Holonomy and parallel transport in the differential geometry of the space of loops and the groupoid of generalized gauge transformations

0.5 MB·English

by  C. A. Rossi

HOLONOMY AND PARALLEL TRANSPORT IN THE DIFFERENTIAL
GEOMETRY OF THE SPACE OF LOOPS AND THE GROUPOID OF
4
0 GENERALIZED GAUGE TRANSFORMATIONS
0
2
CARLOA.ROSSI
n
a
J
ABSTRACT. Themotivationforthispaperstems[4]fromtheneedtoconstructexplicit
5 isomorphismsof(possiblynontrivial)principalG-bundlesonthespaceofloopsor,more
1 generally, ofpathsinsomemanifoldM,overwhichIconsiderafixedprincipalbundle
P;theaforementionedbundlesarethenpull-backsofP w.r.t.evaluationmapsatdifferent
] points.
G
Theexplicitconstructionoftheseisomorphismsbetweenpulled-backbundlesrelieson
D thenotionofparalleltransport.Iintroduceanddiscussextensivelyatthispointthenotion
ofgeneralizedgaugetransformationbetween(apriori)distinctprincipalG-bundlesover
.
h thesamebaseM; onecanseeimmediately thattheparalleltransportcanbeviewedas
t ageneralizedgaugetransformationfortwospecialkindofbundlesonthespaceofloops
a
orpaths;atthispoint,itispossibletogeneralizethepreviousargumentsformoregeneral
m
pulled-backbundles.
[ Finally,Idiscusshowflatnessofthereferenceconnection,w.r.t.whichIconsiderholo-

nomyandparalleltransport,isrelatedtohorizontalityoftheassociatedgeneralizedgauge
1
transformation.
v
0
8
1
1
CONTENTS
0
4 1. Introduction 2
0
2. Backgrounddefinitions 3
/
h 2.1. Thebridgebetweenconnection1-formsandhorizontalbundle 4
t 3. Holonomyandparalleltransport 5
a
m 3.1. Holonomy:definitionandmainproperties 6
3.2. Paralleltransport:definitionandmainproperties 7
:
v 4. Isomorphismsofprincipalbundlesandgeneralizedgaugetransformations 9
i
X 4.1. Thegroupoidofgeneralizedgaugetransformations 14
5. Somegeneralconstructionsforgroupoids 18
r
a 5.1. TheproductgroupoidoftwogroupoidsG,H 19
5.2. Morphismsofgroupoids 19
5.3. Theoppositegroupoid 20
5.4. Left-andrightG actionsforthegroupoidG 20
5.5. ThegeneralizedconjugationofG 21
5.6. Equivariantmapsbetweengroupoid-spaces 22
5.7. Someexplicitcomputationsforthegroupoidofgeneralizedgaugetransformations 23
6. Applicationstobundlesonthespaceofloopsorpaths 23
6.1. RestrictiontotheboundaryofI 27
7. Someconsequencesofflatnessrelatedtoholonomyandparalleltransport 28
7.1. Connectionsonthefibredproductoftwoprincipalbundles 29
C.A.RossiacknowledgespartialsupportofAlyKaufmanFellowship.
1
2 C.A.ROSSI
7.2. Compositionpropertiesandinversionoftheparalleltransport 30
7.3. Holonomyandflatness 32
7.4. Flatnessandparalleltransport 37
8. Applicationsofthegeneralizedgaugetransformationassociatedtotheparalleltransport:iteratedChenintegrals 41
8.1. Paralleltransportandsimplices 42
8.2. Flatnessandparalleltransportonsimplices 43
8.3. IteratedChen-typeintegrals 44
8.4. Restrictionstoboundaryfacesofsimplices 48
8.5. ThegeneralizedholonomyandChen’siteratedintegrals 48
AppendixA. Theorientationofthenthsimplex△ 54
n
References 55
1. INTRODUCTION
Inthelastsectionof[4],wheretheTopologicalQuantumFieldtheoreticalbackground
behindthe higher-ordercohomologygroupsof the space of knotsin Rm, for m ≥ 3, of
Cattaneo–Cotta-Ramusino–Longoni[2]wasexplicitlyconstructed,somehintsweremen-
tioned towards possible generalizations of the computationsin [4], dealing with iterated
Chen-typeintegrals,toanontrivialprincipalG-bundleP;infact,themainobjectthrough-
outthe paper was the so-called generalizedholonomy, viewed as an iterated integral, on
somefixedprincipalG-bundleP,whichweassumedtobetrivialinordertosimplifythe
computations.
Inthelastsection, weaddressedinformallytheproblemofdefiningiteratedintegrals,
asourexpressionfortheparalleltransport,foranontrivialbundleP. Thesameproblem
arisesalsoinSection2of[3]: theauthorsdiscussthenotionofiteratedChenintegralsin
relationshipwithso-called“specialconnections”onthespaceofhorizontalpathsina(not
necessarilytrivial)principalbundleovera4-manifoldM,namelytheyneediteratedChen
integralsof formsof the adjointtype on M in order to computethe curvatureof special
connections. AnexplicitformulaisdisplayedtherefortheChenbracket;still, astheau-
thorsthemselvespointedout,itiswithoutdetails,towhichtheyplannedtodedicatesome
furtherpaper.Oneofthemainsubtletiesofthetaskin[3]and[4]liestherein,thatweneed
to identifypulled-backbundlesof P on the spaceof loops(or, moregenerally,of paths)
in a generalmanifoldM w.r.t. evaluationmaps at differentpoints. We sketched without
detailsthereinsomeargumentsleadingtotheanswer;inparticular,wepointedouttheim-
portanceinthistaskoftheholonomyand,moregenerally,oftheparalleltransport.Infact,
theparalleltransport,whichdependsexplicitlybyitsveryconstructiononaconnectionA
onP,definesanisomorphismbetweentwoparticulartypeofpulled-backbundlesonthe
spaceofloopsorpathsinM.
Inthepresentpaper,Iexplainalldetailsofthisconstruction,performingthecomplete
computationswehintedatinthelastsectionof[4]. Thepaperisorganisedasfollows: in
Section2, Irecallthemainnotions,namelyconnectionsonprincipalG-bundlesoverthe
manifoldM,andIdiscussthetwoequivalentwaysofdefiningaconnection,sinceIwill
makeuseofthembothinthesubsequentcomputations.
In Section 3, I recall the notions of holonomy w.r.t. a chosen connection A of a loop
γ in M and parallel transport w.r.t. A of a path γ; I then state and prove two technical
lemmatacontaininginformationaboutsomesortofequivariancedisplayedbyholonomy
andparalleltransportw.r.t.theactionsofthestructuregroupGofP andthegaugegroup
HOLONOMYANDPARALLELTRANSPORT... 3
G of P. The contentsof the first two sections are standardfacts in gaugetheory; I will
P
nonethelessreviewtheminsomedetailtofixconventionsandnotations.
InSection4,IdiscussthenotionofisomorphismsofG-principalbundles(aprioridis-
tinct) over M.It is well known that fibre-preserving, G-equivariant automorphisms of a
fixed G-principalbundles are in one-to-onecorrespondenceto maps from P to G, equi-
variantw.r.t.conjugationonG;similarly,fibre-preserving,G-equivariant(iso)morphisms
between (a priori distinct) principal G-bundles over the same base space M are in one-
to-onecorrespondencetowhatIcallgeneralizedgaugetransformations. Thesearemaps
fromthefibredproductoftheaforementionedbundles(aconceptsomehowmimickingthe
notion of Whitney sum of vector bundles) with values in G, equivariant w.r.t. an action
of the product group G×G on G, which, when restricted to the diagonal subgroup G,
restrictstotheconjugationonG. Thekeyelementoftheabovecorrespondenceliesinthe
canonicalmapfromthefibredproductofabundleP withitselfassociatedtotheidentity
mapofP;Ialsodiscussitsproperties. Ithenproceedwiththediscussionofthestructure
ofthe setofgeneralizedgaugetransformations;itturnsouttobe (obviously)a groupoid
over the category of principal G-bundles over M; this notion generalizes that of gauge
group,andIthereforespeakofthe groupoidofgeneralizedgaugetransformationsofM
andG.
InSection5, I partlyreview,partlyexplainsome generalconstructionsforgroupoids;
inparticular,IrecalltheconceptofactionsofgroupoidsonsetsandconstructwhatIcall
thegeneralizedconjugationforgroupoids,whichisanactionoftheproductofagroupoid
with itself on the groupoiditself, which mimicssomehowthe conjugationof a groupon
itself, a notionwhichno longermakessense fora groupoid. I also discuss the notionof
equivariantmorphismsfrom(leftorright)groupoidactionstoother(leftorright)groupoid
actions.
InSection6,Iinterpretholonomyandparalleltransportasgeneralizedgaugetransfor-
mationsforaparticulartypeofpulled-backbundlesoverthespaceofloopsorpathsinM
of a fixed principalbundleP over M w.r.t. evaluationmaps. Thus, there is a map asso-
ciating to a connection on P a generalized gauge transformation; interpreting the gauge
groupofP asagroupoid,hencethespaceofconnectionsonP asaG -space,thetechni-
P
callemmataofSection3maybeinterpretedinthesensethatthereisanequivariantmap
of groupoidactions from the space of connectionsto the groupoidof generalized gauge
transformations.
In Section 7, I discuss the important consequences of the restriction of the map dis-
cussed in Section 6 between the space of connections and the groupoid of generalized
gauge transformationsto the space of flat connections. Namely, the flatness (which has
a non-abelian cohomologicalinterpretation) implies in a highly nontrivial way the hori-
zontality of a the corresponding generalized gauge transformation; the key step for this
achieving this result lies in the well-known fact that flat connections induce representa-
tionsofthehomotopygroupofM.
Acknowledgment. I thank A. S. Cattaneo and G. Felder for many inspiring suggestions,
mainlyonthesubjectofgroupoidsandforconstantsupport;Ialsoacknowledgethepleas-
antatmosphereat the Departmentof Mathematicsof the Technion,where thiswork was
accomplished.
2. BACKGROUND DEFINITIONS
Inthissection, I introducethe mainnotionsandnotationsI use throughoutthepaper;
amongthem,Iwanttodiscussindetailthenotionofconnectiononprincipalbundles.Let
4 C.A.ROSSI
π
menoticethatIworkonageneralprincipalbundleP 7→M overarealmanifoldM (ifnot
otherwise stated, M is assumed to be connectedand paracompact);I do notassume any
particularpropertyonthestructuregroupG. BygIdenotethecorrespondingLiealgebra.
Definition2.1. A connection1-formontheprincipalbundleP isa 1-formAonP with
valuesing,satisfyingthetwofollowingrequirements:
i) (Equivariance)
(2.1) R∗A=Ad(g−1)A,
g
wherebyR Ihavedenotedthe(free)rightactionofGonP.
g
ii) (Verticality)Foranyξ ∈g,
(2.2) A (T L(ξ))=ξ, ∀p∈P;
p e p
here,IhavedenotedbyR thefibreinjectionG→P givenbyg 7→pg.
p
Forourpurposes,itisbettertointroduceaslightlydifferentcharacterisationofconnec-
tions;later,Iwilldiscusstherelationshipbetweenthem.
Firstofall,atangentvectorX toP atthepointpissaidtobevertical,ifitsatisfiesthe
equation
T π(X)=0.
p
TheverticalspaceV P, consistingofallverticaltangentvectorsatp, foranyp ∈ P, is
p
isomorphictotheLie-algebragvia
ξ 7→T L (ξ), ξ ∈g.
e p
It turns out that the vertical spaces V P can be glued together to give a smooth vector
p
bundle, the so-called vertical bundle VP (whose typical fibre is isomorphic to the Lie
algebrag);itisclearlyasubbundleofTP.
Alas,thereisinprinciplenocanonicalwaytodefineacomplementofVP w.r.t.Whit-
neysum,i.e.thereisnocanonicalbundleHP suchthatVP ⊕HP =TP. Itturnsoutthat
thechoiceofsuchacomplementarybundlereliesonthechoiceofaconnection1-formA
onP,asitismotivatedbythefollowingdefinition:
Definition2.2. Givenaconnection1-formAonP,atangentvectorX toP atpissaid
p
tobeA-horizontalifthefollowingequationholds:
A (X )=0.
p p
(Forthesakeofbrevity,whentheconnection1-formAisclearfromthecontext,Isimply
speakofhorizontalvectors.)
2.1. Thebridgebetweenconnection1-formsandhorizontalbundle. Idiscuss,forthe
sakeofcompleteness,howaconnection1-formAonP givesrisetoasmoothassignment
toanyp∈P ofasubspaceH P ⊂T P,suchthatthefollowingtworequirementshold:
p p
i) T P =V P ⊕H P;
p p p
ii) H P =T R (H P),foranyg ∈G,
pg p g p
andviceversa. Infact,thetworequirementslistedabovearealsoanalternativedefinition
of connectionas a way of splitting the tangentbundle of P into the Whitney sum of the
verticalbundle,whichisclearlyG-invariantasaconsequenceoftheidentity
R ◦L =L ◦c g−1 , ∀p∈P,g ∈G,
g p pg
andsomehorizontalbundle. (cid:0) (cid:1)
HOLONOMYANDPARALLELTRANSPORT... 5
Given such an assignment, it is possible to define a correspondingconnection1-form
via
(2.3) A (X ): =ξ ,
p p Xp
where ξ is the uniqueelementof g which correspondsto the verticalpartof X w.r.t.
Xp p
theabovesplitting,i.e.
T L ξ =Xv,
e p Xp p
whereXv istheverticalpartofX w.r.t.theabovesplitting. Itfollowsimmediatelythat
p p (cid:0) (cid:1)
thehorizontalspaceatp ∈ P isexactlythe kernelofA . Theverydefinitionofvertical
p
space and Equation (2.3) imply together that A satisfies (2.2). The invariance given by
Conditionii)aboveensuresthatbothprojectionsintoverticalandhorizontalsubspaceare
G-invariant,whichinturnleadsto(2.1).
On the other hand, given a connection1-formA on P, the correspondingsplitting of
TP atsomepointp∈P isgivenby
X = X −XA +XA, XA: =T L (A (X )), p∈P.
p p p p p e p p p
Infact,by(2.2),iti(cid:0)seasytopr(cid:1)ovethatthelinearoperatorTeLp◦Apistheprojectiononto
theverticalsubspace;itskernelisthereforethehorizontalsubspacecorrespondingtothe
choiceofA. Condition(2.1),inturn,alongsidetheidentity
L ◦c g−1 =R ◦L , ∀p∈P,g ∈G,
pg g p
ensures that the correspondin(cid:0)g dis(cid:1)tribution is G-equivariant (c denotes the conjugation
on G). Finally, let me spend some words on the concept of gauge transformation of a
principalG-bundleP overM. AgaugetransformationσofP isa(smooth)mapfromP
toP,enjoyingthefollowingtwoproperties:
π◦σ =π, σ(pg)=σ(p)g, ∀p∈P,g ∈G.
ThefirstconditionmeansthatanygaugetransformationrespectsthefibresofP;thesecond
onemeansthatσisequivariantw.r.t.the(right)G-actiononP.Later,wewillseethatthere
isanotherwayofdefininggaugetransformationsofabundleP, but, forthemoment,let
meskiptheproblem.
AnotherproblemIwilladdressto lateristhat, infact, anygaugetransformationisan
isomorphism;theproofofthisfactiseasy,butIprefertopostponeittoSection4,deserving
itatreatmentinamoregeneralcontext.
Let me notice that this last fact that means that the set of gauge transformationsis a
groupw.r.t.theproductoperationgivenbycomposition;hence,itmakessensetospeakof
thegaugegroupofaprincipalbundleP,whichIdenotebyG .
P
Finally,thegaugegroupG operatesonthespaceofconnectionsonP,whichIdenote
P
byA = A ,tomakemanifestthedependenceonthechosenbundleP;the(right)action
P
isgivenexplicitlyby
Aσ: =σ∗A, A∈A ,σ ∈G .
P P
3. HOLONOMY AND PARALLELTRANSPORT
Let me considera curveγ on M; by the word “curve”, I mean in this context(if not
otherwisestated)apiecewisesmoothmapfromtheunitintervalItoM.
Definition3.1. Givenaconnection1-formAonP,ahorizontalliftofγbasedatp∈P is
acurveonP,lyingoverγ,basedatthepointpandsuchthatallitstangentdirectionsare
(A-)horizontal.
6 C.A.ROSSI
Iquotewithoutprooffrom[7],Chapter2,Section3,thefollowingTheorem,whichis
themainingredientofmanyofthesubsequentconstructions:
Theorem3.2. Givenaconnection1-formAonP andacurveγ inM,thereisaunique
horizontalliftofγbasedatthepointp,whichIdenotebyγ .
A,p
Now,itispossibletodisplaytwoimportantconsequencesofTheorem3.2.
e
3.1. Holonomy: definitionandmainproperties. Iconsideraloopγ,i.e.acurveinM
satisfying γ(0) = γ(1); I choose additionally a point p ∈ P lying over γ(0) = γ(1).
GivenaconnectionA,byTheorem3.2,thereisauniquehorizontalliftγ ofγbasedat
A,p
p. Sinceγ isaliftofγ,γ (1)alsoliesoverγ(0),and,asGactstransitivelyoneach
A,p A,p
fibreofP,itmakessensetoproposethefollowing
e
Definitioen3.3. Theholonoemyofγ w.r.t. theconnectionAandbasepointp overγ(0)is
theuniqueelementofG,usuallydenotedbyH(A;γ;p),satisfying
(3.1) γ (1)=pH(A;γ;p),
A,p
whereγ istheuniquehorizontalliftbasedatp∈P overγ(0),oftheloopγ.
A,p
e
The structuregroupG and the gaugegroupG operate fromthe righton P, resp. on
P
the spaece of connectionsA. The holonomydependsby its very constructionon a given
loopγ,aconnectionAonP andabasepointp∈P ;hence,itmakessensetoconsider
γ(0)
itasafunctiononthecartesianproductofthespaceofconnectionsAonP,thespaceof
loopswithvaluesinthestructuregroupGandthebundleP itself;later,wewillseethatit
ismorepreciselyasectionofsomeprincipalbundleoverit.
ThenextLemmashowshowtheholonomybehavesw.r.t.theactionofG,resp.ofG,on
P,resp.A.
Lemma 3.4. I assume g, resp. σ, to be an element of G, resp. a gauge transformation;
I denoteby g the functionfrom P with valuesin G canonicallyassociatedto σ via the
σ
formula
σ(p)=pg (p).
σ
(ThisformulamakessensebythetransitivityoftheactionofGonanyfibreofP.)
Then,thefollowingformulaehold:
(3.2) H(A;γ;pg)=c g−1 H(A;γ;p);
(3.3) H(Aσ;γ;p)=c(cid:0)gσ(p(cid:1))−1 H(A;γ;p).
Proof. ByDefinition3.3,theidentityhold(cid:0)s (cid:1)
γ (1)=pg H(A;γ;pg),
A,pg
whereγ istheuniquehorizontalliftofγbasedatpggivenbyTheorem3.2. Iconsider
A,pg
thecurveR (γ ): itisclearleybasedatpg,and(2.1)impliesthat
g A,p
e d d
A R (γ (t)) =A T R γ (t) =
Rg(γA,p(et)) dt g A,p Rg(γA,p(t)) γA,p(t) g dt A,p
(cid:18) (cid:19) (cid:20) (cid:18) (cid:19)(cid:21)
e e e d
e =Ad g−1 A γ (et) =0.
γA,p(t) dt A,p
(cid:20) (cid:18) (cid:19)(cid:21)
(cid:0) (cid:1)
Hence,thecurveR (γ )isalsohorizontal,itliesoveerγ (bytheG-invarianceofπ)and
g A,p e
isbasedatpg.BytheuniquenessinTheorem3.2,itfollowsR (γ )=γ ,whence
g A,p A,pg
e γ (1)=γ (1)g =p H(A;γ;p)g.
A,pg A,p
e e
e e
HOLONOMYANDPARALLELTRANSPORT... 7
Finally,thefreenessoftheactionofGimplies(3.2).
Similarly,againbyDefinition3.3,itholds
γ (1)=p H(Aσ;γ;p).
Aσ,p
Iclaimnow
e
(3.4) σ(γ )=γ .
Aσ,p A,σ(p)
Both curves σ(γAσ,p) and γA,σ(p) lie over γ, as σ is a gauge transformation, and both
e e
areclearlybasedatσ(p). Toproveequation(3.4),itsufficestoshowthatbothcurvesare
A-horizontalbyuniquenessofhorizontallifts. Adirectcomputationshows:
e e
d d
Aσ(γAσ,p(t)) dtσ(γAσ,p(t)) =(Aσ)γAσ,p(t) dtγAσ,p(t) =
(cid:18) (cid:19) (cid:18) (cid:19)
=0,
e e e e
wherethelastidentityisaconsequenceoftheAσ-horizontalityofγ . Hence,Identity
Aσ,p
(3.4)holdstrue.Therefore,onegets
e
σ(p) H(Aσ;γ;p)By=(3.4)σ(p) H(A;γ;σ(p))
SincetheactionofGisfree,itfollows
H(Aσ;γ;p)=H(A;γ;σ(p))=H(A;γ;pg (p))By=(3.2)
σ
=c g (p)−1 H(A;γ;p).
σ
(cid:0) (cid:1) (cid:3)
3.2. Paralleltransport: definitionandmainproperties. ThenextobjectIwanttode-
fineistheparalleltransportw.r.t.aconnectionAalongageneralcurveγ.
Definition3.5. Letγ bea curveinM, notnecessarilyclosed, andlett ∈ I andp ∈ P,
resp.q ∈P,besuchthat
π(p)=γ(0), resp. π(q)=γ(t).
Idefinetheparalleltransportfromptoq alongγ from0totw.r.t.A,usuallydenotedby
H(A;γ;t;p,q),astheuniqueelementofGobeyingtherule
(3.5) γ (t)=q H(A;γ;t;p,q),
A,p
whereγ isthehorizontalliftofγ.
A,p
e
Theparalleltransportdefinedbyequation(3.5)satisfiestwoidentitiessimilarinspirit
e
to(3.2)and(3.3).
Lemma3.6. GivenaconnectionAonP,acurveγ inM,p ∈ P andq ∈ P , for
γ(0) γ(t)
some t in the unit interval, general elements h and k of G and a gauge transformation
σ ∈G,theparalleltransportofγfromptoqalongγfrom0totsatisfiesthetwofollowing
identities:
(3.6) H(A;γ;t;pg,qh)=h−1H(A;γ;t;p,q)g;
(3.7) H(Aσ;γ;t;p,q)=g (q)−1H(A;γ;t;p,q)g (p).
σ σ
8 C.A.ROSSI
Proof. ByDefinition3.5,onehas
γ (t)=qh H(A;γ;t;p,qh)=q H(A;γ;t;p,q).
A,p
BythefreenessoftheactionofGonP,onegets
e
(3.8) H(A;γ;t;p,qh)=h−1H(A;γ;t;p,q), ∀h∈G.
Ontheotherhand,theidentityholds
γ (t)=q H(A;γ;t;pg,q)=
A,pg
=γ (t)g =
A,p
e
=q H(A;γ;t;p,q)g,
e
wherethesecondidentitywasshownintheproofofLemma3.4. Itfollowstherefore
(3.9) H(A;γ;t;pg,q)=H(A;γ;t;p,q)g, ∀g ∈G.
Combining(3.8)and(3.9),onegets(3.6).
Asforthesecondidentity,Imakeuseagainof(3.4):itthenholds
γ (t)=σ(q)H(A;γ;t;σ(p),σ(q))=
A,σ(p)
=σ(γAσ,p(t))=
e =σ(qH(Aσ;γ;t;p,q))=
e
=σ(q)H(A;γ;t;σ(p),σ(q)).
ThefreenessoftheactionofGimpliesthen
H(Aσ;γ;t;p,q)=H(A;γ;t;σ(p),σ(q))=
=H A;γ;t;pg ,qg (q) =
σ(p) σ
=gσ(cid:0)(q)−1H(A;γ;t;p,q)gσ(cid:1)(p),
wherethelastidentityisaconsequenceof(3.6). (cid:3)
Afterhavingintroducedtheparalleltransportofacurveγ from0to apointt w.r.t.A
andhavingdiscussedsomeofitsproperties,Imayalsointroduceanotherobject,namely
theparalleltransportofthecurveγfromstot,wheres,t∈Isatisfys<t.
Firstofall,givenacurveγfromtheunitintervalItoM,andgivens∈I,letmedefine
anewcurveγ fromtheinterval[0,1−s]bytheassignment
s
γ(t): =γ(t+s), ∀t∈[0,1−s].
s
Definition3.7. Letγ be acurveinM, definedon theunitintervalI; letthens < ttwo
pointsinI,andpandqtwopointsinP satisfying
π(p)=γ(s), π(q)=γ(t).
The parallel transport of γ from p to q w.r.t. the connection A, which I denote by
H(A;γ;s,t;p,q),istheuniqueelementofGobeyingtherule
(3.10) γ (t−s)=qH(A;γ;s,t;p,q).
sA,p
RecallingEquation(3.5),Equation(3.10)isequivalentto
e
(3.11) H(A;γ;s,t;p,q)=H(A;γ ;t−s;p,q).
s
Lemma3.6togetherwithEquation(3.11)impliestheuseful
HOLONOMYANDPARALLELTRANSPORT... 9
Corollary3.8. GivenaconnectionAonP,acurveγ inM,twopointss,t ∈ Iobeying
s<t,p∈P andq ∈P ,generalelementshandkofGandagaugetransformation
γ(s) γ(t)
σ ∈ G, the paralleltransportofγ from p to q alongγ from s to t satisfiesthe following
identities:
H(A;γ;s,t;pg,qh)=h−1H(A;γ;s,t;p,q)g;
H(Aσ;γ;s,t;p,q)=g (q)−1H(A;γ;s,t;p,q)g (p).
σ σ
4. ISOMORPHISMS OF PRINCIPAL BUNDLES AND GENERALIZEDGAUGE
TRANSFORMATIONS
π π
LetP → M,P → M twoprincipalbundlesoverthesamemanifoldM andwiththe
samestructuregroupeG; therightactionofGonP,resp.P, isdenotedbyR•, resp.R•,
orsimplyby(p,ge)7→pg,resp.(p,g)7→pg,giventhecase.
e e
Definition4.1. GiventwoprincipalbundlesP,P asabove,aG-equivariantmapσ from
e e
P toP isasmoothmapfromP toP satisfyingthetwoproperties:
e
π◦σ =π,
(4.1) e e
σ◦R =R ◦σ, ∀g ∈G.
g g
e
ThesetofallsuchmapsisdenotedbyG .
P,P
e
Remark4.2. ThesetG ,foragivenprincipalbundleP,isthesetofgaugetransforma-
P,P e
tionsofP.
GiventwoprincipalbundlesP,P overM asabove,itispossibletoformoutofthema
manifoldasfollows:
e
Definition4.3. GiventwoprincipalG-bundlesP andP overthesamebasespaceM,their
fibredproduct,denotedusuallybyP ⊙P,isdefinedas
e
P ⊙P: = (p,p)∈P ×P: π(p)=π(p) .
e
n o
ThereisanaturalmapπefromthefiberedproduectP ⊙P toeMe,whichissimply
π(p,p): =π(p)=π(p), (p,p)∈P ⊙P.
e
Additionally,P ⊙P receivesarightG×G-action:
e e e e e
(p,p;(g,h))7→(pg,ph), ∀(p,p)∈P ⊙P,(g,h)∈G×G.
e
Itisclearthattheaboveactionisfree,asbothactionsofGonP andP arefree;moreover,
consideringthefiebre e e e
P ⊙P : =π−1({x}), e
x
itfollowsimmediatelythatthe(cid:16)actionof(cid:17)G×Gistransitiveonit.
e
Thesetwofactsarenotincidental,becauseofthefollowing
Proposition4.4. The fibred product P ⊙P is a principal G×G-bundle over M, with
projectionπ.
e
Proof. If U is an open set of M, let me denote by ϕ , resp. ϕ , the trivialization of P
U U
overU,resp.ofP overU. Atrivializationϕ ofP ⊙P overU maythusbedefinedvia
U
ϕ: π−1(U)−→U ×(G×G) e
e e
(p,p)7−→(π(p);(pr ◦ϕ )(p),(pr ◦ϕ )(p)).
2 U 2 U
e e e
10 C.A.ROSSI
where pr denotes the projection from U ×G onto G. These maps are invertible, their
2
inversesbeinggivenby
ϕ−1: U ×(G×G)−→π−1(U)
U
(x;g,h)7−→ ϕ−1(x,g),ϕ−1(x,h) .
U U
Itisclearfromtheirdefinitionthatthemaps(cid:0)ϕ andtheirinversesa(cid:1)resmooth. Hence,I
U
e
haveobtainedatrivializationofthefibredproductP ⊙P.
For the sake of completeness, let me write down explicitly the transition maps of the
G×G-principalbundleP ⊙P: e
ϕ : U ∩V −→Diff(G×G)
U,V
e
x7−→R ×R ,
ϕU,V(x) ϕU,V(x)
whereϕU,V,resp.ϕU,V,arethetransitionmapsofP,reespe.P w.r.t.thetrivializationsϕU,
ϕ ,resp.ϕ ,ϕ ,forU,V anytwoopensubsetsofM withnontrivialintersection. (cid:3)
V U V
e e
Remark4.5. LetmenoticethatthefibredproductmaybealsoseenasaprincipalG-bundle
e e
overP:infact,byitsverydefinition,
P ⊙P =π∗(P),
and the latter manifold inherits clearly a principal G-bundle structure over P. (Equiva-
e e
lently,thefibredproductmaybealsoviewedasaG-bundleoverP,asP ⊙P =π∗(P).)
Remark 4.6. The fibred product of two principal bundles over the same base space and
e e e
with thesame structuregroupmaybeseen asananalogueoftheWhitneysumof vector
bundlesforprincipalbundles.
Furthermore,itisclearthatthereisacanonicalisomorphism
P ⊙P ∼=P ⊙P,
foranytwoprincipalbundlesP,P overthesamebasespaceM.
e e
Now,thereisaleftactionofG×GonGspecifiedbytherule:
e
c: G×G−→Diff(G)
(g,h)7−→c(g,h)k: =hkg−1.
ItisquiteevidentthattherestrictionofctothediagonalsubgroupGofG×Ggivesthe
usual conjugation of G on itself; therefore, one can speak of the above action as of the
generalizedconjugationinG.
Definition4.7. UnderthesamehypothesesasinDefinition4.3,thesetofsmoothG×G-
G×G
equivariant maps from P ⊙ P to G, denoted by C∞ P ⊙P,G , is the subset of
C∞ P ⊙P,G ofthosemapsK satisfyingtheequiva(cid:16)riancew.r.t(cid:17).thegeneralizedconju-
e e
gati(cid:16)on (cid:17)
e
(4.2) K(pg,ph)=c g−1,h−1 K(p,p), ∀(p,p)∈P ⊙P,(g,h)∈G×G.
ToagivenprincipalG(cid:0)-bundleP(cid:1)overM,onecanassociate thecanonicalmapφ on
e e e e P
P ⊙P withvaluesinGbytherule
(4.3) q =pφ (p,q),
P