CHAPTER

Principal bundles and connections

Motivation Gauge theory

The simplest example of a gauge theory in physics is electromagnetism Recall

Maxwells equations for an electromagentic eld

B homogeneous equations r B r E

t

and

r E r B E j inhomogeneous equations

t

Here we have chosen units so that and c are equal to E is the electric eld

B the magnetic eld j the current density and the electric charge density All of

these elds are functions on spacetime R with co ordinates x t and x x x x

One can rewrite Maxwells equations in more concise co ordinate free form Let the

electromagnetic eld strength F b e the following form on R

F dx E dx E dx E dx B dx dx B dx dx B dx dx

and dene a chargecurrent density form by

J d x dx dx dx j dx dx j dx dx j dx d x

k k

Also R R b e the star op erator with resp ect to the standard ori

entation and the Minkowski metric on R Then Maxwells equations are equivalent

to

dF homogeneous equations

and

d F J inhomogeneous equations

The inhomogeneous equations require the integrability condition dJ which is easily

recognized as charge conservation Note that the Equations are not quite as natural

as the homogenous Maxwellequations The homogeneous Maxwell equations make sense

1 k 

For any pseudoRiemannian manifold M g of dimension n g extends to a metric on T M

2

If M is oriented one therefore obtains a unique oriented volume form such that jjjj The

k nk 4

Ho dge star op erator M M is dened by the equation For R with

0 1 2 3 0 1 2 3

Minkowski metric one has dx dx dx dx thus eg dx dx dx dx

PRINCIPAL BUNDLES AND CONNECTIONS

on any manifold M while the inhomogeneous ones involve the choice of a pseudo

Riemannian metric g More precisely since multiplying g by a p ositive function do es

not change the star op erator on forms in R they dep end on the conformal structure

of g

The homogeneous Maxwell equations just say that F is a closed form Of course

on R any closed form is exact so F dA for some form A R called the

electromagnetic p otential The form A is dened up to a closed form On R any

closed form is exact so that any two p otentials A A for F are related by A A df

for some function f R One says that f denes a gauge transformation of the

p otential A At this stage the p otential A on R itself do es not have physical meaning

Only F dA ie the gauge equivalence class of A can b e interpreted in terms of B

and E and can b e measured in exp eriments

Using A the inhomogeneous Maxwell equations can b e obtained as EulerLagrange

equations for a Lagrange density a form

L d A dA A J

The rst term in this expression can also b e written jjF jj dx dx dx dx by

A

denition of the star op erator In terms of B and E jjF jj jjB jj jjE jj The

A

functional

R

A dA dA

4

R

dened on p otentials A which decrease suciently fast at innity is a sp ecial case of

the YangMil ls functional

Gauge transformations b ecome more interesting if the electromagnetic eld is coupled

to some particle eld Consider for example an electron describ ed according to Diracs

theory by a wave function R C Here C accomo dates b oth the electron

particle and its antiparticle the p ositron Diracs equation for a free as a spin

P

m where the gammamatrices are matrices electron reads i

giving a representation of the Cliord algebra of R The equation for in a non

P

A dx are obtained by minimal coupling quantized electromagnetic eld A

ie replacing derivatives by covariant derivatives eA

i i

X

i eA m

This equation is invariant under the gauge transformations

if

A e A df

2 0

Note that on a nonsimply connected space time M it is p ossibly for two forms A A to have the

same dierential without b eing gauge equivalent Thus gauge equivalence of A is a ner invariant than

eld strength F dA The famous AharanovBohm exp eriment shows that the gauge equivalence

A

class does have physical meaning

PRINCIPAL BUNDLES AND CONNECTIONS

if

for f R R Note that only g e R U really enters the gauge transfor

mations In terms of g

A g A idg g

In this sense the theory of an electromagnetic eld is a gauge theory with gauge group

G fg R U Note that the theory of an electron without an EM eld only has

a global U gauge symmetry Electromagnetism is necessary to turn the global U

gauge symmetry into a lo cal gauge symmetry

The idea of YangMills theory is to replace the ab elian gauge group U by non

commutative Lie groups G The gauge elds A are now forms with values in the Lie

algebra g of G Again there is a notion of curvature form F which is interpreted as

A

a eld strength The coupling of this gauge eld to a particle eld with values in some

complex inner pro duct space V dep ends on the choice of a unitary representation of G

on V and replacing derivatives by covariant derivatives in such a way that the coupled

theory b ecomes gauge invariant

Principal bundles and connections

In this section we review basic material on principal bundles connections curvature

and parallel transp ort

Fib er bundles Let F b e a given manifold A smo oth b er bundle with

standard ber F is a smo oth map E B from a manifold E the total space to a

manifold B called base with the following prop erty called local triviality There exists

an op en covering U of B and dieomorphisms

U U F

such that pr x x That is intertwines with pro jection to the rst

U

factor We will denote the b ers by E b

b

A morphism of b er bundles E B with b ers F is a smo oth map

j j j j

E E taking b ers to b ers In particular induces a map B B on the base

Conversely supp ose E B is a b er bundle with b er F and Y B is a smo oth

map Then one can dene a pullback bundle E Y as a b ered pro duct

E fy x Y E j y xg

By construction one has a bundle map E E The bundle E is called trivializable if

there exists a bundle map E B F the choice of is called a trivialization

A sp ecial case of this construction is the bered product of two b er bundles E E

B over the same base Let E E B B b e the direct pro duct and diag B

B

B B the diagonal map One denes

B

E E diag E E

B

B

Thus E E B has b ers

B

E E E E

b b b

PRINCIPAL BUNDLES AND CONNECTIONS

Additional structures on F give rise to sp ecial typ es of b er bundles

If F V is a vector space one denes a vector bund le with standard b er V to b e

a b er bundle E B where all b ers b are vector spaces and the lo cal

trivializations can b e chosen to b e b erwise linear A homomorphism of two

vector bundles is a b er bundle homomorphism that is b erwise linear The b ered

pro duct of vector bundles E E is a vector bundle also called Whitney sum and

denoted E E One has natural bundle maps E E E b erwise addition

and a map R E E b erwise scalar multiplication

If F G has the structure of a Lie group one denes a group bund le G B with

standard b er G to b e a b er bundle where all b ers carry group structures and

the lo cal trivializations can b e chosen to b e b erwise group homomorphisms A

group bundle homomorphism is a b er bundle homomorphism which is b erwise

a group homomorphism The b ered pro duct of group bundles is a group bundle

B

One has natural bundle maps G G G b erwise group multiplication and

G G b erwise inversion

Similarly one denes algebra bundles Lie algebra bundles An action of a group

B

bundle G B on a b er bundle E B is a smo oth map G E E preserving b ers

and dening a group action b erwise Similarly one can dene b erwise linear actions

of group or algebra bundles on vector bundles

Example Given a vector bundle E B one obtains a group bundle

GL E B with b er over b the invertible transformations automorphisms of the

vector space E b and an algebra bundle EndE B with b ers the en

b

B

domorphisms of E There are natural b erwise linear maps GL E E E and

b

B

End E E E

Principal bundles Let G b e a Lie group A principal homogeneous Gspace

is a manifold X together with a transitive free Gaction For example G itself with

action g a ag is a principal homogeneous space We could also use the leftaction

of course Any principal homogeneous Gspace X is isomorphic to this example Given

x X there is a unique Gequivariant isomorphism X G taking x to e The only

reason for intro ducing the p edantic notion of principal homogeneous Gspace is that

G

in general there is no distinguished choice of x Note that the group Di X of G

equivariant dieomorphisms of X to itself is dieomorphic to G but not canonically

G

so and that the space of Gequivariant dieomorphisms Di X G is a principal

homogeneous Gspace with G acting on the target G by left multiplication which is

canonically isomorphic to X itself

Example If V is a real vector space of dimension n the space FrV of linear

n

isomorphisms frames V R is a principal homogeneous GLn Rspace The

abstract vector space V can b e recovered from X as a quotient

n

V FrV R GL n R

PRINCIPAL BUNDLES AND CONNECTIONS

by the diagonal action In fact all the natural that is GL n Requivariant con

n

structions with R give rise to the corresp onding constructions of V by a similar quo

n

tient For instance P V FrV RP GL n R is the pro jectivization of V

k k n

V FrV R GL n R is the exterior algebra and so on

A principal Gbund le is a b er bundle P B where each b er P carries the

b

structure of a principal homogeneous Gspace and such that the lo cal trivializiations

can b e chosen to b e b erwise Gequivariant

Examples a Let S RZ the standard circle The k fold covering

S S is a principal bundle with structure group Z Zk Z

k

b Let E B b e a real vector bundle of rank n Then the frame bund le FrE

with b ers FrE FrE is a principal GLn R bundle

b b

c If in addition the b ers E carry inner pro ducts and orientations dep ending

b

smo othly on b B one can dene the bundle Fr E Fr E of sp ecial

SO b SO b

orthogonal frames which is a principal SOn bundle

A homomorphism of principal Gbund les P B is a Gequivariant smo oth

j j j

map P P taking b ers to b ers Any such map induces a smo oth map

B B on the base Conversely given a smo oth map Y B and given a principal

Gbundle P B one can form the pul lback bundle P Y using the b ered

pro duct of and That is

P fy p Y P j y pg

The bundle pro jection P Y is induced by pro jection Y P Y to the rst

factor and pro jection to the second factor Y P P gives rise to a principal bundle

homomorphism P P In fact any principal bundle homomorphism P P

identies P with the pullback of P by the map B B on the base

Asso ciated bundles Let P B b e a principal Gbund le Given a

Gmanifold F one denes the associated ber bund le by

F P P F P F G

G

The space P F is a b er bundle over B P G with standard b er F The sections

G

G

B P F of this b er bundle are naturally identied with the space C P F

G

of equivariant maps P F

Examples a If V is a vector space on which G acts linearly then P V

G

is a vector bundle Taking V g with the adjoint representation one obtains the

adjoint bund le gP P g

G

b If K is a Lie group on which G acts by automorphisms P K is a group bundle

G

Taking K G with G acting by the adjoint action one obtains a group bundle

GP P G which is also called the adjoint bundle It has gP as its Lie

G

algebra bundle

PRINCIPAL BUNDLES AND CONNECTIONS

c If F is a principal homogeneous H space on which G acts H equivariantly the

asso ciated bundle P F is a principal H bundle For example if F H G

G

with G acting by left multiplication one recovers P itself

d Let E b e a real vector bundle of rank n and FrE its Gln frame bundle The

n

bundle asso ciated to the dening representation of GL n on R is E itself

n

FrE R E

GLn

k n k n

The representation on R resp S R gives the antisymmetric resp symmetric

k k

p owers E and S E The bundle asso ciated to the conjugation action of GLn

on itself is the group bundle GL E For s R the real line bundle FrE R

GLn

s

dened by the representation A j detAj of GL n R is a real line bundle called

the bundle of sdensities The line bundle for the representation A detA is the

determinant line bundle detE The bundle to the contragredient representation

t n

A A on R is the dual bundle E

Gequivariant maps F F give rise to b er bundle homomorphisms F P

F P For instance if V is a Grepresentation the map

G V V a v av

is equivariant for the Gaction g a v g ag g v on G V It follows that one

obtains an action of the group bundle GP on V P

Sections Let E B b e a b er bundle with b er F A smo oth map

B E is called a section of E if Id The set of sections will b e denoted

B

B E For a vector bundle it is a vector space for an algebra bundle it is an

algebra for a group bundle it is a group Let P B b e a principal Gbundle and

F a Gmanifold The pullback bundle F P is canonically isomorphic to the trivial

bundle

F P P F

Hence pullback of sections gives a canonical isomorphism

G

C P F B F P

G

For sections B F P we will denote by C P F the corresp onding

equivariant function

Group bundles and vector bundles always have a distinguished section the identity

section resp zero section In general b er bundles need not admit any sections at all

A principal b er bundle has a section if and only if it is trivializable and the choice of a

section is equivalent to a choice of trivialization Indeed given P B G one can

dene a section b b e Conversely given one denes B G P by

b g g b

For vector bundles E B one can dene a space of E valued dierential forms

k k

B E B T M E

PRINCIPAL BUNDLES AND CONNECTIONS

It is a mo dule for the algebra B but do es not carry a natural dierential Supp ose

E P V is an asso ciated bundle Then there is a canonical identication

G

k G k k

P V P V B E

hor basic

The isomorphism is obtained as follows By construction we have a bundle homomor

phism P V E p v Gp v covering the bundle pro jection P B This

identies the pullback E P with the trivial bundle P V It follows that

induces a map

k k k

B E P E P V

The image takes values in horizontal forms since the pullback of a form under vanishes

on vectors tangent to the b ers which are Ginvariant since is Ginvariant It is

k

easy to check that this map is and onto P V

basic

We will often use this isomorphism since it is usually easier to work with vector

space valued forms rather than vectorbundle valued forms If B V P we will

G

denote by the corresp onding form in P V

hor

Gauge transformations A principal bundle automorphism is a G

equivariant dieomorphism P P taking b ers to b ers The group of principal

bundle automorphisms will b e denoted AutP The gauge group GauP AutP

consists of automorphisms P P inducing the identity map on the base B That

is Gau P is dened by an exact sequence of groups

Gau P AutP Di B

In general the last map is not onto

Any Gau P gives rise to a map P G by

p p p

The map is equivariant since

g p p g p g p g p g p

Conversely given any equivariant map P G the map dened by p

p p is a gauge transformation The map is a group homomorphism since

p p p p p p p p p p

It therefore denes a canonical isomorphism

G

C P G B GP GauP

G

The Lie algebra of the group AutP is the Lie algebra aut P XP of Ginvariant

vector elds on P and the Lie algebra of the gauge group is the subspace gau P of

vertical invariant vector elds One has an exact sequence of Lie algebras

gau P autP XB

PRINCIPAL BUNDLES AND CONNECTIONS

Here surjectivity of the map aut P XB can b e proved using a trivializing op en

cover U of B and a partition of unity sub ordinate to that cover gau P can b e identied

with sections of gP

B gP gau P

The group GauP acts on all asso ciated bundles F P since id P F P F

descends to quotients by the Gaction Furthermore it acts on the space of sections

B F P by b b One has

f

If F V is a vector space on which G acts linearly this action extends uniquely to

B V P in such a way that for B and B V P

One has

g

B V P

Innitesimally one has a Lie algebra action of gauP B gP this action extends

to an action of the graded Lie algebra B gP

Connections

Connections on b er bundles For any b er bundle E B the tangent

bundle TE of the total space has a distinguished subbundle the vertical bundle VE

TE The b er V E for x b is the image of T F under the natural inclusion TF

x x b b

TE An Ehresmann connection on E is the choice of a complementary horizontal

subbund le HE such that TE VE HE Equivalently a connection is a bundle

pro jection TE VE which is leftinverse to the inclusion VE TE one denes HE

as the kernel of this pro jection Ehresmann connections always exist For instance one

may take HE to b e the orthogonal complement of VE for some Riemannian metric on

the total space of E If Y B is a smo oth map and HE TE a given connection

one obtains a connection on the pullback bundle E Y by dening

H E H E

Here E E is the bundle map covering and T E TE its dierential

For any smo oth path t t B from b t to b t the connection

denes a paral lel transport

E E

b b

0 1

as follows Given p E let t t E b e the unique path with and

b

0

d

t H E Put p t The denition extends uniquely to piecewise

t

dt

smo oth paths in such a way that for any two paths such that the end p oint of

equals the initial p oint of

2 1 2 1

where is the concatenation of and This can b e nicely phrased in terms of

group oid homomorphisms but we wont do this here If one xes b b and considers

CONNECTIONS

only lo ops based at b one obtains a map from the group LM b of piecewise smo oth

lo ops based at b into Di b called holonomy of the connection

Hol LM b Di E

b

If B is connected the holonomy subgroups with resp ect to dierent base p oints b b are

isomorphic each choice of a path from b b denes an isomorphism A connection is

called at if the holonomy for any contractible lo op is trivial

Principal connections If F has additional structure one is interested in

connections such that parallel transp ort preserves that structure For example if E is a

vector bundle each should b e a linear map for group bundles it should b e a b erwise

group homomorphism and so on

To construct such connections it is most convenient to work with the corresp onding

principal bundle

For a principal Gbundle P B the map

d

P g V P p p j expt p

P t

dt

denes a canonical trivialization of the vertival bundle V P Since g p

P

Ad g p the trivialization is Gequivariant for the adjoint action of G on its Lie

g P

algebra

Hence an Ehresmann connection on P is equivalent to a vector bundle homomorphism

T P P g taking p to Equivalently a connection is a Liealgebra valued form

P

P g with

for all g

P

The connection is Gequivariant if and only if the pro jection map T P P g is

Gequivariant that is

g Ad for all g G

g

Definition A principal connection on a b er bundle is an equivariant Lie

G

algebra valued form P g such that for all g The space of

P

principal connections will b e denoted AP

For later reference let us note the innitesimal version of the invariance condition

L

P

Proposition The space AP of principal connections has a natural ane

structure with underlying vector space the space B gP of forms on B with values

in the adjoint bund le

Proof We rst show that AP is nonempty This is obvious if P is a trivial

bundle P B G If P is nontrivial cho ose an op en cover U of B with trivializations

U G and a sub ordinate partition of unity Over each U th U

trivialization denes a principal connection A global connection is given by

PRINCIPAL BUNDLES AND CONNECTIONS

P

The dierence b etween any two connections satises

P

G

B gP Conversely adding such a It is thus a form in P g

hor

form to a principal connection pro duces a new principal connection

By construction parallel transp ort with resp ect to a principal connection is a

Gequivariant map That is the holonomy at b B b ecomes a group homomorphism

G

Hol LB b Di P

b

G

G G and therefore Di P Any choice of a base p oint p P identies P

b b b

Connections on asso ciated bundles If F is any Gmanifold the principal

connection on P denes a connection on P F as follows Let q P F P F

G G

the quotient map and note that

H P TF kerq T P F

Hence q TF V P F and

G

H P F q H P

G

denes a complementary subbundle If G acts by automorphisms of a given structure

on F parallel transp ort on the asso ciated bundle preserves that structure

Say F is a vector space V with G acting linearly View P V as a Gequivariant b er

bundle over P The pullback of the connection on P V is given by the horizontal

G

subbundle H P V H P kerq the horizontal subspace of P together with

orbit directions Vectorelds taking values in H P preserve the linear structure as do

generating vector eld for the diagonal Gaction By equivariance parallel transp ort on

P V descends to parallel transp ort on the asso ciated bundle

k

Covariant derivative A form P is called horizontal if for

P

all and basic if in addition is Ginvariant Denote the space of horizontal kforms by

k k

P and the space of basic kforms by P Pullback induces an isomorphism

hor basic

k k

P A principal connection gives rise to a Gequivariant pro jection B

basic

op erator

k k

Hor P P

hor

The covariant derivative dened by is the comp osition

k k

d Hor d P P

hor

More generally let V b e a Grepresentation and E P V the corresp onding asso

G

ciated vector bundle As we remarked earlier pullback denes an isomorphism

k G k G k

P E P V B E

hor hor

since E P V canonically The dierential d extended to V valued forms pre

serves this space so it denes an op erator

k k

d B V P B V P

CONNECTIONS

Often one denotes this op erator by r or simply r It resp ects the B mo dule

structure in the sense that

k

d d d

k l

for all B B V P

G

Lemma For al l P V

hor

d d

Here the denotes the Lie algebra representation of g on V

Proof The right hand side is horizontal since

d L d

P P

P

by invariance Here we have used that by horizontality and L

P

P

It remains to show that the two sides agree on horizontal vectors But this is clear since

vanishes on any k horizontal vector elds and d Hor d and d agree on

any k horizontal vector elds by denition

Curvature Let b e a principal connection on P B The curvature of

is the gvalued form

F d

By denition F is Ginvariant and horizontal so it can also b e viewed as a form

F B gP One has the alternative expression

F d

To see this note that d agrees with d on horizontal vectors and that it is

horizontal since

d L

P P

P

Proposition The curvature satises the Bianchi identity

d F

G

Proof Since F P g Lemma applies and we can calculate

hor

d d d F dF F

The rst two terms cancel and the last vanishes by the Jacobi identity for g

There are many dierent interpretations of what curvature measures The following

Prop osition says that curvature measures the extent to which the bracket of horizontal

vector elds fails to b e horizontal Equivalently by Frob enius theorem curvature mea

sures the extent to which the horizontal distribution ker T P fails to b e integrable

Proposition If X Y XP are horizontal vector elds on P

X Y F X Y

PRINCIPAL BUNDLES AND CONNECTIONS

Proof Using that

X Y

X Y L Y L L L d F X Y

X X Y Y X Y X Y X

One can lo ok at this result from a slightly dierent angle The connection determines

a horizontal lift of vector elds Lift XB XP such Lift X is horizontal and

pro jects down to X ie is related to X In general the lift is not a Lie algebra

homomorphism However since with any two pairs of related vector elds their bracket

is also related the dierence Lift X Y Lift X Lift Y is vertical By the

prop osition if X Y are any two vector elds on B

F Lift X Lift Y Lift X Y Lift X Lift Y

Thus curvature measures the extent to which the horizontal lift Lift fails to b e a Lie

algebra homomorphism Finally curvature measures the extent to which the covariant

dierential d fails to dene a dierential

Lemma Let V be a linear Grepresentation In the notation of Lemma if

k

B P V

G

d F

Proof Using Lemma d d d d

Gauge transformations of connections The group of automorphisms

AutP acts on the space AP of principal connections by pullback by the inverse For

the action of the subgroup Gau P there is an alternative formula for this action which

L R

we derive b elow Let G g the left and rightinvariant Maurer Cartan forms

L R

Thus if are the left and rightinvariant vector elds on G equal to g T G at

e

L L R R L

the group unit and In a matrix representation of G g dg

R R L

and dg g One has Ad Under the inversion map G G g g

g

L R

one has

G

Proposition Let Gau P and C P G the corresponding equivari

G

ant function For al l X autP XP

G

X X gauP X P

vert

gau P is given by The corresponding section B gP

R

X

Proof The vector eld X is invariant since commutes with the Gaction and

under pro jects to the same vector eld as X since This shows that

CONNECTIONS

G

X X X P Let denote the ow of X Then

vert t

d

X p

p t

dt t

d

p p

t t

dt

t

d d

p p p p

t t

t t dt d t

The rst term is X since

p

p p p p p

t t t

For the second term dene a curve on G starting at e by

p g p

t t

Then

d d d

p p p p g g p

t t t P p

t t t dt d t dt

with

d

L L R

g X

t p X X

p

p p

t dt

Proposition The natural action of GauP on AP is given by the formula

R

Ad

L

It suces to Proof We will prove the equivalent prop erty Ad

1

G

check this identity on invariant vector elds X XP By Prop osition and since

G

f Ad f for all f C P g

1

L

X X Ad X X Ad

1 1

as desired

Let us informally regard AP as an innite dimensional manifold equipp ed with

an action of an innitedimensional Lie group What are the generating vector elds and

gau P and identifying what are the stabilizer subgroups Given P gP

P gP one has T AP

Corollary The generating vector eld for the action of on AP is given by

the covariant dierential

d

AP

G

Proof Let P g b e the equivariant function corresp onding to Then

d d

R

expt Ad expt

expt

t t dt dt

d d

PRINCIPAL BUNDLES AND CONNECTIONS

where we have used Lemma

The stabilizer subgroups Gau P can b e characterized in terms of the holonomy

Let us rst describ e how gauge transformations act on parallel transp ort For any path

t t B let

P P

t t

0 1

denote parallel transp ort with resp ect to We will need

Lemma For al l Gau P

t t

We leave the pro of as an exercise Hint Pulling back P under we may assume B

is dimensional

Supp ose B is connected and let b B Recall the holonomy homomorphism

Hol LB b AutP

b

G of automorphisms of the from the group of lo ops based at b into the group AutP

b

b er Let the holonomy subgroup G AutP b e the image of the holonomy map and

b

the restricted holonomy subgroup G the image of the subgroup L B b of contractible

lo ops G is a closed subgroup hence Lie subgroup of AutP while G need not b e

b

a closed subgroup

Proposition Let P B be a principal Gbund le over a connected base b

B and AP The evaluation map Gau P AutP b is injective

b

and denes an isomorphism of Gau P with the centralizer in AutP of the holonomy

b

group

Gau P Z G

AutP

b

Proof A gauge transformation GauP preserves if and only if it preserves

the horizontal subbundle H P T P Equivalently by the Lemma with in place of

Gau P if and only if for all paths t t B commutes with parallel

transp ort

t t

Taking paths with t b it follows that if is trivial at b then is trivial

everywhere That is the evaluation map b is injective Taking LB b

shows that b centralizes G Conversely if b Z G then we can dene a gauge

transformation Gau P by putting

b b

where is any path from b to b the condition Gau P guarantees that the right

hand side do es not dep end on the choice of

The prop osition shows in particular that all stabilizer subgroups are nite dimensional

Lie groups Moreover

CONNECTIONS

Corollary The based gauge group Gau P b f Gau P j b eg acts

freely on AP

Corollary If G is abelian every principal connection has stabilizer equal to

G

G viewed as constant gauge transformations in Gau P B GP C B G

The quotient AP Gau P is called the mo duli space of connections It is still

innitedimensional To obtain nite dimensional mo duli spaces one has to imp ose

additional gaugeinvariant constraints on Eg that is a at connection or more

generally a YangMills connection See b elow

Reducible connections Let P B b e a principal Gbundle

Definition A reduction of the structure group from G to a Lie subgroup H

G is a principal H bundle Q B together with an H equivariant b er bundle map

Q P making Q into an H invariant submanifold of P A principal connection

on P is called reducible to H if the horizontal distribution H P for is tangent to Q A

connection on P B is called irreducible if it is not reducible to a subgroup H of G

Thus is a reducible connection if and only if its pullback takes values in h

and is a principal connection for Q The principal bundle P can b e recovered from Q

Q G where H acts on G from the left The reducible as an asso ciated bundle P

H

connections on P are those which come from connections on Q using the construction

for asso ciated bundles

Examples A choice of an inner pro duct on a real vector bundle is equivalent

to a reduction of the structure group of the frame bundle FrE from GL n R to O n

a choice of an orientation is equivalent to a reduction of the structure group to the

identity comp onent GL n R Similarly the choice of a complex structure symplectic

structure Hermitian structure etc are equivalent to various reductions of the structure

group Connections which are reducible to these subgroups have parallel transp orts

preserving these extra structures

Reducible connections can b e recognized from their holonomy groups Indeed if is

reducible to an H G connection on Q P then its holonomy subgroup is contained

in AutQ AutP Conversely if the holonomy subgroup G AutP at b B

b b b

is a prop er subgroup of AutP pick p P and let Q b e the set of all p oints in P

b b

that can b e reached from p by parallel transp ort Then Q is a principal H bundle for

G given by the the subgroup H corresp onding to G under the identication AutP

b

choice of p

We have shown ab ove that the stabilizer group of under gauge transformations is

isomorphic to the centralizer of G Hence for an irreducible connection it is isomorphic

to the center Z G The converse is not true cf Corollary

Reducibility can also b e detected from the curvature using that the holonomy group

and restricted holonomy group coincide in this case together with

PRINCIPAL BUNDLES AND CONNECTIONS

Theorem AmbroseSinger theorem Let P B be a principal bund le over a

connected base B and a principal connection Given p P let Q B be the set of

al l points which can be connected to p by paral lel transport Then the Lie algebra of the

holonomy subgroup relative to p is equal to the set of al l F X Y g with q Q

q

For a pro of see eg KobayashiNomizu I p

The universal connection Consider AP B as an innitedimensional

manifold which is the base for a universal principal bundle

AP P AP B

not of course to b e confused with the classifying bundle EG BG which often is

also called universal bundle The tangent space at any p oint p is the direct sum

of B gP and of T P Let AP P g b e the form such that

p p

vanishes on B gP and equals on T P Clearly is a connection form It is

p p

of this called the universal connection Let us calculate its curvature F d

connection For a b B gP and X T P we have

p

d

abd a

p tbp

dt t

d

X ad X X a

p

p tap

dt t

and nally X Y d X Y d Thus

F a b F X Y F X Y F X a aX

The group Gau P acts on AP P by automorphisms preserving the universal con

nection hence also preserving the curvature F

YangMills connections Supp ose P B is a principal Gbundle over a

compact oriented Riemannian manifold B The inner pro duct on TB gives rise to an

k

inner pro duct on T M and on all T M Taking the inner pro duct of dierential form

followed by integration over B with resp ect to the Riemannian volume form denes an

inner pro duct on B In terms of the Ho dge star op erator

Z

h i

B

This inner pro duct extends to gO valued forms using the inner pro duct on the vector

bundle gP

Z

h i

B

Let jj jj b e the norm corresp onding to h i The YangMills functional on AP is the

functional

Z

YM jjF jj F F B

CONNECTIONS

It is invariant under the action of the gauge group

YM YM

hence all its critical p oints called YangMills connections are invariant as well If P

admits at connections then A P is the absolute minimum of YM

at

Proposition A connection is a critical point of the YangMil ls functional

if and only if it satises the YangMil ls equation

d F

G

Proof We use the formula F d For any P g

hor

F d F F d

Hence

Z Z

d F F d YM YM

B B

Z

d F

B

where denotes terms which are at least quadratic in Thus is a stationary p oint

R

d F for all that is d F if and only if

B

The quotient of the space of YangMills connections by the action of the gauge group is

called the YangMills mo duli space

The Yang Millsequations dep end up on the Riemannian metric on B only via the

star op erator on B The case dim B is sp ecial in that takes B to itself

since We mentioned already that in this case the YangMills equations are

conformally invariant Multiplying the metric by a p ositive function do es not change the

star op erator in middle dimension hence do es not change the YangMills equations A

sp ecial typ e of YangMills connections in dimensions are those satisfying one of the

equations

F F or F F

selfduality resp antiselfduality b ecause for such connections the YangMills equa

tions are a consequence of the Bianchy identity d F A change of orientation of

B changes the sign of the op erator and therefore exchanges the notion of duality and

antiself duality The value of the YangMills functional for an ASD connection is

Z

F F YM

B

From the theory of charactersitic classes one knows that the right hand side is inde

p endent of it is a multiple of the second Chern numb er c P Using this fact and

decomp osing the curvature of a connection into its selfdual ann ASD part one nds that

for c P ASD connections give the absolute minimum of the YangMills functional

PRINCIPAL BUNDLES AND CONNECTIONS

Antiself dual connections over S are also called instantons Up to gauge trans

formation they can b e viewed as connections on the trivial bundle over R which are

at outside a compact set The famous ADHM construction AtiyahDrinfeldHitchin

Manin gives a complete description of such instantons

The mo duli space for antiself dual YMconnections for G SU is the starting

p oint for Donaldson theory of manifolds As realized by Donaldson they contain infor

mation not only ab out the top ology but also the dierentiable structure of manifolds

In this course we will b e concerned with the very dierent case dim B and in

fact mostly with at connections