Table Of ContentMEMOIRS
of the
American Mathematical Society
Number 1031
Elliptic Integrable Systems:
A Comprehensive Geometric
Interpretation
Idrisse Khemar
September 2012 • Volume 219 • Number 1031 (fourth of 5 numbers) • ISSN 0065-9266
American Mathematical Society
Number 1031
Elliptic Integrable Systems:
A Comprehensive Geometric
Interpretation
Idrisse Khemar
September2012 • Volume219 • Number1031(fourthof5numbers) • ISSN0065-9266
Library of Congress Cataloging-in-Publication Data
Khemar,Idrisse,1979-
Ellipticintegrablesystems: acomprehensivegeometricinterpretation/IdrisseKhemar.
p.cm. —(MemoirsoftheAmericanMathematicalSociety,ISSN0065-9266;no. 1031)
“September2012,volume219,number1031(fourthof5numbers).”
Includesbibliographicalreferencesandindex.
ISBN978-0-8218-6925-3(alk. paper)
1.Geometry,Riemannian. 2.Hermitianstructures. I.Title.
QA649.K44 2011
516.3(cid:2)73—dc23 2012015562
Memoirs of the American Mathematical Society
Thisjournalisdevotedentirelytoresearchinpureandappliedmathematics.
Publisher Item Identifier. The Publisher Item Identifier (PII) appears as a footnote on
theAbstractpageofeacharticle. Thisalphanumericstringofcharactersuniquelyidentifieseach
articleandcanbeusedforfuturecataloguing,searching,andelectronicretrieval.
Subscription information. Beginning with the January 2010 issue, Memoirs is accessi-
ble from www.ams.org/journals. The 2012 subscription begins with volume 215 and consists of
six mailings, each containing one or more numbers. Subscription prices are as follows: for pa-
per delivery, US$772 list, US$617.60 institutional member; for electronic delivery, US$679 list,
US$543.20institutional member. Uponrequest, subscribers topaper delivery ofthis journalare
also entitled to receive electronic delivery. If ordering the paper version, subscribers outside the
United States and India must pay a postage surcharge of US$69; subscribers in India must pay
apostagesurchargeofUS$95. ExpediteddeliverytodestinationsinNorthAmericaUS$61;else-
whereUS$167. Subscriptionrenewalsaresubjecttolatefees. Seewww.ams.org/help-faqformore
journalsubscriptioninformation. Eachnumbermaybeorderedseparately;pleasespecifynumber
whenorderinganindividualnumber.
Back number information. Forbackissuesseewww.ams.org/bookstore.
Subscriptions and orders should be addressed to the American Mathematical Society, P.O.
Box 845904, Boston, MA 02284-5904USA. All orders must be accompanied by payment. Other
correspondenceshouldbeaddressedto201CharlesStreet,Providence,RI02904-2294USA.
Copying and reprinting. Individual readers of this publication, and nonprofit libraries
actingforthem,arepermittedtomakefairuseofthematerial,suchastocopyachapterforuse
in teaching or research. Permission is granted to quote brief passages from this publication in
reviews,providedthecustomaryacknowledgmentofthesourceisgiven.
Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication
is permitted only under license from the American Mathematical Society. Requests for such
permissionshouldbeaddressedtotheAcquisitionsDepartment,AmericanMathematicalSociety,
201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by
[email protected].
MemoirsoftheAmericanMathematicalSociety(ISSN0065-9266)ispublishedbimonthly(each
volume consisting usually of more than one number) by the American Mathematical Society at
201CharlesStreet,Providence,RI02904-2294USA.PeriodicalspostagepaidatProvidence,RI.
Postmaster: Send address changes to Memoirs, American Mathematical Society, 201 Charles
Street,Providence,RI02904-2294USA.
(cid:2)c 2012bytheAmericanMathematicalSociety. Allrightsreserved.
Copyrightofindividualarticlesmayreverttothepublicdomain28years
afterpublication. ContacttheAMSforcopyrightstatusofindividualarticles.
(cid:2)
ThispublicationisindexedinMathematicalReviewsR,Zentralblatt MATH,ScienceCitation
(cid:2) (cid:2) (cid:2) (cid:2)
IndexR,SciSearchR,ResearchAlertR,CompuMath Citation IndexR,Current
(cid:2)
ContentsR/Physical,Chemical& Earth Sciences. Thispublicationisarchivedin
Portico andCLOCKSS.
PrintedintheUnitedStatesofAmerica.
(cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines
establishedtoensurepermanenceanddurability.
VisittheAMShomepageathttp://www.ams.org/
10987654321 171615141312
Contents
Introduction 1
0.1. The primitive systems 1
0.2. The determined case 2
0.2.1. The minimal determined system 2
0.2.2. The general structure of the maximal determined case 3
0.2.3. The model system in the even case 3
0.2.4. The model system in the odd case 3
0.2.5. The coupled model system 7
0.2.6. The general maximal determined odd system (k(cid:2) =2k+1,m=2k) 7
0.2.7. The general maximal determined even system (k(cid:2) =2k,m=2k−1) 9
0.2.8. The intermediate determined systems 11
0.3. The underdetermined case 11
0.4. In the twistor space 12
0.5. Related subjects and works, and motivations 13
0.5.1. Relations with surface theory 13
0.5.2. Relations with mathematical physics 13
0.5.3. Relations of F-stringy harmonicity and supersymmetry 14
Notation, conventions and general definitions 15
0.6. List of notational conventions and organisation of the paper 15
0.7. Almost complex geometry 16
Chapter 1. Invariant connections on reductive
homogeneous spaces 19
1.1. Linear isotropy representation 19
1.2. Reductive homogeneous space 19
1.3. The (canonical) invariant connection 20
1.4. Associated covariant derivative 20
1.5. G-invariant linear connections in terms of equivariant bilinear maps 21
1.6. A family of connections on the reductive space M 23
1.7. Differentiation in End(T(G/H)) 24
Chapter 2. m-th elliptic integrable system associated to a k(cid:2)-symmetric space 27
2.0.1. Definition of Gτ (even when τ does not integrate in G) 27
2.1. Finite order Lie algebra automorphisms 28
2.1.1. The even case: k(cid:2) =2k 28
2.1.2. The odd case: k(cid:2) =2k+1 30
2.2. Definitions and general properties of the m-th elliptic system 31
iii
iv CONTENTS
2.2.1. Definitions 31
2.2.2. The geometric solution 33
2.2.3. The increasing sequence of spaces of solutions: (S(m))m∈N 36
2.2.4. The decreasing sequence (Syst(m,τp))p/k(cid:2) 38
2.3. The minimal determined case 38
2.3.1. The even minimal determined case: k(cid:2) =2k and m=k 39
2.3.2. The minimal determined odd case 42
2.4. The maximal determined case 45
Adding holomorphicity conditions; the intermediate determined systems 46
2.5. The underdetermined case 47
2.6. Examples 47
2.6.1. The trivial case: the 0-th elliptic system associated to a Lie group 47
2.6.2. Even determined case 47
2.6.3. Primitive case 48
2.6.4. Underdetermined case 48
2.7. Bibliographical remarks and summary of the results 48
Chapter 3. Finite order isometries and twistor spaces 51
3.1. Isometries of order 2k with no eigenvalues =±1 52
3.1.1. The set of connected components in the general case 52
3.1.2. Study of AdJ, for J ∈Za (R2n) 54
2k
3.1.3. Study of AdJj 56
3.2. Isometries of order 2k+1 with no eigenvalue =1 58
3.3. The effect of the power maps on the finite order isometries 58
3.4. The twistor spaces of a Riemannian manifolds and its reductions 60
3.5. Return to an order 2k automorphism τ: g→g 60
3.5.1. Case r =k 60
3.5.2. Action of Adτ|m on adgCj 61
3.6. The canonical section in (Z (G/H))2, the canonical embedding, and
2k
the twistor lifts 62
3.6.1. The canonical embedding 62
3.6.2. The twistor lifts 63
3.7. Bibliographical remarks and summary of the results 64
Chapter 4. Vertically harmonic maps and harmonic sections of submersions 65
4.1. Definitions, general properties and examples 65
4.1.1. The vertical energy functional 65
4.1.2. Examples 65
4.1.3. Ψ-torsion, Ψ-difference tensor, and curvature of a Pfaffian system 70
4.2. Harmonic sections of homogeneous fibre bundles 72
4.2.1. Definitions and geometric properties 73
4.2.2. Vertical harmonicity equation 76
4.2.3. Reductions of homogeneous fibre bundles 78
4.3. Examples of homogeneous fibre bundles 81
4.3.1. Homogeneous spaces fibration 81
4.3.2. The twistor bundle of almost complex structures Σ(E) 86
4.3.3. The twistor bundle Z (E) of a Riemannian vector bundle 89
2k
4.3.4. The twistor subbundle Zα (E) 92
2k,j
4.4. Geometric interpretation of the even minimal determined system 99
CONTENTS v
4.4.1. The injective morphism of homogeneous fibre bundle 99
4.4.2. Conclusion 102
4.5. Bibliographical remarks and summary of the results 103
Chapter 5. Generalized harmonic maps 105
5.1. Affine harmonic maps and holomorphically harmonic maps 105
5.1.1. Affine harmonic maps: general properties 105
5.1.2. Holomorphically harmonic maps 106
5.2. The sigma model with a Wess-Zumino term in nearly K¨ahler
manifolds 112
5.2.1. Totally skew-symmetric torsion 112
5.2.2. The general case of an almost Hermitian manifold 114
5.2.3. The example of a 3-symmetric space 116
5.2.4. The good geometric context/setting 117
5.2.5. J-twisted harmonic maps 119
5.3. The sigma model with a Wess-Zumino term in G -manifolds 119
1
5.3.1. TN-valued 2-forms 119
5.3.2. Stringy harmonic maps 121
5.3.3. Almost Hermitian G -manifolds 122
1
5.3.4. Characterization of Hermitian connections in terms of their torsion 126
5.3.5. The example of a naturally reductive homogeneous space 127
5.3.6. Geometric interpretation of the maximal determined odd case 128
5.4. Stringy harmonicity versus holomorphic harmonicity 129
5.5. Bibliographical remarks and summary of the results 130
Chapter 6. Generalized harmonic maps into f-manifolds 133
6.1. f-structures: General definitions and properties 133
6.1.1. f-structures, Nijenhuis tensor and natural action on the space of
torsions T 133
6.1.2. Introducing a linear connection 134
6.2. The f-connections and their torsion 135
6.2.1. Definition, notation and first properties 135
6.2.2. Characterization of metric connections preserving the splitting 136
6.2.3. Characterization of metric f-connections. Existence of a
characteristic connection 140
6.2.4. Precharacteristic and paracharacteristic connections 146
6.2.5. Reductions of f-manifolds 148
6.3. f-connections on fibre bundles 151
6.3.1. Riemannian submersion and metric f-manifolds of global type G 151
1
6.3.2. Reductions of f-submersions 155
6.3.3. Horizontally K¨ahler f-manifolds and horizontally projectible
f-submersions 157
6.3.4. The example of a naturally reductive homogeneous space 158
6.3.5. The example of the twistor space Zα(M) 159
2k
6.3.6. The example of the twistor space Zα (M,J ) 160
2k,j j
6.3.7. The reduction of homogeneous fibre bundle I : G/G (cid:3)→
J0 0
Zα0 (G/H,J ) 161
2k,2 2
6.4. Stringy harmonic maps in f-manifolds 161
6.4.1. Definitions 161
vi CONTENTS
6.4.2. The closeness of the 3-forms F •T and F (cid:4)T 163
6.4.3. The sigma model with a Wess-Zumino term in reductive metric
f-manifold of global type G 167
1
6.4.4. The example of a naturally reductive homogeneous space 168
6.4.5. Geometric interpretation of the maximal determined even case 168
6.4.6. Twistorial geometric interpretation of the maximal determined even
case 170
6.4.7. About the variational interpretation in the twistor spaces 171
6.5. Bibliographical remarks and summary of the results 172
Chapter 7. Generalized harmonic maps into reductive homogeneous spaces 175
7.1. Affine harmonic maps into reductive homogeneous spaces 176
Affine harmonic maps into symmetric spaces 178
7.2. Affine/holomorphically harmonic maps into 3-symmetric spaces 178
7.3. (Affine) vertically (holomorphically) harmonic maps 179
7.3.1. Affine vertically harmonic maps: general properties 179
7.3.2. Affine vertically holomorphically harmonic maps 180
7.4. Affine vertically harmonic maps into reductive homogeneous space 180
The Riemannian case 182
7.5. Harmonicity vs. vertical harmonicity 184
7.6. (Affine) vertically (holomorphically) harmonic maps into reductive
homogeneous space with an invariant Pfaffian structure 186
7.7. The intermediate determined systems 192
7.7.1. The odd case 192
7.7.2. The even case 193
7.7.3. Sigma model with a Wess-Zumino term 194
7.8. Some remarks about the twistorial interpretation 195
7.8.1. The even case 195
7.8.2. The odd case 195
7.9. Bibliographical remarks and summary of the results 196
Chapter 8. Appendix 197
8.1. Vertical harmonicity 197
8.2. G-invariant metrics 199
8.2.1. About the natural reductivity 199
8.2.2. Existence of an AdH-invariant inner product on k for which τ|m is
an isometry 200
8.2.3. Existence of a naturally reductive metric for which J is an isometry,
resp. F is metric 201
Bibliography 203
Index 207
List of symbols 209
Section 1 209
Section 2 210
Section 3 212
Section 4 214
Section 5 215
CONTENTS vii
Section 6 216
Section 7 217
Abstract
In this paper, we study all the elliptic integrable systems, in the sense of C.
L. Terng [66], that is to say, the family of all the m-th elliptic integrable systems
associated to a k(cid:2)-symmetric space N = G/G . Here m ∈ N and k(cid:2) ∈ N∗ are
0
integers. For example, it is known that the first elliptic integrable system asso-
ciated to a symmetric space (resp. to a Lie group) is the equation for harmonic
maps into this symmetric space (resp. this Lie group). Indeed it is well known
that this harmonic maps equation can be written as a zero curvature equation:
dα + 1[α ∧ α ] = 0, ∀λ ∈ C∗, where α = λ−1α(cid:2) + α + λα(cid:2)(cid:2) is a 1-form
λ 2 λ λ λ 1 0 1
on a Riemann surface L taking values in the Lie algebra g. This 1-form α is
λ
obtained as follows. Let f: L → N = G/G be a map from the Riemann sur-
0
face L into the symmetric space G/G . Then let F: L → G be a lift of f, and
0
consider α = F−1.dF its Maurer-Cartan form. Then decompose α according to
the symmetric decomposition g = g ⊕g of g : α = α +α . Finally, we define
0 1 0 1
α :=λ−1α(cid:2)+α +λα(cid:2)(cid:2),∀λ∈C∗,whereα(cid:2),α(cid:2)(cid:2)aretheresp.(1,0)and(0,1)partsof
λ 1 0 1 1 1
α . Thenthezerocurvatureequationforthisα ,forallλ∈C∗,isequivalenttothe
1 λ
harmonic maps equation for f: L→N =G/G , and is by definition the first ellip-
0
tic integrable system associated to the symmetric space G/G . Thus the methods
0
of integrable system theory apply to give generalised Weierstrass representations,
algebro-geometric solutions, spectral deformations, and so on. In particular, we
can apply the DPW method [23] to obtain a generalised Weierstrass representa-
tion. More precisely, we have a Maurer-Cartan equation in some loop Lie algebra
Λg ={ξ: S1 →g|ξ(−λ)=τ(ξ(λ))}. Thenwecanintegrateitinthecorresponding
τ
loopgroupandfinallyapplysomefactorizationtheoremsinloopgroupstoobtaina
generalised Weierstrass representation: this is the DPW method. Moreover, these
methodsofintegrablesystemtheoryholdforallthesystemswrittenintheformsof
azerocurvatureequationforsomeαλ =λ−mαˆ−m+···+αˆ0+···+λmαˆm. Namely,
these methods apply to construct the solutions of all the m-th elliptic integrable
systems. So it is natural to ask what is the geometric interpretation of these sys-
tems. Do they correspond to some generalisations of harmonic maps? This is the
problem that we solve in this paper: to describe the geometry behind this family
ReceivedbytheeditorApril8,2010and,inrevisedform,March31,2011.
ArticleelectronicallypublishedonFebruary22,2012;S0065-9266(2012)00651-4.
2010 MathematicsSubjectClassification. Primary53B20,53B35;Secondary53B50.
Research supported successively by the DFG-Schwerpunkt Globale Differentialgeometrie,
grantDO776(inT.U.Munich),andthenbytheSFB-TR71(inUniversita¨tTu¨bingen).
Affiliationat time ofpublication: Institut E´lie Cartande Nancy, Universit´e Henri Poincar´e
Nancy1,B.P.70239,54506Vandoeuvre-l`es-NancyCdex,France;email: [email protected].
(cid:2)c2012 American Mathematical Society
ix
