Marathe K. Topics in Physical Mathematics

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228 7 Theory of Fields, II: Quan t um and Topological

−

(M,ad(P )). Thus F

−

is the inﬁnite-dimensional analogue of a vector ﬁeld

and we may hope to obtain topological information about the base space B by

studying the set of zeros of this vector ﬁeld. In the generic ﬁnite-dimensional

case, one can deﬁne an index of a vector ﬁeld by studying its zero set and

thus obtain a topological invariant, namely the Euler characteristic. In our

case, the zeros of F

−

in A are precisely the Yang–Mills instantons and the

zero set of the associated section s

−

is the moduli space M

⊂B.Thus,

=(F

−

)

−1

({0})/G. (7.40)

In this general setting there is no obvious way to deﬁne the index of s

−

and

consider its relation with the analytic deﬁnition of the Euler characteristic,

which should correspond to an integral of some “curvature form” analogous

to the Chern–Gauss–Bonnet integrand.

Once again we consider the ﬁnite-dimensional case. Here the Thom class,

which lies in the cohomology of the vector bundle over M with compact

support, is used to obtain the desired relation between the index and the Euler

characteristic as follows. We assume that there exists a section s of the vector

bundle E.Usings, we pull back the Thom class α to a cohomology class on the

base M. In fact, we have a family of homologous sections λs, λ ∈ R, and the

corresponding family of pull-backs (λs)

∗

α which, for λ = 0, gives the Euler

class of M and, for large λ (λ →∞) gives, in view of the compact support of

α, the index of s. Thus, one would like to obtain a suitable generalization of

the Thom class in the inﬁnite-dimensional case. We shall refer to this idea, of

interpolating between two diﬀerent deﬁnitions of the Euler characteristic, as

the Mathai–Quillen formalism. On the principal G-bundle A = P (B, G),

we can deﬁne a natural connection as follows. For ω ∈A, the tangent space

A is identiﬁed with Λ

(M,ad(P )) and hence carries the inner product as

deﬁned in Deﬁnition 6.1. With respect to this inner product we have the

orthogonal splitting

A = V

⊕ H

, (7.41)

where the vertical space V

is identiﬁed as the tangent space to the ﬁber G or,

alternatively, as the image of the covariant diﬀerential d

: Λ

(M,ad(P)) →

(M,ad(P )). The horizontal space H

can be identiﬁed with ker δ

,where

is the formal adjoint of d

. The horizontal distribution is equivariant

with respect to the action of G on A and deﬁnes a connection on A.IfLG(=

(M,ad(P ))) denotes the Lie algebra of the inﬁnite-dimensional Hilbert Lie

group G then the connection form ˆω : T A→LGof this connection is given

ˆω(X)=G

(δ

X),X∈ T

A, (7.42)

where G

is the Green operator, which inverts the Laplacian

= δ

: Λ

(M,ad(P )) → Λ

(M,ad(P )).

7.5 Topological Quantum Field Theories (TQFTs) 229

The curvature Ω

ˆω

of this natural connection form ˆω is the 2-form with values

in Λ

(M,ad(P )) given by the usual formula

ˆω

= d

ˆω

and can be expressed locally, in terms of the Green operator, by

ˆω

(X, Y )=−2G

]), (7.43)

where locally X = X

,Y= Y

,andg

= g(dx

,dx

). We remark that

ˆω

should be considered as deﬁned on a local slice at [α] ∈Btransversal to

the vertical gauge orbit through α.LetP denote the Poincar´e bundle over

M ×B. Pulling back the natural connection ˆω to the Poincar´e bundle P,we

get a connection which coincides with the one deﬁned in [23]. The Poincar´e

bundle with this connection can be regarded as a universal bundle with re-

spect to deformations of the bundle P (M, G). If F denotes the curvature of

the canonical connection on P, then the K¨unneth formula applied to F gives

its (i, j)component

(i,j)

∈ Λ

(P ) ⊗ Λ

(A),i+ j =4.

Let (p, ω) ∈ P ×A.ThenforX, Y ∈ T

P and α, β ∈ T

A = Λ

(M,ad(P ))

we have

(2,0)

(X, Y )=Ω

(X, Y ) (7.44)

(1,1)

(X, α)=α(X) (7.45)

(0,2)

(α, β)=(α, β)

(7.46)

where (α, β)

is the inner product on forms induced by the metric and bracket

on the Lie algebra g (identiﬁed with the ﬁber of ad(P )). Let c

=tr(F∧F)

denote the second Chern class of the canonical connection on P; then the

K¨unneth formula applied to c

gives its (i, j)component

∈ Λ

(M) ⊗ Λ

(A),i+ j =4.

Explicit expressions for c

in terms of the components of the curvature are

given by (see [33] for details)

=tr



(0,2)

∧F

(0,2)



, (7.47)

=2tr



(0,2)

∧F

(1,1)



, (7.48)

=tr



(1,1)

∧F

(1,1)



+2tr



(2,0)

∧F

(0,2)



, (7.49)

=2tr



(2,0)

∧F

(1,1)



, (7.50)

=tr



(2,0)

∧F

(2,0)



=tr(F

∧F

). (7.51)

230 7 Theory of Fields, II: Quan t um and Topological

The TQFT interpretation of the Donaldson polynomials makes essential

use of the Mathai–Quillen formalism to obtain the expectation value of a

quantum observable η as

η = lim

t→∞



ηω

where M

is the moduli space of SU(2)-instantons on M , s is the self-dual

part of the curvature, and ω

is a form that is, essentially, the exponential

of Witten’s Lagrangian. At least on a formal level, the Mathai–Quillen for-

malism can be used to obtain an expression for the equivariant Euler class

e of the Poincar´e bundle P.In[404] Witten constructs a Lagrangian out of

bosonic and fermionic variables and deﬁnes a set of forms W

, 0 ≤ i ≤ 4. The

Lagrangian L is a function of the bosonic ﬁelds φ, λ ∈ Λ

(M,ad(P )) and A ∈

(M,ad(P )) and the fermionic ﬁelds ψ ∈ Λ

(M,ad(P )), η ∈ Λ

(M,ad(P ))

and χ ∈ Λ

(M,ad(P )) and their covariant derivatives ∇ with respect to the

connection ω (corresponding to the local gauge potential A)onP as well as

the curvature F

and the metric g. In local coordinates L has the expression

L =tr



φ∇

∇

λ + i∇

− iη∇

−

φ[χ

,χ

] −

λ[ψ

,ψ

φ[η, η]+

[φ, λ]



where c is a real coupling constant. The TQFT action S is deﬁned by

S(g,ω,φλ,B,ψ,η,χ)=







det(g)dx

where



= L +

tr(F ∧ F ).

The additional term is a multiple of the instanton number and is a topological

invariant. It is inserted so that the action becomes invariant under an odd

supersymmetry operator Q. When restricted to gauge-invariant functionals,

Q is a boundary operator. The Lagrangian and the corresponding energy-

momentum tensor turn out to be Q-exact. In fact, this property plays a

fundamental role in the TQFT interpretation of the Donaldson invariants.

The forms W

are deﬁned by

tr(φ

), (7.52)

=tr(φ ∧ψ), (7.53)

=tr(

ψ ∧ ψ + iφ ∧F ), (7.54)

= i tr(ψ ∧ F ), (7.55)

= −

tr(F ∧ F ). (7.56)

Then, the expectation value of Witten’s observables

7.5 Topological Quantum Field Theories (TQFTs) 231



in the path integral formalism in the corresponding TQFT coincides with the

integral of the Euler class e against the j-form



∈ Λ

(A),

where γ

is an i-cycle. More generally, we can deﬁne observables O by

O := Π

i=1



where γ

is a k

-cycle and



i=1

(4 −k

)=2d(k)=dim(M

Then one can show that the expectation value of O is a topological invariant.

If we set k

=2,∀i, then this is, essentially, Donaldson’s deﬁnition of his

polynomial invariants. A similar interpretation of the Casson invariant is

also given in [21].

Another important topic that gave an early indication of the importance

of topological methods in ﬁeld theory is that of anomalies. In the physical

literature the word anomaly is used in a generic sense as denoting an ob-

struction to the lifting of a given structure to another structure. For example,

we already referred to anomalies in our discussion of the BRST cohomolgy

in the previous section. The term quantum anomaly in ﬁeld theory refers

to the situation in which a certain symmetry or invariance of the classical

action is not preserved at the quantum level. The requirement of anomaly

cancellation imposes strong restrictions on the construction of ﬁeld theoretic

models of gauge and associated ﬁelds. A detailed discussion of anomalies from

a physical point of view and their geometrical interpretation can be found in

[187,325, 349].

As we have indicated earlier, there are formidable mathematical diﬃculties

in obtaining physically signiﬁcant results from evaluating the full Feynman

path integral. To overcome these diﬃculties, physicists have developed sev-

eral approximation procedures, which allow them to extract some information

from such integrals. We remark that the vacuum expectation values of Wilson

loop observables in the Chern–Simons theory have been computed recently

up to second order of the inverse of the coupling constant. These calcula-

tions have provided a quantum ﬁeld-theoretic deﬁnition of certain invariants

of knots and links in 3-manifolds [87, 178]. A geometric formulation of the

quantization of Chern–Simons theory is given in [25]. In the next subsection

we discuss the Atiyah–Segal axioms for TQFT, which arose out of attempts

232 7 Theory of Fields, II: Quan t um and Topological

to obtain a mathematical formulation of the physical methods of QFT as

applied to topological calculations.

7.5.1 Atiyah–Segal Axioms for TQFT

The Atiyah–Segal axioms for TQFT [16, 346] arose from an attempt to give

a mathematical formulation of the non-perturbative aspects of quantum ﬁeld

theory in general and to develop, in particular, computational tools for the

Feynman path integrals that are fundamental in the Hamiltonian approach

to QFT. They generalize the formulation given earlier by Segal for conformal

ﬁeld theories. The most spectacular application of the non-perturbative meth-

ods has been in the deﬁnition and calculation of the invariants of 3-manifolds

with or without links and knots. In most physical applications however, it

is the perturbative calculations that are predominantly used. Recently, per-

turbative aspects of the Chern–Simons theory in the context of TQFT have

been considered in [30]. For other approaches to the invariants of 3-manifolds

see [145,156,221, 223, 294,380, 383].

Let C

denote the category of compact, oriented, smooth n-dimensional

manifolds with morphism given by oriented cobordism. Let V

denote the

category of ﬁnite-dimensional complex vector spaces. An (n + 1)-dimensional

TQFT is a functor T from the category C

to the category V

that satisﬁes

the following axioms.

A1. Let −Σ denote the manifold Σ with the opposite orientation of Σ and

let V

∗

be the dual vector space of V ∈V

.Then

T (−Σ)=(T (Σ))

∗

, ∀Σ ∈C

A2. Let  denote disjoint union. Then

T (Σ

 Σ

)=T (Σ

) ⊗T(Σ

), ∀Σ

,Σ

∈C

A3. Let Y

: Σ

→ Σ

i+1

,i=1, 2, be morphisms. Then

T (Y

)=T (Y

)T (Y

) ∈ hom(T (Σ

), T (Σ

)),

where Y

denotes the morphism given by composite cobordism Y

∪

A4. Let φ

be the empty n-dimensional manifold. Then

T (φ

)=C.

A5. For every Σ ∈C

T (Σ × [0, 1]) : T (Σ) →T(Σ)

is the identity endomorphism.

7.5 Topological Quantum Field Theories (TQFTs) 233

We note that if Y is a compact, oriented, smooth (n + 1)-manifold with

compact, oriented, smooth boundary Σ;then

T (Y ):T (φ

) →T(Σ)

is uniquely determined by the image of the basis vector 1 ∈ C ≡T(φ

In this case the vector T (Y ) · 1 ∈T(Σ) is often denoted also by T (Y ). In

particular, if Y is closed, then

T (Y ):T (φ

) →T(φ

)andT (Y ) · 1 ∈T(φ

) ≡ C

is a complex number, which turns out to be an invariant of Y .AxiomA3

suggests a way of obtaining this invariant by a cut and paste operation on Y

as follows. Let Y = Y

∪

so that Y

(resp., Y

) has boundary Σ (resp.,

−Σ). Then we have

T (Y ) · 1=< T (Y

) ·1, T (Y

) · 1 >, (7.57)

where <, >is the pairing between the dual vector spaces T (Σ)and

T (−Σ)=(T (Σ))

∗

.Equation(7.57) is often referred to as a gluing formula.

Such gluing formulas are characteristic of TQFT. They arise in Fukaya–Floer

homology theory of 3-manifolds (see Chapter 10), Floer–Donaldson theory of

4-manifold invariants (see Chapter 9), as well as in 2-dimensional conformal

ﬁeld theory. For speciﬁc applications the Atiyah axioms given above need to

be reﬁned, supplemented and modiﬁed. For example, one may replace the

category V

of complex vector spaces by the category of ﬁnite-dimensional

Hilbert spaces. This is in fact the situation of the (2 + 1)-dimensional Jones–

Witten theory. In this case it is natural to require the following additional

axiom.

A6. Let Y be a compact oriented 3-manifold with ∂Y = Σ

(−Σ

). Then

the linear transformations

T (Y ):T (Σ

) →T(Σ

)andT (−Y ):T (Σ

) →T(Σ

)

are mutually adjoint.

For a closed 3-manifold Y axiom A6 implies that

T (−Y )=

T (Y ) ∈ C.

It is this property that is at the heart of the result that in general the Jones

polynomials of a knot and its mirror image are diﬀerent, i.e.,

(t) = V

(t),

where κ

is the mirror image of the knot κ.

An important example of a (3 + 1)-dimensional TQFT is provided by the

Floer–Donaldson theory. The functor T goes from the category C of compact,

234 7 Theory of Fields, II: Quan t um and Topological

oriented homology 3-spheres to the category of Z

-graded Abelian groups. It

is deﬁned by

T : Y → HF

∗

(Y ),Y∈C.

For a compact, oriented, 4-manifold M with ∂M = Y , T (M) is deﬁned to

be the vector q(M,Y )

q(M,Y ):=(q

(M,Y ),q

(M,Y ),...),

where the components q

(M,Y ) are the relative polynomial invariants of

Donaldson deﬁned on the relative homology group H

(M,Y ; Z).

The axioms also suggest algebraic approaches to TQFT. The most widely

studied of these approaches are those based on quantum groups, oper-

ator algebras, and Jones’s theory of subfactors. See, for example, books

[227, 229, 209, 381, 119] and articles [380, 383, 384]. Turaev and Viro have

given an algebraic construction of such a TQFT. Ocneanu’s method [304]

starts with a special type of subfactor to generate the data, which can be

used with the Turaev and Viro construction.

The correspondence between geometric (topological) and algebraic struc-

tures has played a fundamental role in the development of modern mathe-

matics. Its roots can be traced back to the classical work of Descartes. Recent

developments in low-dimensional geometric topology have raised this corre-

spondence to a new level bringing in ever more exotic algebraic structures

such as quantum groups, vertex algebras, and monoidal and higher cate-

gories. This broad area is now often referred to as quantum topology. See,

for example, [414,259].

Chapter 8

Yang–Mills–Higgs Fields

8.1 Introduction

Yang–Mills equations, originally derived for the isospin gauge group SU(2),

provide the ﬁrst example of gauge ﬁeld equations for a non-Abelian gauge

group. This gauge group appears as an internal or local symmetry group

of the theory. In fact, the theory can be extended easily to include the other

classical Lie groups as gauge groups. Historically, the classical Lie groups ap-

peared in physical theories, mainly in the form of global symmetry groups

of dynamical systems. Noether’s theorem established an important relation

between symmetry and conservation laws of classical dynamical systems. It

turns out that this relationship also extends to quantum mechanical systems.

Weyl made fundamental contributions to the theory of representations of

the classical groups [401] and to their application to quantum mechanics.

The Lorentz group also appears ﬁrst as the global symmetry group of the

Minkowski space in the special theory of relativity. It then reappears as the

structure group of the principal bundle of orthonormal frames (or the inertial

frames) on a space-time manifold M in Einstein’s general theory of relativity.

In general relativity a gravitational ﬁeld is deﬁned in terms of the Lorentz

metric of M and the corresponding Levi-Civita connection on M.Thus,a

gravitational ﬁeld is essentially determined by geometrical quantities intrin-

sically associated with the space-time manifold subject to the gravitational

ﬁeld equations. This geometrization of gravity must be considered one of

the greatest events in the history of mathematical physics.

Weyl sought a geometric setting for a uniﬁed treatment of electromag-

netism and gravitation. He proposed local scale invariance as the origin of

electromagnetism. While this proposal was rejected on physical grounds, it

contained the fundamental idea of gauge invariance and local symmetry. In-

deed, replacing the local scale invariance by a local phase invariance leads to

the interpretation of Maxwell’s equations as gauge ﬁeld equations with gauge

group U (1). In Section 8.2 we discuss the theory of electromagnetic ﬁelds

K. Marathe, Topics in Physical Mathematics, DOI 10.1007/978-1-84882-939-8 8, 235

 Springer-Verlag London Limited 2010

236 8 Yang–Mills–Higgs Fields

from this point of view. This theory is the prototype of Abelian gauge

theories, i.e., theories with Abelian gauge group. Here, a novel feature is

the discussion of the geometrical implications of Maxwell’s equations and the

use of universal connections in obtaining their solutions. This last method

also yields solutions of pure Yang–Mills ﬁeld equations, which are described

in Section 8.3. We also introduce here the instanton equations, which corre-

spond to self-dual Yang–Mills ﬁelds; however a detailed discussion of these

equations and their applications is postponed until Chapter 9. The removable

nature of isolated singularities in the solutions of Yang–Mills equations is also

discussed in this section. In Section 8.4 we give a brief discussion of the non-

dual solutions of the Yang–Mills equations. Yang–Mills–Higgs equations with

various couplings to potential are studied in Section 8.5. The Higgs mecha-

nism of symmetry-breaking is introduced in Section 8.6, and its application

to the electroweak theory is discussed in Section 8.7. The full standard model

including QCD is also considered here. Invariant connections are important

in physical theories where both local and global symmetries are present. We

discuss them brieﬂy in Section 8.8.

8.2 Electromagnetic Fields

A source-free,orsimplyfree, electromagnetic ﬁeld is the prototype of

Yang–Mills ﬁelds. We will show that a source-free electromagnetic ﬁeld is a

gauge ﬁeld with gauge group U (1). Let P (M

,U(1)) be a principal U (1)-

bundle over the Minkowski space M

. Any principal bundle over M

is triv-

ializable. We choose a ﬁxed trivialization of P (M

,U(1)) and use it to write

P (M

,U(1)) = M

× U(1). The Lie algebra u(1) of U(1) can be identiﬁed

with iR. Thus a connection form on P may be written as iω, ω ∈ Λ

(P ), by

choosing i as the basis of the Lie algebra iR. The gauge ﬁeld can be written

as iΩ,whereΩ = dω ∈ Λ

(P ). The Bianchi identity dΩ = 0 is an immedi-

ate consequence of this result. The bundle ad(P ) is also trivial and we have

ad(P)=M

× u(1). Thus, the gauge ﬁeld F

∈ Λ

, ad(P )) on the base

canbewrittenasiF , F ∈ Λ

). Using the global gauge s : M

→ P

deﬁned by s(x)=(x, 1), ∀x ∈ M

, we can pull the connection form iω on P

to M

to obtain the gauge potential iA = is

∗

ω.Thusinthiscasewehavea

global potential A ∈ Λ

) and the corresponding gauge ﬁeld F = dA.The

Bianchi identity dF =0forF follows from the exactness of the 2-form F .

The ﬁeld equations δF = 0 are obtained as the Euler–Lagrange equations

minimizing the action



|F |

,where|F | is the pseudo-norm induced by the

Lorentz metric on M

and the trivial inner product on the Lie algebra u(1).

We note that the action represents the total energy of the electromagnetic

ﬁeld. The two equations

dF =0,δF=0 (8.1)

8.2 Electromagnetic Fields 237

are Maxwell’s equations for a source-free electromagnetic ﬁeld.

A gauge transformation f is a section of Ad(P )=M

× U(1). It is com-

pletely determined by the function ψ ∈F(M

) such that

f(x)=(x, e

iψ(x)

) ∈ Ad(P ), ∀x ∈ M

If iB denotes the potential obtained by the action of the gauge transforma-

tion f on iA,thenwehave

iB = e

−iψ

(iA)e

iψ

+ e

−iψ

iψ

or B = A + dψ,

which is the classical formulation of the gauge transformation f.Weobserve

that the group G of gauge transformations acts transitively on the solution

space of gauge connections A of equation (8.1) and hence the moduli space

A/G reduces to a single point. This observation plays a fundamental role in

the path integral approach to QED (quantum electrodynamics).

The above considerations can be applied to any U (1)-bundle over an arbi-

trary pseudo-Riemannian manifold M. We now consider the Euclidean ver-

sion of Maxwell’s equations to bring out the relation of diﬀerential geometry,

topology and analysis with electromagnetic theory as an example of gauge

theory. Let (M,g) be a compact, simply connected, oriented, Riemannian

4-manifold with volume form v

. A connection ω on P (M, U(1)) is called a

Maxwell connection or potential if it minimizes the Maxwell action

(ω) deﬁned by

(ω)=

8π



. (8.2)

The corresponding Euler–Lagrange equations are

=0,δF

=0. (8.3)

A solution of equations (8.3) is called a Maxwell ﬁeld or a source-free Eu-

clidean electromagnetic ﬁeld on M.Wenotethatequations(8.3)areequiv-

alent to the condition that F

be harmonic. Thus, a Maxwell connection is

characterized by its curvature 2-form being harmonic.

Now from topology we know that

(M,Z)=[M,CP

∞

where [M,CP

∞

] is the set of homotopy classes of maps from M to CP

∞

But, as discussed earlier in Chapter 5, CP

∞

is the classifying space for U(1)-

bundles. Thus, each element of [M, CP

∞

] determines a principal U (1)-bundle

over M by pulling back the universal U (1)-bundle S

∞

over CP

∞

. Hence each

element α ∈ H

(M,Z) corresponds to a unique isomorphism class of U(1)-

bundles P

over M. Moreover, the ﬁrst Chern class of P

equals α.Note

that the natural embedding of Z into R induces an embedding of H

(M,Z)