CAMPINAS
Instituto de Matemática, Estatística e
Computação Científica
LUIS ERNESTO PORTILLA PALADINES
Instantons on Sasakian 7–manifolds
Instantons sobre 7–variedades sasakianas
Campinas
2020
Instantons on Sasakian 7–manifolds
Instantons sobre 7–variedades sasakianas
Tese apresentada ao Instituto de Matemática, Estatística e Computação Científica da Univer-sidade Estadual de Campinas como parte dos requisitos exigidos para a obtenção do título de Doutor em Matemática.
Thesis presented to the Institute of Mathemat-ics, Statistics and Scientific Computing of the University of Campinas in partial fulfillment of the requirements for the degree of Doctor in Mathematics.
Supervisor: Henrique Nogueira de Sá Earp
Este exemplar corresponde à versão
fi-nal da Tese defendida pelo aluno Luis
Ernesto Portilla Paladines e orientada
pelo Prof. Dr. Henrique Nogueira de Sá
Earp.
Campinas
2020
Biblioteca do Instituto de Matemática, Estatística e Computação Científica Ana Regina Machado - CRB 8/5467
Portilla Paladines, Luis Ernesto,
Sa11i PorInstantons on Sasakian 7–manifolds / Luis Ernesto Portilla Paladines. – Campinas, SP : [s.n.], 2020.
PorOrientador: Henrique Nogueira de Sá Earp.
PorTese (doutorado) – Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica.
Por1. Variedades sasakianas. 2. Variedades complexas. 3. Campos de calibre (Física). 4. Instantons. 5. Teoria de módulos. I. Sá Earp, Henrique Nogueira de, 1981-. II. Universidade Estadual de Campinas. Instituto de Matemática,
Estatística e Computação Científica. III. Título.
Informações para Biblioteca Digital
Título em outro idioma: Instantons sobre 7-variedades Sasakianas Palavras-chave em inglês:
Sasakian manifolds Complex manifolds Gauge fields (Physics) Instantons
Moduli theory
Área de concentração: Matemática Titulação: Doutor em Matemática Banca examinadora:
Henrique Nogueira de Sá Earp [Orientador] Lino Anderson da Silva Grama
Marcos Benevenuto Jardim Maurício Barros Corrêa Júnior Lázaro Orlando Rodríguez Díaz
Data de defesa: 03-03-2020
Programa de Pós-Graduação: Matemática
Identificação e informações acadêmicas do(a) aluno(a)
- ORCID do autor: https://orcid.org/0000-0002-5468-0377 - Currículo Lattes do autor: http://lattes.cnpq.br/3933727619225265
pela banca examinadora composta pelos Profs. Drs.
Prof(a). Dr(a). HENRIQUE NOGUEIRA DE SÁ EARP
Prof(a). Dr(a). LINO ANDERSON DA SILVA GRAMA
Prof(a). Dr(a). MARCOS BENEVENUTO JARDIM
Prof(a). Dr(a). MAURÍCIO BARROS CORRÊA JÚNIOR
Prof(a). Dr(a). LÁZARO ORLANDO RODRÍGUEZ DÍAZ
A Ata da Defesa, assinada pelos membros da Comissão Examinadora, consta no SIGA/Sistema de Fluxo de Dissertação/Tese e na Secretaria de Pós-Graduação do Instituto de Matemática, Estatística e Computação Científica.
I would like to express my gratitude to my advisor Prof. Henrique Sá Earp for his advice and continued support. I am thankful to Prof. Lino Grama, Professor Marcos Jardim, Prof. Lazaro Rodriguez and Prof. Maurício Corrêa, for their contributions to this work.
I would like to thank my colleagues for the useful discussions and all the moments that we shared. Also, I am grateful to the administrative staff from IMECC for their kind-hearted help. Finally, this thesis would not be possible with the understanding and the constant encour-agement from my family, wife and daughter.
This study was financed in part by the Coordenação de Aperfeiçoamento de Pes-soal de Nível Superior - Brasil (CAPES) - Finance Code 001 and by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) processo no141215/2019-4
y del muladar exalta al necesitado; los hace sentarse con príncipes, con los príncipes de su pueblo.” (Sagrada Bíblia, Salmos113 : 7 ´ 8)
Neste trabalho, nós estudamos uma equação natural de instantons de contato sobre campos de calibre em variedades Sasakianas de dimensão 7, esta equação está relacionada à equação dos G2-instantons e à condição Hermite Yang-Mills transversal (tHYM). Nós provamos por teoria
de Fredholm, que o espaço de moduli de soluções irreducíveis da equação dos instantons de contato tem um modelo local de dimensão finita. Por outro lado, usando os métodos propostos por Baraglia e Hekmati no caso de 5 dimensões [BH16] nós mostramos condições nas quais este modelo local é uma variedade suave. Como um exemplo de interesse, nós focamos ao caso particular em que o fibrado tem uma estrutura transversalmente holomorfa sobre a variedade Sasakiana de dimensão 7, dada por um Link de Calabi-Yau, como estudado por Calvo-Andrade, Rodríguez e Sá Earp [CARSE16]. Nós conseguimos demonstrar que, neste contexto as noções de instantons de contato, G2-instantons compatíveis e conexões tHYM coincidem. Finalmente nos
provamos que o espaço de moduli, herda uma estrutura de variedade Kähler no caso Sasakiano.
Palavras-chave: Variedades Sasakiana, variedade complexa, campo de Calibre (física), teoría de modulos de instantons
We study a natural contact instanton equation on gauge fields over 7-dimensional Sasakian man-ifolds, which is closely related to both the G2-instanton equation and the transverse Hermitian
Yang-Mills (tHYM) condition. We prove by standard Fredholm theory that the moduli space of irreducible solutions has a finite-dimensional local description, following the approach by Baraglia and Hekmati in 5 dimensions [BH16]. As an instance of concrete interest, we specialise to transversely holomorphic Sasakian bundles over 7-dimensional Calabi-Yau links, as studied by Calvo-Andrade, Rodríguez and Sá Earp [CARSE16], and we show that in this context the notions of contact instanton, compatible G2-instanton and tHYM connection coincide.
Further-more we show that the moduli space of contact instantons inherits a Kähler structure if M is a Sasakian manifold.
Keywords: Sasakian manifolds, Complex manifolds, Gauge fields (Physics), Instantons Moduli theory.
i Imaginary unity of the complex numbers field 1 Identity map in the appropriate space
Sn The round sphere of radius 1 inside Rn`1
8 Infinity
EndpEq Endomorphism of the vector bundle E ΓpEq Sections of vector bundle E
TM Tangent vector bundle of the manifold M
VC Complexification of the vector space V
ΛpV p-forms on vector space V
Λp,qV pp, qq-forms on complex vector space V
ΩkpM q Differential k-forms on M ΩkpEq Differential E-valued k-forms Ωp,qpM q Differential pp, qq-forms on TMC
Ωp,qpEq Differential E-valued pp, qq-forms on TMC
} ¨ }p Sobolev p-norm on the space of sections ΓpEq of the bundle E.
ApEq Space of all connections of the bundle E g, h, . . . Lie algebra of the Lie group G, H, . . .
pM, gq Riemannian manifold with Riemannian metric g
pM, J q Almost complex manifold with with almost complex structure J pM, ηq Contact manifold with contact structure η
Introduction . . . . 13
1 PRELIMINARIES: GENERALITIES ON SASAKIAN GEOMETRY . . 19
1.1 Foliations. . . 19 1.1.1 Transverse geometry . . . 19 1.1.2 Reimannian Foliations . . . 21 1.1.3 Flows . . . 22 1.2 Contact structures . . . 23 1.3 Sasakian Manifolds . . . 25
1.3.1 Transverse Kähler geometry on Sasakian manifolds . . . 26
1.3.2 Curvature of Sasakian manifolds. . . 28
1.4 Holomorphic vector bundles on Sasakian manifolds . . . 29
2 PRELIMINARIES: GAUGE THEORY AND G2-GEOMETRY ON CALABI-YAU LINKS . . . . 32
2.1 Links as Sasakian and Contact Calabi-Yau 7-manifolds . . . 32
2.2 Gauge theory on Contact Calabi-Yau manifolds . . . 33
2.2.1 Yang-Mills connections and G2-instantons . . . 34
2.2.2 Topological energy bounds . . . 34
2.2.3 G2-Instantons and the Hermitian Yang-Mills condition . . . 34
3 PRELIMINAIRES: MODULI SPACE OF CONTACT INSTANTON IN 5-DIMENSIONS . . . . 36
3.1 The contact instanton equation . . . 36
3.1.1 Deformation theory . . . 37
3.1.2 Smoothness and geometry of the the moduli space . . . 38
3.1.3 Vanishing of the obstruction . . . 39
3.1.4 Geometry of the moduli space of contact instantons in 5-dimensions . 39 4 CONTACT INSTANTONS IN SASAKIAN 7–MANIFOLDS . . . . 41
4.1 Local structure of the space of differential forms on Sasakian manifolds. . . 41
4.1.1 Eigenspaces of Lσ from the contact structure . . . 43
4.1.2 Compatible splitting from the transverse complex structure. . . 45
4.2 Gauge theory on 7-dimensional Sasakian manifolds . . . 47
4.3 Contact instantons, G2-instantons and Yang-Mills theory . . . 54
4.3.1 G2-instantons on contact Calabi-Yau (cCY) manifolds . . . 54
4.3.2 Yang-Mills theory and Chern-Simons action in Sasakian setting . . . . 56
5 THE MODULI SPACE OF CONTACT INSTANTONS IN SASAKIAN 7-MANIFOLDS . . . . 60
5.1 The associated elliptic complex of a contact instanton . . . 60
5.2 Deformation theory of selfdual contact instantons . . . 66
5.3 Obstruction and smoothness . . . 74
5.3.1 The obstruction map . . . 74
5.3.2 Cohomological vanishing of obstruction . . . 79
5.4 Geometry on the moduli space . . . 81
5.4.1 Decomposition of the tangent space of the moduli space. . . 81
5.4.2 Hermitian metric on the moduli Space of SD contact instantons . . . . 82
5.4.3 Almost complex structure on M˚ . . . 84
5.5 The transverse Calabi-Yau case . . . 88
Introduction
We describe the local model of the moduli space of solutions to a natural gauge-theoretic equation, on a suitable class of vector bundles E Ñ M over a Sasakian manifold. In [DK90], Donaldson studied solutions of the instanton equation ˘FA“ ˚FAto understand the
geometry of 4-manifolds, where FAis the curvature of a connection A and ˚ is the Hodge star.
On manifolds of dimension d ě 4, this equation can be generalised relatively to an appropriate pd ´ 4q-form σ [DT98,Tia00a], by
λFA“ ˚pσ ^ FAq, (1)
for eigenvalues λ of ˚pσ ^ ¨q : Ω2pM q Ñ Ω2pM q. We shall be particularly interested in Sasakian 7-manifolds, on which the contact structure η induces a natural 3-form σ “ η ^ dη and λ P t˘1, ´2u (see Section4.1.1). We adopt the approach of Baraglia and Hekmati [BH16], who described the moduli space of contact instantons on contact metric 5-manifolds, providing sufficient conditions for smoothness away from reducibles, and determining its dimension as the index of an elliptic operator transverse to the Reeb foliation. We will see that these results admit precise analogues in the appropriate 7-dimensional context, while some new distinct gauge-theoretic phenomena also occur.
Three natural notions of instanton
Let pM2n`1, η, ξq denote a contact manifold, with contact form η and Reeb vector
field ξ [BG08,Bla10]. Then the natural p2n ´ 3q-form σ “ η ^ pdηqn´2provides an instance of (1):
˘FA “ ˚pη ^ pdηqn´2^ FAq. (2)
Solutions of (2) are said to be selfdual contact instantons if λ “ 1 (respectively anti-selfdual for λ “ ´1). When the contact manifold is endowed in addition with a Sasakian structure, namely an integrable transverse complex structure Φ and a compatible metric g, [BS10] proposes a natural notion of Sasakian holomorphic structure for complex vector bundles E Ñ M (see Section1.4). We can extend the notion Hermitian Yang-Mills (HYM) to Sasakian bundles [see Definition47], namely, a connection A on a complex vector bundle over a Kähler manifold is said to be Hermitian Yang-Mills (HYM) if
ˆ
FA :“ pFA, ωq “ 0 and F0,2A “ 0. (3)
by taking ω :“ dη P Ω1,1pM q as a “transverse Kähler form” [see Definition47], and defining HYM connections to be the solutions of (3) in that sense. Given a Hermitian bundle metric,
the well-known concept of Chern connection also extends, namely as a connection mutually compatible with the holomorphic structure and the Hermitian metric [BS10, Section 3].
An important class of Sasakian manifolds are those endowed with a so-called contact Calabi-Yau (cCY) structure (see Definition41), the Riemannian metrics of which have transverse holonomy SUp2n ` 1q, in the sense of foliations, corresponding to the existence of a global transverse holomorphic volume form P Ωn,0pM q [HV15]. Furthermore, when n “ 3, such cCY 7-manifolds are naturally endowed with a G2-structure defined by the 3-form
ϕ :“ η ^ dη ` Impεq, (4) which is cocalibrated, in the sense that its Hodge dual ψ :“ ˚gϕ is closed under the de Rham
differential. When a 3-form ϕ on a 7-manifold defines a G2-structure, the instanton condition
(1) for σ “ ϕ and λ “ 1 is referred to as the G2-instanton equation. On holomorphic Sasakian
bundles over closed cCY 7-manifolds, it has the distinctive feature that integrable solutions are indeed Yang-Mills critical points, even though the G2-structure has torsion [CARSE16].
Throughout this work, we will be concerned with Sasakian 7-manifolds, on which the three above concepts of instanton interrelate. The contact instanton equation (2) in this case is determined by the natural 3-form
σ :“ η ^ dη. (5)
The operator ˚pσ ^ ¨q splits the space of 2-forms into t˘1, ´2u-eigenspaces (see Section4.1.1). However, the p´2q-eigenspace is unidimensional spanned by dη and rather uninteresting, so we will focus on the p˘1q-eigenspaces, which in some sense still signify the instanton equation (1) as an (anti-) selfduality condition. On a contact Calabi-Yau 7-manifold, the three notions of instanton are related in the following ways:
Theorem 1. Let E be a holomorphic Sasakian bundle over a 7-dimensional cCY manifold pM, η, ξ, g, Φq endowed with its natural G2-structure(4); then the following hold:
(i) Every solution of the contact instanton equation ˘FA“ ˚pσ ^ FAq is also a solution of
˚pϕ ^ FAq “ ˘FA, i.e., every contact instanton is aG2-instanton [Proposition67].
(ii) A Chern connection is aG2-instanton if, and only if, it is a contact instanton [Proposition
68].
(iii) A Chern connection is tHYM if, and only if, it is aG2-instanton [CARSE16, Lemma 21].
Local model of the moduli space
Our main result is a complete description of the local deformation theory of 7-dimensional Sasakian contact instantons. Let E Ñ M be a complex vector bundle with compact, connected, semi-simple structure group G, and denote by gE its adjoint bundle and by ΩkpgEq
the gE-valued k-forms on M . The operator [cf. Definition56]
Lσ :“ ˚pσ ^ ¨q : Ω2pgEq Ñ Ω2pgEq (6)
induces the irreducible splitting (4.7)
Ω2pgEq “ pΩ26‘ Ω 2 8‘ Ω 2 1 looooooomooooooon Ω2 H ‘ Ω2VqpgEq
where Ω26pgEq, Ω28pgEq and Ω21pgEq are the eigenspaces associated to ´1, 1 and ´2, respectively.
Definition 1. A connection A P ApEq is called anti-selfdual (ASD) contact instanton if its curvature FAis in Ω26pgEq and it is called selfdual (SD) contact instantons, if FA P Ω28pgEq.
In Proposition62, we show that Chern connections (compatible with both a Her-mitian bundle metric and a Sasakian holomorphic structure) being ASD contact instantons are necessarily flat. Hence the meaningful notion in our case is that of SD contact instantons, i.e., in the kernel of the projection map
p : Ω2HpgEq Ñ Ω26‘1pgEq :“ pΩ26‘ Ω 2
1qpgEq. (7)
The Hilbert Lie group G of smooth gauge transformations acts smoothly on the space A of connections on E, and the topological quotient B :“ A{G is a Hausdorff space. We denote by B˚
Ă B the open subspace of irreducible connections, and by M Ă B the set of gauge equivalence classes of solutions to the selfdual contact instanton equation:
M :“ trAs P B | ppFAq “ 0u (8)
and, accordingly, M˚
Ă M for its irreducible part. Linearising the selfdual contact instanton condition, in terms of the projection p in (7), we introduce:
d7 :“ p ˝ dA: Ω1HpgEq Ñ Ω26‘1pgEq. (9)
In Section4.2, we will see that a local model for the moduli space of selfdual contact instan-tons M˚ is given by the cohomology group H1
pCq :“ kerpd7q ImpdAq
of the deformation complex [Proposition63]:
C‚: 0 Ω0
HpgEq Ω1HpgEq Ω26‘1pgEq 0
dA d7
Ωp`1,qH pgCEq Ωp,qH pgCEq Ω p,q H pgCEq Ωp,q`1H pgCEq XXz B∇˚ : B∇ XXz ¯ B∇ : ¯ B∇˚ .
This complex, however, is not elliptic, and in order to compute the dimension of H1pCq we resort to an auxiliary construction [cf. Section5]. We introduce the quotient spaces of
k-forms modulo the Lie algebra ideal I generated by Ω28pgEq (In the same fashion than [Bar09]):
Lk:“ Ω kpg Eq I , k “ 0, 1, 2, 3, with I :“@Ω 2 8pgEq D Ă pΩ‚pgEq, ^q .
For a natural choice of differentials Dk, the Lkspaces fit in a complex [Proposition73]:
L‚: 0 L0 D0 L1 D1 L2 D2 L3 0
. (11)
We denote by H‚ :“ H‚
pLq the cohomology of (11), and indeed, H1 is isomorphic to the infinitesimal deformations of rAs P M as a contact instanton. Relatively to the Reeb orbits, a
S1-invariant differential form α P Ω‚
pM q is called basic (see Definition9). The graded ring Ω‚
BpM q of basic forms inherits a natural basic de Rham differential
dB :“ d|Ωk BpM qΩ k BpM q Ñ Ω k`1 B pM q,
and the cohomology of dB is referred to as the basic de Rham cohomology. Restricting the
differentials Dk in (11) to basic forms in Lk “ ΩkpM q{xΩ28y, we obtain a basic complex
[Proposition73]: L‚ B: 0 Ω 0 BpgEq Ω1BpgEq pΩ26‘1qBpgEq 0 DB DB . (12) We denote by H‚B :“ H‚
pLBq the corresponding basic cohomology. If A is a contact instanton,
the transverse index of A is defined as the index of the basic complex (12): indexTpAq “ dimpH0Bq ´ dimpH
1
Bq ` dimpH 2 Bq.
In particular, when A is irreducible, indexTpAq “ dimpH2Bq ´ dimpH1Bq. Our main theorem
provides a local model for the moduli space M˚in terms of basic cohomology:
Theorem 2. Let E Ñ M be a G-bundle over a closed, connected Sasakian 7-manifold pM, η, ξ, g, Φq, with adjoint bundle gE, and denote byM˚ the moduli space(8) of irreducible
selfdual contact instantons, i.e., solutions of (1) for λ “ 1:
FA“ ˚pη ^ dη ^ FAq.
(i) The tangent spaceTrAsM˚ofM˚
at rAs, i.e. the space of infinitesimal deformations of rAs as a contact instanton, is isomorphic to the finite-dimensional cohomology group H1pCq :“ kerpd7q
ImpdAq
of the complex(10), and
dimpTrAsM˚q ´ dimpH2pCqq “ 0.
(ii) The dimension ofM˚
near rAs can be computed from the cohomology of the basic complex (12), which is elliptic transversely to the Reeb foliation, namely there is an isomorphism H1 – H1B, whereH1Bis the cohomology of (12):
dimpTrAsM˚q “ dimpH2Bq ´ indexTpAq.
(iii) The local model of M˚ is cut out as the zero set of an obstruction map (Definition
5.33), which vanishes precisely whenH2B “ 0 [Proposition88]. Thus, for an irreducible contact instanton A such that H2B “ 0, M˚ is smooth near A with finite dimension
dim M˚
“ ´ indexTpAq [Corollary86].
Remark2. Parts piq and piiq in Theorem2establish somewhat independently that the tangent space near an irreducible contact instanton is finite-dimensional, since it occurs as the first cohomology group in both complexes (11) and (12). However, in terms of the obstruction theory, we learn something finer from piiq and piiiq. In the context of piq, the moduli space near an acyclic point, i.e. h0pCq “ h2pCq “ 0 in (10), would be necessarily 0-dimensional, whereas the complex (12) in terms of basic cohomology is merely transverse-elliptic, hence the moduli space near an acyclic smooth point, with h0B “ h2B “ 0, can in principle have nonzero dimension
´ indexTpAq.
Remark3. For most steps in our argument, it suffices to assume M compact and connected, with possibly nontrivial boundary. However, in Theorem1, piiiq, taken from [CARSE16, Lemma 21], and in Propositions12and77, one actually needs M to be closed.
Remark4. For a Sasakian 7-manifold with positive transverse scalar curvature, the second basic cohomology group HB2 should vanish, and therefore M˚ is smooth. This is announced here as a
Conjecture. In particular, indexTpAq “ ´dimpH1Bq, see e.g. part (ii) of Theorem2.
Regarding the various instanton notions related by Theorem1, our main Theorem2
has the following significance:
Corollary 5. Let E Ñ M be a holomorphic Sasakian bundle (Definition 36) over a 7-dimensional cCY manifold pM7, η, ξ, g, Φq, endowed with its natural G2-structure(4). Among
Chern connections inApEq, the three notions coincide: contact, HYM and G2-instantons. The
Outline
Preliminaries: In Chapters 1, 2 and 3 we present the theoretical context where we was inspired to performance this work. We begin with a exposition on Sasakian geometry in Chapter
1and this part can be omitted by expert readers, we present definitions and results appearing along all text, the main references are [BG08,Bla10,Bla06]. We finish this Section discussing another important concept in this work, the so named Sasakian holomorphic vector bundles, the main reference is [BS10].
Chapter 3 is devoted to point out the result and techniques used in [BH16] for contact instantons in the 5-dimensional case, principally that ones that we get to generalise to the 7-dimensional case. Finally in Chapter2, We present the results and definitions relevant in this work on G2-geometry and G2-instantons on Calabi-Yau links following [CARSE16].
Results: In Chapters4and5we present the results in 7-dimensions that we get generalise from the 5-dimensional case [cf. Chapter3] and that ones that we can establish independently for this case. Also we study the relationship between, contact instantons in 7-dimensions and
G2-instantons [cf. Chapter2] in the following way:
In Section 4.1.1we describe a local splitting (4.7) of Ω2pM q under the contact structure and the operator Lσ :“ ˚pσ ^ ¨q (6). Another natural decomposition of Ω2pgEq comes
from the transverse complex structure induced by Φ P EndpT M q [cf. Section 5.4.1], and both are related by (4.13). Furthermore, by imposing unitary and integrable condition on the connections, we show in Proposition62that the curvature component in Ω26pgEq vanishes, so
we focus our attention on the selfdual contact instanton case.
Parts piq and piiq of Theorem1are proven respectively in Propositions67and68. The proof of Theorem2is organised as follows: part piq is the content of Proposition64, which uses an auxiliary elliptic complex [Proposition65] to establish that this local model has finite dimension; part piiq is an immediate consequence [Corollary79] of Proposition78; and part piiiq requires a thorough study of the moduli space of the obstruction theory of selfdual contact instantons, under the 5-dimensional paradigm from [BH16], we study this in Section 5.3.2. Finally in Section5.4we study the geometry of the moduli space.
1 Preliminaries:
Generalities
on
Sasakian geometry
Instead to prove results, in this section we pretend introduce notions and definitions necessary in this text but that are scattered in the literature. Readers who know the general theory on Sasakian Geometry can completely avoid this Section.
1.1
Foliations
A classical reference for this section is [Ton12], although all definitions are presented, in general, for a p-dimensional foliation F on a n-dimensional, compact, oriented smooth manifold M and co-dimension q :“ n ´ p, for the purpose of this work, it will be enough take a 1-dimensional foliation, because later in Section1.3, we will consider the Reeb foliation generated by a Killing vector ξ in a Sasakian manifold pM, η, ξ, g, Φq. More precisely, we can take p “ 1 and q “ 6 for a Sasakian 7-manifold in which many concepts take a particularly simple form.
1.1.1
Transverse geometry
A foliation atlas is denoted by U “ tUα, ϕαu yields local foliation coordinates
px, yq “ px1, ¨ ¨ ¨ , xq, y1, ¨ ¨ ¨ , ypq such that the first q components of the transformation γαβ
satisfy
Bγj Bxi
“ 0, i “ 1, ¨ ¨ ¨ , p j “ 1 ¨ ¨ ¨ , q
We denote by L “ T F and H “ T M {L to be the normal bundle of L.
Definition 6. A foliated principal bundle is a G principal bundle P Ñ M together with a lifted foliation FP of F , precisely, on all p P P in the fiber of x P M , FP is isomorphic to TxF ,
TpFP X TpPz “ 0 and T FP is invariant by the action of the structure group G´action on P .
In particular, T FP can be viewed as the horizontal distribution of a partial connection on P , since T FP is involutive, this partial connection is flat.
Definition 7. A foliated vector bundle E Ñ M is a vector bundle such that is frame bundle
PE is foliated principal bundle, alternatively, a foliated vector bundle E is given by a partial
connection r∇, namely r∇Xs is defined for all sections s P ΓpEq but for tangent vector X P ΓpLq
The curvature rRpX, Y q :“ r∇X∇rY ´ r∇Y∇rX ´ r∇rX,Y s is defined for sections in ΓpLq.
Definition 8. A section s P ΓpEq is basic if r∇sX “ 0, for every X P ΓpLq, follows of the
Leibnitz rule that this concept can be localized. Let ΓBpEq denote the set of all basic sections.
Definition 9. Basic differential forms ΩrBpM q :“ ΓBpΛrH˚q are characterised by iXα “
0, iXdα “ 0 @ X P ΓpLq where iX is the contraction operator.
The first condition above means that α is reducible to ΓpΛrH˚
q and the second one means that α is basic on ΛrH˚
, locally in a chart px, yq “ px1, ¨ ¨ ¨ , xq, y1, ¨ ¨ ¨ , ypq we have the
expression
α “ ÿ
|J |“r
fJdxJ
where fJpxq is independent of y, in particular the basic functions are constant on the leaf of the
foliation F .
Definition 10. The differential operator preserves basic forms, giving rise to the basic complex pΩkBpF q, dq and its associated De Rhan cohomology HrBpF q, r “ 1, ¨ ¨ ¨ , q is called basic cohomology of F .
It is known that H0BpF q “ R and the natural inclusion H1BpF q Ñ H1pM q is injective.
Basic sections ΓpEq of a foliated vector bundle pE, r∇q is a module over basic functions, under a foliation atlas, the transition functions are basic and vice versa. The bundle ΛrH˚
b E it is foliated by the connection given by tensor connection. Set
ΩrBpEq “ ΓBpΛrH˚b Eq Ă ΩrpEq
to be the space of basic forms with values in E, this space is a module over the space C8 BpM q
of basic functions. In this sense all s P ΩrBpEq can be written as s “ ÿαi b si, for some
α P ΩrBpM q and si P ΓBpEq.
Definition 11. Let P be a principal foliated bundle with lifted foliation FP, a basic connection
α on P is a usual connection whose connection form on P is basic with respect to FP. i.e., the horizontal distribution of α contains T FP.
In this context a basic connection ∇ on E is associated with a basic connection in the frame bundle of E, directly on E, a connection ∇ is basic if, and only if, it is adapted to F , ∇X “ r∇X and its curvature satisfies iXR “ 0 for all X P ΓpLq. The following proposition is
shown in [Wan12]
(i) Locally under a basic frame ofE, ∇ can be written ∇ “ d ` A with A P Ω1BpEndpEqq so, the curvatureR is a global basic form R P ΩeBpEndpEqq.
(ii) ∇ preserves basic sections, i.e., ∇ : ΓBpEq Ñ ΓBpEq b Ω1BpM q, more generally the
extended differential operatord∇: ΩrBpEq Ñ Ω r`1
B pEq preserves basic forms.
In particular, the set ABpEq of basic connections forms is an affine space modelled
on Ω2BpEndpEqq.
1.1.2
Reimannian Foliations
Any metric g on M , splits T M “ L ‘ H, where H “ LK, in turn H inherits a metric
gH “ gLK, fix a metric g and suppose that F is oriented
Definition 13. Let pe1, ¨ ¨ ¨ , eq, ¨ ¨ ¨ , enq Ă H ‘ L be an oriented orthonormal basis
(i) At any point, the characteristic form X “ XF P ΩppM q of F is defined by
X pX1, ¨ ¨ ¨ , Zpq “ detpgpei, Zjq1ďi,jďpq, @Zi P ΓpT M q
(ii) The transverse volume form ν P ΓpΛqH˚
q Ă ΩqpM q is the volume form of gHviewed
as a q-form on M through the inclusion.
(iii) Let ∇M the Levi-Civita connection on M , the mean curvature field τ P ΓpHq is defined by τ n ÿ i“1 p∇eieiq K .
(iv) The mean curvature form κ P ΓpH˚q Ă Ω1pM q is the dual 1-form of τ . (v) An F -trivial form α is one of degree ě p and such that
iX1 ˝ ¨ ¨ ¨ ˝ iXpα “ 0, @ X1, ¨ ¨ ¨ , Xp P ΓpLq.
A important formula is a so called Rummler formula dX “ ´κ ^ X ` α, for some F -trivial pp ` 1q-form α.
Definition 14.
(i) If there exists a metric g on M such that the mean curvature form κ “ 0, F is called Tau foliation, i.e., leaves of F are minimal submanifolds.
(i) A metric g on M is called bundle-like if the restriction gH satisfies LXgH “ 0 for all
X P ΓpLq. This condition is called holonomy invariance
Suppose F Reimannian and fix a bundle-like metric g “ gH‘ gLon T M “ H ‘ L,
since gHis holonomy invariant, we can define
Definition 15. The transverse Hodge start ˚T: ΛrH˚ Ñ Λq´rH˚is defined like the ˚ operator
associated to gH. It is related with the usual Hodge star operator as follow
˚α “ ˚Tα ^ X , ˚Tα “ p´1qppq´rq˚ pα ^ X q @ α P ΩrBpM q. (1.1)
In particular X “ ˚Tν, and ν “ p´1qpq ˚ X . Note that ν is basic and closed form, so generated
a top degree basic cohomology class rνs P HqBpM q.
˚T preserves basic sections, hence it yield ˚T: ΩrBpM q Ñ Ω q´r
pM q for r “ 1, ¨ ¨ ¨ , q. Not that ˚T no preserves basic forms if gH is not holonomy invariant, also ˚T is
not defined in the complement of ΩrBpM q Ă ΩrpM q.
1.1.3
Flows
Now, we concentrate in a particular case in which p “ 1, such one foliation are called flows, i.e., F “ Fξis generated by a non singular vector field ξ. We can assume that ξ is
an unit vector field, the closure of expptξq in the compact isometry group of gM is denoted by T
and it is a abelian, compact Lie group, so a torus. For the basic forms of the foliation, we have ΩBpF q Ă ΩpM qT, where ΩpM qT are the T-invariant form on M . Note ξ provides a degree ´1,
surjective map iξ: Ω‚pM qTÑ ΩÝ ‚´1pF q, so there exist an exact sequence of complexes
0 Ñ Ω‚
BpM q Ñ Ω‚pM q
T iξ
ÝÑ Ω‚´1pF q Ñ 0. For any compact Lie group acting on M , it is known that HpΩ‚
pM qT – HpΩ‚pM qq.
Theorem 3. [Ton12, Theorem 10.13] Letξ be a non-singular Killing vector field on the closed
manifold pM, gq and Fξ generated byξ.
(i) There is a long exact cohomology
¨ ¨ ¨ Ñ HrBpF q Ñ HrDRpM qÝpiÝÝξqÑ H˚ r´1B pF qÑ HÝδ r`1B pF q Ñ ¨ ¨ ¨ with connecting homomorphismδ.
(ii) The groupsHrBpF q are all finite-dimensional for 0 ď r ď n ´ 1 and zero otherwise.
(iii) The Euler characteristicXBpF q “ n´1
ÿ
r“0
p´1qrdimpHrBpF qq is a well-defined integer.
In particular 0 Ñ HnDRpM q piξq˚ ÝÝÝÑ
– H
n´1
B pF q Ñ 0; and the co-boundary map δ is given by
1.2
Contact structures
For a detailed treatment of results in this section the reader is referred to [BG08,
Bla10,Bla06]. For R2n`1we can define
α “ dz ´
n
ÿ
i“1
yi^ dxi (1.2)
and one can show that α ^ pdαqn ‰ 0, this motivate the following
Definition 16. Let M a p2n ` 1q-dimensional manifold, a nowhere vanishing α P Ω1pM q satisfying η ^ pdηqn ‰ 0 is called a contact form , the pair pM, ηq is called contact manifold. Theorem 4. [Bla10, Theorem. 3.1] Let α P Ω1pR2n`1q be, such that α ^ pdαqn ‰ 0, them there exist an open setU Ă R2n`1 and local coordinates px1, ¨ ¨ ¨ , xn, y1¨ ¨ ¨ , yn, zq such that α
has the form(1.2) in U .
In view of Theorem4, which is an analogous to Darboux’ Theorem, the 1–form α in (1.2) will be called the standard contact form.
Example 17. Let M “ S2n`1 Ă R2n`2 be the p2n ` 1q-dimensional sphere, on R2n`2 consider the 1-form given in coordinates px0, ¨ ¨ ¨ , xn, y0, ¨ ¨ ¨ , ynq by β “
n
ÿ
i“0
pxidyi´ yidxiq, let α denote
the restriction of β to the sphere S2n`1, then η ^ pdηqn‰ 0 on S2n`1, hence η define a contact form on S2n`1[BG08, Example 6.1.16]. Furthermore, notice η is invariant under the reflection pxi, yiq ÞÑ p´xi, ´yiq, for all i “ 0, ¨ ¨ ¨ , n. Thus we get an induced contact structure on the real
projective space RP2n`1as well.
Example17in a special case of the next more general result.
Theorem 5. [Bla06] Leti : M2n`1 Ñ R2n`2 be a smooth hypersurface immersed on R2n`2 and suppose that no tangent space ofM2n`1contains the origin of R2n`2, thenM2n`2 has a contact structure.
Lemma 1. [BG08, Lemma 6.1.24] Let pM, ηq be a contact manifold , there exist a unique vector fieldξ, called Reeb vector field satisfying: iξη “ 1 and iξdη “ 0.
We let Nξdenote the trivial line bundle consisting of tangent vectors that are tangent
to the leaves of Fξgenerated by ξ. This splits the tangent bundle as
TM “ H ‘ Nξ (1.3)
The foliation Fξ of a contact structure is said quasi-regular if there is k P N, such that for each
x P M exists a foliated coordinate chart pU, xq such that each leaf of Fξ passes through U at
most k times, if k “ 1 the foliation is called regular. We will also say that the Reeb vector field
Proposition 18. [BG08, Corollary 6.2.4] LetM2n`1be a contact manifold. Then the structural group ofT M can be reduce to U pnq ˆ Z2.
It is said that a manifold M2n`1 is an almost contact structure if the structural group of its tangent bundle is reducible to U pnq ˆ 1. The following is an equivalent definition. Definition 19. [Bla06, § 4.1] An almost contact structure on M is a triple pξ, η, Φq, where Φ P EndpT M q is a tensor of type p1, 1q, ξ P XpM q is nowhere zero vector field, and η is a 1-form which satisfy: ηpξq “ 1 and Φ ˝ Φ “ ´1 ` ξ b η.
Proposition 20. [Bla10, Theorem 4.1] Suppose M2n`1 has an almost contact structure pξ, η, Φq, then Φξ “ 0 and η ˝ Φ “ 0, moreover, the endormorphism Φ has rank 2n.
Let pξ, η, Φq be an almost contact structure on M2n`1, if there exist a Riemannian metric g on M such that
gpΦX, ΦY q “ gpX, Y q ´ ηpXqηpY q, @ X, Y P XpM q (1.4) pM, η, ξ, g, Φq is called an almost contact metric structure, or we say that g is a Riemannian metric compatible with the almost contact structure. If we set Y “ ξ in (1.4) since ηpξq “ 1 and Φξ “ 0, we obtain
gpΦX, Φξq “ gpΦX, 0q “ 0 “ gpX, ξq ´ ηpXqηpξq
so, we have that η is the covariant form of ξ, i.e., ηpXq “ gpX, ξq. The requirement for g in (1.4) is justified by the following result, whose proof can be find in [Bla06, p. 21]
Proposition 21. All almost contact manifold M2n`1 admits a Riemannian metric that makes it into an almost contact metric manifold, i.e., pξ, η, Φ, gq is an almost contact metric structure on
M2n`1.
Given a almost contact metric structure on M2n`1, define ω on M2n`1 the so called the fundamental 2-form of the almost contact metric structure pξ, η, Φ, gq by
ωpX, Y q “ gpX, ΦY q (1.5) notice that ω is skew-symmetry. Since RankpΦq “ 2n, we have that η ^ ω2 ‰ 0.
Proposition 22. [Bla10, Theorem 4.4] LetM2n`1 be a contact manifold with1-form η and characteristic vectorξ, then there exist an almost contact metric structure pξ, η, Φ, gq such that
the fundamental2-form ω is dη.
An almost contact metric structure with dη “ ω is called an associated almost contact metric structure for the contact structure η. There exist examples (pag. 27 [Bla06]) of almost contact metric structures pξ, η, Φ, gq with η ^ pdηqn ‰ 0 but ω ‰ dη.We now consider those almost contact manifolds which are, in some sense, analogous to the complex manifolds.
Definition 23. Let M2n`1be a contact metric manifold with associated almost contact metric structure pξ, η, Φ, gq. If ξ is a Killing vector field with respect g, then the contact metric structure is called K-contact structure.
Notice that for a K-contact structure, using gpξ, Xq “ ηpXq and the Koszul formula, we have
gpX, ΦY q :“ ωpX, Y q “ 1
2pXpηpY qq ´ Y pηpXqq ´ ηprX, Y sqq “ gpX, ´∇Yξq for all X, Y P XpM q, so
ΦX “ ´∇Xξ. (1.6)
Conversely, as Φ is a skew-symmetric, for a contact metric structure satisfying the equation (1.6) we obtain that ξ is a Killing vector. In fact, [Bla06, p.51] the contact metric structure pξ, η, Φ, gq is a K-contact structure if, and only if, for all X P XpM2n`1q the Lie derivative pLξΦqX “ 0.
1.3
Sasakian Manifolds
All next sections are concerned with Sasakian manifolds, we can take the following Theorem as a definition
Theorem 6. [BG07, Theorem 10] pM, gq is called Sasakian manifold, if any of the following equivalent conditions is satisfied:
(i) There is a unit Killing vector fieldξ P XpM q, such that the section Φ P ΓpT M b T˚M q,
defined byΦpXq “ ´∇Xξ satisfies the identity:
p∇XΦqpY q “ gpX, Y qξ ´ gpξ, Y qX (1.7)
(ii) There is a unit killing vector fieldξ P XpM q, such that the Riemann curvature tensor R of
pM, gq satisfies:
RpX, ξqY “ gpξ, Y qX ´ gpX, Y qξ :“ ´p∇XΦqpY q (1.8)
(iii) The metric cone pR`ˆ M, dr2` r2¨ gq is Kähler.
A very important example of Sasakian manifolds are the following
Definition 24. Let pM, gq be a Riemannian manifold of real dimension m. We say that pM, gq is 3–Sasakian if the holonomy group of the metric cone on M , pCpM q, ¯gq “ pR`ˆ M, dr2` r2gq
reduces to a subgroup of Spˆ m ` 1 4
˙
. In particular, m “ 4n ` 3, n ě 1 and pCpM q, ¯gq is
1.3.1
Transverse Kähler geometry on Sasakian manifolds
We introduce the transverse Kähler structure both globally and locally, these two description are equivalent and readers are referred to [BG08, Section 2.5] for a detailed treatment on this subject. The action of the Reeb vector ξ on CpM q has local orbits that defines a transverse holomorphic structure on the Reeb foliation Fξin the following sense. Let tUα:“ Vαˆ Iu be a
foliated chart covering M , where Vα Ă Cnand I Ă R are open sets. Let πα: Uα Ñ Vα Ă Cn
be a submersion such that if UαX Uβ :“ Uαβ ‰ H
πα˝ πβ´1: πβpUαβq Ñ παpUαβq
is biholomorphism. παprovides a canonical isomorphism dπα: Hp Ñ Tπαppq(1.3) for every p P
Uα, since ξ generates isometries, the restriction g|Hgives a well defined Hermitian metric gαT on
Vα, this structure is in fact Kähler. Let pz1, ¨ ¨ ¨ , zn, xq be coordinates for Uα, the complexification
pH b Cq0,1is generates by elements of the form Bzi´ ηpBziξq, where Bzi :“ B
Bzi. Note that, from
iξdη “ 0 and iξη “ 1 follows
dηpBzi´ ηpBziqξ, Bzi´ ηpBziqξq “ dηpBzi, Bziq,
i.e., the fundamental 2-form ωαis the restriction of dη|tx“constantuXUα and ωαis closed, hence
πα ˝ πβ´1: πβpUαβq Ñ παpUαβq gives and isometry of Kähler manifolds. The fundamental
2-form ωαT for the Hermitian metric gαT on each Vα Ă Cn is the restriction 1{2dη|tx“constantu.
Hence ωαT is closed and gTα is Kähler on Vα. The metric gT “ tgαTu is defined as a collection of
metrics in each coordinate chart with can be identified with the global tensor on M via
gTpX, Y q “ 1
2dηpX, ΦY q, X, Y P ΓpHq. (1.9) i.e., the restriction gT|tx“constantuin each foliated chart is gαT, analogously the transverse Kähler
form ωT is globally identified with 1{2dη. The metric gT is related to the Sasakian metric g by:
g “ gT ` η b η. (1.10) From the transverse metric gT (1.9), the connection ∇T on H (1.3) is defined by
∇T “ $ & % ∇T XY “ p∇XY qp, X, Y P ΓpHq ∇TξY “ rξ, Y sp, Y P ΓpHq. (1.11)
where p : T M Ñ H is the natural projection, ∇T is the unique torsion free connection satisfying ∇TgT “ 0, also it is shown that that ∇TJ “ 0. Here we use the identification dπα: Hp Ñ TπαppqVα. Let T MC, NξC and HCdenote the complexification of T M , Nξ and H
respectively.
Definition 25. α P ΩkpM q is called transverse (with respect to the foliation Fξ) if iξα “ 0. If
Let ΩkHpM q “ ΓpΛkH˚q denote the space of transverse k-forms. Let α P ΩpHpM q a
locally defined complex form and x P M , since αxpξq “ 0, αxis determined by the values of
αx in ΛpppHCqxq Ă Λ p
ppT MCqxq. We denote the complexification of Φ P EndpT M q by
ΦC x :“ Φ|HxbRC : pHCqx Ñ pHCqx. (1.12) Since pΦ|Hxq 2 “ ´1, the eigenvalues of ΦC x are ˘ ?
´1 :“ ˘i, this induces a decomposition pHCqx “ H
1,0 x ‘ H
0,1
x (1.13)
hence, for p, q ě 0 we set Hp,qx :“ Λ p pH1,0x q ˚ b ΛqpH0,1x q ˚ Ă Λp`qpHxq˚bRC, therefore ΛdpHCq ˚ x “ d à i“0 pHi,d´ix q ˚ @ 0 ă d ă 2n. (1.14)
α is said to be of type pa, d ´ aq, if the evaluation of α on Hi,d´ix is zero for all i ‰ a, in fact,
the decomposition given in (1.14) gives a decomposition into a direct sum of vector bundles ΩdH CpM q “ d à i“0 Ωi,d´iH C pM q, where Ω pq HCpM q denotes ΓpΛ p
pHCq˚bΛqpHCq˚q. Taking into account the dimension of Nξand H, it follows that ΛdpT MCq
˚ “ ΛdpHCq ˚ ‘`η b Λd´1pHCq ˚˘ hence, from the decomposition in (1.14) we have that
ΩdpM q “ ˜ d à i“0 Ωi,d´iH C pM q ¸ ‘ ˜ η b ˜ d´1 à j“0 Ωj,d´j´1H C pM q ¸¸ . (1.15)
Proposition 26. [BS10, Corollary. 3.1] The transversal formω :“ dη|H (1.5) is of type p1, 1q.
Let α a basic p-form and Uα “ VαˆI a foliated chart with coordinates pzi, ¨ ¨ ¨ , zn, xq,
if UαX Uβ ‰ H and pwi, ¨ ¨ ¨ , wn, yq are coordinates for Uβ, then
Bzi B ¯wi “ 0 and Bzi By “ 0, hence, α “ ai1,¨¨¨ip,¯j1¨¨¨¯jqdz i1 ^ ¨ ¨ ¨ ^ dzip ^ dzj1
^ ¨ ¨ ¨ ^ dzjq a form of type pp, qq is also of type pp, qq with respect to the coordinates pw1, ¨ ¨ ¨ , wn, yq, furthermore, the functions ai1,¨¨¨ip,¯j1¨¨¨¯jq does not depend of the coordinate x. Thus we have the well-defined operators
BB : Ωp,qB Ñ Ω p`1,q
B and ¯BB : Ωp,qB Ñ Ω p,q`1
B . (1.16)
If α is basic, dα is basic, so we set dB “ d|ΩpB, then we have the split dB “ BB` ¯BB. Relatively
to the transverse Hodge operator ˚T (see Definition15), for (1.16) we have adjoins
BB˚: Ω p,q B Ñ Ω p´1,q B BB˚ :“ ´ ˚T B˚B˚T and ¯ B˚B: Ω p,q B Ñ Ω p,q´1 B ¯ B˚B :“ ´ ˚T ¯BB˚˚T, (1.17) We set ω :“ dη, denote the multiplication by ω by:
Lω: α P ΩkpFξq ÞÑ α ^ ω P Ωk`2pFξq, (1.18)
the adjoin L˚ ωis
Λ :“ L˚
ω “ ´ ˚T Lω˚T . (1.19)
Since M is Sasakian and the Kähler identities are local, we have the following holds Lemma 2. [BG08, Lemma 7.2.7]
(i) rΛ, BBs “ ´i¯BB˚ (ii) rΛ, ¯BBs “ iB˚B (iii) BBB¯B˚ ` ¯B ˚ BBB“ B˚B¯BB` ¯BBB˚B “ 0 (iv) B˚ BBB` BBB˚B“ ¯BB˚¯BB` ¯BB¯B˚B (v) If we define ∆BB :“ B ˚ BBB ` BBBB˚ and ∆¯BB :“ ¯BB ˚ ¯ BB` ¯BBB¯B ˚ , then on a Sasakian manifold one has∆dB “ 2∆¯BB.
1.3.2
Curvature of Sasakian manifolds
We define the transverse curvature operator to be the curvature of the transverse Levi-Civita connection defined in (1.11), namely
RTpX, Y qZ “ ∇TX∇TYZ ´ ∇TY∇T
XZ ´ ∇ T
rX,Y sZ. (1.20)
One can check that RTpX, ξqY “ 0. When X, Y, Z, W P ΓpHq, the following relation holds RpX, Y, Z, W q “ RTpX, Y, Z, W q ´ gpΦY, W qgpΦX, Zq ` gpΦX, W qgpΦY, Zq. (1.21) Proposition 27. [BG08, Lemma 7.3.8] On a Sasakian manifold pM, η, ξ, g, Φq the curvature tensorR satisfy:
(i) RpX, ξqξ “ X ´ ηpXqξ
(ii) RpX, ξqY “ gpξ, Y qX ´ gpX, Y qξ
(iii) RpX, Y qξ “ ηpY qX ´ ηpXqY
Definition 28. The transverse Ricci curvature is defined by the trace of the transverse curvature tensor RT (1.20), i.e., in an orthonormal basis teiu of H (1.3), the transverse Ricci
curvature is defined by RicTpX, Y q “ ÿ i gpRTpX, eiqei, Y q “ ÿ i gTpRTpX, eiqei, Y q.
Proposition 29. [BG08, Proposition 7.3.12] Let pM, ξ, η, g, Φq be a K-contact manifold of dimension2n ` 1 [cf. Definition23], then
(i) RicpX, ξq “ 2nηpXq, for all X P T M
(ii) RicTpX, Y q “ RicpX, Y q ` 2gpX, Y q “ RicpX, Y q ` 2gTpX, Y q, @ X, Y P ΓpHq
The transverse scalar curvature sT is defined by the trace of RicT. Hence using
iξη “ 1, iξdη “ 0 and (1.10), for the transverse scalar curvature we have
sT “ s ` 2n (1.22)
Definition 30. A Sasakian manifold is called η-Einstein if there exists constants λ and µ, such that Ric “ λg ` µη b η.
By definition of Sasakian manifold (Theorem6item piiq), we have Ricpξ, ξq “ 2n, hence for a η-Einstein manifold we have the relation λ ` µ “ 2n.
Definition 31. A Sasakian manifold pM, ξ, η, g, Φq is called transverse Kähler-Einstein if there exists a constant λ such that RicT “ λgT.
So by Proposition29a Sasakian manifold is η-Einstein if and only if it is a transverse Kähler-Einstein.
Definition 32. A Sasakian manifold pM, ξ, η, g, Φq is Sasakian-Einstein if Ric “ 2ng. In particular Sasakian-Einstein manifolds are necessary Ricci positive. This can be summarized in the following result
Proposition 33. [Spa10, Proposition 1.9] Let pM, ξ, η, g, Φq a Sasakian manifold of dimension 2n ` 1, the following are equivalent
(i) pM, gq is Sasaki-Einstein with Ricg “ 2ng.
(ii) The Kähler cone pCpM q, ¯gq in Theorem6is Ricci-flat,Ricg“0¯ .
(iii) The transverse Kähler structure to the Reeb foliationFξis Kähler-Einstein withRicT “
p2n ` 2qgT.
1.4
Holomorphic vector bundles on Sasakian manifolds
All the results and definitions in this Section are taken from [BS10]. Consider
S Ă T MCa positive rank subbundle, let S also be an integrable subbundle, i.e., the sections of S are closed by the Lie brake. We define the projection qS: pT MCq
˚
Ñ S˚ as the dual of inclusion S ãÑ T MC.
Definition 34. Consider a complex vector bundle E Ñ M , a partial connection on E in the direction of S is a smooth operator D : ΓpEq Ñ ΓpΛ1S˚
q b ΓpEq, satisfying the Leibniz rule
Dpf sq “ f Dpsq ` qSpdf q b s.
Since the distribution S is integrable, smooth sections of kerpqSq are closed under
the exterior derivations, therefore, we have an induce exterior derivation on the smooth sections of S˚ which is an differential operator of order one. ˆd : Λ1S˚
Ñ Λ2S˚. Consider a partial
connection D on S and the operator D1: ΓpΛ1S˚q b ΓpEq Ñ ΓpΛ2S˚q b ΓpEq by the formula
D1pθ b sq “ ˆdθ b s ´ θ ^ Dpsq, so we can consider the composition
ΓpEq ΓpEq b ΓpΛ1S˚
q ΓpEq b ΓpΛ2S˚q
to define D1˝ D :“ KpDq P Γ`Λ2S˚b EndpEq
˘
. This section is called the curvature of D, if a connection is such that, KpDq ” 0, we will say that D is a flat connection. Let pM, η, ξ, g, Φq be a Sasakian manifold, we denote the extended anti-holomorphic pn ` 1q–dimensional foliation
r
H0,1 :“ H0,1‘ pNC
ξq Ă T M bRC, (1.23)
where H0,1is defined in (1.13), the distribution in (1.23) is integrable (see [BS10, Lemma 3.2]). Definition 35. A complex Sasakian vector bundle on a Sasakian manifold pM, g, ξq is a pair pE, D0q. Where D0is a partial connection in the direction of subbundle NξĂ T M and E Ñ M
is a complex vector bundle. Since NC
ξ Ă rH
0,1, we can consider partial connections in the direction of rH0,1, and by
restriction, partial connections in the direction of NC
ξ. Furthermore, since Nξis a 1´dimensional,
any partial connection on the direction of Nξis flat. When the context is clear, we denote by
¯
B a flat partial connection D in the direction of rH0,1 such that ¯B|Nξ “ D0. In the same sense, we abbreviate the notation pE, D0q just denoting E. We define a Hermitian structure on E, as a
smooth Hermitian structure h in the usual fashion on E which is preserved by D0, in the sense
that if s1, s2 are sections of E
dphps1, s2qq|Nξ “ hpD0ps1q, s2q ` hps1, D0ps2qq.
A unitary connection on pE, hq is a connection A on E such that dApreserves h in the usual
sense.
Definition 36. A holomorphic vector bundle on a Sasakian manifold pM, g, ξq is a pair ppE, D0q, ¯Bq, where pE, D0q is a Sasakian complex vector bundle and ¯B is defined as we have
just done above.
If ppE, D0q, Dq is Sasakian holomorphic vector bundle, then the dual E˚ also has
a natural structure of holomorphic Sasakian vector bundle, also if we have another Sasakian holomorphic vector bundle on M , namely ppE1, D1
0q, D1q, the tensor product E b E1 has a
natural structure of Sasakian holomorphic vector bundle, consequently, EndpEq – E b E˚has
a Sasakian holomorphic structure.
Definition 37. Let ppE, D0q, Dq and ppE1, D10q, D1q be Sasakian vector bundles on pM, g, ξq, a
morphism f : E Ñ E1 of vector bundles is called a Sasakian morphism, if for all s P ΓpEq
pD1˝ f qpsq “ pf ¨ Dqpsq.
Definition 38. Let E :“ pE, ¯Bq be a holomorphic Sasakian vector bundle, a connection A P ApEq induces a partial connection along rH0,1given by DHr0,1 :“ dA|Hr0,1 if this induced partial
Let ApEq Ă ApEq denote the subset of integrable connections on E, the class of connections mutually compatible with the holomorphic structure and the Hermitian metric is the natural extension of concept of Chern connection, the following proposition holds.
Proposition 39. Let pE, ¯Bq be a holomorphic Sasakian bundle with Hermitian structure, then there exist a unique unitary and integrable Chern connectionAhonE with HAh P Ω
1,1, moreover
the following expression defines closed Chern formscjpE , hq P Ωj,jpM q.
det ˆ 1E ` i 2πHAh ˙ “ n ÿ j“0 cjpE , hq.
2 Preliminaries: Gauge theory and G
2
-geometry on Calabi-Yau links
We are concerned with Sasakian 7-manifolds, however, in 7-dimensional manifolds other structures may be possible, in stance G2-structures. An special case of these is studied
in [CARSE16], namely, a 7-dimensional link K of a weighted homogeneous hypersurface on the sphere S9 Ă C5inherits a Sasakian structure which is transverse contact Calabi-Yau [HV15]. This structure admits a co-calibrated G2-structure ϕ, i.e., ˚gϕ is closed under the de Rham
differential. There, they describe a natural Yang-Mills theory on such spaces and a Chern-Simons formalism as well as topological energy bounds. We obtain analogous results in Section4.3. A remarkable result in [CARSE16] is that compatible G2-instantons on holomorphic Sasakian
bundles [cf. Definition 35] over K are the transversely Hermitian Yang-Mills connections. We show the equivalence between G2-instantons and contact instantons for transverse contact
Calabi-Yau case in Section4.3.
This Section is not intended to be a monograph of the original document [CARSE16], in which details are studied in depth. We are only interested in contextualizing the approach that we will give to our case in Section4.3, so details are referred to this work bellow or to the original reference whenever it is necessary.
2.1
Links as Sasakian and Contact Calabi-Yau 7-manifolds
Definition 40. Let f : Cn`1 Ñ Cna w-weighted homogeneus polynomial with an isolated criti-cal point at 0, so that each sphere Sn`1 “ BBp0q intersects V :“ f´1p0q Ă Cn`1transversely.
Then K :“ V X Sn`1is called a weighted link of degree degpf q and weight w.
For a link K we have the following commutative diagram, in which, the upper horizontal arrow is an embedding and the down horizontal arrow is a Kählerian embedding, the left hand vertical arrow is a principal S1-orbibundle and the right hand arrow is an orbifold Riemannian submersion [CARSE16, Definition 5]
K S9
V P4
S2n`1 has a natural Sasakian structure [see Example 17], so K inherits a natural Sasakian structure induced by restriction of that one of S2n`1 (see [BG08, Proposition 9.2.2]). We
now examine the contact Calabi-Yau case, a Sasakian 7-manifold pM, η, ξ, g, Φq may be also equipped with a transverse SUp3q-structure, and hence with a natural G2-structure [HV15], as
somewhat of an interpolation between CY3-geometry and G2-geometry.
Definition 41. A Sasakian manifold pM, η, ξ, Φ, εq [cf. Definition 6] is said to be a contact Calabi-Yau manifold (cCY) if ε is a nowhere vanishing transverse form of horizontal type pn, 0q [cf. Definition25] such that ε ^ ¯ε “ p´1qnpn`1q2 inωnand dε “ 0, where i :“
? ´1. A remarkable number of facts from [HV15, Section 6.2.1] can be summarised as the following:
Proposition 42. Let pM, η, ξ, Φ, εq be a cCY 7-manifold. Then M carries a cocalibrated G2
-structure defined by
ϕ “ η ^ dη ` Impq (2.1) with torsiondϕ “ dη ^ dη and Hodge dual 4-form ψ “ ˚ϕ “ 1
2ω ^ ω ` η ^ <pεq.
For a Link K, Definition41is equivalent to the following one [CARSE16, Defini-tion 12]
Definition 43. A weighted link K of degree d and weight w “ pw0, ¨ ¨ ¨ , wnq is said to be
Calabi-Yau(CY) link if d “ÿw1
In fact, we have from [HV15, Proposition 6.7] that every Calabi-Yau link admits a S1-invariant contact Calabi-Yau structure. Hence, every Calabi-Yau link has a cocalibrated S1-invariant G2-structure of the form of (2.1) [HV15, Corollary 6.8].
2.2
Gauge theory on Contact Calabi-Yau manifolds
Let E Ñ M a Sasakian bunble [cf. Definition35] and M closed, compact Calabi-Yau manifold, we also assume E to be a G bundle with G a compact, semisimple Lie group, g the Lie algebra of G and gE the adjoint bundle, we also recall that ApEq denotes the space of all
connections on E. The Yang-Mills functional is defined by YM : A P ApEq ÞÑ }FA}2 :“
ż
M
xFA^ ˚FAygE. (2.2)
Critical points of the above functional are solutions to the Yang-Mills equation
d˚
2.2.1
Yang-Mills connections and G
2-instantons
An instance of the instanton equation generalized (1) in presence of a G2-structure
ϕ is the so called G2-Instanton equation [Tia00b], which is equivalently formulated as
FA^ ϕ “ ˚FA ðñ FA^ ψ “ 0 where ψ “ ˚ϕ. (2.4)
Solutions of (2.4) are called G2-instantons, and (2.4) is the Euler-Lagrage equation for the
Chern-Simons action defined as follow
Definition 44. Fix a reference connection A0 P ApEq. The Chern-Simons action is defined by:
CS : A0` α P A – A0` Ω1pgEq ÞÑ CSpA0` αq :“ 1 2 ż M TrpdA0α ^ α ` 2 3α ^ α ^ αq ^ ψ P R and CSpA0q “ 0.
For the cocalibrated case, i.e., dψ “ 0, the Chern-Simons action is well defined and
G2-instantons are critical points of CS.
Remark 45. If ϕ was closed, i.e., dϕ “ 0, then all G2-instanton would be critical point of
the Yang-Mills functional. However, the implication (2.4) ñ (2.3) fails in general for merely cocalibrate G2-structures (see [CARSE16, Section 4.1]).
2.2.2
Topological energy bounds
Critical points of the Chern-Simon functional saturate the Yang-Mills energy, just like in the classical 4-dimensional case [DK90].
Definition 46. Let E Ñ M be a Sasakian bundle [cf. Definition 35] over a 7-dimensional closed contact Calabi-Yau link K [cf. Definition43] and ϕ the G2-structure given in (42), the
topological charge of A P ApEq is defined by: κpAq “ ż
M
TrpFA2q ^ ϕ.
Lemma 3. [CARSE16, Lemma 24] Fix a Hermitian metrich on the holomorphic Sasakian
vector bundleE [cf. Definition36] with Chern connectionA “ Ah P ApE q [cf. Proposition39],
then the topological chargeκpAq assessed among Chern connections in ApEq is independent of
the Hermitian structure and it defines a topological chargeκpEq.
It is shown in [CARSE16, Section 4.3] that among compatible connections, G2
-instantons (2.4) over a contact Calabi-Yau 7-manifold are Yang-Mills connections (2.3) even thought the natural G2-structure given by (42) is not closed (see Remark45)
2.2.3
G
2-Instantons and the Hermitian Yang-Mills condition
On a Kähler manifold X with symplectic form ω, a connection A on a holomorphic vector bundle E Ñ X is called Hermitian Yang-Mills (HYM) if ˆFA :“ xFA, ωy “ 0 and
Definition 47. If E Ñ M is a Sasakian holomorphic vector bundle [cf. Definition36], the above HYM condition generalises identically, reinterpreting ω :“ dη P Ω1,1pM q as the transverse Kähler form [cf. Proposition26], and we say that the connection A is transverse Hermitian Yang-Mills (tHYM).
In fact, fixing a holomorphic structure on E, tHYM-connections are exactly G2
-instantons
Lemma 4. [CARSE16, Lemma 21] Let E be a holomorphic Sasakian bundle over a 7-dimensional cCY manifold pM, η, ξ, g, Φq endowed with its natural G2-structure (42); then a
3 Preliminaires: Moduli space of
con-tact instanton in 5-dimensions
This work is, in part, motivated by the work of Baraglia and Hekmati in [BH16] on 5-dimensional Sasakian manifolds, which in turn is a generalization of the classical Donaldson theory on 4-manifolds [DK90] using the methods of [Ito88]. This chapter is devoted to stand the main results and techniques used by them. Since almost any result in this Section has a generalization on the 7-dimensional case, that we study closely in Section5, details and proofs are referred to this work bellow or to the main reference for this section [BH16]. I do not show any proof in this Section.
3.1
The contact instanton equation
Baraglia and Hekmati construct the moduli space M˚ (8) of irreducible contact
instantons in 5-dimensional contact metric manifold M , an analogue of Yang-Mills connections in 4-dimensional case and they study the geometrical structure of M˚. They give sufficient conditions for smoothness of M˚in the K-contact case [cf. Definition23]. They show that the dimension of the moduli space is given by the index of an operator transversely elliptic to the Reeb foliation. Furthermore, for Sasakian manifolds it is shown that the moduli space is Kähler and hyperKähler if M is transverse Calabi-yau [cf. Definition41].
In this Section, unless otherwise stated, pM, ηq denotes a 5-dimensional contact manifold, pM, η, gq and pM, η, ξ, g, Φq denote a contact metric manifold and a Sasakian manifold respectively [cf. Section1.2]. Consider a complex vector bundle E Ñ M on contact metric manifold M , and let A P ApEq be a connection on E. An instance of the generalised instantons equation (1) for 5-dimensional contact manifolds becomes
˚pFAq “ ˘η ^ FA (3.1)
Solutions of ˚pFAq “ η ^ FAand ˚pFAq “ ´η ^ FAare called selfdual (SD) contact instantons
and anti-selfdual (ASD) contact instantons respectively. Note that we can define an analogous equation to (3.1) for any Sasakian p2n ` 1q–manifold in the following way
˚pFAq “ ˘`η ^ pdηqn´2^ FA
˘
By differentiation in (3.1) and using the Bianchi identity, a solution of (3.1) satisfy the Yang-Mills equation dAp˚FAq “ 0, since one can show that dη ^ FA “ 0.
3.1.1
Deformation theory
From now, let M be a compact, connected 5-manifold with contact metric structure pξ, η, Φ, gq [cf. Section1.2]. Let ΩkBpM q denote the space of basic k-forms [cf. Definition9], the differential operator d preserves basic forms, denote by dB: ΩkBpM q Ñ Ω
k`1
B pM q the restriction
of d|Ωk
BpM q
. The cohomology of the complex pΩ‚
BpM q, dBq is called the basic cohomology and
it is denoted H‚BpM q. Since the horizontal distribution H (1.3) is 4-dimensional, (3.1) provides a decomposition (compare with decomposition (4.7) in the 7–dimensional case) into SD and ASD transverse 2-forms as follow
Ω2HpM q “ Ω`HpM q loomoon SD ‘ Ω´HpM q loomoon ASD (3.2)
Remark48. In the classical 4-dimensional case, there is no difference between the SD and ASD concepts, since these are interchanged by reversing the orientation of the 4-manifold. In the case of contact instantons, a choice of orientation is distinguished by the contact structure. One can see that any ASD contact instanton satisfies the Yang–Mills equations d∇p˚F∇q “ 0, while this
is generally not the case for SD contact instantons.
Let ∇ be a ASD contact instanton, i.e., F∇P Ω´HpgEq, where gE is the adjoin bundle
with fiber g. With aid of the extended differential operator d∇: ΩkpgEq Ñ Ωk`1pgEq, define the
operators dV : ΩkHpgEq Ñ ΩkHpgEq dVpαq :“ iξpdAαq and dT : Ω k HpgEq Ñ Ωk`1H pgEq dTpαq :“ dAα ´ η ^ dVα (3.3)
Note that kerpdVq “ ΩkBpM q, d∇descends to the quotient L
‚ of the graded Lie algebra Ω‚
pgEq
modulus the algebraic ideal I generated by Ω´
HpM q (see [BH16, Lemma 3.1]), and it defines in
that way a complex pL‚, Dq, in which the following identification holds [BH16, Proposition 3.2]
L0 – Ω0pgEq, L2 – Ω`HpgEq ‘ η ^ Ω1HpgEq,
L1 – Ω1pgEq, L3 – η ^ Ω`HpgEq.
(3.4) Analogous results follow in 7-dimensional case [cf. Proposition73]. With the identification (3.4) the complex pL‚, Dq is as follow
0 L0 D0 L1 D1 L2 D2 L3 0
. (3.5)
denote by Hk the cohomology groups of (3.5), H0 represents the Lie algebra of infinitesimal automorphisms of ∇. H1represents infinitesimal deformations of ∇ as contact instanton and H2 may be used to describe the obstruction to extending an infinitesimal deformation to a genuine deformation. Basic forms in Lk are defined in the natural way [cf. Definition 9], we denote by LkB the space of basic forms in Lk. The operator D (3.5) restricts to LkB, i.e.
