Derived Geometry and The Integrability Problem for G-Structures by Anthony Ryan McCormick

134 Pages·2017·0.68 MB·English

Derived Geometry and
The Integrability Problem
for G-Structures
by
Anthony Ryan McCormick
Athesis
presentedtotheUniversityofWaterloo
infulfilmentofthe
thesisrequirementforthedegreeof
MasterofMathematics
in
PureMathematics
Waterloo,Ontario,Canada,2017
(cid:13)c AnthonyRyanMcCormick2017
AUTHOR’S DECLARATION
IherebydeclarethatIamthesoleauthorofthisthesis. Thisisatruecopyofthethesis,including
anyrequiredfinalrevisions,asacceptedbymyexaminers.
Iunderstandthatmythesismaybemadeelectronicallyavailabletothepublic.
ii
Abstract
In this thesis, we study the integrability problem for G-structures. Broadly speaking, this is the
problemofdeterminingtopologicalobstructionstotheexistenceofprincipalG-subbundlesofthe
framebundleofamanifold,subjecttocertaindifferentialequations. Webeginthisinvestigationby
introducinggeneralmethodsfromhomologicalalgebrausedtoobtaincohomologicalobstructions
totheexistenceofsolutionstocertaingeometricproblems. Thisleadsustoapreciseanalogybe-
tweendeformationtheoryandtheformalintegrabilitypropertiesofpartialdifferentialequations.
Alongtheway,weprovethefollowingdifferential-geometricanalogueofawell-knownresultfrom
derivedalgebraicgeometry:
H∗(N•(S1⊗C∞(M)))∼= Ω−∗(M)
aswellastheidentificationoftheinfinitesimalgeneratorofthenaturalS1-actioncorrespondingto
thedeRhamdifferential. Asashortcorollaryweobtainanaturalisomorphism
H∗(Hom(N•(S1⊗C∞(M)),C∞(M)))∼= Γ(M,Λ−∗T )
M
leading to a comparison between the standard Gerstenhaber bracket on the left-hand-side of the
above equation and the Schouten bracket on the right. These two results are well-known in de-
rivedalgebraicgeometryandarefolk-loreindifferentialgeometry,wherewewereunabletofind
an explicit proof in the literature. In the end, this machinery is used to provide what the author
believesisanewperspectiveontheintegrabilityproblemforG-structures.
iii
Acknowledgements
First and foremost I would like to thank my supervisor Spiro Karigiannis for spending so much
timeansweringmyquestionseveryweek,forgoingthroughthismonstrosityofadocumentwith
afine-toothcomb,andforrunningtheGeometricAnalysisWorkingSeminarinwhichIlearnedso
much.
I also want to thank Anton Borissov, Henry Liu and Raymond Cheng for running the Geometry
WorkingSeminaraswellastheMirrorSymmetrySeminarinthesummerof2016.Muchofthisthe-
siswouldhavebeenfarmoredifficultformewithoutsomeoftheintuitionforgeometryIgainedin
thatseminar. Furthermore,Iextendmygratitudetomyfriendsandfamilyforbearingwithmein
thiscrazyyear. EspeciallymyparentsAngelinaandJim,andmysisterMichelle. Aspecialthanks
toGrantandIanfororganizingboardgamenights,Brandonfororganizingtrivianights,Samand
IliaforFriday,Parhamfortheofficeantics,andOsmanandDannyformakingtimespentinsome
ofthegrossestapartmentsbearable,ifnotenjoyable.
Finally,I’dliketothankBenoitCharbonneauforsparkingmyinitialinterestindifferentialgeome-
try,JasonBellforteachingmemostofthealgebraIknow,andAnduNicaforshowingmehowto
integrateininfinitedimensions(somethingwhichblowsmymindtothisday).
iv
Table of Contents
1 Preliminaries 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 TopologyandPDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3 HomotopicalAlgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.4 HomologicalAlgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2 AlgebraicMethodsinDifferentialGeometry 51
∞
2.1 C -Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.2 CofiberandFiberSequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.3 Chern-WeilTheoryandtheDerivedLoopSpace . . . . . . . . . . . . . . . . . . . . . 82
3 Applications 94
3.1 Prolongation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.2 DeformationTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.3 FormalIntegrabilityandtheSpencerComplex . . . . . . . . . . . . . . . . . . . . . . 110
3.4 TheFrölicher-NijenhuisandSchoutenBrackets . . . . . . . . . . . . . . . . . . . . . . 118
Bibliography 123
A CategoryTheoryReview 127
v
Chapter 1
Preliminaries
Throughoutthisthesisallofourcomplexeswillbecohomological. Givenacomplex A,we’llwrite
A[n]forthecomplexwith A[n]i = Ai+n anddifferentialmultipliedby(−1)n. Notationalconven-
tionsareintheappendix.
1.1 Introduction
In this section I’ll be describing the basic motivating problem for this thesis. Our main goal will
betoobtainathoroughunderstandingoftheintegrabilityproblemfor G-structures. Todothis, let’s
beginbydevelopinganintuitiveunderstandingofwhataG-structureis.
Let’sthinkabouttheuniversewelivein. Ahypotheticalhigh-schoolstudent,whenaskedwhatan
appropriate mathematical model for space is, might respond with R3. Indeed, the 3-dimensional
spaceweliveinisoftenhowwefirstthinkofthevectorspaceR3. Butthisisinaccurate. Forex-
ample, onecouldaskthishigh-schoolstudent: whereisthepoint (0,0,0) ∈ R3 locatedinspace?
In which directions do the three standard coordinate axes point? This line of questioning leads
ustopositthatperhapsspaceisbestrepresentedbyanaffineR3. Soonecanchooseanypointin
spaceandanythreeindependentdirectionvectoremanatingfromthatpointandfromthisonewill
obtainanidentificationofspacewithR3.
Butthisdescriptionalsofallsshort. Indeed,usingtheverysamethoughtprocess,onecouldarrive
at the conclusion that the surface of the earth is modelled by an affine R2, i.e. that the earth is
flat. Inthismoderneraweknowthistobefalse. Thepointbeingthatwhilelocallytheearthdoes
indeedappeartobeflat,globallythisisnotthecase. Forexample,weknowthatthesurfaceofthe
earthiscompact(atopologicalproperty)whileaffineR2isnot.
Assuch,wecannotassumethatspaceisanaffineR3apriori. Locallyitcertainlyseemstolooklike
onebutgloballytheremaybetopologicalnon-trivialities.Anotherwayinwhichspacediffersfrom
affineR3iscurvature.OnemeasuresdistanceonanaffineR3usingthestandardEuclideandistance
function. Returningtotheexampleofthesurfaceoftheearth,wecanrecallthatifonedrawsasuf-
ficientlylargetriangleonthegroundandsumstheinteriorangles, itispossibletoobtainavalue
largerthan180◦. ThisisnottrueforaffineR3. Although,ifoneisonlyallowedtodrawtriangles
on the ground in such a way that by standing at any corner one can see the whole triangle, then
thesumoftheinterioranglesshouldbeareasonablygoodapproximationof180◦. Thepointis,the
distance function on the surface differs non-trivially from the usual Euclidean distance function.
Thisdifferenceiscalledthecurvatureofthedistancefunction. Inthesamewaythatweshouldnot
haveassumedthesurfaceoftheearthwasflat(hadnocurvature)weshouldn’tassumespaceisflat.
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Sowenowhavetwopotentialwaysinwhichspacecoulddiffernon-triviallyfromaffineR3: the
topology and the curvature. It turns out that these two properties are distinct but are related in
subtleways. Let’snowdescribethetypeofmathematicalobjectwewillusetomodelspace.
Definition1.1.1. AsmoothmanifoldisaHausdorffsecondcountabletopologicalspaceMtogether
withanopencoverbyUi’sandcontinuousmaps ϕi : Ui → Rni whicharehomeomorphismsonto
their(open)imagessuchthat
ϕ ◦ϕ−1 : ϕ (U ∩U ) → ϕ (U ∩U )
i j j i j i i j
isadiffeomorphismforalli,jsuchthatU ∩U (cid:54)= ∅. Furthermore,wewillassumethatthecollec-
i j
tionofsuchpairs(U,ϕ )ismaximalandwecallthiscollectiona(maximal)atlas.Thepairs(U,ϕ )
i i i i
themselvesarecalledcharts.
A standard fact from differential geometry is that the n’s appearing in the above definition are
i
locallyconstantwithrespecttothetopologyon M. However,itispossibletohavenon-connected
manifolds for which the n’s tend to infinity since second-countability allows the number of con-
i
nectedcomponentstobecountablyinfinite.
We should mention a couple of things regarding the above definition. The assumption that our
spacebeHausdorffcanbethoughtofassayingthatitispossibletodistinguishbetweenanytwo
pointsinspace.Thesecondcountabilityassumptionisequivalent,fortopologicalspacessatisfying
all of the other assumptions in our definition, to having countably many connected components
eachofwhichismetrizable.
The last point in our definition was that the transition functions between charts (i.e. changes of
coordinates)wereassumedtobediffeomorphisms. Thisisessentiallysayingthattherulesofcal-
culusworkthesameinanycoordinatechart.Animportantadvantageofthisisthatitallowsoneto
performmultivariablecalculusgloballyonmanifoldsbysimplydoingconstructionsineachchart.
One thing worth observing is that we defined manifolds in dimensions other than just 3. This is
becauseotherobjectsthanjustspaceendupbeingmodelledwellbymanifolds. Forexample,the
surfaceoftheearthisa2-dimensionalmanifold, space-timeisa4-dimensionalmanifold, andthe
spaceofpossibleconfigurationsofarigidbodyinspacelookslike(atleastlocally)
R3(cid:111)SO(3).
This is a 6-dimensional manifold (and, in fact, a Lie group). The most important example of a
higher dimensional manifold for us will be the frame bundle of a manifold. Recall that the whole
pointofthinkingofspace-timeasamanifoldisthatitis, insomesense, themostgeneraltypeof
reasonable object for which one can choose a reference frame at each point. As it turns out, the
space of all reference frames at any point is itself a manifold. We describe this object in general
below.
Definition1.1.2. ForasmoothmanifoldMofdimensionn,theframebundleofMistheprincipal
GL(Rn)-bundleF → Mwhosefibreaboveapoint p ∈ Misgivenby
M
(F ) := {f : T M →Rn : f isalinearisomorphism}.
M p p
TheleftactionofGL(Rn)onF isgivenbypost-compositionoffunctions.
M
InthecasethatMisthemanifolddescribingspace,theframebundleF becomesimportantwhen
M
attempting to describe notions such as inertia and the introduction of units of measurement. In-
deed,afterabitofthoughtonecanrealizethatitisnighimpossibletodefineareasonablenotion
ofastationaryreferenceframe. Whileweoftenthinkofourselvesasbeing“stationary”whenlying
2
inourbeds,weknowthatwithrespecttothesunweareactuallymovingquitefastinanon-trivial
way. The point is: being stationary is a relation. One can only say something is stationary with
respecttosomethingelse.
Suppose I’m given a reference frame in space, thought of as a chart ϕ : U → R3, and let p ∈ U
be the point corresponding to 0 ∈ R3 (we say the chart is centered at p). The velocities of objects
withrespecttothisreferenceframeinvariousdirectionsaretypicallymeasuredwithrespecttothe
tangentvectors ϕ−1(e) ∈ T M wherethe e arethestandardbasisvectorsforR3. Weintuitively
∗,p i p i
want two reference frames at p to be called inertial with respect to one another if and only if the
differentialequationsgoverningourmeasurementsofthemotionofpointparticleswithrespectto
theabovevelocityvectorsshouldbethesame. Inparticular, measurementsofspeedmadeusing
thedotproductonR3andthevectors ϕ−1(e)shouldyieldthesameresults.
∗,p i
In order to relate reference frames at various points in space one would need to make use of the
frameLiegroupoid [42,43,55]howeverwewon’tbeconcerningourselveswiththis. Despitethis,I
believetheseobjectsareincrediblyimportanttowardsprogressindifferentialgeometryasIhope
willbecomeclearinsection2.3.
Speedissupposedtobethelengthofthevelocitybut,asofyet,wehavenocoordinate-independent
way of measuring the length of tangent vectors to M. However, speeds in reality can indeed be
measuredandsowewillassumethatourspacemanifold M comesequippedwithaRiemannian
metric
g ∈ Γ(M,Sym2T∗ )
M
i.e. a smoothly varying inner product on each tangent space. Notice that by multiplying g by a
positivegloballydefinedfunctionef on Mwechangethelengthscale(i.e. units)inwhichweare
measuringspeedswithoutaffectinghowanglesbetweenvelocityvectorsaremeasured.
With respect to the usual Fréchet topology, the space of all Riemannian metrics on M is an open
coneinΓ(M,Sym2T∗ ). Now,anyreferenceframe f ∈ (F ) atparisesfromalocalchartcentered
M M p
at pbytheinversefunctiontheorem,theexistenceofsolutionstoODEs,andourassumptionthat
ouratlasesaremaximal. Indeed,givenaframe f : T M →Rn weuselocalcoordinatescenteredat
p
ptolocallyextendtheinversesimagesofthestandardbasisvectors f−1(e) ∈ T Mtovectorfields
i p
X onourcoordinatepatchwith [X,X ] = 0. Thenoneobtainsourdesiredlocalchartbysucces-
i i j
sivelyflowingalongthesevectorfields. It’sworthmentioning,however,thatdifferentchartsat p
willoftenyieldthesameframeat p.
Given two frames f,g ∈ (F ) it then follows that measurements performed at p using these
M p
localcoordinateswillagreeifandonlyiftheyarerelatedbyanorthogonaltransformation,i.e. an
elementofthegroup
O(T M,g ) = {A ∈GL(T M) : g (AX ,AY ) = g (X ,Y ) forallX ,Y ∈ T M}.
p p p p p p p p p p p p
Referenceframesϕwithorthogonalvelocityvectorsϕ−1(e)ofunitlengtharedistinguishedinthe
∗,p i
sensethattoobtainthecorrectresultswhenperformingcomputationsinlocalcoordinatesweare
requiredtoworkinthem. Furthermore, bydistinguishingacompatiblecollectionofsuchframes
one actually specifies a metric g given by using the standard dot product on R3 in each distin-
guishedreferenceframe.
Thisisallrelatedtothenotionofinertiainthefollowingway. Firstofall, aRiemannianmetric g
determines a notion of parallel transport along curves in space, replacing the notion of translation
by vectors in R3. This allows us to relate choices of units at different points in space and is seen
mostnaturallyfromtheformalismofLiegroupoids,mentionedabove. Whilethisgeneralizesthe
3
actionofR3(cid:110)SO(3)onspace,thenotionofintertiacorrespondstoananalogousgeneralizationof
theactionoftheGalileangroup,whoseconnectedcomponentoftheidentityis
R4(cid:110)(R3(cid:110)SO(3)),
onspace-time. Allofthisgeneralizesfurtherasfollows.
Thisnotionofadistinguishedcollectionof“orthonormalframes”onspace Miswhatiscalledan
O(3)-structre(sinceeachofthegroupsO(T M,g ) isisomorphictoO(3)). Wenowlistexamples
p p
wherestructuresonmanifoldsarisingfromLiegroupsotherthanG =O(3)arisenaturally.
1. Inordertomeasurebothtranslationalandrotationalspeedsontheconfigurationspace Mof
arigidbody(recall:thislocallylookslikeR3(cid:111)SO(3))oneneedsaO(6)-structureonM. This
specifiesunitsforbothtranslationalandangularspeed;
2. Since the change of variables formula for integration involves an absolute value of the deter-
minantoftheJacobianmatrix,itfollowsthatifonehopestointegratetopdegreeformsona
manifold(asopposedtotheless-talked-aboutdensities)onerequiresaGL+(Rn)structure.
3. Hamilton’s equations in classical mechanics come from the existence of a globally defined
pairing of sorts between the generalized position and momentum coordinates on the phase
spaceofaclassicalmechanicalsystem.Mathematically,suchapairingispreciselyaSp(2n,R)-
structure.
4. When compactifying string theories, initially defined on R10 or R11, down to R4×M one
mustusea(possiblynon-flat)metriconMtogetherwiththeusualLorenzianmetriconR4in
ordertowritedowntheactionfunctional. Ifonehopestheassociatedquantumfieldtheory
toadmitacertainamountofsupersymmetry(anextensionofthecanonicallydefinedrepre-
sentation of the Poincaré algebra to some Lie superalgebra extension of it by Clifford mod-
ules) then the action functional, which is used to describe the inner product on the Hilbert
spacetheseLiealgebrasarebeingrepresentedasself-adjointoperators,needstohavecertain
symmetries. Thisamountstothemetricon MhavingholonomycontainedinU(3)orG re-
2
spectively,whichinturngivesrisetoU(3)andG -structuresonM. Furthermore,intheU(3)
2
case,fortheresultingtheorytoberenormalizableweendupactuallyneedingthemetricon
MtohaveholonomyinSU(3),givingrisetoaSU(3)-structure. [3,13,15,31,44]
Now that we have provided (hopefully) sufficient motivation, I will present a definition of a G-
structure. Thisformaldefinitionwillprobablybefairlynon-transparentandsoIwillfollowitup
withadiscussionattemptingtogivemoreintuition.
Definition 1.1.3. Fix a Lie group G together with a faithful representation G → GL(Rn). Then a
G-structure on a n-dimensional manifold M is a principal G-subbundle P of the frame bundle F
(wheretheG-equivarianceoftheinclusionP ⊆ Fisdescribedusingourgiveninjectivehomomor-
phismG →GL(Rn)).
Earlierwementionedtwodistinct,butrelated,waysinwhichthesurfaceoftheearthdifferedfrom
affineR2. Oneofthemwastopological,namelythefactthatthesurfaceoftheearthwascompact,
andtheotherwaswithregardstothemetric, i.e. O(2)-structure. Let’snowseeourfirstexample
of the interplay between these two properties. The claim is: the existence of a G-structure on a
manifoldimposesrestrictionsonthetopologyofthemanifold.
Indeed, what is a G-structure? A principal G-bundle on M is a surjective submersion P → M
together with a free and proper smooth left action of G on P which preserves the fibres and has
P/G ∼= M via the submersion P → M. Now, a G-structure is a principal G-subbundle P of the
framebundleFinsuchawaythattheG-equivarianceoftheinclusionP ⊆ Fisgivenbyaspecified
4
faithfulrepresentationG ⊆GL(Rn). Inotherwords,GisnotmerelyaLiegroupbutisinfactaLie
subgroupofGL(Rn)togetherwithachoiceofinclusionG ⊆GL(Rn).
Now,givensuchasubgroupG ⊆GL(Rn)wecanlookatalocaltrivializationoftheframebundle
F| Rn×GL(Rn)
U
U Rn
Bypulling-backtheinclusionRn×G ⊆Rn×GL(Rn)wegetaprincipalG-subbundleofF| (only
U
defined over U). In other words, via the action of GL(Rn) on F| we also have an action of G
U
on F| . However, while the action of GL(Rn) is transitive on each fibre, this is not the case for
U
theactionof G andsothequotient F| /G willnotbediffeomorphictoU ⊆ M viatheprojection
U
F| →U. Instead,thequotientisgivenbytheassociatedfibrebundle
U
F|U/G ∼= F|U×GL(Rn)(GL(Rn)/G).
NoticethatthequotientGL(Rn)/GisindeedasmoothmanifoldsincetheactionofGonGL(Rn)by
leftmultiplicationisfree(ifgh = hthenwesimplyright-cancelh)andifwehavesequencesg ∈ G,
i
h ∈ GL(Rn)suchthath → hand gh → athenbycontinuityof(−)−1 wehaveh−1 → h−1 and
i i i i i
furthermorebythecontinuityofmultiplicationwehave
g = (gh )h−1 → ah−1
i i i i
and hence g (and, a fortiori, some subsequence of it) converges. Thus the action is also proper
i
since the connected components of a manifold are metrizable and on metric spaces compactness
andsequentialcompactnessagree. ThusthequotientGL(Rn)/Gisnaturallyasmoothmanifold.
So, by pulling back Rn×G ⊆ Rn×GL(Rn) to obtain a principal G-subbundle of F| , we have
U
really chosen a global section of the above associated fiber bundle. Namely, a smoothly varying
choiceof G-orbitineachfiberoftheframebundle. Inquotientingthissubmanifoldbytheaction
of G, we then obtaina diffeomorphism withU ⊆ M whereU is identified withthe image of our
globallychosensectionoftheassociatedbundleinthequotientofoursubmanifold. Fromthiswe
obtainthefollowingpropositionwhichcanbefoundin [51].
Proposition 1.1.4. Let M be a smooth n-dimensional manifold and G ⊆ GL(Rn) a Lie subgroup. Then
G-structuresonMareinanaturalbijectivecorrespondencewithglobalsectionsoftheassociatedfiberbundle
F×GL(Rn)(GL(Rn)/G).
Proof. Via the identification of the associated fiber bundle with F/G we see that given a global
sectionof F/G,thepreimageofitsimagein F/Gthrough F → F/GisaprincipalG-subbundleof
F. Conversely,givenaprincipal G-subbundle P ⊆ F wenotethatthefiberof Poveranypointof
M is sent to a single point of F/G under the quotient map. This gives a globally defined section
whichcanbeshowntobesmoothusinganatlasof MwhichtrivializesPlocally.
Some first examples of Lie groups G for which there are topological obstructions to the existence
of G-structures on a manifold would be any Lie subgroup of GL+(Rn) ⊆ GL(Rn). Any such
structuredetermines,inparticular,aGL+(Rn)-structureandhenceanorientation. Indeed,ifG ⊆
H ⊆GL(Rn)isachainofLiesubgroupsthenthequotientmap
GL(Rn)/G →GL(Rn)/H
yieldsamorphismoffiberbundles
F/G F/H
M
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