Marathe K. Topics in Physical Mathematics

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188 6 Theory of Fields, I: Classical

f ·ω := R

−1

(ω)=(f

−1

)

∗

ω, f ∈G,ω∈A(P ). (6.17)

The second is the left action L

obtained by pushing forward the horizontal

distribution deﬁned by the given connection by f

∗

.Itcanbeshownthatthe

two are related by L

= R

−1

. Thus we see that there is essentially only one

natural action of G on A. We shall use both forms of this action as convenient.

We now derive a local expression for this action when G is one of the classical

matrix groups. Let A

(resp., A

) be a local gauge potential in local gauge t

(resp., t

)overU

(resp. U

), corresponding to the gauge connection ω.Let

= ψ

,ψ

: U

∩ U

→ G

be the local expression of the gauge transformation f .Thenequation(6.17)

becomes

= ψ

−1

+ ψ

−1

dψ

It is customary to write g for ψ

and A, A

for A

, A

, respectively. Then

the local expression of equation (6.17) becomes

= g

−1

Ag + g

−1

dg. (6.18)

Equation (6.4) for the gauge transformation of an electromagnetic potential

is a special case of equation (6.18) corresponding to G = U(1). In many

applications it is the local form (6.18) of the equation (6.17)thatisusedfor

calculating the gauge transforms of potentials and ﬁelds. For example, using

equation (6.18) we obtain the following expression for the gauge transform

of the gauge ﬁeld F

= g

−1

g. (6.19)

From equation (6.19) it follows that |F

| is a gauge invariant function on M ,

i.e.,

| = |F

|∈F(M ).

It is this gauge invariant function that is used in deﬁning the Yang–Mills

action functional (see Chapter 8).

We say that connections α, β ∈Aare gauge equivalent if there exists a

gauge transformation f ∈Gsuch that β = f · α. From the deﬁnition of the

action of G on A given above, it follows that each equivalence class of gauge

equivalent connections is an orbit of G in A. The orbit space O = A/G thus

represents gauge inequivalent connections and is called the moduli space

of gauge potentials on P (M,G).

Gauge ﬁeld equations that arise in physical applications are partial diﬀer-

ential equations on manifolds. Their analysis is greatly facilitated by consid-

ering Sobolev completions of the various inﬁnite-dimensional spaces involved

in the formulation and study of these equations. Now we deﬁne Sobolev

norms and completions of some of the structures used in this chapter. We

use the assumptions and notation of Deﬁnition 6.1.LetE be a Rieman-

6.4 The Space of Gauge Potentials 189

nian vector bundle over the manifold M associated to P (M,G). Recall that

(M,E)=Γ (A

(M) ⊗ E)isthespaceofp-forms on M with values in E.

Fixingaconnectionα on P we get the induced covariant derivative ∇

(M,E). If λ is the Levi-Civita connection on (M,g), then we can deﬁne

the covariant derivative

∇

(ω,λ)

: Λ

(M,ad P ) → Γ (T

∗

M ⊗ A

∗

M ⊗ ad P )

or simply ∇ by

∇≡∇

(ω,λ)

:= 1 ⊗∇

+ ∇

⊗ 1.

Using the covariant derivative ∇ we deﬁne a Sobolev k-norm on Λ

(M,E)

φ

⎛

⎝



j=0



|(∇)

φ|

⎞

⎠

1/2

,φ∈ Λ

(M,E).

The completion of Λ

(M,E) in this norm is a Hilbert space (under the as-

sociated bilinear form) denoted by H

(Λ

(M,E)). A diﬀerent choice of the

connection on P and metrics on M and E gives an equivalent norm. The map

: Λ

(M,E) → Λ

p+1

(M,E) extends to a smooth bounded map of Hilbert

spaces (also denoted by d

)

: H

(Λ

(M,E)) → H

k−1

(Λ

p+1

(M,E)).

For p = 0, this map has ﬁnite dimensional kernel and closed range. In general,

the sequence

0 −→ H

(Λ

(M,E))

−→ H

k−1

(Λ

(M,E))

−→···

fails to be a complex, the obstruction being provided by the curvature of α.

In particular, ﬁxing a connection α on P gives an identiﬁcation of A with

(M,ad P ) and we denote the corresponding Sobolev completion of A in

the k-norm by H

(A). This Sobolev k-norm deﬁnes a strong Riemannian

metric on A. Strong Riemannian metrics and the weak or L

metric on

A deﬁned above are used in studying the geometry of A and other related

spaces in Chapter 9. The curvature map

F : ω → F

of A→Λ

(M,ad P )

extends to a smooth bounded Hilbert space map from H

k+1

(A)into

(Λ

(M,ad P )) for k ≥ 1. The Sobolev completion of the gauge group

G is obtained by considering G as a subset of Λ

(M,P ×

End(V )), where

ρ : G → End(V ) is a faithful representation of the gauge group G. We deﬁne

(G) to be the closure of G in H

(Λ

(M,P ×

End(V ))). The Lie alge-

bra structure of LG = Λ

(M,ad P ) extends to a Lie algebra structure on

190 6 Theory of Fields, I: Classical

its Sobolev completion H

(Λ

(M,ad P )). The above discussion leads to the

following theorem.

Theorem 6.5 Let k>

(m +1); then we have the following:

1. The Sobolev completion H

(G), is an inﬁnite-dimensional Lie group mod-

eled on a separable Hilbert space with Lie algebra H

(LG).

2. The action of G on A deﬁned by equation (6.17) extends to a smooth action

of H

k+2

(G) on H

k+1

(A).

3. The curvature map F : A→Λ

(M,ad P ) extends to a smooth,

bounded, H

k+2

(G)-equivariant, Hilbert space map from H

k+1

(A) into

(Λ

(M,ad P )), i.e.,

F (g.ω)=g

−1

F (ω)g, ∀g ∈ H

k+2

(G).

From now on we consider that all objects requiring Sobolev completions

have been completed in appropriate norms and drop the H

from H

(object).

In many applications one is interested in the orbit space O = A/G whose

points correspond to equivalence classes of gauge connections. The orbit space

is given the quotient topology and is a Hausdorﬀ topological space. However,

in general, the action of G on A is not free and O fails to be a manifold. We

now discuss two methods of suitably modifying A or G to obtain orbit spaces

with nice mathematical structure.

1. Let G

⊂Gdenote the group of based gauge transformations, deﬁned

= {f ∈G|f (u

)=u

for some ﬁxed u

∈ P }.

The group G

is also called the restricted group of gauge transforma-

tions. A based gauge transformation that ﬁxes a connection is the identity.

Therefore G

acts freely on A and the orbit space O

= A/G

is an inﬁ-

nite dimensional Hilbert manifold. A(O

, G

) is a principal ﬁber bundle with

canonical projection

: A→A/G

= O

The relation between G, G

and the gauge group G is given by the following

proposition.

Proposition 6.6 The group G

is a normal subgroup of G and the quotient

G/G

is isomorphic to the gauge group G.

Proof :Wenotethatifπ(u

)=x

then a based gauge transformation f

ﬁxes every point of the ﬁber π

−1

). If h ∈G,thenh(u

) ∈ π

−1

)and

hence f(h(u

)) = h(u

); i.e., (h

−1

fh)(u

)=u

. This proves that G

is a

normal subgroup of G. Now deﬁne

h to be the unique element of G such that

h(u

)=(u

)

h. It is now easy to verify that the map

: h →

h of G→G

6.4 The Space of Gauge Potentials 191

is a surjective, homomorphism with kernel G

. From this it follows that the

map

→

h of G/G

→ G

is well deﬁned and is an isomorphism of groups. 

2. Let Z(G) denote the center of the gauge group G.DenotebyZ the group

Γ (P ×

Z(G))

∼

Z(G). Then G

= G/Z is called the group of eﬀec-

tive gauge transformations. By an argument similar to that in the above

proposition we can show that Z

∼

Z(G). Let A

⊂Adenote the space of

irreducible connections.ThenG

acts freely on A

and we denote by O

the orbit space A

. O

is an inﬁnite-dimensional Hilbert manifold and

, G

) is a principal ﬁber bundle with canonical projection

: A

→A

= O

We now discuss the topology of the various spaces and ﬁbrations intro-

duced above and use it to study the problem of global gauge ﬁxing. We begin

by considering a ﬁber bundle E over base B and ﬁber F as follows:

Given a point x

∈ B and a point u

∈ π

−1

)

∼

F , there exists a natural

group homomorphism ∂ : π

(B,x

) → π

n−1

(F, u

). This homomorphism

together with the homomorphisms induced by the inclusion ι : F → E and

by the bundle projection p leads to the following long exact sequence of

homotopy groups

···→π

k+1

(E,u

) → π

k+1

(B,x

) → π

(F, u

) → π

(E,u

) →···.

Applying the above homotopy sequence to the various ﬁber bundles related

to the based and eﬀective gauge transformations we obtain the following

theorem.

Theorem 6.7 Let G be a compact, simply connected, non-trivial Lie group

and let P(M, G) be a principal bundle over a closed, simply connected base

manifold M. Then we have

1. π

(G)

∼

), for j =0, 1;

2. π

(G)

∼

), for j ≥ 2;

3. π

)

∼

), ∀j;

4. π

)

∼

), ∀j.

Furthermore, if M is orientable (resp., spin) then there exists a non-negative

integer j such that π

) (resp., π

))isnon-trivial.

192 6 Theory of Fields, I: Classical

Now we use these results and related constructions to study the Gribov

ambiguity.

6.5 Gribov A mbiguity

Recall that the orbit space O whose points represent gauge equivalent con-

nections (gauge potentials), is deﬁned by

O = A/G.

We denote by

p : A→O= A/G

the inﬁnite-dimensional principal G-bundle with natural projection p.The

group of gauge transformations G acts on A by

(g,ω) → g · ω, g ∈G,ω∈A.

The induced action on the curvature Ω is given by g.Ω = gΩg

−1

.These

transformation properties are used to construct gauge invariant functionals

on A. In the Feynman integral approach to quantization one must integrate

these gauge invariant functions over the orbit space O to avoid the inﬁnite

contribution coming from gauge equivalent ﬁelds. However, the mathematical

nature of this space is essentially unknown. Physicists often try to get around

this diﬃculty by choosing a section s : O→Aand integrating over its

image s(O) ⊂Awith a suitable weight factor such as the Faddeev–Popov

determinant, which may be thought of as the Jacobian of the change of

variables eﬀected by p

|s(O)

: s(O) →O. This procedure amounts to a choice

of one connection in A from each equivalence class in O andisreferredtoas

gauge ﬁxing. The question of the existence of such sections is thus crucial for

this approach. For the trivial SU(2)-bundle over R

, Gribov showed that the

so called Coulomb gauge fails to be a section, i.e., the Coulomb gauge is not

a true global gauge. Gribov showed that the local section corresponding to

the Coulomb gauge at the zero connection if extended (under some boundary

conditions) intersects the orbit through zero at large distances and thus fails

to be a section. The boundary conditions imposed by Gribov amount to the

gauge potential being deﬁned over the compactiﬁcation of R

to S

.Healso

discussed a similar problem for R

. This non-existence of a global gauge is

referred to as the Gribov ambiguity. In view of this negative result, it

is natural to ask if any true gauge exists under these boundary conditions.

Without any boundary conditions it is possible to exhibit a global gauge,

but it does not seem to have any physical meaning. We show that, in fact,

the Gribov ambiguity is present in all physically relevant cases, so that no

global gauge exists. The existence of Gribov ambiguity can be proven for

6.5 Gribov Ambiguity 193

a number of diﬀerent base manifolds and gauge groups. For examples and

further details see [166, 167,220, 351].

The Gribov ambiguity is a consequence of the topology of the principal

bundle A over O as we now explain. Let P (S

,SU(2)) be a principal bundle.

Recall that A is isomorphic to the vector space of 1-forms on S

with values

in the vector bundle ad P = P ×

g once a connection is ﬁxed. If α ∈Ais a

ﬁxed connection we can write

∼

{α + A | A ∈ Λ

, ad P )},

where the pull-back of A to P is understood. Consider the slice S

deﬁned

:= {α + A | δ

A =0}⊂A.

We call this the generalized Coulomb gauge. In particular, if α =0then

= {A | δ

A =0} .

Locally the condition δ

A = 0 can be written as



∂A

∂x

=0.

Locally (or on R

as a base) one can ﬁnd a connection with zero time com-

ponent that is gauge equivalent to the given connection. The gauge condition

then reduces to the classical Coulomb gauge condition

div A =0.

It is convenient to reformulate the deﬁnition of the group G of gauge

transformations as follows. Let

= E(M,C

,r,P)

be the vector bundle associated to P with ﬁber C

,wherer is the deﬁning

or standard representation of SU(2) on C

.Then

∼

{h ∈ Γ (Hom(E

)) | h(x) ∈ SU(2) , ∀x ∈ M}.

Deﬁne the isotropy group G

of a ﬁxed connection α by

= {g ∈G|g · α = α}.

It is easy to see that g ∈G

if and only if

g =0.

194 6 Theory of Fields, I: Classical

In particular, g is completely determined by specifying its value at a single

point, say x

∈ M. To study the question of the irreducibility of α,we

consider the holonomy group H

⊂ SU(2) of the connection α at x

.We

observe that g ∈G

if and only if g(x

) ∈ SU(2) and [g(x

),H

]=0.If

α is irreducible then this is equivalent to requiring that g(x

) ∈ Z(SU(2))

(the center of SU(2), which is isomorphic to Z

). Recall that the group G

eﬀective gauge transformations is the quotient of G by the center Z(G),

which in this case is isomorphic to Z

and hence

= G/Z

Let A

⊂Abe the set of irreducible connections. G

acts on A

freely

and the quotient O

is the orbit space of irreducible connections. A

is a

principal G

-bundle over O

, i.e.,

= P (O

, G

We now compute the homotopy groups of the space A

.Letf : S

→A

⊂

A be a continuous map. Regarding f as a map of the boundary of a (k +1)-

simplex Δ

k+1

we can extend f linearly to a map of Δ

k+1

to A.Itcanbe

shown that the extended map is homotopic to a map g, which actually lies in

. The construction uses the fact that the set A

of reducible connections

is a closed, nowhere dense, stratiﬁed subset of A. A simple argument then

shows that f ∼ g ∼ c

,wherec

is the constant map c

(x)=α, ∀x ∈ S

Thus

)=0. (6.20)

Under certain topological conditions it can be shown that A

= A.For

example, if P (M, SU(2)) is a non-trivial bundle and the second cohomology

group H

(M,Z)=0,thenA

= A. In particular, this second condition is

satisﬁed by M = S

and we have the following proposition.

Proposition 6.8 Let P be a ﬁxed non-trivial SU(2)-bundle over S

,then

there exists some k such that

) =0. (6.21)

Proof : By deﬁnition of G

we have the following short exact sequence of

groups

0 → Z

→G→G

→ 0.

Therefore, if G is connected then π

) is non-trivial. For k>1wehave

(G)=π

). Recall that the group G

is a group of based gauge trans-

formations over some ﬁxed point of the manifold, which we may take to be

the point at inﬁnity (i.e., the north pole) on S

. We have the following short

exact sequence of groups

0 →G

→G→SU(2) → 0.

6.5 Gribov Ambiguity 195

It induces the following long exact sequence in homotopy

···→π

k+1

(SU(2)) → π

) → π

(G) → π

(SU(2)) →···

We shall use the following part of the above sequence

(G) → π

(SU(2)) → π

) → π

(G) → π

(SU(2)). (6.22)

In the present case of an SU(2)-bundle over S

it can be shown that

)

∼

k+4

(SU(2)).

In particular,

)

∼

(SU(2)) = Z

We also know that

(SU(2)) = Z and π

(SU(2)) = 0.

Thus, (6.22) becomes

···→π

(G) → Z → Z

→ π

(G) → 0.

Hence, if π

(G)=0thenπ

(G) =0.Thuseither

)=π

(G) =0 or π

)=π

(G) =0.

Thus, there exists some k such that π

) =0. 

We note that equation (6.21) is a consequence of Theorem 6.7 applied

to the case when M = S

and G = SU(2). However, we have given the

above proof to illustrate the kind of computations involved in establishing

such results. Using the result (6.21) we can prove that no continuous global

gauge s : O→Aexists in this case. For if such an s exists then it induces a

map s

,thatis,

: O

→A

which is a section of the principal G

-bundle A

and we have a corresponding

trivialization of the bundle A

= O

×G

. Now choose some k such that

) =0.Thenwehave

0=π

)=π

) ×π

) =0.

This contradiction shows that a continuous global gauge s does not exist in

this case.

The non-existence of a global gauge ﬁxing need not prevent an application

of path-integral methods. For example, one may use the fact that the orbit

space O

and the space A

are paracompact. Thus we may be able to ﬁnd a

suitable locally ﬁnite covering and a subordinate partition of unity for O

and

196 6 Theory of Fields, I: Classical

construct local gauges and local weights to deﬁne the path integrals. However,

for this procedure to work we need an explicit description of the locally ﬁnite

covering and local weights for the Faddeev–Popov approach. More generally,

one can consider physical conﬁguration spaces that are subspaces or quotient

spaces of the space of gauge potentials A. This observation is the starting

point of the development of a geometric setting for ﬁeld theories in [276],

on which part of our treatment of classical and quantum ﬁeld theories and

coupled ﬁelds is based.

In [167] Gribov proposed another interesting approach that we now dis-

cuss. It is based on the assumption that the physically relevant part of the

conﬁguration space can be deﬁned by the following two conditions:

1. local transversality to the gauge orbits,

2. positivity of the Faddeev–Popov determinant.

The second condition insures that the eﬀective contributions at various orders

in perturbation theory do not tend to inﬁnity at ﬁnite distances. The region

of the conﬁguration space satisfying the above two conditions is called the

Gribov region. It is denoted by Ω. The boundary ∂Ω of the Gribov region

is called the ﬁrst Gribov horizon. It has been conjectured by Gribov that

the problem of conﬁnement of gluons may be due to the restriction of the

conﬁguration space to the physically allowable region Ω.Itcanbeshown

that the two conditions deﬁning the Gribov region Ω are the conditions for

a local minimum of the L

norm on each gauge orbit. In [97]itisshownthat

the L

norm attains its absolute minimum on each gauge orbit and hence

each gauge orbit intersects the Gribov region at least once at the absolute

minimum. It would be interesting to study the structure of the set of absolute

minima and to check if this set is the appropriate region of integration for

the Feynman path integral method of quantization.

6.6 Matter Fields

Let E(M, F,r,P) be a vector bundle associated to the principal bundle

P (M, G). We call a section φ ∈ Γ (E)anE-ﬁeld (or a generalized Higgs

ﬁeld)onM .IfE =ad(P ):=P ×

g then φ is called the Higgs ﬁeld (in

the adjoint representation) on M. In general there are several ﬁelds that can

be deﬁned on bundles associated with a given manifold. For example, on a

Lorentz 4-manifold the Levi-Civita connection is interpreted as representing

a gravitational potential. Recall that the Levi-Civita connection is the unique

torsion-free, metric connection deﬁned on the bundle of orthonormal frames

O(M )ofM. In general if M is a pseudo-Riemannian manifold, we can deﬁne

the space of linear connections A(O(M)) on M by

A(O(M )) := {α ∈ Λ

(O(M ),so(m)) | α is a connection on O(M)}.

6.6 Matter Fields 197

Generalized theories of gravitation often use this space as their conﬁgura-

tion space. If M is a spin manifold and S(M )(M,Spin(m)) is the Spin(m)-

principal bundle, then one can consider the space A(S(M)) of spin connec-

tions on the bundle S(M), deﬁned by

A(S(M )) := {β ∈ Λ

(S(M),spin(m)) | β is a connection on S(M)}.

For a given signature (p, q) we may consider the space RM

(p,q)

(M)ofall

pseudo-Riemannian metrics on M of signature (p, q). The space RM

(m,0)

(M)

of Riemannian metrics on M is denoted simply by RM(M). Recall that there

is a canonical principal GL(m, R)-bundle over M,namelyL(M) the bundle

of frames of M.Ifρ : GL(m, R) → End V is a representation of GL(m, R)

on V and E(M,V, ρ, L(M)) is the corresponding associated bundle of L(M),

then we denote by W the space of E-ﬁelds Γ (E), i.e.,

W = Γ (E(M,V,ρ,L(M ))).

Thus we see that we have an array of ﬁelds on a given base manifold M and we

must specify the equations governing the evolution and interactions of these

ﬁelds and study their physical meaning. There is no standard procedure for

doing these things. In many physical applications one obtains the coupled

ﬁeld equations of interacting ﬁelds as the Euler–Lagrange equations of a

variational problem with the Lagrangian constructed from the ﬁelds. For

any given problem the Lagrangian is chosen subject to certain invariance

or covariance requirements related to the symmetries of the ﬁelds involved.

We now discuss three general conditions that are frequently imposed on the

Lagrangians in physical theories. In what follows we restrict ourselves to a

ﬁxed, compact 4-manifold M as the base manifold, but the discussion can

be easily extended to apply to an arbitrary base manifold. Let P(M,G)bea

principal bundle over M whose structure group G carries a bi-invariant metric

h. For example if G is a semisimple Lie group, then a suitable multiple of the

Killing form on g pr

vides a bi-invariant metric on G.

We want to consider coupled ﬁeld equations for a metric g ∈RM(M ),

aﬁeldφ ∈W(M) (a section of the bundle associated to the frame bundle

L(M)), a connection ω ∈A(P ) and a generalized Higgs ﬁeld ψ ∈H=

ΓE(M,V

,r,P), where r : G → End(V

) is a representation of the gauge

group G.Thus,ourconﬁguration space is deﬁned by

C := RM×W×A×H.

We assume that the ﬁeld equations are the variational equations of an action

integral deﬁned by a Lagrangian L on the conﬁguration space with values in

(M). When a ﬁxed volume form such as the metric volume form is given,

we may regard L as a real-valued function. We shall use any one of these

conventions without comment. The action E is given by