Quantized Abelian principal connections on Lorentzian manifolds

32 Pages·2013·0.32 MB·English

Quantized Abelian principal connections on Lorentzian manifolds
Marco Benini1,3,a, ClaudioDappiaggi1,b andAlexanderSchenkel2,c
1DipartimentodiFisica
Universita`diPavia&INFN,sezionediPavia-ViaBassi6,27100Pavia,Italia.
2FachgruppeMathematik
BergischeUniversita¨tWuppertal,Gaußstraße20,42119Wuppertal,Germany.
3
3II.Institutfu¨rTheoretischePhysik
1
0 Universita¨tHamburg,LuruperChausse149,22761Hamburg,Germany.
2
r a [email protected],[email protected],c [email protected]
a
M
March12, 2013

1
1

Abstract
]
h
p WeconstructacovariantfunctorfromacategoryofAbelianprincipalbundlesovergloballyhyper-
- bolicspacetimestoacategoryof∗-algebrasthatdescribesquantizedprincipalconnections. We work
h
within an appropriatedifferentialgeometricsetting by using the bundleof connectionsand we study
t
a thefullgaugegroup,namelythegroupofverticalprincipalbundleautomorphisms. Propertiesofour
m
functorareinvestigatedindetailand,similartoearlierworks,itisfoundthatduetotopologicalobstruc-
[ tionsthelocalitypropertyoflocallycovariantquantumfieldtheoryisviolated.Furthermore,weprove

1 that,forAbelianstructuregroupscontaininganontrivialcompactfactor,thegaugeinvariantBorchers-
v Uhlmannalgebraofthevectordualofthebundleofconnectionsisnotseparatingongaugeequivalence
5 classes of principalconnections. We introducea topologicalgeneralizationof the conceptof locally
1
covariantquantumfields. Asexamples,weconstructforthefullsubcategoryofprincipalU(1)-bundles
5
twonaturaltransformationsfromsingularhomologyfunctorstothequantumfieldtheoryfunctorthat
2
. canbeinterpretedastheEulerclassandtheelectriccharge. Inthiscasewealsoprovethattheelectric
3
chargescanbeconsistentlysettozero,whichyieldsanotherquantumfieldtheoryfunctorthatsatisfies
0
allaxiomsoflocallycovariantquantumfieldtheory.
3
1
:
v Keywords: locally covariant quantum field theory, quantum field theory on curved spacetimes, gauge
Xi theoryonprincipal bundles
r
a
MSC2010: 81T20,81T05,81T13,53Cxx
1 Introduction
ThealgebraictheoryofquantumfieldsonLorentzianmanifoldshasmadetremendousdevelopmentssince
theintroduction oftheprinciple ofgenerallocalcovariancebyBrunetti, FredenhagenandVerch[BFV03],
seealso[FV12]. Mathematically,thisprinciplestatesthatanyreasonablequantumfieldtheoryhastobefor-
mulated by a covariant functor from a category of globally hyperbolic Lorentzian manifolds (spacetimes)
toacategoryofunital(C)∗-algebras,subjecttocertainphysicalconditions. Manyexamplesoflinearquan-
tumfieldtheoriessatisfyingtheaxiomsoflocallycovariantquantumfieldtheoryhavebeenconstructed in
the literature, see e.g. [BGP07, BG11] and references therein. The mathematical tools which are used in
these constructions is the theory of normally hyperbolic and Dirac-type operators on vector bundles over
spacetimes together with the CCR and CAR quantization functors. In our previous work [BDS12] we
1
havegeneralized theseconstructions toclasses ofoperators onaffinebundles overspacetimes. Inaddition
to these exactly tractable models, the techniques of locally covariant quantum field theory are essential
for the perturbative construction of interacting quantum field theories, see for example [BDF09], and the
generalization ofthespin-statistics theoremfromMinkowskispacetimetogeneral spacetimes[Ver01].
One of the weak points of the current status of algebraic quantum field theory is our incomplete un-
derstanding of the formulation of gauge theories. Even though there exist important results on the quan-
tization of electromagnetism [Dim92, Pfe09, DL12, DS13, SDH12], linearized general relativity [FH12]
and generic linear gauge theories [HS12], as wellas on the perturbative quantization ofinteracting gauge
theories[Hol08,FR13],therearestillopenproblemsthatdeserveadetailedstudy. Inparticular,thereisup
to now no satisfactory formulation of quantized electromagnetism for the following two reasons: Firstly,
applying canonical quantization techniques it has been found that electromagnetism violates the locality
axiom of locally covariant quantum field theory. This has been shown for the field strength algebra in
[DL12] and for the vector potential algebra in [SDH12]. The latter reference also gives an interpretation
of this feature in terms of Gauss’ law. Secondly, the differential geometric developments over the past
decades indicate that the natural language for formulating gauge theories of Yang-Mills type is that of
principal connections on principal G-bundles, which includes electromagnetism by choosing G = U(1).
Taking into account the principal bundle structure has far reaching consequences for the very principle
of general local covariance: Since principal connections can not be associated to spacetimes, but only to
principalbundlesoverspacetimes,thecategoryofspacetimesin[BFV03]shouldbereplacedbyacategory
of principal bundles over spacetimes. This notion of general local covariance for gauge theories of Yang-
Millstype appeared recently inthediscussion ofthelocally covariant charged Diracfield[Zah12],where
howevertheprincipalconnectionswereassumedtobenon-dynamicalbackgroundfields. Besidesthisnew
notionofgenerallocalcovarianceingaugetheoriesofYang-Millstype,theclassicalconfigurationspaceis
different totheoneusedinprevious works: Thesetofprincipal connections doesnotcarryavectorspace
structure, butitisanaffinespaceoverthevectorspaceofgaugepotentials. Thevectorspacestructureem-
ployedintheworks[Dim92,Pfe09,DS13,SDH12]comesfroma(necessarilynon-unique) fixingofsome
reference connection, whichisunnatural indifferential geometryandleadstotheunnecessary question of
independence ofthetheoryonthischoice[Hol08].
We outline the structure of our paper: In Section 2 we fix the notations and review some aspects
of the theory of Abelian principal bundles and principal connections. This material is essentially well-
knowninthedifferentialgeometryliterature,butwerequiresomedetailsthatgobeyondstandardtextbook
presentations and hence are worth for being discussed. In particular, we need a full-fledged study of the
bundleofconnections [Ati57]togetherwiththeactionofprincipalbundlemorphismsandthegaugegroup
(thegroupofverticalprincipalbundleautomorphisms)definedonit. Sectionsofthebundleofconnections,
thatisanaffinebundleoverthebasespace,areinbijectivecorrespondencewithprincipalconnectionforms
onthetotalspace, buttheyhavetheadvantage ofbeing fieldsonthebasespaceandnotonthetotalspace.
This has far reaching consequences when one studies dynamical equations of connections and causality
properties, sincethetotalspaceisnotequipped withaLorentzian metric.
InSection3weassociatetoanyAbelianprincipalbundleagaugeinvariantphasespaceforitsprincipal
connections by extending ideas from [BDS12] and [HS12]. Our notion of gauge invariance is dictated
by the principal bundle and in the general case differs from the one employed in [Dim92, Pfe09, DS13,
SDH12]. Thephasespaceisnotsymplectic,butonlyapresymplecticvectorspace,whoseradicalcontains
topological information tobediscussed inSection6.
We characterize explicitly the gauge invariant phase space and its radical in Section 4 by using Cˇech
cohomology. This leads to two interesting observations: Firstly, the gauge invariant phase space and its
radical for theories with a compact Abelian structure group exhibit a different structure with respect to
their counterparts with a non-compact Abelian structure group. Secondly, if the Abelian structure group
contains a compact factor, then the gauge invariant phase space is not separating on gauge equivalence
classes of principal connections. In particular, gauge non-equivalent flat connections can not be resolved.
The reason for this feature is that our gauge invariant phase space consists of affine functionals, but for
Abelianstructuregroupswithcompactfactorsthesetofgaugeequivalenceclassesofprincipalconnections
2
isingeneralnolongeranaffinespace. Thisshowsthatinthesecasesthestandard phasespaceofsections
of the vector dual of affine bundles introduced in [BDS12] has to be extended in order to be separating.
NaturalcandidatesforthisextensionareWilsonloops,whicharehowevertoosingularforastraightforward
description inalgebraic quantum fieldtheory. Wehopetocomebacktothisissueinfutureinvestigations.
TheresultsabovearecombinedinSection5toconstructacovariantfunctorfromacategoryofAbelian
principal bundles over spacetimes to a category of presymplectic vector spaces. Composing this functor
with the usual CCR-functor we obtain a quantum field theory functor that satisfies the causality property
andthetime-sliceaxiom. However,thelocalitypropertyof[BFV03]isviolated,confirmingthattheresults
of[DL12,SDH12]alsoholdtrueinourprincipal bundlegeometricapproach. Thisresultwasnotobvious
from the beginning, since our concept ofmorphisms and configuration space is different from the ones in
earlierinvestigations.
InSection6weextendtheconceptofalocallycovariantquantum fielddevelopedin[BFV03]towhat
wecall a ‘generally covariant topological quantum field’. Bythis wemean anatural transformation from
afunctor describing topological information to the quantum fieldtheory functor. Forthe full subcategory
of principal U(1)-bundles we provide two explicit examples where the functor describing topological in-
formationisasingularhomologyfunctor. Thenaturaltransformations arethenthecoherentassociation of
observables that measure the Euler class of the principal bundle and the electric charge, that is a certain
cohomology class.
Following the electric charge interpretation of the previous section (see also [SDH12] for an earlier
account) weshow in Section 7 that the electric charges can be consistently set to zero. This is physically
motivatedsinceinpureelectromagnetism, withoutthepresenceofcharged fields,therecannotbeelectric
charges. Theresulting quantum fieldtheory functor thensatisfiesinaddition tothecausality property and
the time-slice axiom also the locality property. With this we succeed in constructing a locally covariant
quantum fieldtheory.
2 Geometric preliminaries
In this work all manifolds will be of class C∞, Hausdorff and second-countable. If not stated otherwise,
mapsbetweenmanifolds areC∞.
2.1 Spacetimes
We briefly review some standard notions of spacetimes, see [BGP07, BG11, Wal12] for a more detailed
discussion.
LetM beamanifoldthatforlaterconvenienceweassumetobeoffinitetype,i.e.M possessesafinite
goodcoverU = {U } ,withI finite. ALorentzianmanifoldisatriple(M,o,g),whereM isamani-
α α∈I
fold(offinitetype),oisanorientationonM andgisaLorentzianmetriconM ofsignature(−,+,...,+).
Given atime-orientation t on aLorentzian manifold (M,o,g), wecall the quadruple (M,o,g,t) aspace-
time. Let(M,o,g,t)beaspacetimeandS ⊆M beasubset. Wedenotethecausalfuture/pastofS inM
byJ±(S). Furthermore, J (S) := J+(S)∪J−(S). Thesubset S ⊆ M iscalled causally compatible,
M M M M
if J±({x}) = J±({x}) ∩ S, for all x ∈ S. A Cauchy surface in a spacetime (M,o,g,t) is a subset
S M
Σ ⊆ M, which is met exactly once by every inextensible causal curve. A spacetime (M,o,g,t) is called
globallyhyperbolic,ifitcontains aCauchysurface.
2.2 Abelianprincipal bundles
Webrieflyreviewstandard notionsofprincipal bundlesandrefertothetextbook[KN96]formoredetails.
Definition 2.1. LetM be a manifold and G aLie group. A principal G-bundleover M isa pair (P,r),
whereP isamanifoldandr :P ×G → P , (p,g) 7→ r (p) =:pg isasmoothrightG-action, suchthat
g
(i) therightG-actionr isfree,
3
(ii) M = P/Gisthequotient oftheG-action randthecanonical projection π : P → M issmooth,
(iii) P is locally trivial, that is, there exists for every x ∈ M an open neighborhood U ⊆ M and a
diffeomorphism ψ : π−1[U] → U ×G, whichisG-equivariant, i.e.,forallp ∈ π−1[U]andg ∈ G,
ψ(pg) = ψ(p)g, and fibre preserving, i.e. pr ◦ψ = π. The right G-action on U ×G is given by,
1
for all x ∈ U and g,g′ ∈ G, (x,g)g′ := (x,gg′) and pr : U × G → U denotes the canonical
1
projection onthefirstfactor.
WecallP thetotalspace,M thebasespace,Gthestructuregroupandπ theprojection.
Definition2.2. LetM beamanifold,G aLiegroupand(P ,r )aprincipalG -bundleoverM ,i= 1,2.
i i i i i i
A principal bundlemap is a pair of smooth maps F = f : P → P ,φ : G → G , such that φ is a
1 2 1 2
homomorphism ofLiegroupsandf satisfies, forallp ∈ P andg ∈ G ,f(pg) = f(p)φ(g).
1 1
(cid:0) (cid:1)
Remark 2.3. For every principal bundle map F = f : P → P ,φ : G → G there exists a unique
1 2 1 2
smoothmapf : M → M ,suchthatthefollowingdiagram commutes:
1 2
(cid:0) (cid:1)
P f //P (2.1)
1 2
π1 π2
M(cid:15)(cid:15) f //M(cid:15)(cid:15)
1 2
Wenowdefineasuitablecategory ofAbelianprincipal bundlesoverspacetimes.
Definition2.4. ThecategoryPrBuGlobHypconsistsofthefollowingobjects andmorphisms:
• AnobjectinPrBuGlobHypisatripleΞ = (M,o,g,t),(G,h),(P,r) ,where(M,o,g,t) isaglob-
allyhyperbolicspacetime,GisaconnectedAbelianLiegroupwithbi-invariantpseudo-Riemannian
(cid:0) (cid:1)
metrichand(P,r)isaprincipal G-bundle overM.
• Amorphism betweentwoobjects Ξ ,i = 1,2,inPrBuGlobHypisaprincipal bundle mapF = f :
i
P → P ,φ : G → G ,suchthatφ :G → G isanisometryandf :M → M isanorientation
1 2 1 2 1 2 1 2
(cid:0)
and time-orientation preserving isometric embedding with f[M ] ⊆ M causally compatible and
1 2
(cid:1)
open.
Remark 2.5. The category PrBuGlobHyp is quite big in the sense that it contains principal bundles for
all possible connected Abelian structure groups. In physics it might be of interest to study only the case
G = U(1)whichcorrespondstoelectromagnetism. Thiscanbeachievedbyrestrictingallfunctorsthatwe
willconstruct inthispaper tothefullsubcategory PrBuGlobHypG definedbythesubcollection ofobjects
Ξ = (M,o,g,t),(G,h),(P,r) where G is fixed. We have decided to include a bi-invariant pseudo-
Riemannian metric hon the structure group G in the data of the category. This datum is equivalent to an
(cid:0) (cid:1)
ad-invariant innerproduct (possibly indefinite) ontheLiealgebra gofG,whichisrequired tospecify the
actionfunctional andtherewithacovariant Poissonbracket forthegaugetheory.
Let M be a manifold, G a Liegroup and (P,r) a principal G-bundle over M. Forevery manifold N
withasmoothleft G-action ρ : G×N → N , (g,ξ) 7→ gξ there existsafibrebundle overM associated
to (P,r) with N as typical fibre: Consider the Cartesian product P × N and define the following right
G-action P ×N ×G → P ×N , (p,ξ,g) 7→ (pg,g−1ξ). Denote by P := (P ×N)/G the quotient
N
of this right G-action and define the map π : P → M , [p,ξ] 7→ π(p), which is well-defined since
N N
π(pg) = π(p), for all p ∈ P and g ∈ G. The data (P ,M,π ,N) specifies a fibre bundle (the local
N N
trivialization isshowntoexistin[KN96]),whichwecallthe(N,ρ)-associated bundleto(P,r).
Ofparticular relevance forusisthecase whereN istheLiealgebra goftheLiegroup Gand ρisthe
adjointactionad : G×g → g. The(g,ad)-associated bundleiscalledtheadjointbundleoftheprincipal
bundle (P,r)andwedenoteitalsoby ad(P),M,π ,g . Wenotice thatthemetrichontheLiegroupG
g
specifiesafibremetricontheadjointbundle
(cid:0) (cid:1)
ad(P)× ad(P) → M ×R, ([p,ξ],[p′,ξ′]) 7→ (π(p),h(ξ,ξ′)). (2.2)
M
4
Lemma2.6. LetM beamanifold,GanAbelianLiegroupand(P,r)aprincipalG-bundleoverM. Then
ad(P) = M ×g,i.e.theadjointbundleistrivial.
Proof. SinceGisAbeliantheadjoint actionistrivial, whichimpliesad(P) = (P ×g)/G = P/G×g =
M ×g.
Anyprincipal bundle mapF = f : P → P ,φ : G → G induces avector bundle map F :
1 2 1 2 ad(P)
ad(P ) → ad(P ),f : M → M betweenthecorresponding adjoint bundles, where
1 2 1 2
(cid:0) (cid:1) (cid:0)
(cid:1)
F : ad(P )→ ad(P ), [p,ξ]7→ [f(p),φ (ξ)] (2.3)
ad(P) 1 2 ∗
and φ : g → g denotes the push-forward. Since φ isan isometry this vector bundle mappreserves the
∗ 1 2
fibre metrics. By Lemma2.6 wehave that for Abelian structure groups ad(P ) = M ×g , i = 1,2, and
i i i
thus(2.3)reads
F : M ×g → M ×g , (x,ξ) 7→ (f(x),φ (ξ)). (2.4)
M×g 1 1 2 2 ∗
Wewillnowshowthattheassociation oftheadjoint bundleisfunctorial.
Definition2.7. ThecategoryVeBuGlobHypconsists ofthefollowingobjectsandmorphisms:
• An object in VeBuGlobHyp is a pair (M,o,g,t),(V,M,πV,V) , where (M,o,g,t) is a globally
hyperbolic spacetime and(V,M,πV,V)isavectorbundleoverM.
(cid:0) (cid:1)
• Amorphismbetweentwoobjects (Mi,oi,gi,ti),(Vi,Mi,πiV,Vi) ,i = 1,2,inVeBuGlobHypisa
vectorbundle map f : V → V ,f : M → M ,suchthatf| : V | → V | isavector space
1 2(cid:0) 1 2 x (cid:1)1 x 2 f(x)
isomorphism, for all x ∈ M , and f : M → M is an orientation and time-orientation preserving
1 1 2
(cid:0) (cid:1)
isometricembedding withf[M ]⊆ M causallycompatible andopen.
1 2
Lemma2.8. Thereisacovariant functor Ad : PrBuGlobHyp → VeBuGlobHyp. Itisspecified onobjects
by Ad(Ξ) = (M,o,g,t),(ad(P),M,π ,g) and on morphisms by Ad(F) = (F ,f), with F
g ad(P) ad(P)
givenin(2.3).
(cid:0) (cid:1)
Proof. Let Ξ be an object in PrBuGlobHyp, then Ad(Ξ) = (M,o,g,t),(ad(P),M,π ,g) is an object
g
in VeBuGlobHyp. Let F be a morphism in PrBuGlobHyp, then Ad(F) = (F ,f) is a morphism in
(cid:0) ad(P) (cid:1)
VeBuGlobHyp,sincethepush-forward φ oftheisometryφisavectorspaceisomorphism.
∗
For the identity id = id : P → P,id : G → G we obtain id = id , φ = id and hence
Ξ P G P M ∗ g
by (2.3) it holds Ad(id ) = (id ,id ). For two morphisms F : Ξ → Ξ and F′ : Ξ → Ξ
Ξ (cid:0) ad(P) M (cid:1) 1 2 2 3
in PrBuGlobHyp we obtain Ad(F′ ◦ F) = (F′ ◦ F) ,(f′◦f) = F′ ◦ F ,f′ ◦ f =
ad(P) ad(P) ad(P)
Ad(F′)◦Ad(F).
(cid:0) (cid:1) (cid:0) (cid:1)
Remark 2.9. We can also associate functorially to any object Ξ in PrBuGlobHyp a vector bundle as
in VeBuGlobHyp equipped with the fibre metric (2.2) and to any morphism F in PrBuGlobHyp a vector
bundlemapasinVeBuGlobHypwhichpreservesthefibremetrics. Werefrainfromintroducingyetanother
notation foracategoryofvectorbundles withfibremetricsandrememberthisfactwhennecessary.
2.3 Principal connections
ConnectionsonprincipalbundlesconstitutethefundamentaldegreesoffreedomingaugetheoriesofYang-
Millstype. Inthissubsection wewillreviewtherelevantdefinitionsandproperties following[KN96].
Definition2.10. LetM beamanifold, GaLiegroupand(P,r)aprincipal G-bundle overM. Aconnec-
tionformon(P,r)isag-valuedone-form ω ∈ Ω1(P,g)satisfying:
(i) ω(Xξ)= ξ,forallξ ∈ gandp ∈ P,whereXξ ∈T P isthefundamentalvectoratpcorresponding
p p p
toξ.
5
(ii) r∗(ω) = ad (ω),forallg ∈ G.
g g−1
Wedenote thesetofallconnection formsbyCon(P).
Remark2.11. Dueto[KN96,ChapterII,Theorem2.1]thereexistsaconnection form,i.e.Con(P) 6= ∅.
Definition2.12. LetΩk(P,g)bethevectorspaceofg-valuedk-forms,k = 0,...,dim(P).
(i) Wecallη ∈ Ωk(P,g)G-equivariant, ifr∗(η) = ad (η),forallg ∈ G.
g g−1
(ii) Wecall η ∈ Ωk(P,g) horizontal, ifη(Y ,...,Y ) = 0whenever atleast one Y ∈ T P is vertical,
1 k i p
i.e.π (Y )= 0.
∗ i
ThevectorspaceofG-equivariant andhorizontal g-valuedk-formsisdenotedbyΩk (P,g)eqv.
hor
Proposition2.13. LetM beamanifold,GanAbelianLiegroupand(P,r)aprincipalG-bundleoverM.
Thenthereexistsalinearisomorphism betweenΩk (P,g)eqv andΩk(M,g),forallk = 0,...,dim(M).
hor
Proof. Letη ∈ Ωk (P,g)eqv be arbitrary. Wedefine an element η ∈ Ωk(M,g) by, for allX ,...,X ∈
hor 1 k
T M,x ∈ M,
x
η(X ,...,X ) := η(Y ,...,Y ), (2.5)
1 k 1 k
where Y ,...,Y ∈ T P are tangent vectors at p ∈ π−1[{x}], such that π (Y ) = X , for all i. Since
1 k p ∗ i i
η is horizontal, η does not depend on the choice of such Y . (We can in particular set Y = X ↑ω as the
i i ip
horizontal lift of X with respect to some connection.) Due to G-equivariance the construction does not
i
dependonthechoiceofp ∈ π−1[{x}].
Letnow η ∈ Ωk(M,g) bearbitrary and consider the pull-back η := π∗(η) ∈ Ωk(P,g). This element
is G-equivariant, since, for all g ∈ G, r∗(η) = (π ◦r )∗(η) = π∗(η) = η. Itis also horizontal, since for
g g
allY ,...,Y ∈ T P withatleastonevectorvertical(thisvectorisannihilated byπ )wehave
1 k p ∗
η(Y ,...,Y )= η π (Y ),...,π (Y ) = 0. (2.6)
1 k ∗ 1 ∗ k
Thesetwoidentifications providethedesiredvec(cid:0)torspaceisomorphis(cid:1)m.
Lemma2.14. LetM beamanifold,GanAbelianLiegroupand(P,r)aprincipalG-bundleoverM. Let
usdefinethemap
Φ :Con(P)×Ω1(M,g) → Con(P), (ω,η) 7→ Φ(ω,η) = ω+η . (2.7)
Then Con(P),Ω1(M,g),Φ isanaffine space.
Proof(cid:0). Theone-form ω+η ∈(cid:1) Ω1(P,g)isanelement inCon(P),sinceη ishorizontal andG-equivariant.
Theaction(2.7)isfreeandtransitive.
Definition 2.15. Let M be a manifold, G a Lie group and (P,r) a principal G-bundle over M. The
curvatureisgivenbythefollowingmap
1
F :Con(P) → Ω2 (P,g)eqv , ω 7→ F(ω) = dω+ [ω,ω] , (2.8)
hor 2 g
where d is the exterior differential and [·,·] denotes the Lie bracket on g. In case G is Abelian, the
g
curvature reads,forallω ∈Con(P),F(ω) = dω,sincetheLiebracket istrivial.
Remark 2.16. Let G be an Abelian Lie group. Applying Proposition 2.13 we can consider equivalently
thecurvature asamap
F :Con(P) → Ω2(M,g), ω 7→ F(ω) = F(ω) = dω . (2.9)
As a consequence of the (Abelian) Bianchi identity dF(ω) = ddω = 0, for all ω ∈ Con(P), we obtain
thatF(ω)∈ Ω2(M,g)isclosed, forallω ∈Con(P).
6
Lemma2.17. LetM beamanifold,GanAbelianLiegroupand(P,r)aprincipalG-bundleoverM. The
mapF : Con(P) → Ω2(M,g)isanaffinemapwithlinearpartF : Ω1(M,g) → Ω2(M,g), η 7→ dη.
V
Proof. Letω ∈ Con(P)andη ∈ Ω1(M,g)bearbitrary, then
F(ω+η) = dω+dη = F(ω)+dπ∗(η) =F(ω)+π∗(dη) = F(ω)+dη . (2.10)
2.4 The Atiyahsequence
WepresenttheAtiyahsequenceonlyforAbelianprincipalbundlesandreferto[Ati57]forthegeneralcase.
Letus consider the tangent bundle TP,P,π ,Rdim(P) over P. Onthe total space TP there isa right
TP
G-action intermsofthepush-forward oftangent vectors
(cid:0) (cid:1)
r : TP ×G → TP , (Y,g) 7→ r (Y). (2.11)
∗ g∗
For any Y ∈ T P we have r (Y) ∈ T P and hence π ◦ r = r ◦ π , for all g ∈ G. In
p g∗ pg TP g∗ g TP
other words, π : TP → P is G-equivariant. As a consequence, we can define the quotient bundle
TP
TP/G,P/G,π ◦π ,Rdim(P) ,whichisavectorbundle overM = P/G. Wedenote theprojection of
TP
thisvectorbundlebyπ := π◦π .
(cid:0) TP/G (cid:1) TP
The push-forward of π : P → M gives a vector bundle map from the tangent bundle over P to the
tangentbundle TM,M,π ,Rdim(M) overM,i.e.thefollowingdiagramcommutes:
TM
(cid:0) (cid:1)TP π∗ //TM (2.12)
πTP πTM
(cid:15)(cid:15) (cid:15)(cid:15)
P π //M
Since π◦r = π, forall g ∈ G, andthus also π ◦r = (π ◦r ) = π ,for allg ∈ G, wecan perform
g ∗ g∗ g ∗ ∗
the quotient by G and obtain the vector bundle map (denoted with a slight abuse of notation by the same
symbol):
TP/G π∗ //TM (2.13)
πTP/G πTM
(cid:15)(cid:15) (cid:15)(cid:15)
M idM //M
Thereisalsoavectorbundle mapfromtheadjointbundle (rememberthatad(P) = M ×gsinceGis
Abelian,cf.Lemma2.6) M ×g,M,pr ,g to TP/G,M,π ,Rdim(P) :
1 TP/G
(cid:0) M ×g(cid:1) (cid:0) ι // TP/G (cid:1) (2.14)
pr1 πTP/G
(cid:15)(cid:15) (cid:15)(cid:15)
M idM //M
Themapιisdefinedby,forall(x,ξ) ∈ M×g,ι(x,ξ) := Xξ ,whereXξ ∈ T P isthefundamentalvector
p p p
corresponding to ξ and p ∈ π−1[{x}] is arbitrary. Indeed, the map ι is well-defined, since for any other
(cid:2) (cid:3)
p′ ∈ π−1[{x}] there exists a g ∈ G, such that p′ = pg and hence Xξ = Xξ = r (Xadg(ξ)) =
p′ pg g∗ p
r (Xξ) = Xξ ,wherewehaveagainusedthatGisAbelian.
g∗ p p (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
(cid:2) Consid(cid:3)er n(cid:2)ow(cid:3)also the trivial vector bundle M × {0},M,pr ,{0} and the following two vector
1
bundlemaps
(cid:0) (cid:1)
M ×{0} α // M ×g (2.15a)
pr pr
1 1
(cid:15)(cid:15) (cid:15)(cid:15)
M idM //M
7
withα(x,0) = (x,0),forallx ∈ M,and
TM β //M ×{0} (2.15b)
πTM pr1
(cid:15)(cid:15) (cid:15)(cid:15)
M idM //M
withβ(X) = (π (X),0), forallX ∈ TM.
TM
Composing(2.15a),(2.14),(2.13)and(2.15b)weobtainthefollowingsequenceofvectorbundlemaps
(wecandropthebasespacemapssincetheyareallgivenbyid )
M
M ×{0} α // M ×g ι //TP/G π∗ // TM β //M ×{0}. (2.16)
ThisistheAtiyahsequence[Ati57]. Forcompleteness, wereviewthefollowing
Proposition 2.18. TheAtiyahsequence (2.16)isashortexactsequence.
Proof. First,wehavetoshowthatthecompositionoftwosubsequentmapsisthetrivialmap,i.e.thevector
bundle map which restricted to all fibres is 0. For ι◦α this property holds true due to linearity. Let now
(x,ξ) ∈ M ×g,thenπ (ι(x,ξ)) = π (Xξ) = 0,since Xξ isbyconstruction avertical vector. Forβ ◦π
∗ ∗ p p ∗
thisproperty holdstrivially.
Next, wehave to prove exactness at every step: Let (x,ξ) ∈ M ×g be such that ι(x,ξ) = [Xξ] = 0.
p
This implies that Xξ = 0 and since X• is a vector space isomorphism between g and vertical vectors
p p
at p ∈ P we find ξ = 0. Let now [Y] ∈ TP/G| be such that π ([Y]) = 0. This implies that any
x ∗
representative Y ∈ T P (where p ∈ π−1[{x}]) is vertical and due to the aforementioned isomorphism
p
there exists a ξ ∈ g, such that ι(x,ξ) = [Xξ] = [Y]. For the last step let X ∈ T M be such that
p x
β(X) = (x,0). ThisconditionissatisfiedforallX. UsingalocaltrivializationofP wecanliftX ∈T M
x
toavectorX ∈ T P (wherep ∈ π−1[{x}]),suchthatπ (X) = X. Theequivalenceclass[X] ∈ TP/G|
p ∗ x
istheelementwhichprovesexactness atthisstep.
b b b
SimilartoLemma2.8,onecanshowthatallvectorbundlesappearingintheAtiyahsequence(2.16)are
assignedbyacovariantfunctorfromPrBuGlobHyptoVeBuGlobHyp. Wedonotrepeatallthestepsinthis
proof and just give an explicit expression for the induced maps: Let F = f : P → P ,φ : G → G
1 2 1 2
be a morphism between two objects Ξ = (M ,o ,g ,t ),(G ,h ),(P ,r ) , i = 1,2, in PrBuGlobHyp.
i i i i i i i i i
(cid:0) (cid:1)
Thentheinducedvectorbundlemaps(coveringf)aregivenby
(cid:0) (cid:1)
F :M ×{0} → M ×{0}, (x,0) 7→ (f(x),0), (2.17a)
M×{0} 1 2
F :M ×g → M ×g , (x,ξ) 7→ (f(x),φ (ξ)), (2.17b)
M×g 1 1 2 2 ∗
F : TP /G → TP /G , [Y]7→ [f (Y)], (2.17c)
TP/G 1 1 2 2 ∗
F : TM → TM , X 7→ f (X). (2.17d)
TM 1 2
∗
Noticefurtherthatforα ,ι ,π ,β denotingthevectorbundlemapsintheAtiyahsequence(2.16)forthe
i i i∗ i
objectΞ inPrBuGlobHyp,i = 1,2,weobtainthecommutingdiagram:
i
M ×{0} α1 //M ×g ι1 //TP /G π1∗ //TM β1 // M ×{0} (2.18)
1 1 1 1 1 1 1
FM×{0} FM×g FTP/G FTM FM×{0}
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
M ×{0} α2 //M ×g ι2 // TP /G π2∗ //TM β2 // M ×{0}
2 2 2 2 2 2 2
8
2.5 The bundle ofconnections
We show that the affine space of connections Con(P),Ω1(M,g),Φ constructed in Lemma 2.14 is iso-
morphictotheaffinespaceofsectionsofanaffinebundleoverM.
(cid:0) (cid:1)
Consider the bundle Hom(TM,TP/G),M,πHom(TM,TP/G),HomR(Rdim(M),Rdim(P)) of homo-
morphisms, that is a vector bundle. Sections of this bundle are in bijective correspondence with vector
(cid:0) (cid:1)
bundlemaps λ :TM → TP/G,id :M → M . Wesaythatsuchavectorbundlemapisasplittingof
M
theAtiyahsequence(2.16),ifπ ◦λ = id . Thesesplittings canbedescribed equivalently bysections
∗ TM
(cid:0) (cid:1)
ofasubbundle ofHom(TM,TP/G).
Definition2.19. Thebundleofconnections C(P),M,π ,A isthesubbundleofthehomomorphism
C(P)
bundle Hom(TM,TP/G),M,πHom(TM,TP(cid:0)/G),HomR(Rdim(M(cid:1)),Rdim(P)) specified by the submani-
foldC(P) := λ ∈ Hom(TM,TP/G) :π ◦λ = id .
∗ TM
(cid:0) (cid:1)
Remark 2.20.(cid:8)The typical fibre A is the set of all linear(cid:9)maps L ∈ HomR(Rdim(M),Rdim(P)) satisfying
π∗ ◦ L = idRdim(M), where π∗ ∈ HomR(Rdim(P),Rdim(M)) is given in a basis {ei ∈ Rdim(M) : i =
1,...,dim(M)}and{E ∈ Rdim(P) :a = 1,...,dim(P)}by
a
f f
e , fora = i ∈ {1,...,dim(M)},
i
π (E )= (2.19)
∗ a
(0 , else.
Notice that Ais anaffine sfpace modeled on HomR(Rdim(M),Rdim(P)−dim(M)), whichis the typical fibre
ofthehomomorphism bundleHom(TM,M ×g).
Wedefineaffinebundles following[KMS93,Chapter6.22]and[BDS12].
Definition2.21. Anaffinebundleisatriple M,(A,M,πA,A),(V,M,πV,V) ,whereM isamanifold,
(A,M,πA,A)isafibrebundleoverM and(V,M,πV,V)isavectorbundle overM,suchthat
(cid:0) (cid:1)
(i) forallx ∈ M,thefibreA| isanaffinespacemodeledonV| ,
x x
(ii) thetypicalfibreAisanaffinespacemodeledonthetypicalfibreV,
(iii) for all x ∈ M, there exists a local bundle chart (U,ψ) of (A,M,πA,A) and a local vector bundle
chart (U,ψV) of (V,M,πV,V), such that, for all y ∈ U, ψ|y : A|y → A is an affine space isomor-
phism with linear part ψ|yV = ψV|y : V|y → V. Wecall thetriple (U,ψ,ψV)alocal affine bundle
chart.
Proposition 2.22. Let M be a manifold, G an Abelian Lie group and (P,r) a principal G-bundle over
M. The bundle of connections C(P),M,π ,A is an affine bundle modeled on Hom(TM,M ×
C(P)
g),M,πHom(TM,M×g),HomR(R(cid:0)dim(M),Rdim(g)) . (cid:1) (cid:0)
Proof. Using the vector bundle embedding ι(see(cid:1)(2.16)) wecan consider Hom(TM,M ×g)as avector
subbundle of the homomorphism bundle Hom(TM,TP/G). By definition, the bundle of connections is
also a subbundle of Hom(TM,TP/G). The vector space structure on the fibres of Hom(TM,TP/G)
induces anaffinespace structure onthefibresofC(P)withunderlying vectorspace givenbythefibresof
Hom(TM,M ×g). By Remark 2.20, the typical fibre A is an affine space modeled on the typical fibre
HomR(Rdim(M),Rdim(g)) of Hom(TM,M × g). The local vector bundle charts of Hom(TM,TP/G)
inducetherequired localaffinebundlecharts.
By [BDS12, Lemma 2.20], the set of sections Γ∞(M,C(P)) of the bundle of connections is an
affinespacemodeled ontheC∞(M)-module Γ∞(M,Hom(TM,M ×g)). Thelatterisisomorphic (asa
C∞(M)-module) to the g-valued one-forms on M, i.e. Ω1(M,g). Hence, Γ∞(M,C(P)),Ω1(M,g),Φ
is an affine space, with action Φ : Γ∞(M,C(P)) × Ω1(M,g) → Γ∞(M,C(P)) given by, for all λ ∈
(cid:0) (cid:1)
Γ∞(M,C(P)),η ∈ Ω1(M,g)andX ∈ T M,x ∈ M, e
x
e
Φ(λ,η) (X) := λ(X)+ι x,η(X) . (2.20)
(cid:0) (cid:1) (cid:0) (cid:1)
e
9
Proposition 2.23. Γ∞(M,C(P)),Ω1(M,g),Φ and Con(P),Ω1(M,g),Φ (cf. Lemma 2.14) are iso-
morphicasaffinespaces.
(cid:0) (cid:1) (cid:0) (cid:1)
e
Proof. Let ω ∈ Con(P) be arbitrary. We define an element λ ∈ Γ∞(M,C(P)) by, for all X ∈ T M,
ω x
x ∈ M,
λ (X) := [X↑ω], (2.21)
ω p
where p ∈ π−1[{x}] and X↑ω ∈ T P denotes the horizontal lift at p with respect to ω. By definition we
p p
have that π λ (X) = X. The equivalence class [X↑ω]isindependent on the choice of p,since for any
∗ ω p
other p′ ∈ π−1[{x}] there exists a g ∈ G, such that p′ = pg and hence X↑ω = X↑ω = r (X↑ω). The
(cid:0) (cid:1) p′ pg g∗ p
last equality follows from the G-equivariance of the horizontal subspaces, π (r (X↑ω)) = X and the
∗ g∗ p
uniqueness ofthehorizontal lift.
Let now λ ∈ Γ∞(M,C(P)) and Y ∈ T P with p ∈ π−1[{x}], x ∈ M, be arbitrary. By the splitting
p
lemma,theelement[Y]∈ TP/G| canbedecomposeduniquelyas[Y] = ι(x,ξ)+λ(X) = [Xξ]+λ(X),
x p
where ξ ∈ g and X ∈ T M. For the fixed element p ∈ π−1[{x}], there exist unique representatives
x
Xξ ∈T P of[Xξ]andX↑ ∈ T P ofλ(X),suchthatY = Xξ+X↑. Wedefineω ∈ Ω1(P,g)bysetting
p p p p p p p λ
ω (Y)= ω (Xξ +X↑)= ξ . (2.22)
λ λ p p
Condition (i) of Definition 2.10 is satisfied. Furthermore, ω is G-equivariant (since G is Abelian this
λ
meansG-invariant), forallY ∈ T P,
p
r∗(ω ) (Y) = ω r (Y) = ω (Xξ +X↑ )= ξ = ω (Y). (2.23)
g λ λ g∗ λ pg pg λ
This shows that ω (cid:0)∈ Con(cid:1)(P). The(cid:0)maps d(cid:1)efined above provide a bijection between Con(P) and
λ
Γ∞(M,C(P)).
We now show that they are also affine space isomorphisms. Let ω ∈ Con(P), η ∈ Ω1(M,g) and
consider ω′ := Φ(ω,η) = ω+η ∈ Con(P). Thecorresponding element λ ∈ Γ∞(M,C(P)) isdefined
ω′
by, for allX ∈ T M, λ (X) = [X↑ω′], withp ∈ π−1[{x}] arbitrary. Using that X↑ω′ = X↑ω +Xξ for
x ω′ p p p p
someξ ∈ g,wefind
0 = ω′ X↑ω′ =ω(X↑ω′)+η(X↑ω′) = ω(X↑ω +Xξ)+η(X) = ξ+η(X), (2.24)
p p p p p
henceξ = −η(X). W(cid:0)eobt(cid:1)ain
λ (X) = [X↑ω′]= [X↑ω +Xξ]= λ (X)+[Xξ]
Φ(ω,η) p p p ω p
= λ (X)+ι(x,ξ) = λ (X)−ι(x,η(X))
ω ω
= Φ(λ ,−η) (X), (2.25)
ω
which shows that the isomorphism λ :(cid:0)Con(P) →(cid:1)Γ∞(M,C(P)), ω 7→ λ is an affine space isomor-
• e ω
phismwithlinearpartΩ1(M,g) → Ω1(M,g), η 7→ −η.
Corollary2.24. Themap(denotedwithaslightabuseofnotationbythesamesymbolasthemapinLemma
2.17)
F : Γ∞(M,C(P)) → Ω2(M,g), λ 7→ F(λ) := F(ω ) (2.26)
λ
is an affine differential operator in the sense of [BDS12, Section 3] with linear part F : Ω1(M,g) →
V
Ω2(M,g), η 7→ −dη.
Proof. Combining Proposition 2.23 and Lemma 2.17 we obtain, for all λ ∈ Γ∞(M,C(P)) and η ∈
Ω1(M,g),
F Φ(λ,η) = F(ω )= F(Φ(ω ,−η)) =F(ω )−dη = F(λ)−dη . (2.27)
Φ(λ,η) λ λ
e
(cid:0) (cid:1)
e
10