one can then decompose the Spin representation according to G . We can now check how the elements of the finite subgroup Γ act on ∆ N , and at least some of them will act non-trivially, thus confirming that there cannot be a mapĩ. By observing this action, we can then consider tensoring ∆ N with another representation V of G such that Γ now acts trivially on ∆ N ⊗ V .
In the context of almost-Cl 0 r -Hermitian manifolds, the structure group embeds into the relevant Spin group [1,Theorem 4.1], with the exception of four cases which we will analyze. More precisely, we found that (4) {±(Id 2m 1 ,Id 2m 2 ,1),±(Id 2m 1 ,−Id 2m 2 ,vol 4 )} does not embed into Spin(4(m 1 + m 2 )) if either m 1 or m 2 (or both) are odd; However, by the same calculations in [1] we know that there are homomorphisms In order to analyze this situation and the appropriate twisting bundles for almost-Cl 0 r -Hermitian manifolds in general, we need to set up some notation regarding weights of Lie groups.

Weights of SO(N ) with respect to the structure subgroups
We need to rewrite the weights of SO(N ) in terms of the maximal torus of the relevant structure group. Let (η 1 , . . . , η N/2 ) denote the coordinates of a maximal torus of SO(N ), and (ϕ 1 , . . . , ϕ [ r 2 ] ) denote the coordinates of a maximal torus of SO(r).
For r odd, let , listed in some order such that the first half of weights have an even number of negative signs, and the second half of weights have an odd number of negative signs.

r ≡ 4 mod 8
Let (θ 1 , . . . , θ m 1 ) and (θ 1 , . . . , θ m 2 ) denote the coordinates of maximal tori of Sp(m 1 ) and Sp(m 2 ) respectively. Since we can set ) denote the coordinates of maximal tori of SO(m 1 ) and SO(m 2 ) respectively. Since we can set • if m 1 , m 2 are even, • if m 1 is even and m 2 is odd, • if m 1 is odd and m 2 is even,

The Spin representation when r = 3, 4, 6, 8
The elements of the finite subgroups involved in the structure groups of almost-Cl 0 r -Hermitian manifolds actually belong to maximal tori. Thus we can calculate their effect on representations in terms of the weights we just described. In this subsection, we examine the cases when the structure group does not embed into Spin(N ).
• The element (−Id m , −1) ∈ U(m) × Spin(6) corresponds to the parameters so that its effect on each weight line is e −2iπm = 1.

r = 8
By and its effect on each weight line is mutiplication by −1. Thus, we can have twisted Spin representations Similarly, by (3.4), if m 1 ≡ m 2 + 1 ≡ 0 (mod 2), we can have twisted Spin representations if u 1 + s ≡ 1 (mod 2) and u 2 , t ∈ N.

Twisting representations
For most r, almost-Cl 0 r -Hermitian manifolds are Spin [1,Theorem 4.1]. In particular, this is the case when r ≥ 5 and r = 6, 8. Thus, we only need to choose suitable representations of the structure group G to twist the spinor bundle: They are representations of the structure group when • For r ≡ 3, 5 (mod 8) our candidates are They are representations of the structure group when u + s ≡ 0 (mod 2).
• For r ≡ 0 (mod 8) our candidates are They are representations of the structure group when

Index calculations
In this section, we recall the definition of twisted Dirac operators, how to apply the Atiyah-Singer fixed point formula [4], (infinitesimal) automorphisms of almost-Cl 0 r -Hermitian manifolds and prove the vanishing Theorems 4.7, 4.8 and 4.9.

Twisted Dirac operators
In this subsection, let M be a 4n-dimensional oriented Riemannian manifold. M is Spin if its orthonormal frame bundle P SO(4n) admits a double cover by a principal bundle P Spin(4n) with structure group Spin(4n), which gives rise to the spinor bundle The Levi-Civita connection on P SO(4n) can be lifted to P Spin(4n) to define a covariant differentiation ∇ on ∆ ∇ : Γ(∆) −→ Γ(T * ⊗ ∆), and the (elliptic and self-adjoint) Dirac operator for ψ ∈ Γ(∆), where (e 1 , . . . , e 4n ) is a local orthonormal frame. Since the spin representation decomposes, the Dirac operator can be split into two parts We are interested in Dirac operators with coefficients in auxiliary vector bundles F equipped with a covariant derivative ∇ F : Γ(F ) −→ Γ(T * ⊗ F ). The Dirac operator twisted by F (or with coefficients in F ) where ψ ∈ Γ(∆), f ∈ Γ(F ).
Remark 4.5. If the manifold is not Spin, there may exist well defined twisted spinor bundles (as above), as it happens when the structure group of M reduces to a subgroup of SO(4n) and ∆ 4n ⊗ F is a representation of such subgroup.

Index formula and localization
Let M be an compact 4n-dimensional oriented Riemannian manifold. Let us assume that the bundle ∆ 4n ⊗ F is well defined, where we will use the same symbol to denote the representation and the associated vector bundle, where the dim C (F ) = p. Since ∆ 4n ⊗F is a Clifford bundle, by the Atiyah-Singer index theorem [5,20], the index of the twisted Dirac operators can be computed as where ch(·) denotes the Chern character, A(M ) denotes the A-genus, and [M ] denotes the fundamental cycle of M . In terms of formal roots, If M admits a non-trivial S 1 action that lifts to ∆ 4n ⊗ F , the equivariant version of the index can be written in terms of the local data of the S 1 -fixed point set M S 1 . More precisely, let z ∈ S 1 be a generic element of S 1 . By the Atiyah-Singer fixed point theorem [4,5] ind(/ ∂ ⊗ F ) z = P ⊂M S 1 µ(P, z), where µ(P, z) is the local contribution of the oriented fixed point submanifold P ⊂ M S 1 , which can be computed as follows. The S 1 action on M induces a decomposition of T M over P , where N k is a bundle over P whose fibers are representations of S 1 on which z ∈ S 1 acts as an automorphism with multiple eigenvalue z k , k ∈ Z. Note that P inherits an orientation since M is oriented and the bundles N k for k = 0 are naturally oriented. Formally, by means of the splitting principle, we can write where L corresponds to the standard representation of S 1 on C, so that z ∈ S 1 acts by multiplication by z q i on L q i . The integers q i = q i (P ) ∈ Z are the exponents of the action at P , which correspond to the aforementioned numbers k. Thus, following [14, p. 67], where n k = n k (P ) are the exponents of the action on F restricted to P . The function µ(P, z) is a rational function of the complex variable z with zeroes at 0 and ∞ as long as |n k | < 1 2 (|q 1 (P )| + · · · + |q 2n (P )|) (4.3) for all 1 ≤ k ≤ p. If such a condition is fulfilled for all P ⊂ M S 1 , then ind(/ ∂ ⊗ F ) z is a rational function of z with zeroes at 0 and ∞. Notice that ind(/ ∂ ⊗ F ) z also belongs to the representation ring R(S 1 ) of S 1 , which can be identified with the Laurent polynomial ring Z[z, z −1 ]. Hence, by Lemma 2.5, i.e.,

A-genus of almost-Cl 0 r -Hermitian manifolds
Given a 4n-dimensional Riemannian manifold, according to the splitting principle with respect to the maxinal torus of SO(4n), its complexified tangent bundle splits formally as follows and, therefore, and its Pontrjagin class is p(T M ) = 1 + x 2 1 · · · 1 + x 2 2n , and the A-genus is given by In the following, we will set x i = η i from Section 3.1.

r ≡ 1, 7 (mod 8)
The A-genus is given by if m is even, and The A-genus is given by if r ≡ 6 (mod 8).

r ≡ 3, 5 (mod 8)
The A-genus is given by

r ≡ 4 (mod 8)
The A-genus is given by

r ≡ 0 (mod 8)
We can set • if m 1 , m 2 are even, • if m 1 is even and m 2 is odd, • if m 1 is odd and m 2 is even, • if m 1 , m 2 are odd,

Inf initesimal automorphisms
An automorphism of an almost-Cl 0 r -Hermitian manifold M is an isometry which preserves the almost even-Clifford Hermitian structure. A vector field X on M is an infinitesimal automorphism if it is a Killing vector field that preserves the structure, i.e., locally for some (local) functions α (ij) kl , where L X denotes the Lie derivative in the direction of X. Consider which can be written in terms of the Levi-Civita connection ∇ as follows i.e., Hence, (∇X) p is a skew-symmetric endomorphism such that i.e., (∇X) p belongs to Lie algebra of the structure group of M [1,2].
We will say that a smooth circle action on an almost-Cl 0 r -Hermitian manifold is an action by automorphisms if the corresponding Killing vector field is an infinitesimal automorphism.   (9) has an almost-Cl 0 9 -Hermitian structure admitting S 1 actions by automorphisms [10].

Exponents of the S 1 action
In this section, let M be a compact, rank r almost even-Clifford Hermitian manifold with a nontrivial (effective) S 1 action by automorphisms. Let P ⊂ M S 1 be an S 1 -fixed submanifold. The corresponding infinitesimal isometry X is such that (∇X) p ∈ so(N ) at any fixed point p ∈ P . This corresponds to the induced action of S 1 to T p M , and such a circle lies in a maximal torus. The tangent space at p decomposes as in Section 4.3. In fact, we can now be more precise about these exponents. By Section 4.5, (∇X) p belongs to a Cartan subalgebra of the Lie algebra of the structure group, and we can assume that the decomposition (4.1) is compatible with a decomposition such as (4.2) into complex lines with respect to a maximal torus of such • for r ≡ 4 (mod 8), if 0 ≤ u 1 + s < m 1 and 0 ≤ u 2 + t < m 2 , If the inequalities are not strict, the indices are rigid.
Proof . Since the S 1 action is by automorphisms of the almost even-Clifford Hermitian structure, the action lifts to the bundles associated to the structure group, such as the twisted spin bundles we are considering. Given that the arguments are similar in all cases, we will only describe the calculation for r ≡ 1, 7 (mod 8) and m even. Let P ⊂ M S 1 be an S 1 -fixed submanifold. By Section 4.5, over P the circle group of automorphisms maps non-trivially to the structure group SO(m)Spin(r), so that the fibers of the bundles ∆ N ⊗ u E ⊗ ∆ ⊗s r over points of P decompose as sums of representations of S 1 . Recall that Thus, the exponents of the twist will be of the form where 0 ≤ c ≤ u, ε a , δ b ∈ {0, 1}. There are two points to verify in the proof: firstly, that the contributions µ(P, z) are rational functions and, secondly, that the exponents of the twisting bundles and the tangent space satisfy the inequality (4.3). The first one follows from the fact that the fibers of the bundles ∆ N ⊗ u E ⊗ ∆ ⊗s r over P decompose as sums of representations of S 1 . so that the S 1 -exponents on these lines will be of the form The bundle ∆ N ⊗ u E ⊗ ∆ ⊗s r will have integer exponents over P of the form Proof . We will only describe the relevant changes to the calculation for r ≡ 1, 7 (mod 8) and m even. Let P ⊂ M S 1 be an S 1 -fixed submanifold. Recall that ch S u E = Thus, the exponents of the twist will be of the form where ε a , δ b ∈ {0, 1}. It is sufficient to consider the exponents of the form 1 2 u a=1 (−1) εa t ia ± sh k .
Among them, there are two extreme types, namely the ones equal to exponents of the exterior powers which we already know how to deal with, and the ones such as ut 1 . For such an exponent, consider Theorem 4.9. Let M be a compact N -dimensional almost-Cl 0 r -Hermitian admitting a smooth circle action by automorphisms, r ≥ 3. Let E, E 1 , E 2 be the (locally defined) bundles described in (2.4), m, m 1 , m 2 the corresponding multiplicities and u i , v i , u i , v i , s, t be non-negative integers satisfying analogous conditions to those given in Sections 3.2 and 3.3. Then, S v j E ⊗ (∆ r ) ⊗s A(M ), [M ] = 0; • for r ≡ 3, 5 (mod 8), if