Table Of Content
Anastasios Mallios
Modern Differential Geometry
in Gauge Theories
Maxwell Fields, Volume I
Birkha¨user
Boston • Basel • Berlin
AnastasiosMallios
UniversityofAthensPanepistimioupolis
DepartmentofMathematics
GR-15784,Athens
Greece
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MathematicsSubjectClassification(2000):53C05,53C07,53C80,18F20,53D50,53Z05,55N30,
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16D40,16E99,58C99,55R05
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ModernDifferentialGeometryinGaugeTheories:
Yang–MillsFields,VolumeII ISBN0-8176-4379-6
ModernDifferentialGeometryinGaugeTheories:
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Contents
GeneralPreface.................................................. ix
PrefacetoVolumeI............................................... xi
Acknowledgements............................................... xv
PartI MaxwellFields:GeneralTheory
1 TheRudimentsofAbstractDifferentialGeometry................. 3
1 TheDifferentialSetting ..................................... 3
1.1 LogarithmicDerivation ............................... 7
2 A-Connections............................................. 8
2.1 TheClassicalCase ................................... 10
2.2 LocalDefinitionofanA-Connection.................... 13
2.3 GaugeTransformation................................ 18
3 InducedA-Connections ..................................... 20
4 ExistenceofA-Connections.CriteriaofExistence............... 27
5 TheSpaceofA-Connections................................. 30
6 RelatedA-Connections.ModuliSpaceofA-Connections......... 33
6.1 ModuliSpace ....................................... 34
7 Curvature ................................................. 41
7.1 LocalFormoftheCurvature........................... 44
7.2 TransformationLawofFieldStrength(Curvature) ........ 46
8 FundamentalIdentitiesoftheCurvature(Continued).Torsion ..... 48
8.1 PullbackofCurvature ................................ 52
8.2 Torsion............................................. 53
9 A-ConnectionsCompatiblewithA-Metrics .................... 54
9.1 HermitianA-Connections ............................. 56
9.2 MatricesofA-Metrics ................................ 58
9.3 Ka¨hlerA-Metrics.................................... 61
vi Contents
9.4 EinsteinA-Metrics................................... 62
9.5 LorentzA-Metrics ................................... 63
10 TheHodge∗-Operator.VolumeForm.......................... 64
2 Elementary Particles: Sheaf-Theoretic Classification, by
Spin-Structure,AccordingtoSelesnick’sCorrespondencePrinciple.. 69
1 Preliminaries.BasicNotions ................................. 69
2 ClassificationofElementaryParticles,ThroughVectorSheaves,
AccordingtoTheirSpin-Structures............................ 71
2.1 StandardClassificationofElementaryParticlesbySpin
Number ............................................ 71
2.2 Classification of Elementary Particles Through
Module-Structures(a` laSelesnick)...................... 73
3 QuantumStateModules..................................... 74
4 Free Bosons and Fermions in Terms of Finitely Generated
ProjectiveModules ......................................... 79
5 Finitely Generated Projective Modules and Vector Bundles
(Serre–SwanTheory) ....................................... 81
6 VectorSheavesandElementaryParticles(Continued:Selesnick’s
Correspondence) ........................................... 84
6.1 Smooth(C∞-)Case .................................. 89
7 CohomologicalClassificationofElementaryParticles ............ 91
7.1 VectorSheaves ...................................... 92
7.2 LineSheaves........................................ 95
7.3 ElementaryParticles ................................. 97
8 ElementaryParticlesasPrincipalSheaves ...................... 98
8.1 PrincipalSheaves .................................... 100
9 VectorSheavesAssociatedwithPrincipalSheavesandPhysical
Interpretation .............................................. 102
9.1 PhysicalApplications................................. 109
9.2 InteractingParticles .................................. 110
3 Electromagnetism............................................ 113
1 TheElectromagneticField.TheMaxwellCategory .............. 114
2 CharacterizationoftheMaxwellGroupThroughLocalData ...... 118
2.1 LocalCharacterizationofMaxwellFields................ 120
2.2 LocalCharacterizationof(Gauge)EquivalentMaxwellFields125
3 ANaturalFibration......................................... 128
3.1 TheImageof(theNaturalFibration)τ .................. 130
3.2 Weil’sIntegralityTheorem(Again) ..................... 132
3.3 TheImageoftheMapτ (Continued).................... 138
3.4 CohomologyClassAssociatedwiththeFieldStrengthofa
MaxwellField(Continued)............................ 142
4 TheFibrationτ asaGroupMorphism ......................... 147
.
5 Actionof H1(X,C )ontheMaxwellGroupΦ1(X)∇............ 152
A
Contents vii
.
5.1 FreenessoftheActionof H1(X,C )ontheMaxwellGroup 154
.
5.2 TransitivityoftheActionof H1(X,C )ontheMaxwell
Group.............................................. 157
5.3 Φ1(X)∇,asaPrincipalHomogeneousSpace ............ 164
A R
6 TheHermitianCounterpart .................................. 168
6.1 Actionof H1(X,S1)onΦ1(X)∇ ...................... 171
A
6.2 HermitianMaxwellFields............................. 172
6.3 HermitianLightBundles.............................. 175
6.4 HermitianLightBundlesoverPath-ConnectedSpaces ..... 178
.
7 EquivariantActionsof H1(X,C )(Continued).................. 181
7.1 TheKerneloftheMapτ .............................. 186
7.2 HermitianCounterpart(Continued) ..................... 191
8 TheMaxwellGroupΦ1(X)∇ asaCentralExtension(Continued).. 192
A
8.1 TheHermitianCounterpart(Continued) ................. 194
4 CohomologicalClassificationofMaxwellandHermitianMaxwell
Fields ...................................................... 197
1 HypercohomologywithRespecttoa(Differential)A-Complex .... 197
1.1 SheafCohomology................................... 197
1.2 Hypercohomology ................................... 202
2 CˇechHypercohomology..................................... 206
3 CˇechHypercohomologyRelativetoaTwo-TermA-Complex...... 210
3.1 IdentificationofHˇ1(X,E0 −d→0 E1)...................... 212
4 Cˇech Hypercohomology, with Respect to the Two-Term
. ∂˜
Z-ComplexA −→Ω1 ...................................... 216
4.1 Characterizationofthe(Abelian)CˇechHypercohomology
GroupHˇ1(X,A. −→∂˜ Ω1).............................. 218
5 CohomologicalWordingoftheMaxwellGroup ................. 221
6 AbstractMaxwellEquations ................................. 227
7 TheHermitianAnalogue .................................... 230
5 GeometricPrequantization .................................... 233
1 SymplecticSheaves......................................... 233
2 PrequantizableSymplecticSheaves ........................... 237
3 TheHermitianFramework................................... 242
4 Cohomological Classification of (Abstract) Geometric
PrequantizationsofHermitianMaxwellFieldswithaGivenField
Strength .................................................. 246
5 PrequantizationofElementaryParticles........................ 250
5.1 BosonicCase ....................................... 251
5.2 TheChernIsomorphism(Continued),andConsequences... 252
5.3 GeometricPrequantizationofBosons(Continued) ........ 256
5.4 FermionicCase...................................... 257
5.5 Pull-BackofMaxwellFields........................... 259
viii Contents
5.6 GeometricPrequantizationofFermions(Continued)....... 267
References ...................................................... 273
IndexofNotation ................................................ 281
Index........................................................... 289
General Preface
“Whatcanbesaidatallcanbesaidclearly.”
L.WittgensteininTractatusLogico-Philosophicus(Routledge,1997),p.3.
,
“Σοφο´ντοι´το´ σαφε´ςουτο´ μη´ σαφε´ς.”
, ,
Ευριπ., Ορε´στης
It is nowadays generally accepted that the theory of principal fiber bundles is the
appropriate mathematical framework for describing one of the most beautiful, as
well as important, physical theories, viz. the so-calledgauge field theory, or gauge
theories,being,ineffect,toquote,M.F.Atiyah,“physicaltheoriesofageometrical
character.”
Now,inthiscontext,aprincipalfibrationisdefinedbythe(local)gaugegroup
(orinternalsymmetrygroup)ofthephysicalsystem(particlefield)underconsider-
ation.Yet,theparticularphysicalsystematissueiscarriedby,orliveson,a“space”
(vacuum)thatintheclassicalcaseisusuallyasmooth(viz.C∞-)manifold.Within
our abstract framework, instead, this is, in general,an arbitrary topological space,
beingalsothebasespaceofallthefiberspacesinvolved.
Accordingly,wedonotuseanynotionofcalculus(smoothness)intheclassical
sense,thoughwecanapply,mostofthepowerfulmachineryofthestandarddiffer-
entialgeometry,inparticular,thetheoryofconnections,characteristicclasses,and
the like. However, all this is done abstractly, which constitutes an axiomatic treat-
ment of differential geometry in terms of sheaf theory and sheaf cohomology (see
A. Mallios [VS: Vols I, II]), while, as already noted, no calculus is used at all! So
thepresentstudycanbeconstruedasafurtherapplicationofthatabstract(i.e.,ax-
iomatic) point of view in the realm of gauge theories, given, as mentioned before,
theintimateconnectionofthelattertheorieswith(differential)geometry.
Thus,workingwithintheaforementionedabstractset-up,weessentiallyreplace
allthepreviousfiberspaces(viz.principaland/orvectorbundles)bythecorrespond-
ing sheaves of sections, the former being, of course, just our model (motivation),
whileourstudyisotherwise,ashasalreadyexplainedabove,quiteabstract(!),that
is, axiomatic. Of course, in the classical case the two perspectives are certainly
mathematically speaking (categorically!) equivalent; however, the sheaf-theoretic
language, to which we also stick throughout the present treatment, is even in the
standardcase,incommonusageintherecentphysicsliterature(cf.,forinstance,Yu.
I.Manin[1]orevenS.A.Selesnick[1]).Thus,itprovesthatthesamelanguageisat
least physically more transparent, while, finally, being more practical. In addition,
x GeneralPreface
wavefunctionsareconsideredassections(i.e.,functionswhosedomainisvariedas
wellastheirrange,alongwiththepointofapplication)ofappropriatebundles(loc.
cit. Vol. II: Chapt. IV; Section 10). Furthermore, it is still very likely that the kind
of common base space of the sheaves involved herewith can also be thought of as
correspondingtorecentaspectsofthe“vacuum,”forinstance,“... thestructureof
such spaces is governed by topology, rather than geometry” (cf. P. J. Braam [1: p.
279]).
On the other hand, a significant advantage of the present abstract formulation
of the classical gauge field theory (i.e., the smooth case) lies in the possibility of
employingthestandardconceptualmachineryoftheusual(smooth)differentialge-
ometry, even for basespaces(of the fiber spaces, as above)that (i) are not smooth
enough, (ii) include a large amount of singularities in the classical sense, and (iii)
are not smooth at all (!), but provide the appropriate framework for the exploita-
tion of the axiomatic theory [VS], as this happens in certain important cases (see
concrete examples throughout the sequel). Of course, this potential generalization
of the classical theory might very likely be of a particular significance to (mathe-
matical) physicists who long ago were already aware of, as well as tantalized by,
theaforesaidtypeofspaces.Furthermore,thesameabstractapproach,hascertainly
theoretical/pedagogical advantages, being, namely, greater perspective, clarity and
unification.Itisthusmoreakintothenowadaysgenerallyacceptedaspectthat“the
basicideasofmodernphysicsarequitesimple”(see,forinstance,H.Fritzcsh[1:p.
211]),oreventhat“... theproblemsofquantumgravityaremuchmorethanpurely
technicalones;theytouchuponveryessentialphilosophicalissues”(cf.G.’tHooft
[1:p.2]).Soitisquitenaturaltotrytomanufactureasimilarsituationpertainingto
the mathematics involved; thus, something like this would also be in concord with
theapostrophes,asstatedintheepigraphofthispreface.
Further details about each of the two individual volumes are given by separate
prefaces.
Preface to Volume I
Thetechnicalaspectsofthepresentvolumeareasfollows:InChapterI,wearecon-
cernedwithexactlythosebasicnotionsandresultsofabstractdifferentialgeometry
thatwillbeofusethroughouttherestofthistreatise,includingbothvolumesofit.
Thishasbeendonefortheconvenienceofthereader,whowillfindhereexplanations
andformalstatementsoftherelevantmaterialusedthroughoutwhilewereferto[VS]
forfurtherdetailsorevencompleteproofsofthestatedresults.Yetitmayalsohap-
penthatoccasionallywerefertoexpandedorevennewmaterial,incomparisonwith
ourpreviousaccountonthesubject,asgivenin[VS].
ChapterIIdealswiththeclassificationofelementaryparticlesaccordingtotheir
spinstructure,asisclassicallythecase,howeverintermsnowofsheaftheory;that
is, by means of the notion of vector sheaves (see Chapter I). In other words, one
canidentifythestatesofa(free)elementaryparticlewithsectionsofanappropriate
vector sheaf, where the latter is determined by the spin of the elementary particle
at issue. Here, the relevant argument is based, in fact, on a previous work of S. A.
Selesnick[1],pertainingtothesameclassificationintermsofvectorbundles,which
justifies our choice for the title of the chapter (see Chapter II; (6.29)). Among the
technicaladvantagesofthissortofclassificationis,forinstance,theresultingcoho-
mologicalclassificationofelementaryparticlesonthebasisofthestandardsimilar
situationonehasforvectorsheaves(cf.[VS:SectionV.2]).Thesameclassification
as before helps also to classify in a similar way the so-called Yang–Mills fields, of
whichparticularcasearetheMaxwellfields(photons);seeChapterIIIinthesequel,
alongwithSectionI.9,inVolumeIIofthistreatise.Anapplicationoftheprevious
typeofclassificationinthecaseofMaxwellfieldsisalsoourconsiderationsinChap-
terV,Section4,concerninggeometricprequantization,alongwithitscohomological
classifications.
Chapter III refers to electromagnetism, viz. the electromagnetic field (photon),
fromthepointofviewofgaugetheory,thatis,toparaphraseM.F.Atiyahagain,ofa
physicaltheorywithinadifferential-geometricframework.Ofcourse,thedifferential
geometrythatisappliedistheabstractone,ashasbeenadvocatedby[VS],thisbeing
alsoourgeneralperspectivethroughoutthepresentstudy.Thus,amongotherthings,
one finds here the beginning of a cohomological classification of Maxwell fields
