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.
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2017
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Цитувати: | Rigidity and Vanishing Theorems for Almost Even-Clifford Hermitian Manifolds / A.L. Garcia-Pulido, R. Herrera // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 24 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1485672025-02-09T17:12:40Z Rigidity and Vanishing Theorems for Almost Even-Clifford Hermitian Manifolds Garcia-Pulido, A.L. Herrera, R. 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. The first named author was supported by CONACyT. The second named author was partially supported by a CONACyT grant. The second named author wishes to thank the International Centre for Theoretical Physics and the Institut des Hautes Etudes Scientifiques for their hospi- ´ tality and support. We would like to express our gratitude to the anonymous referees for their careful reading of this manuscript and their helpful comments. 2017 Article Rigidity and Vanishing Theorems for Almost Even-Clifford Hermitian Manifolds / A.L. Garcia-Pulido, R. Herrera // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 24 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53C10; 53C15; 53C25; 58J20; 57S15 DOI:10.3842/SIGMA.2017.027 https://nasplib.isofts.kiev.ua/handle/123456789/148567 en Symmetry, Integrability and Geometry: Methods and Applications application/pdf Інститут математики НАН України |
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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. |
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Article |
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Garcia-Pulido, A.L. Herrera, R. |
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Garcia-Pulido, A.L. Herrera, R. Rigidity and Vanishing Theorems for Almost Even-Clifford Hermitian Manifolds Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Garcia-Pulido, A.L. Herrera, R. |
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Garcia-Pulido, A.L. |
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Rigidity and Vanishing Theorems for Almost Even-Clifford Hermitian Manifolds |
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Rigidity and Vanishing Theorems for Almost Even-Clifford Hermitian Manifolds |
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Rigidity and Vanishing Theorems for Almost Even-Clifford Hermitian Manifolds |
| title_fullStr |
Rigidity and Vanishing Theorems for Almost Even-Clifford Hermitian Manifolds |
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Rigidity and Vanishing Theorems for Almost Even-Clifford Hermitian Manifolds |
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rigidity and vanishing theorems for almost even-clifford hermitian manifolds |
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Інститут математики НАН України |
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2017 |
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https://nasplib.isofts.kiev.ua/handle/123456789/148567 |
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Rigidity and Vanishing Theorems for Almost Even-Clifford Hermitian Manifolds / A.L. Garcia-Pulido, R. Herrera // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 24 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT garciapulidoal rigidityandvanishingtheoremsforalmostevencliffordhermitianmanifolds AT herrerar rigidityandvanishingtheoremsforalmostevencliffordhermitianmanifolds |
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2025-11-28T11:06:08Z |
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2025-11-28T11:06:08Z |
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1850031972359864320 |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 027, 28 pages
Rigidity and Vanishing Theorems
for Almost Even-Clifford Hermitian Manifolds
Ana Lucia GARCIA-PULIDO and Rafael HERRERA
Centro de Investigación en Matemáticas, A. P. 402, Guanajuato, Gto., C.P. 36000, México
E-mail: lucia@cimat.mx, rherrera@cimat.mx
URL: https://sites.google.com/site/algarciapulido
Received October 10, 2016, in final form April 19, 2017; Published online April 23, 2017
https://doi.org/10.3842/SIGMA.2017.027
Abstract. 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.
Key words: almost even-Clifford Hermitian manifolds; index of elliptic operator; twisted
Dirac operators; circle action by automorphisms
2010 Mathematics Subject Classification: 53C10; 53C15; 53C25; 58J20; 57S15
1 Introduction
There are two classical vanishing theorems for the Â-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 Spinc Dirac operators were ex-
plored by Hattori [11] on almost complex manifolds and Spinc manifolds with compatible circle
actions, which have parallels on complex manifolds with ample line bundles (a positivity condi-
tion 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 man-
ifolds 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.
2 Preliminaries
The material presented in this section can be consulted in [1, 7, 9].
mailto:lucia@cimat.mx
mailto:rherrera@cimat.mx
https://sites.google.com/site/algarciapulido
https://doi.org/10.3842/SIGMA.2017.027
2 A.L. Garcia-Pulido and R. Herrera
2.1 Clifford algebra, spin group and representation
Let Cln denote the 2n-dimensional real Clifford algebra generated by the orthonormal vectors
e1, e2, . . . , en ∈ Rn subject to the relations
eiej + ejei = −2δij ,
and Cln = Cln ⊗R C its complexification. The even Clifford subalgebra Cl0r is defined as the
invariant (+1)-subspace of the involution of Clr induced by the map −IdRr .
There exist algebra isomorphisms
Cln ∼=
{
End
(
C2k
)
if n = 2k,
End
(
C2k
)
⊕ End
(
C2k
)
if n = 2k + 1,
(2.1)
and the space of (complex) spinors is defined to be
∆n := C2k = C2 ⊗ · · · ⊗ C2︸ ︷︷ ︸
k times
.
The map
κ : Cln −→ End
(
C2k
)
is defined to be either the aforementioned isomorphism for n even, or the isomorphism followed
by the projection onto the first summand for n odd. In order to make κ explicit, consider the
following matrices
Id =
(
1 0
0 1
)
, g1 =
(
i 0
0 −i
)
, g2 =
(
0 i
i 0
)
, T =
(
0 −i
i 0
)
.
In terms of the generators e1, . . . , en of the Clifford algebra, κ can be described explicitly as
follows
e1 7→ Id⊗ Id⊗ · · · ⊗ Id⊗ Id⊗ g1,
e2 7→ Id⊗ Id⊗ · · · ⊗ Id⊗ Id⊗ g2,
e3 7→ Id⊗ Id⊗ · · · ⊗ Id⊗ g1 ⊗ T,
e4 7→ Id⊗ Id⊗ · · · ⊗ Id⊗ g2 ⊗ T, (2.2)
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
e2k−1 7→ g1 ⊗ T ⊗ · · · ⊗ T ⊗ T ⊗ T,
e2k 7→ g2 ⊗ T ⊗ · · · ⊗ T ⊗ T ⊗ T,
and, if n = 2k + 1,
e2k+1 7→ iT ⊗ T ⊗ · · · ⊗ T ⊗ T ⊗ T.
The vectors
u+1 =
1√
2
(1,−i) and u−1 =
1√
2
(1, i),
form a unitary basis of C2 with respect to the standard Hermitian product. Thus,
B = {uε1,...,εk = uε1 ⊗ · · · ⊗ uεk | εj = ±1, j = 1, . . . , k},
is a unitary basis of ∆n = C2k with respect to the naturally induced Hermitian product.
Rigidity and Vanishing Theorems for Almost Even-Clifford Hermitian Manifolds 3
The Spin group Spin(n) ⊂ Cln is the subset
Spin(n) =
{
x1x2 · · ·x2l−1x2l |xj ∈ Rn, |xj | = 1, l ∈ N
}
,
endowed with the product of the Clifford algebra. It is a Lie group and its Lie algebra is
spin(n) = span{eiej | 1 ≤ i < j ≤ n}.
The restriction of κ to Spin(n) defines the Lie group representation
κn := κ|Spin(n) : Spin(n) −→ GL(∆n),
which is, in fact, special unitary. We have the corresponding Lie algebra representation
κn∗ : spin(n) −→ gl(∆n).
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
λn : Spin(n)→ SO(n) ⊂ GL
(
Rn
)
.
Its differential is given by λn∗(eiej) = 2Eij , where Eij = e∗i ⊗ ej − e∗j ⊗ ei 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 λn the induced representation on the exterior
algebra
∧∗Rn.
By means of κ, we have the Clifford multiplication
µn : Rn ⊗∆n −→ ∆n,
x⊗ φ 7−→ µn(x⊗ φ) = x · φ := κ(x)(φ).
The Clifford multiplication µn is skew-symmetric with respect to the Hermitian product
〈x · φ1, φ2〉 = 〈µn(x⊗ φ1), φ2〉 = −〈φ1, µn(x⊗ φ2)〉 = −〈φ1, x · φ2〉,
is Spin(n)-equivariant and can be extended to a Spin(n)-equivariant map
µn :
∧∗(Rn)⊗∆n −→ ∆n,
ω ⊗ ψ 7−→ ω · ψ.
When n is even, we define the following involution
∆n −→ ∆n,
ψ 7−→ (−i)
n
2 voln · ψ,
where voln = e1 · · · en. 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 u1,...,1 to be always positive. In this case, we will denote the two representations by
κ±n : Spin(n) −→ GL
(
∆±n
)
.
4 A.L. Garcia-Pulido and R. Herrera
For future use, let us recall the effect of voln on ∆n = ∆+
n ⊕∆−n when n is even:
n (mod 8) ∆+
n ∆−n
0 1 −1
2 i −i
4 −1 1
6 −i i
Furthermore, for n ≡ 0 (mod 4), n 6= 4,
ker(κ+
n ) =
{
{1, volr} if r ≡ 0 (mod 8),
{1,−volr} if r ≡ 4 (mod 8),
and
ker(κ−n ) =
{
{1,−volr} if r ≡ 0 (mod 8),
{1, volr} if r ≡ 4 (mod 8).
For r even, let
PSO(r) :=
SO(r)
{±Idr}
∼=
Spin(r)
{±1,±volr}
,
and for r ≡ 0 (mod 4) let
Spin±(r) ∼=
Spin(r)
{1,±volr}
.
Note that we will always denote by 1 and Idr the identity elements of Spin(r) and SO(r)
respectively.
2.2 Maximal tori
2.2.1 SO(n)
Recall that a maximal torus of SO(n) is given by
cos(η1) − sin(η1)
sin(η1) cos(η1)
. . .
cos(ηn/2) − sin(ηn/2)
sin(ηn/2) cos(ηn/2)
if n is even, and
cos(η1) − sin(η1)
sin(η1) cos(η1)
. . .
cos(η[n/2]) − sin(η[n/2])
sin(η[n/2]) cos(η[n/2])
1
if n is odd.
Rigidity and Vanishing Theorems for Almost Even-Clifford Hermitian Manifolds 5
2.2.2 Spin(n)
Each one of the 2× 2 rotation blocks is a transformation that can be achieved by using Clifford
product. For instance the rotation
cos(ϕ1) − sin(ϕ1)
sin(ϕ1) cos(ϕ1)
1
. . .
1
can be achieved by using the element
e1(− cos(ϕ1/2)e1 + sin(ϕ1/2)e2) = cos(ϕ1/2) + sin(ϕ1/2)e1e2 ∈ Spin(n)
as follows
(cos(ϕ1/2) + sin(ϕ1/2)e1e2)y(cos(ϕ1/2) + sin(ϕ1/2)e2e1)
= (y1 cos(ϕ1)− y2 sin(ϕ1))e1 + (y1 sin(ϕ1) + y2 cos(ϕ1))e2 + y3e3 + · · ·+ ynen,
for y = y1e1 + · · ·+ ynen ∈ Rn. Thus, we see that the corresponding elements in Spin(n) are
±(cos(ϕ1/2) + sin(ϕ1/2)e1e2).
Furthermore, we see that a maximal torus of Spin(n) is given by elements of the form
t(ϕ1, . . . , ϕk) =
k∏
j=1
(cos(ϕj/2) + sin(ϕj/2)e2j−1e2j).
By using the explicit description (2.2) of the isomorphisms (2.1), we can check that
t(ϕ1, . . . , ϕk) · uε1,...,εk = e
i
2
k∑
j=1
εk+1−jϕj
· uε1,...,εk ,
i.e., the basis vectors uε1,...,εk are weight vectors of the spin representation with weight
1
2
k∑
j=1
εk+1−jϕj ,
which in coordinate vectors with respect to the basis {ϕj} give the well known expressions(
±1
2
,±1
2
, . . . ,±1
2
)
.
Indeed, in terms of the (appropriately ordered) basis B, the matrix associated to an element
t(ϕ1, . . . , ϕ[n
2
]) is
e
i
2
(ϕ1+ϕ2+···+ϕ[n2 ])
e
i
2
(−ϕ1+ϕ2+···+ϕ[n2 ])
e
i
2
(ϕ1−ϕ2+···+ϕ[n2 ])
. . .
e
i
2
(−ϕ1−ϕ2+···+ϕ[n2 ])
. . .
e
i
2
(−ϕ1−ϕ2−···−ϕ[n2 ])
.
6 A.L. Garcia-Pulido and R. Herrera
Note that, when n is even, ∆+
n is generated by the basis vectors uε1,...,εn
2
with an even number
of εj equal to −1, and ∆−n is generated by the basis vectors uε1,...,εn
2
with an odd number of εj
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
e
i
2
(ϕ1+ϕ2+···+ϕn
2
)
e
i
2
(−ϕ1−ϕ2+···+ϕn
2
)
e
i
2
(−ϕ1+ϕ2−···+ϕn
2
)
. . .
,
and another block in which the exponents contain an odd number of negative signs
e
i
2
(−ϕ1+ϕ2+···+ϕn
2
)
e
i
2
(ϕ1−ϕ2+···+ϕn
2
)
e
i
2
(ϕ1+ϕ2−ϕ3+···+ϕn
2
)
. . .
.
2.2.3 U(m)
The standard maximal torus of U(m) is
eiθ1
eiθ2
. . .
eiθm
.
2.2.4 Sp(m)
The standard maximal torus of Sp(m) is
eiθ1
e−iθ1
. . .
eiθm
e−iθm
.
2.3 Almost even-Clifford Hermitian structures
Definition 2.1. Let N ∈ N and (e1, . . . , er) an orthonormal frame of Rr.
• A linear even-Clifford structure of rank r on RN is an algebra representation
Φ: Cl0r −→ End
(
RN
)
.
• A linear even-Clifford Hermitian structure of rank r on RN (endowed with a positive
definite inner product) is a linear even-Clifford structure of rank r such that each bivec-
tor eiej , 1 ≤ i < j ≤ r, is mapped to a skew-symmetric endomorphism Φ(eiej) = Jij .
Rigidity and Vanishing Theorems for Almost Even-Clifford Hermitian Manifolds 7
Remark 2.2.
• Note that J2
ij = −IdRN .
• Given a linear even-Clifford structure of rank r on RN , we can average the standard inner
product 〈 , 〉 on RN as follows
(X,Y ) =
[r/2]∑
k=1
[ ∑
1≤i1<···<i2k<r
〈Φ(ei1...i2k)(X),Φ(ei1...i2k)(Y )〉
]
,
where (e1, . . . , er) is an orthonormal frame of Rr, 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−(RN ).
Definition 2.3. Let r ≥ 2.
• 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-Cl0r-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].
2.3.1 Structure groups of almost even-Clifford manifolds
Thanks to [2], we know that the complexification of the tangent space of an almost-Cl0r-Hermitian
manifold decomposes as follows
r (mod 8) RN ⊗ C
0 Cm1 ⊗∆+
r ⊕ Cm2 ⊗∆−r
1, 7 Cm ⊗∆r
2 Cm ⊗∆+
r ⊕ Cm ⊗∆−r
6 Cm ⊗∆+
r ⊕ Cm ⊗∆−r
3, 5 C2m ⊗∆r
4 C2m2 ⊗∆+
r ⊕ C2m1 ⊗∆−r
(2.3)
where the different Cp denote the corresponding standard complex representations of the clas-
sical 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 struc-
ture and the multiplicity m or multiplicities m1, m2.
The structure groups of the aforementioned manifolds, for r ≥ 3, are given as follows (see [1]):
8 A.L. Garcia-Pulido and R. Herrera
• For r 6≡ 0 (mod 4)
r (mod 8)
m (mod 2)
0 1
1, 7 SO(m)×Spin(r)
{±(Idm,1)} SO(m)× Spin(r)
2, 6 U(m)×Spin(r)
{±(Idm,1),±(iIdm,−volr)}
3, 5 Sp(m)×Spin(r)
{±(Idm,1)}
• For r ≡ 0 (mod 8)
m1
m2 0 0 (mod 2) 1 (mod 2)
0 SO(m2)×Spin(r)
{±(Idm2 ,1),±(Idm2 ,−volr)}
SO(m2)×Spin(r)
〈(Idm2 ,−volr)〉
0 (mod 2) SO(m1)×Spin(r)
{±(Idm1 ,1),±(Idm1 ,volr)}
SO(m1)×SO(m2)×Spin(r)
{±(Idm1 ,Idm2 ,1),±(Idm1 ,−Idm2 ,volr)}
SO(m1)×SO(m2)×Spin(r)
〈(−Idm1 ,Idm2 ,−volr)〉
1 (mod 2) SO(m1)×Spin(r)
〈(Idm1 ,volr)〉
SO(m1)×SO(m2)×Spin(r)
〈(Idm1 ,−Idm2 ,volr)〉 SO(m1)× SO(m2)× Spin(r)
• For r ≡ 4 (mod 8)
m1,m2 > 0 m1 > 0, m2 = 0 m1 = 0, m2 > 0
r = 4 Sp(m1)×Sp(m2)×Spin(r)
{±(Id2m1 ,Id2m2 ,1),±(Id2m1 ,−Id2m2 ,volr)}
Sp(m1)×Spin(3)
{±(Id2m1 ,1)}
Sp(m2)×Spin(3)
{±(Id2m2 ,1)}
r > 4 Sp(m1)×Spin(r)
{±(Id2m1 ,1),±(Id2m1 ,volr)}
Sp(m2)×Spin(r)
{±(Id2m2 ,1),±(Id2m2 ,−volr)}
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 6≡ 0 (mod 4)
r (mod 8)
m (mod 2)
0 1
1, 7 SO(r) Spin(r)
2, 6 PSO(r)
3, 5 SO(r)
• For r ≡ 0 (mod 8)
m1
m2 0 (mod 2) 1 (mod 2)
0 (mod 2) PSO(r) Spin−(r)
1 (mod 2) Spin+(r) Spin(r)
• For r ≡ 4 (mod 8)
r ≡ 4 (mod 8)
m1, m2 m1,m2 > 0 m1 > 0, m2 = 0 m1 = 0, m2 > 0
r = 4 PSO(r)
SO(3) SO(3)
r > 4 PSO(r) PSO(r)
Rigidity and Vanishing Theorems for Almost Even-Clifford Hermitian Manifolds 9
This can be summarized roughly as follows: the structure group of an almost-Cl0r-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-Cl0r-Hermitian manifold:
r (mod 8) TM ⊗ C
0 E1 ⊗∆+
r ⊕ E2 ⊗∆−r
1, 7 E ⊗∆r
2 E ⊗∆+
r ⊕ E ⊗∆−r
6 E ⊗∆+
r ⊕ E ⊗∆−r
3, 5 E ⊗∆r
4 E2 ⊗∆+
r ⊕ E1 ⊗∆−r
(2.4)
where E, E1, E2 are locally defined vector bundles with fibre Cp which correspond to the
standard complex representation of the different Lie groups mentioned in (2.3).
2.4 A useful lemma
Lemma 2.5. Let x ∈ C and k,m ∈ Z/2 such that k +m ∈ Z. If |k| < |m|, then
G(z) =
zk
z−mex − zme−x
is a rational function on C and
lim
z→0
G(z) = 0 = lim
z→∞
G(z).
3 Twisted spinor bundles on almost-Cl0r-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 G-
principal bundle PG by means of the representations of G. If the manifold is Spin, one can ask
if there exists a lifting map ĩ making the following diagram commute
Spin(N)
2:1
��
G
i //
ĩ
;;
SO(N)
in which case, the Spin representation ∆N may decompose according to G.
Even when such map ĩ does not exist (necessarily π1(G) 6= {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 map ĩ. 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-Cl0r-Hermitian manifolds, the structure group embeds into the rele-
vant Spin group [1, Theorem 4.1], with the exception of four cases which we will analyze. More
precisely, we found that
10 A.L. Garcia-Pulido and R. Herrera
• Sp(m)×Spin(3)
{±(Id2m,1)} does not embed into Spin(4m) if m is odd;
• Sp(m1)×Sp(m2)×Spin(4)
{±(Id2m1 ,Id2m2 ,1),±(Id2m1 ,−Id2m2 ,vol4)} does not embed into Spin(4(m1 + m2)) if either m1
or m2 (or both) are odd;
• U(m)×Spin(6)
{±(Idm,1),(iIdm,−vol6)} does not embed into Spin(8m) if m is odd;
• SO(m1)×SO(m2)×Spin(8)
{(Idm1 ,Idm2 ,1),(Idm1 ,−Id2m2 ,vol8)} if m1 +1 ≡ m2 ≡ 0 (mod 2), SO(m1)×SO(m2)×Spin(8)
{(Idm1 ,Idm2 ,1),(−Idm1 ,Id2m2 ,−vol8)}
if m1 ≡ m2 + 1 ≡ 0 (mod 2) do not embed into Spin(8(m1 +m2)).
However, by the same calculations in [1] we know that there are homomorphisms
• Sp(m)× Spin(3) −→ Spin(4m);
• Sp(m1)× Sp(m2)× Spin(4) −→ Spin(4(m1 +m2));
• U(m)× Spin(6) −→ Spin(8m);
• SO(m1)× SO(m2)× Spin(8) −→ Spin(8(m1 +m2)).
In order to analyze this situation and the appropriate twisting bundles for almost-Cl0r-
Hermitian manifolds in general, we need to set up some notation regarding weights of Lie
groups.
3.1 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 (η1, . . . , ηN/2) denote the coordinates of a maximal torus of SO(N), and (ϕ1, . . . , ϕ[ r
2
])
denote the coordinates of a maximal torus of SO(r).
For r odd, let λ1, . . . , λ2[
r
2 ] denote the weights of ∆r
±1
2
ϕ1 ± · · · ±
1
2
ϕ[ r
2
],
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.
For r even, let λ±1 , . . . , λ
±
2
r
2−1 denote the weights of ∆±r
±1
2
ϕ1 ± · · · ±
1
2
ϕ r
2
,
which have an even and odd number of negative signs respectively. If r ≡ 0 mod 4 we will
be considering λ±1 , . . . , λ
±
2
r
2−1 to be listed in some order so that the first and second halves are
interchanged by reflection (changing all the signs),
3.1.1 r ≡ 1, 7 mod 8
Let (θ1, . . . , θ[m
2
]) denote the coordinates of maximal tori of SO(m). Since
CN = Cm ⊗∆r,
we can set
η
(j−1)2[
r
2 ]+k
= θj + λk
if m is even, and
η
(j−1)2[
r
2 ]+k
= θj + λk, η
[m
2
]2[
r
2 ]+l
= λl
if m is odd, where 1 ≤ j ≤ [m2 ], 1 ≤ k ≤ 2[ r
2
] and 1 ≤ l ≤ 2[ r
2
]−1 in both cases.
Rigidity and Vanishing Theorems for Almost Even-Clifford Hermitian Manifolds 11
3.1.2 r ≡ 2, 6 mod 8
Let (θ1, . . . , θm) denote the coordinates of maximal tori of U(m). Since
CN =
{
Cm ⊗∆+
r ⊕ Cm ⊗∆−r if r ≡ 2 (mod 8),
Cm ⊗∆+
r ⊕ Cm ⊗∆−r if r ≡ 6 (mod 8),
we can set
η
(j−1)2
r
2−1+k
=
{
θj + λ+
k if r ≡ 2 (mod 8),
θj + λ−k if r ≡ 6 (mod 8),
(3.1)
where 1 ≤ j ≤ m and 1 ≤ k ≤ 2
r
2
−1.
3.1.3 r ≡ 3, 5 mod 8
Let (θ1, . . . , θm) denote the coordinates of maximal tori of Sp(m). Since
CN = C2m ⊗∆r,
we can set
η
(j−1)2[
r
2 ]+k
= θj + λk, (3.2)
where 1 ≤ j ≤ m and 1 ≤ k ≤ 2[ r
2
].
3.1.4 r ≡ 4 mod 8
Let (θ1, . . . , θm1) and (θ′1, . . . , θ
′
m2
) denote the coordinates of maximal tori of Sp(m1) and Sp(m2)
respectively. Since
CN = C2m1 ⊗∆+
r ⊕ C2m2 ⊗∆−r ,
we can set
η
(j1−1)2
r
2−1+k
= θj1 + λ+
k , η
m12
r
2−1+(j2−1)2
r
2−1+k
= θ′j2 + λ−k , (3.3)
where 1 ≤ j1 ≤ m1, 1 ≤ j2 ≤ m2 and 1 ≤ k ≤ 2
r
2
−1.
3.1.5 r ≡ 0 mod 8
Let (θ1, . . . , θ[
m1
2
]) and (θ′1, . . . , θ
′
[
m2
2
]
) denote the coordinates of maximal tori of SO(m1) and
SO(m2) respectively. Since
CN = Cm1 ⊗∆+
r ⊕ Cm2 ⊗∆−r ,
we can set
• if m1, m2 are even,
η
(j1−1)2
r
2−1+k
= θj1 + λ+
k , η
m12
r
2−1+(j2−1)2
r
2−1+k
= θ′j2 + λ−k ,
where 1 ≤ j1 ≤ m1
2 , 1 ≤ j2 ≤ m2
2 and 1 ≤ k ≤ 2
r
2
−1;
12 A.L. Garcia-Pulido and R. Herrera
• if m1 is even and m2 is odd,
η
(j1−1)2
r
2−1+k
= θj1 + λ+
k , η
m12
r
2−1+(j2−1)2
r
2−1+k
= θ′j2 + λ−k ,
η
m12
r
2−1+[
m2
2
]2
r
2−1+l
= λ−l , (3.4)
where 1 ≤ j1 ≤ m1
2 , 1 ≤ j2 ≤ [m2
2 ], 1 ≤ k ≤ 2
r
2
−1 and 1 ≤ l ≤ 2
r
2
−2;
• if m1 is odd and m2 is even,
η
(j1−1)2
r
2−1+k
= θj1 + λ+
k , η
[
m1
2
]2
r
2−1+l
= λ+
l ,
η
[
m1
2
]2
r
2−1+2
r
2−2+(j2−1)2
r
2−1+k
= θ′j2 + λ−k , (3.5)
where 1 ≤ j1 ≤ [m1
2 ], 1 ≤ j2 ≤ m2
2 , 1 ≤ k ≤ 2
r
2
−1 and 1 ≤ l ≤ 2
r
2
−2;
• if m1, m2 are odd,
η
(j1−1)2
r
2−1+k
= θj1 + λ+
k , η
[
m1
2
]2
r
2−1+l
= λ+
l ,
η
[
m1
2
]2
r
2−1+2
r
2−2+(j2−1)2
r
2−1+k
= θ′j2 + λ−k , η
[
m1
2
]2
r
2−1+2
r
2−2+[
m2
2
]2
r
2−1+l
= λ−l ,
where 1 ≤ j1 ≤ [m1
2 ], 1 ≤ j2 ≤ [m2
2 ], 1 ≤ k ≤ 2
r
2
−1 and 1 ≤ l ≤ 2
r
2
−2.
3.2 The Spin representation when r = 3, 4, 6, 8
The elements of the finite subgroups involved in the structure groups of almost-Cl0r-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).
3.2.1 r = 3
Recall (3.2), which in this case is
η2j−1 = θj +
ϕ1
2
, η2j = θj −
ϕ1
2
,
so that the weights of the spin representation are
±η1
2
± · · · ± η2m
2
=
∑
j∈I1
θj + ϕ1
2
2
−
∑
j∈Ī1
θj + ϕ1
2
2
+
∑
j∈I2
θj − ϕ1
2
2
−
∑
j∈Ī2
θj − ϕ1
2
2
,
where I1, I2 ⊆ {1, . . . ,m}, and Īj = {1, . . . ,m} − Ij denote their complements, j = 1, 2. The
element (−Id2m,−1) ∈ Sp(m)× Spin(3) corresponds to the parameters
θj = π, ϕ1 = 2π,
for 1 ≤ j ≤ m, so that such a sum is equal to
2|I1|π −mπ
and the effect of (−Id2m,−1) on each weight line is
e−imπ = (−1)m.
Thus, (−Id2m,−1) ∈ Sp(m) × Spin(3) acts trivially on ∆4m if m is even and as multiplication
by (−1) if m is odd.
Thus, in order to have a twisted Spin representation
∆4m ⊗
∧uC2m ⊗ (∆3)⊗s
of Sp(m)×Spin(3)
{±(Id2m,1)} , the exponents must satisfy m+ u+ s ≡ 0 (mod 2), which is a well known fact
for almost quaternion-Hermitian manifolds [21].
Rigidity and Vanishing Theorems for Almost Even-Clifford Hermitian Manifolds 13
3.2.2 r = 4
By (3.3), the weights of the spin representation are
±η1
2
± · · · ± η2m1+2m2
2
=
∑
j1∈I1
θj1 + ϕ1+ϕ2
2
2
−
∑
j1∈Ī1
θj1 + ϕ1+ϕ2
2
2
+
∑
j1∈I2
θj1 + −ϕ1−ϕ2
2
2
−
∑
j1∈Ī2
θj1 + −ϕ1−ϕ2
2
2
+
∑
j2∈I′1
θ′j2 + −ϕ1+ϕ2
2
2
−
∑
j2∈Ī′1
θ′j2 + −ϕ1+ϕ2
2
2
+
∑
j2∈I′2
θ′j2 + ϕ1−ϕ2
2
2
−
∑
j2∈Ī′2
θ′j2 + ϕ1−ϕ2
2
2
,
where I1, I2 ⊆ {1, . . . ,m1} and I ′1, I
′
2 ⊆ {1, . . . ,m2}.
• The element (−Id2m1 ,−Id2m2 ,−1) ∈ Sp(m1) × Sp(m2) × Spin(4) corresponds to the pa-
rameters
θj1 = π, θ′j2 = π, ϕ1 = 2π, ϕ2 = 0,
so that such a sum is equal to
π(2|I1| −m1 + 2|I ′2| −m2)
and the effect of (−Id2m1 ,−Id2m2 ,−1) on each weight line is
e−iπ(m1+m2) = (−1)m1+m2 .
• The element (Id2m1 ,−Id2m2 , vol4) ∈ Sp(m1) × Sp(m2) × Spin(4) corresponds to the pa-
rameters
θj1 = 0, θ′j2 = π, ϕ1 = π, ϕ2 = π,
so that the effect of (Id2m1 ,−Id2m2 , vol4) on each weight line is
e−iπm2 = (−1)m2 .
• The element (−Id2m1 , Id2m2 ,−vol4) ∈ Sp(m1) × Sp(m2) × Spin(4) corresponds to the
parameters
θj1 = π, θ′j2 = 0, ϕ1 = π, ϕ2 = −π,
so that the effect of (−Id2m1 , Id2m2 ,−vol4) on each weight line is
e−iπm1 = (−1)m1 .
Thus, in order to have a twisted Spin representation
∆4(m1+m2) ⊗
∧u1C2m1 ⊗
∧u2C2m2 ⊗ (∆+
4 )⊗s ⊗ (∆−4 )⊗t
of
Sp(m1)× Sp(m2)× Spin(4)
{±(Id2m1 , Id2m2 , 1),±(Id2m1 ,−Id2m2 , vol4)}
,
the exponents must satisfy
m1 + u1 + t ≡ 0 (mod 2), m2 + u2 + s ≡ 0 (mod 2).
14 A.L. Garcia-Pulido and R. Herrera
3.2.3 r = 6
By (3.1), the weights of the spin representation are
±η1
2
± · · · ± η8m
2
=
∑
j∈I1
θj + −ϕ1+ϕ2+ϕ3
2
2
−
∑
j∈Ī1
θj + −ϕ1+ϕ2+ϕ3
2
2
+
∑
j∈I2
θj + ϕ1−ϕ2+ϕ3
2
2
−
∑
j∈Ī2
θj + ϕ1−ϕ2+ϕ3
2
2
+
∑
j∈I3
θj + ϕ1+ϕ2−ϕ3
2
2
−
∑
j∈Ī3
θj + ϕ1+ϕ2−ϕ3
2
2
+
∑
j∈I4
θj + −ϕ1−ϕ2−ϕ3
2
2
−
∑
j∈Ī4
θj + −ϕ1−ϕ2−ϕ3
2
2
,
where I1, I2, I3, I4 ⊆ {1, . . . ,m1}.
• The element (−Idm,−1) ∈ U(m)× Spin(6) corresponds to the parameters
θj = π, ϕ1 = 2π, ϕ2 = 0, ϕ3 = 0,
so that its effect on each weight line is
e−2iπm = 1.
• The element (iIdm,−vol6) ∈ U(m)× Spin(6) corresponds to the parameters
θj =
π
2
, ϕ1 = −π, ϕ2 = π, ϕ3 = π,
so that its effect on each weight line is
e−iπm = (−1)m.
Thus, in order to have a twisted Spin representation
∆8m ⊗
∧u1Cm ⊗
∧u2Cm ⊗ (∆+
6 )⊗s ⊗ (∆−6 )⊗t
of U(m)×Spin(6)
{±(Idm,1),±(iIdm,−vol6)} , the exponents must satisfy
u1 + u2 + s+ t ≡ 0 (mod 2), 2m+ u1 + 3u2 + s+ 3t ≡ 0 (mod 4),
2m+ 3u1 + u2 + 3s+ t ≡ 0 (mod 4).
3.2.4 r = 8
By (3.5), if m1 + 1 ≡ m2 ≡ 0 (mod 2), the element (Idm1 ,−Idm2 , vol8) ∈ SO(m1) × SO(m2) ×
Spin(8) corresponds to the parameters
θj1 = 0, θ′j2 = π, ϕ1 = ϕ2 = ϕ3 = ϕ4 = π,
and its effect on each weight line is mutiplication by −1. Thus, we can have twisted Spin
representations
∆8(m1+m2) ⊗
∧u1Cm1 ⊗
∧u2Cm2 ⊗ (∆+
8 )⊗s ⊗ (∆−8 )⊗t
of
SO(m1)× SO(m2)× Spin(8)
{(Idm1 , Idm2 , 1), (Idm1 ,−Idm2 , vol8)}
if u2 + t ≡ 1 (mod 2) and u1, s ∈ N.
Rigidity and Vanishing Theorems for Almost Even-Clifford Hermitian Manifolds 15
Similarly, by (3.4), if m1 ≡ m2 + 1 ≡ 0 (mod 2), we can have twisted Spin representations
∆8(m1+m2) ⊗
∧u1Cm1 ⊗
∧u2Cm2 ⊗ (∆+
8 )⊗s ⊗ (∆−8 )⊗t
of
SO(m1)× SO(m2)× Spin(8)
{(Idm1 , Idm2 , 1), (−Idm1 , Idm2 ,−vol8)}
if u1 + s ≡ 1 (mod 2) and u2, t ∈ N.
3.3 Twisting representations
For most r, almost-Cl0r-Hermitian manifolds are Spin [1, Theorem 4.1]. In particular, this is the
case when r ≥ 5 and r 6= 6, 8. Thus, we only need to choose suitable representations of the
structure group G to twist the spinor bundle:
• For r ≡ 1, 7 (mod 8) our candidates are∧uCm ⊗∆⊗sr , SuCm ⊗∆⊗sr .
They are representations of the structure group when
u+ s ≡ 0 (mod 2) if m is even,
u, s ∈ N if m is odd.
• For r ≡ 3, 5 (mod 8) our candidates are∧uC2m ⊗∆⊗sr , SuC2m ⊗∆⊗sr .
They are representations of the structure group when
u+ s ≡ 0 (mod 2).
• For r ≡ 4 (mod 8) our candidates are∧u1C2m1 ⊗
∧u2C2m2 ⊗ (∆+
r )⊗s ⊗ (∆−r )⊗t,
Su1C2m1 ⊗ Su2C2m2 ⊗ (∆+
r )⊗s ⊗ (∆−r )⊗t.
They are representations of the structure group when
u2 + s ≡ 0 (mod 2), u1 + t ≡ 0 (mod 2).
• For r ≡ 2 (mod 8), r 6= 2, our candidates are∧u1Cm ⊗
∧u2Cm ⊗ (∆+
r )⊗s ⊗ (∆−r )⊗t, Su1Cm ⊗ Su2Cm ⊗ (∆+
r )⊗s ⊗ (∆−r )⊗t.
They are representations of the structure group when
u1 + u2 + s+ t ≡ 0 (mod 2), u1 + 3u2 + 3s+ t ≡ 0 (mod 4),
3u1 + u2 + s+ 3t ≡ 0 (mod 4).
16 A.L. Garcia-Pulido and R. Herrera
• For r ≡ 6 (mod 8) our candidates are∧u1Cm ⊗
∧u2Cm ⊗ (∆+
r )⊗s ⊗ (∆−r )⊗t, Su1Cm ⊗ Su2Cm ⊗ (∆+
r )⊗s ⊗ (∆−r )⊗t.
They are representations of the structure group when
u1 + u2 + s+ t ≡ 0 (mod 2), u1 + 3u2 + s+ 3t ≡ 0 (mod 4),
3u1 + u2 + 3s+ t ≡ 0 (mod 4).
• For r ≡ 0 (mod 8) our candidates are∧u1Cm1 ⊗
∧u2Cm2 ⊗ (∆+
r )⊗s ⊗ (∆−r )⊗t,
Su1Cm1 ⊗ Su2Cm2 ⊗ (∆+
r )⊗s ⊗ (∆−r )⊗t.
They are representations of the structure group when{
u2 + t ≡ 0 (mod 2)
u1 + s ≡ 0 (mod 2)
if m1 ≡ m2 ≡ 0 (mod 2),{
u2 + t ≡ 0 (mod 2)
u1, s ∈ N
if m1 + 1 ≡ m2 ≡ 0 (mod 2),{
u2, t ∈ N
u1 + s ≡ 0 (mod 2)
if m1 ≡ m2 + 1 ≡ 0 (mod 2),
u1, u2, s, t ∈ N if m1 ≡ m2 ≡ 0 (mod 2).
4 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-Cl0r-Hermitian manifolds
and prove the vanishing Theorems 4.7, 4.8 and 4.9.
4.1 Rigidity of elliptic operators
Definition 4.1. Let D : Γ(E) −→ Γ(F ) be an elliptic operator acting on sections of the vector
bundles E and F over a compact manifold M . The index of D is the virtual vector space
ind(D) = ker(D) − coker(D). If M admits a circle action preserving D, i.e., such that S1 acts
on E and F , and commutes with D, ind(D) admits a Fourier decomposition into complex 1-di-
mensional irreducible representations of S1 ind(D) =
∑
amL
m, where am ∈ Z and Lm is the
representation of S1 on C given by λ 7→ λm. The elliptic operator D is called rigid if am = 0 for
all m 6= 0, i.e., ind(D) consists only of the trivial representation with multiplicity a0.
Let us recall three examples.
Example 4.2. The deRham complex
d+ d∗ : Ωeven −→ Ωodd
from even-dimensional forms to odd-dimensional ones, where d∗ denotes the adjoint of the
exterior derivative d, is rigid for any circle action on M by isometries since by Hodge theory
the kernel and the cokernel of this operator consist of harmonic forms, which by homotopy
invariance stay fixed under the circle action.
Rigidity and Vanishing Theorems for Almost Even-Clifford Hermitian Manifolds 17
Example 4.3. The signature operator on an oriented manifold
ds : Ω+
c −→ Ω−c
from even to odd complex forms under the Hodge ∗ operator is rigid for any circle action on M
by isometries since the kernel and cokernel of this operator consist of harmonic forms.
Example 4.4. The Dirac operator on a Spin manifold is rigid for any circle action by isomet-
ries [3].
4.2 Twisted Dirac operators
In this subsection, let M be a 4n-dimensional oriented Riemannian manifold. M is Spin if its
orthonormal frame bundle PSO(4n) admits a double cover by a principal bundle PSpin(4n) with
structure group Spin(4n), which gives rise to the spinor bundle
PSpin(4n) ×κ ∆4n.
The Levi-Civita connection on PSO(4n) can be lifted to PSpin(4n) to define a covariant differentia-
tion ∇ on ∆
∇ : Γ(∆) −→ Γ(T ∗ ⊗∆),
and the (elliptic and self-adjoint) Dirac operator
/∂(ψ) =
4n∑
i=1
ei · ∇eiψ
for ψ ∈ Γ(∆), where (e1, . . . , e4n) is a local orthonormal frame. Since the spin representation
decomposes, the Dirac operator can be split into two parts
/∂ : Γ(∆+) −→ Γ(∆−), /∂∗ : Γ(∆−) −→ Γ(∆+).
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 )
(/∂ ⊗ F ) : Γ(∆+ ⊗ F ) −→ Γ(∆− ⊗ F )
is defined by
(/∂ ⊗ F )(ψ ⊗ f) =
(
4n∑
i=1
ei · ∇eiψ
)
⊗ f +
4n∑
i=0
µ
(
ei ⊗ ψ
)
⊗∇Feif,
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.
18 A.L. Garcia-Pulido and R. Herrera
4.3 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 dimC(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
ind(/∂ ⊗ F ) =
〈
Â(M)ch(F ), [M ]
〉
,
where ch(·) denotes the Chern character, Â(M) denotes the Â-genus, and [M ] denotes the
fundamental cycle of M . In terms of formal roots,
c(TM ⊗ C) = (1 + η1)(1− η1) · · · (1 + η2n)(1− η2n),
p(TM) =
(
1 + η2
1
)
· · ·
(
1 + η2
2n
)
,
c(F ) = (1 + ν1) · · · (1 + νp),
ch(F ) =
∑
l=1
eνl ,
ind(/∂ ⊗ F ) =
〈∑
l=1
eνl ·
2n∏
i=1
ηi
e
ηi
2 − e−
ηi
2
, [M ]
〉
.
If M admits a non-trivial S1 action that lifts to ∆4n ⊗ F , the equivariant version of the index
can be written in terms of the local data of the S1-fixed point set MS1
. More precisely, let
z ∈ S1 be a generic element of S1. By the Atiyah–Singer fixed point theorem [4, 5]
ind(/∂ ⊗ F )z =
∑
P⊂MS1
µ(P, z),
where µ(P, z) is the local contribution of the oriented fixed point submanifold P ⊂MS1
, which
can be computed as follows. The S1 action on M induces a decomposition of TM over P ,
TM |P =
∑
k
Nk, (4.1)
where Nk is a bundle over P whose fibers are representations of S1 on which z ∈ S1 acts as an
automorphism with multiple eigenvalue zk, k ∈ Z. Note that P inherits an orientation since M
is oriented and the bundles Nk for k 6= 0 are naturally oriented. Formally, by means of the
splitting principle, we can write
TM |P = Lq1 + · · ·+ Lq2n , (4.2)
where L corresponds to the standard representation of S1 on C, so that z ∈ S1 acts by multi-
plication by zqi on Lqi . The integers qi = qi(P ) ∈ Z are the exponents of the action at P , which
correspond to the aforementioned numbers k. Thus, following [14, p. 67],
µ(P, z) =
〈∑
z−nkeνk
∏
qi=0
ηi
e
ηi
2 − e−
ηi
2
∏
qj 6=0
1
z−
qj
2 e
ηj
2 − z
qj
2 e−
ηj
2
, [P ]
〉
,
Rigidity and Vanishing Theorems for Almost Even-Clifford Hermitian Manifolds 19
where nk = nk(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
|nk| <
1
2
(|q1(P )|+ · · ·+ |q2n(P )|) (4.3)
for all 1 ≤ k ≤ p. If such a condition is fulfilled for all P ⊂MS1
, 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(S1) of S1, which can be identified with the Laurent polynomial ring Z[z, z−1]. Hence,
by Lemma 2.5,
ind(/∂ ⊗ F ) = ind(/∂ ⊗ F )1 = 0,
i.e., 〈
Â(M)ch(F ), [M ]
〉
= 0.
4.4 Â-genus of almost-Cl0r-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
TM ⊗ C = L1 ⊕ L−1
1 ⊕ · · · ⊕ L2n ⊕ L−1
2n
and, therefore,
c(TM ⊗ C) = (1 + x1)(1− x1) · · · (1 + x2n)(1− x2n) =
(
1− x2
1
)
· · ·
(
1− x2
2n
)
and its Pontrjagin class is
p(TM) =
(
1 + x2
1
)
· · ·
(
1 + x2
2n
)
,
and the Â-genus is given by
Â(M) =
2n∏
j=1
xi/2
sinh(xi/2)
=
2n∏
j=1
xi
exi/2 − e−xi/2
.
In the following, we will set xi = ηi from Section 3.1.
4.4.1 r ≡ 1, 7 (mod 8)
The Â-genus is given by
Â(M) =
[m
2
]∏
j=1
2[
r
2 ]∏
k=1
θj + λk
e
θj+λk
2 − e−
θj+λk
2
if m is even, and
Â(M) =
[m
2
]∏
j=1
2[
r
2 ]∏
k=1
θj + λk
e
θj+λk
2 − e−
θj+λk
2
2[
r
2 ]−1∏
l=1
λl
e
λl
2 − e−
λl
2
if m is odd.
20 A.L. Garcia-Pulido and R. Herrera
4.4.2 r ≡ 2, 6 (mod 8)
The Â-genus is given by
Â(M) =
m∏
j=1
2
r
2−1∏
k=1
θj + λ+
k
e
θj+λ
+
k
2 − e−
θj+λ
+
k
2
if r ≡ 2 (mod 8),
m∏
j=1
2
r
2−1∏
k=1
θj + λ−k
e
θj+λ
−
k
2 − e−
θj+λ
−
k
2
if r ≡ 6 (mod 8).
4.4.3 r ≡ 3, 5 (mod 8)
The Â-genus is given by
Â(M) =
m∏
j=
2[
r
2 ]∏
k=1
θj + λk
e
θj+λk
2 − e−
θj+λk
2
.
4.4.4 r ≡ 4 (mod 8)
The Â-genus is given by
Â(M) =
m1∏
j1=1
2
r
2−1∏
k=1
θj1 + λ+
k
e
θj1
+λ+
k
2 − e−
θj1
+λ+
k
2
m2∏
j2=1
2
r
2−1∏
k=1
θ′j2 + λ−k
e
θ′
j2
+λ−
k
2 − e−
θ′
j2
+λ−
k
2
.
4.4.5 r ≡ 0 (mod 8)
We can set
• if m1, m2 are even,
Â(M) =
m1
2∏
j1=1
2
r
2−1∏
k=1
θj1 + λ+
k
e
θj1
+λ+
k
2 − e−
θj1
+λ+
k
2
m2
2∏
j2=0
2
r
2−1∏
k=1
θ′j2 + λ−k
e
θ′
j2
+λ−
k
2 − e−
θ′
j2
+λ−
k
2
;
• if m1 is even and m2 is odd,
Â(M) =
m1
2∏
j1=1
2
r
2−1∏
k=1
θj1 + λ+
k
e
θj1
+λ+
k
2 − e−
θj1
+λ+
k
2
[
m2
2
]∏
j2=1
2
r
2−1∏
k=1
θ′j2 + λ−k
e
θ′
j2
+λ−
k
2 − e−
θ′
j2
+λ−
k
2
2
r
2−2∏
l=1
λ−l
e
λ−
l
2 − e−
λ−
l
2
;
• if m1 is odd and m2 is even,
Â(M) =
[
m1
2
]∏
j1=1
2
r
2−1∏
k=1
θj1 + λ+
k
e
θj1
+λ+
k
2 − e−
θj1
+λ+
k
2
2
r
2−2∏
l=1
λ+
l
e
λ+
l
2 − e−
λ+
l
2
m2
2∏
j2=1
2
r
2−1∏
k=1
θ′j2 + λ−k
e
θ′
j2
+λ−
k
2 − e−
θ′
j2
+λ−
k
2
;
• if m1, m2 are odd,
Â(M) =
[
m1
2
]∏
j1=1
2
r
2−1∏
k=1
θj1 + λ+
k
e
θj1
+λ+
k
2 − e−
θj1
+λ+
k
2
2
r
2−2∏
l=1
λ+
l
e
λ+
l
2 − e−
λ+
l
2
×
[
m2
2
]∏
j2=1
2
r
2−1∏
k=1
θ′j2 + λ−k
e
θ′
j2
+λ−
k
2 − e−
θ′
j2
+λ−
k
2
2
r
2−2∏
l=1
λ−l
e
λ−
l
2 − e−
λ−
l
2
.
Rigidity and Vanishing Theorems for Almost Even-Clifford Hermitian Manifolds 21
4.5 Infinitesimal automorphisms
An automorphism of an almost-Cl0r-Hermitian manifold M is an isometry which preserves the al-
most 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
LXJij =
∑
k<l
α
(ij)
kl Jkl,
for some (local) functions α
(ij)
kl , where LX denotes the Lie derivative in the direction of X.
Consider
LX(Jij(Y )) = (LXJij)(Y ) + Jij(LXY ),
which can be written in terms of the Levi-Civita connection ∇ as follows
∇X(Jij(Y ))−∇Jij(Y )X =
∑
k<l
α
(ij)
kl Jkl(Y ) + Jij(∇XY −∇YX).
Now, if p ∈M is such that Xp = 0,
−∇Jij(Y )X =
∑
k<l
α
(ij)
kl Jkl(Y )− Jij(∇YX),
i.e.,
[Jij ,∇X](Y ) =
∑
k<l
α
(ij)
kl Jkl(Y ).
Hence, (∇X)p is a skew-symmetric endomorphism such that
[Jij ,∇X] =
∑
k<l
α
(ij)
kl Jkl.
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-Cl0r-Hermitian manifold is an action by
automorphisms if the corresponding Killing vector field is an infinitesimal automorphism.
Example 4.6. The 16-dimensional symmetric space
F4
Spin(9)
has an almost-Cl09-Hermitian structure admitting S1 actions by automorphisms [10].
4.6 Exponents of the S1 action
In this section, let M be a compact, rank r almost even-Clifford Hermitian manifold with a non-
trivial (effective) S1 action by automorphisms. Let P ⊂MS1
be an S1-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 S1 to TpM , 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
22 A.L. Garcia-Pulido and R. Herrera
a group. Hence, we can read off the exponents of the action with respect to the weights given
in Section 3.1:
r (mod 8) ±qi
0
tj1±h
+
k
2 ,
t′j2
±h−k
2 1 ≤ j1 ≤ [m1
2 ], 1 ≤ k ≤ 2[ r
2
]−2(
hl
2 ,
h
2
r
2−1
+l
2
)
1 ≤ j2 ≤ [m2
2 ]
(
1 ≤ l ≤ 2[ r
2
]−3
)
1, 7
tj±hk
2
(
hl
2
)
1 ≤ j ≤ [m2 ] 1 ≤ k ≤ 2[ r
2
]−1(
1 ≤ l ≤ 2[ r
2
]−2
)
2, 6
tj±h+k
2 ,
−tj±h−k
2 1 ≤ j ≤ m 1 ≤ k ≤ 2[ r
2
]−2
3, 5
tj±hk
2 1 ≤ j ≤ m 1 ≤ k ≤ 2[ r
2
]−1
4
tj1±h
+
k
2 ,
t′j2
±h−k
2 1 ≤ j1 ≤ m1, 1 ≤ j2 ≤ m2 1 ≤ k ≤ 2[ r
2
]−2
where m, m1, m2 denote the corresponding multiplicities. Here, the numbers
tj
2 ,
t′j
2 are the
exponents corresponding to the complex representations E, E1, E2 described in Section 2.3.1,
fj are the exponents for the SO(r) representation for r odd or PSO(r) representation for r even,
hk denote the numbers
±f1 ± · · · ± f[ r
2
]
in some order for r odd, and h±k denote the numbers
±f1 ± · · · ± f r
2
with an even or odd number of negative signs respectively, listed in some order for r even.
4.7 Vanishing theorems
In this section, we give the main details of the proofs of the vanishing theorems.
Theorem 4.7. Let M be a compact N -dimensional almost-Cl0r-Hermitian admitting a smooth
circle action by automorphisms, r ≥ 3. Let E, E1, E2 be the (locally defined) bundles described
in (2.4), m, m1, m2 the corresponding multiplicities and u, u1, u2, s, t be non-negative integers
satisfying the conditions given in Sections 3.2 and 3.3. Then,
• for r ≡ 1, 7 (mod 8), if 0 ≤ u+ s < [m2 ],〈
ch(
∧u
E)ch(∆r)
sÂ(M), [M ]
〉
= 0;
• for r ≡ 3, 5 (mod 8), if 0 ≤ u+ s < m,〈
ch(
∧u
E)ch(∆r)
sÂ(M), [M ]
〉
= 0;
• for r ≡ 0 (mod 8), if 0 ≤ u1 + s < [m1
2 ], 0 ≤ u2 + t < [m2
2 ],〈
ch(
∧u1E1)ch(
∧u2E2)ch(∆+
r )sch(∆−r )tÂ(M), [M ]
〉
= 0;
• for r ≡ 2, 6 (mod 8), if 0 ≤ u1 + s < m and 0 ≤ u2 + t < m, or if 0 ≤ u1 + t < m and
0 ≤ u2 + s < m,〈
ch(
∧u1E)ch(
∧u2E)ch(∆+
r )sch(∆−r )tÂ(M), [M ]
〉
= 0;
Rigidity and Vanishing Theorems for Almost Even-Clifford Hermitian Manifolds 23
• for r ≡ 4 (mod 8), if 0 ≤ u1 + s < m1 and 0 ≤ u2 + t < m2,〈
ch(
∧u1E1)ch(
∧u2E2)ch(∆+
r )sch(∆−r )tÂ(M), [M ]
〉
= 0.
If the inequalities are not strict, the indices are rigid.
Proof. Since the S1 action is by automorphisms of the almost even-Clifford Hermitian struc-
ture, 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 ⊂ MS1
be an S1-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 ⊗ ∆⊗sr over points of P decompose as sums of representations of S1.
Recall that
ch
(∧u
E
)
=
∑
1≤i1<···<iu≤2[m
2
]
eϑi1+···+ϑiu ,
where
ϑj = θj , ϑ[m/2]+j = −θj , j = 1, . . . , [m/2].
Thus, the exponents of the twist will be of the form
1
2
(
c∑
a=1
(−1)εatia +
s∑
b=1
(−1)δbhlb
)
,
where 0 ≤ c ≤ u, εa, δb ∈ {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 ⊗∆⊗sr over P
decompose as sums of representations of S1. Formally, according to the splitting principle, if
TMc = L1 ⊕ L−1
1 ⊕ · · · ⊕ LN/2 ⊕ L
−1
N/2,
then
∆N =
(
L
1/2
1 ⊕ L−1/2
1
)
⊗ · · · ⊗
(
L
1/2
N/2 ⊕ L
−1/2
N/2
)
= L
1/2
1 ⊗ · · · ⊗ L1/2
N/2 ⊕ · · · ⊕ L
−1/2
1 ⊗ · · · ⊗ L−1/2
N/2 ,
so that the S1-exponents on these lines will be of the form∑(
± tj ± hk
4
)
+
∑(
±hl
4
)
.
The bundle ∆N ⊗
∧u
E ⊗∆⊗sr will have integer exponents over P of the form
1
2
(
c∑
a=1
(−1)εatia +
s∑
b=1
(−1)δbhlb
)
+
∑
qj 6=0
(
(−1)γj
tj + hk
4
)
+
∑
qj′ 6=0
(
(−1)γj′
t′j − hk
4
)
+
∑
ql 6=0
(
(−1)ζl
hl
4
)
.
24 A.L. Garcia-Pulido and R. Herrera
Thus, the powers of z in each summand of µ(P, z) can be rearranged in order to show that such
a summand is a product of rational functions such as the one described in Lemma 2.5.
For the second point, it is sufficient to consider the exponents of the form
1
2
(
u∑
a=1
(−1)εatia ± shk
)
.
Since u + s < [m2 ], there exists an s-tuple of indices j1 < · · · < js such that {j1, . . . , js} ⊂
{1, . . . , [m2 ]} − {i1, . . . , iu}. Thus,∣∣∣∣∣
u∑
a=1
(−1)εatia ± shk
∣∣∣∣∣ =
∣∣∣∣∣
u∑
a=1
(−1)εa
(
tia + hk
2
+
tia − hk
2
)
±
s∑
b=1
(
hk + tjb
2
+
hk − tjb
2
)∣∣∣∣∣
≤
u∑
a=1
(∣∣∣∣ tia + hk
2
∣∣∣∣+
∣∣∣∣ tia − hk2
∣∣∣∣)+
s∑
b=1
(∣∣∣∣hk + tjb
2
∣∣∣∣+
∣∣∣∣hk − tjb2
∣∣∣∣)
≤
m∑
i=1
∣∣∣∣ ti + hk
2
∣∣∣∣+
∣∣∣∣ ti − hk2
∣∣∣∣ < 2[r/2]−1∑
l=1
m∑
i=1
∣∣∣∣ ti + hl
2
∣∣∣∣+
∣∣∣∣ ti − hl2
∣∣∣∣
≤
N/2∑
c=1
|qc|,
which is the corresponding version of the inequality (4.3) in Section 4.3 �
Theorem 4.8. Let M be a compact N -dimensional almost-Cl0r-Hermitian admitting a smooth
circle action by automorphisms, r ≥ 3. Let E, E1, E2 be the (locally defined) bundles described
in (2.4), m, m1, m2 the corresponding multiplicities and u, u1, u2, s, t be non-negative integers
satisfying the conditions given in Sections 3.2 and 3.3. Then,
• for r ≡ 1, 7 (mod 8), if 0 ≤ u+ s < [m2 ] and u ≤ 2[ r
2
]−1,〈
ch(SuE)ch(∆r)
sÂ(M), [M ]
〉
= 0;
• for r ≡ 3, 5 (mod 8), if 0 ≤ u+ s < m and u ≤ 2[ r
2
]−1,〈
ch(SuE)ch(∆r)
sÂ(M), [M ]
〉
= 0;
• for r ≡ 0 (mod 8), if 0 ≤ u1 + s < [m1
2 ], 0 ≤ u2 + t < [m2
2 ] and u1, u2 ≤ 2[ r
2
]−2,〈
ch(Su1E1)ch(Su2E2)ch(∆+
r )sch(∆−r )tÂ(M), [M ]
〉
= 0;
• for r ≡ 2, 6 (mod 8), if u1, u2 ≤ 2[ r
2
]−2 and one of 0 ≤ u1 + s < m, 0 ≤ u2 + t < m or
0 ≤ u1 + t < m, 0 ≤ u2 + s < m,〈
ch(Su1E)ch(Su2E)ch(∆+
r )sch(∆−r )tÂ(M), [M ]
〉
= 0;
• for r ≡ 4 (mod 8), if 0 ≤ u1 + s < m1, 0 ≤ u2 + t < m2 and u1, u2 ≤ 2[ r
2
]−2,〈
ch(Su1E1)ch(Su2E2)ch(∆+
r )sch(∆−r )tÂ(M), [M ]
〉
= 0.
If the inequalities are not strict, the indices are rigid.
Rigidity and Vanishing Theorems for Almost Even-Clifford Hermitian Manifolds 25
Proof. We will only describe the relevant changes to the calculation for r ≡ 1, 7 (mod 8) and m
even. Let P ⊂MS1
be an S1-fixed submanifold. Recall that
ch
(
SuE
)
=
∑
1≤i1≤···≤iu≤2[m
2
]
eϑi1+···+ϑiu ,
where
ϑj = θj , ϑ[m/2]+j = −θj , j = 1, . . . , [m/2].
Thus, the exponents of the twist will be of the form
1
2
(
u∑
a=1
(−1)εatia +
s∑
b=1
(−1)δbhlb
)
,
where εa, δb ∈ {0, 1}. It is sufficient to consider the exponents of the form
1
2
(
u∑
a=1
(−1)εatia ± shk
)
.
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 ut1. For such an exponent,
consider
|ut1| =
∣∣∣∣∣
u∑
l=1
t1 + hl
2
+
t1 − hl
2
∣∣∣∣∣ ≤
u∑
l=1
∣∣∣∣ t1 + hl
2
∣∣∣∣+
∣∣∣∣ t1 − hl2
∣∣∣∣ ≤ 2[
r
2 ]−1∑
l=1
∣∣∣∣ t1 + hl
2
∣∣∣∣+
∣∣∣∣ t1 − hl2
∣∣∣∣
<
[m
2
]∑
j=1
2[
r
2 ]−1∑
l=1
∣∣∣∣ t1 + hl
2
∣∣∣∣+
∣∣∣∣ t1 − hl2
∣∣∣∣ ≤ N/2∑
c=1
|qc|,
if u < 2[r/2]−1. �
Theorem 4.9. Let M be a compact N -dimensional almost-Cl0r-Hermitian admitting a smooth
circle action by automorphisms, r ≥ 3. Let E, E1, E2 be the (locally defined) bundles described
in (2.4), m, m1, m2 the corresponding multiplicities and ui, vi, u
′
i, v
′
i, s, t be non-negative
integers satisfying analogous conditions to those given in Sections 3.2 and 3.3. Then,
• for r ≡ 1, 7 (mod 8), if
0 ≤
b∑
i=1
ui +
b∑
j=1
vj + s <
[m
2
]
and
a+
b∑
i=1
vi ≤ 2[ r
2
]−1,
〈
ch
( a⊗
i=1
∧uiE ⊗
b⊗
j=1
SvjE ⊗ (∆r)
⊗s
)
Â(M), [M ]
〉
= 0;
26 A.L. Garcia-Pulido and R. Herrera
• for r ≡ 3, 5 (mod 8), if
0 ≤
b∑
i=1
ui +
b∑
j=1
vj + s < m
and
a+
b∑
i=1
vi ≤ 2[ r
2
]−1,
〈
ch
( a⊗
i=1
∧uiE ⊗
b⊗
j=1
SvjE ⊗ (∆r)
⊗s
)
Â(M), [M ]
〉
= 0;
• for r ≡ 0 (mod 8), if
0 ≤
b∑
i=1
ui +
b∑
j=1
vj + s <
[m1
2
]
, 0 ≤
c∑
i=1
u′i +
d∑
j=1
v′j + t <
[m2
2
]
and
a+
b∑
i=1
vi, c+
d∑
i=1
v′i ≤ 2[ r
2
]−2,
〈
ch
( a⊗
i=1
∧uiE1 ⊗
b⊗
j=1
SvjE1 ⊗
c⊗
k=1
∧u′kE2 ⊗
d⊗
l=1
Sv
′
lE2
⊗ (∆+
r )⊗s ⊗ (∆−r )⊗t
)
Â(M), [M ]
〉
= 0;
• for r ≡ 2 (mod 8), if
a+
b∑
i=1
vi, c+
d∑
i=1
v′i ≤ 2[ r
2
]−2,
and
0 ≤
b∑
i=1
ui +
b∑
j=1
vj + s < m, 0 ≤
c∑
i=1
u′i +
d∑
j=1
v′j + t < m
or
0 ≤
b∑
i=1
ui +
b∑
j=1
vj + t < m, 0 ≤
c∑
i=1
u′i +
d∑
j=1
v′j + s < m,
〈
ch
( a⊗
i=1
∧uiE ⊗
b⊗
j=1
SvjE ⊗
c⊗
k=1
∧u′kE ⊗
d⊗
l=1
Sv
′
lE
⊗ (∆+
r )⊗s ⊗ (∆−r )⊗t
)
Â(M), [M ]
〉
= 0;
Rigidity and Vanishing Theorems for Almost Even-Clifford Hermitian Manifolds 27
• for r ≡ 4 (mod 8), if
0 ≤
b∑
i=1
ui +
b∑
j=1
vj + s < m1, 0 ≤
c∑
i=1
u′i +
d∑
j=1
v′j + t < m2
and
a+
b∑
i=1
vi, c+
d∑
i=1
v′i ≤ 2[ r
2
]−2,
〈
ch
( a⊗
i=1
∧uiE1 ⊗
b⊗
j=1
SvjE1 ⊗
c⊗
k=1
∧u′kE2 ⊗
d⊗
l=1
Sv
′
lE2
⊗ (∆+
r )⊗s ⊗ (∆−r )⊗t
)
Â(M), [M ]
〉
= 0.
If the inequalities are not strict, the indices are rigid.
Remark 4.10. When r = 3, Theorems 4.7 and 4.8 return the vanishings for almost quaternion-
Hermitian manifolds proved in [12].
Remark 4.11. Theorems 4.7, 4.8 and 4.9 do not restrict to the well known vanishings for almost
Hermitian manifolds proved in [11], which require a divisibility condition on c1(M). This is due
to the fact that the structure group of a 2m-dimensional almost Hermitian manifold is U(m)
instead of
U(m)× Spin(2)
{±(Id2, 1),±(iId2,−vol2)}
.
Remark 4.12. For r 6= 3, 4, 6, 8, an almost-Cl0r-Hermitian manifold is Spin (see [1, Theo-
rem 4.1]). Thus, for u = u1 = u2 = s = t = 0, the vanishings in the theorems restrict to
Atiyah–Hirzebruch’s vanishing.
Acknowledgements
The first named author was supported by CONACyT. The second named author was partially
supported by a CONACyT grant. The second named author wishes to thank the International
Centre for Theoretical Physics and the Institut des Hautes Études Scientifiques for their hospi-
tality and support. We would like to express our gratitude to the anonymous referees for their
careful reading of this manuscript and their helpful comments.
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https://doi.org/10.1007/BFb0078045
1 Introduction
2 Preliminaries
2.1 Clifford algebra, spin group and representation
2.2 Maximal tori
2.2.1 SO(n)
2.2.2 Spin(n)
2.2.3 U(m)
2.2.4 Sp(m)
2.3 Almost even-Clifford Hermitian structures
2.3.1 Structure groups of almost even-Clifford manifolds
2.4 A useful lemma
3 Twisted spinor bundles on almost-Clr0-Hermitian manifolds
3.1 Weights of SO(N) with respect to the structure subgroups
3.1.1 r1, 7 12mumod8
3.1.2 r2,6 12mumod8
3.1.3 r3,5 12mumod8
3.1.4 r4 12mumod8
3.1.5 r0 12mumod8
3.2 The Spin representation when r=3,4,6,8
3.2.1 r=3
3.2.2 r=4
3.2.3 r=6
3.2.4 r=8
3.3 Twisting representations
4 Index calculations
4.1 Rigidity of elliptic operators
4.2 Twisted Dirac operators
4.3 Index formula and localization
4.4 A"0362A-genus of almost-Clr0-Hermitian manifolds
4.4.1 r1,7 (mod8)
4.4.2 r2,6 (mod8)
4.4.3 r3,5 (mod8)
4.4.4 r4 (mod8)
4.4.5 r0 (mod8)
4.5 Infinitesimal automorphisms
4.6 Exponents of the S1 action
4.7 Vanishing theorems
References
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