Marathe K. Topics in Physical Mathematics

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208 7 Theory of Fields, II: Quan t um and Topological

theories,orTFT. They are subdivided into topological classical ﬁeld

theories,orTCFT and topological quantum ﬁeld theories,orTQFT.

The earliest result in TCFT is the Gauss formula for the linking number.

We discuss this in Section 7.4. Donaldson’s instanton invariants are also dis-

cussed here. Topological quantum ﬁeld theories are discussed in Section 7.5.

Interpretation of the Donaldson’s polynomial invariants and the Jones poly-

nomial via TQFT is then given. The Atiyah–Segal axioms for TQFT are also

considered in this section. These topics are of independent interest and we

will return to them in later chapters. They are included here because of their

recent connections with topological quantum ﬁeld theories, which now form

an important branch of quantum ﬁeld theory (or QFT for short). Ideas from

QFT have already led to new ways of looking at old topological invariants of

low-dimensional manifolds as well as to surprising new invariants.

7.2 Non-perturbative Methods

From the mathematical point of view, gauge ﬁeld theories are classical ﬁeld

theories formulated on the inﬁnite-dimensional space of connections on a prin-

ciple bundle over a 4-dimensional space-time manifold M [273]. In physical

applications the manifold M is usually a non-compact, Lorentzian 4-manifold.

In many cases, M is taken to be the ﬂat Minkowski space of special relativ-

ity. The corresponding quantum ﬁeld theory is constructed by considering the

space of classical ﬁelds as a conﬁguration space C and deﬁning the quantum

expectation values of gauge invariant functions on C by using path integrals.

This is usually referred to as the Feynman path integral method of quan-

tization. Application of this method together with perturbative calculations

has yielded some interesting results in the quantization of gauge theories. The

starting point of this method is the choice of a Lagrangian deﬁned on the con-

ﬁguration space of classical gauge ﬁelds. This Lagrangian is used to deﬁne the

action functional that enters in the integrand of the Feynman path integral.

Two important examples of the action functional considered are the Yang–

Mills action and the Yang–Mills–Higgs action (see Chapter 8). Dimensional

reduction allows us to think of a Yang–Mills–Higgs ﬁeld as a Yang–Mills ﬁeld

on a higher dimensional manifold which is invariant under a certain sym-

metry group. Thus we may restrict our attention to the Yang–Mills case.

Furthermore, we wish to consider only the Euclidean quantum ﬁeld theory,

where the pseudo-Riemannian space-time manifold is replaced by a Rieman-

nian manifold. In physical literature the passage from a Lorentzian manifold

to a Riemannian manifold is often referred to as a Wick rotation of the

time coordinate. The Euclidean quantum ﬁeld theory plays an important

role in the calculation of tunneling amplitudes in quantum ﬁeld theories (see

Coleman [80] for a discussion of this and related aspects of quantization).

7.2 Non-perturbative Methods 209

We consider the following mathematical setting for Euclidean quantum

ﬁeld theory. Let (M,g) be a compact, connected, oriented, Riemannian 4-

manifold and let G be a compact, semisimple Lie group with Lie algebra

g. From the physical point of view, the group G is the gauge group of the

classical gauge ﬁeld that we wish to quantize. Let P (M, G) be a principal G-

bundle. If G = SU(n), then the isomorphism class of P is determined by its

topological quantum number or the instanton number k = −c

(P ) ∈ Z.We

introduce a free parameter μ in the theory by considering the one-parameter

family of G-invariant inner products X, Y 

on the Lie algebra g deﬁned by

X, Y 

:= −

K(X, Y ), 0 <μ∈ R, (7.1)

where K is the Killing form on G.

The inner product deﬁned by (7.1)andthemetricg induce inner products

on each ﬁber of the space Λ

(M,ad(P )) = Λ

(M) ⊗ ad(P)ofp-forms on M

with values in the bundle ad(P )=P ×

g. We denote these inner products

by  , 

or simply by  , 

when the metric g is ﬁxed. The corresponding

pointwise norm is denoted by ||

,orsimplyby||

. By integration over M

these deﬁne the L

inner product

α, β



α, β

, ∀α, β ∈ Λ

(M,ad(P )), (7.2)

where dv

is the volume form on M determined by the metric g. The norm of

α corresponding to this L

inner product  , 

is denoted by α

.Thus,

α



|α|

,α∈ Λ

(M,ad(P )). (7.3)

In classical gauge ﬁeld theories, the norms corresponding to diﬀerent coupling

constants μ lead to the same ﬁeld equations up to trivial rescaling. However,

as we shall show below, on quantization the diﬀerent coupling constants μ

lead to a one-parameter family of quantum ﬁeld theories. By taking μ to be

a suitable function of the other parameters of the given theory it may be

possible to construct a renormalized version of the theory.

Recall that a connection ω on P determines the exterior covariant diﬀer-

ential d

, its formal adjoint δ

, and the covariant derivative ∇

on the full

tensor algebra with values in ad(P ). These in turn determine the Laplacians

on the various spaces Λ

(M,ad(P )). The ± 1 eigenspaces Λ

(M,ad(P )) of

the Hodge operator give a decomposition of the curvature F

into its self-dual

and anti-dual parts, i.e.,

= F

+ F

−

. (7.4)

In terms of these parts we can write the instanton number k and the Yang–

Mills action S

as follows:

210 7 Theory of Fields, II: Quan t um and Topological

k =

8π



||F

−||F

−



8π





−|F

−



, (7.5)

(ω)=

8π

[||F

+ ||F

−

8π



[|F

+ |F

−

]dv

. (7.6)

From equation (7.6) it follows that μ rescales the Yang–Mills action, i.e.,

(ω)=

(ω). We denote by A

the space of connections on P and by

the group of gauge transformations. The Yang–Mills action S

deﬁned on

is gauge invariant (i.e., invariant under the natural action of G

on A

We now deﬁne A(M) to be the disjoint union of the A

over all equivalence

classes of principal G-bundles over M. The Euclidean quantum ﬁeld theory

may be considered an assignment of the quantum expectation Φ

each gauge invariant function Φ : A(M) → R. A gauge invariant function

Φ : A(M ) → R is called an observable in quantum ﬁeld theory. In the

Feynmanpathintegralapproach to quantization, the quantum expecta-

tion Φ

of an observable is given by the following expression

Φ



A(M)

−S

(ω)

Φ(ω)DA



A(M)

−S

(ω)

, (7.7)

where DA denotes a measure on A(M) whose precise deﬁnition is not known,

in general. It is customary to express the quantum expectation <Φ>

terms of the partition function Z

deﬁned by

(Φ):=



A(M)

−S

(ω)

Φ(ω)DA. (7.8)

Thus, we can write

Φ

(Φ)

(1)

. (7.9)

In the above equations we have written the quantum expectation as Φ

indicate explicitly that, in fact, we have a one-parameter family of quantum

expectations indexed by the coupling constant μ in the Yang–Mills action.

In what follows we drop this subscript with the understanding that we do

have a one-parameter family of quantum ﬁeld theories when dealing with

non-perturbative aspects of the theory. A mathematically precise deﬁnition

of the Feynman path integral is not available at this time. Feynman (1918–

1988) developed his diagrams and a set of rules to extract physically relevant

information from the path integral in speciﬁc applications. He successfully

applied these methods to his study of quantum electrodynamics. Feynman

had a wide range of interests in and out of science. He received the Nobel

Prize for physics for 1965, jointly with Schwinger and Tomonaga for their

work in quantum electrodynamics.

7.2 Non-perturbative Methods 211

There are several examples of gauge-invariant functions. For example, pri-

mary characteristic classes evaluated on suitable homology cycles give an

important family of gauge invariant functions. The instanton number k of

P (M, G) belongs to this family, as it corresponds to the second Chern class

evaluated on the fundamental cycle of M representing the fundamental class

[M]. The pointwise norm |F

of the gauge ﬁeld at x ∈ M,theabsolute

value |k| of the instanton number k, and the Yang–Mills action S

are also

gauge-invariant functions. We now give two other important examples of

gauge-invariant functions.

Example 7.1 (Instanton scale size λ) In Chapter 9 we will show that an

instanton solution of Yang–Mills equations on R

(or on its compactiﬁcation

) is characterized by three parameters, namely the instanton number k,

the center x ∈ R

,andthescale size λ. We can extend the deﬁnition given

there to the entire conﬁguration space A(R

) as follows:

λ(ω)=inf{ρ(x)|x ∈ R

}, (7.10)

where

ρ(x)=sup







(x,r)

≤



Thus, λ(ω) is the radius of the smallest sphere that contains half the Yang–

Mills action. It is ea sy to see that the function λ : A(R

) → R is gauge-

invariant. An expression for the semiclassical expectation of λ is given in the

next section.

In the next example we introduce two families of gauge invariant func-

tions that generalize the Wilson loop functional well-known in the physics

literature.

Example 7.2 (Wilson loop functional) Let ρ denote a representation of G

on V .Letα ∈ Ω(M, x

) denote a loop at x

∈ M. Let π : P (M,G) → M

be the canonical projection and let p ∈ π

−1

). If ω is a connection on P

then the parallel translation along α maps the ﬁber π

−1

) into itself. Let

ˆα

: π

−1

) → π

−1

) denote this map. Since G acts transitively on t he

ﬁbers, ∃g

∈ G such that ˆα

(p)=pg

. Now deﬁne

ρ,α

(ω):=Tr[ρ(g

)]. (7.11)

We note that g

and hence ρ(g

) change by conjugation if instead of p we

choose another point in the ﬁber π

−1

), but the trace remains unchanged.

Alternatively, we can consider the ve ctor bundle P ×

V associated to the

principal bundle P and parallel displacement of its ﬁbers induced by α.Letπ :

P ×

V → M be the canonic al projection. We note that in this c ase π

−1

)

∼

V . Now the map ˆα

: π

−1

) → π

−1

) is a linear transformation and we

can deﬁne

ρ,α

(ω):=Tr[ˆα

]. (7.12)

212 7 Theory of Fields, II: Quan t um and Topological

We call these W

ρ,α

the Wilson loop functionals associated to the repre-

sentation ρ and the loop α. In the particular case when ρ =Ad, the adjoint

representation of G on g, our constructions reduce to those considered in

physics.

We note that the gauge invariance of Φ makes the integral deﬁning Z di-

vergent, due to the inﬁnite contribution coming from gauge-equivalent ﬁelds.

To avoid this diﬃculty observe that the integrand is gauge-invariant and

hence Z descends to the orbit space O = A(M)/G and can be evaluated by

integrating over this orbit space O. However, the mathematical structure of

this space is essentially unknown at this time. Physicists have attempted to

get around this diﬃculty by choosing a section s : O→Aand integrating

over its image s(O) with a suitable weight factor such as the Faddeev–Popov

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

ables eﬀected by p

|s(O)

: s(O) →O. As we saw in Chapter 6, this gauge

ﬁxing procedure does not work in general, due to the presence of the Gri-

bov ambiguity. Also the Faddeev–Popov determinant is inﬁnite-dimensional

and needs to be regularized. This is usually done by introducing the anti-

commuting Grassmann variables called the ghost and anti-ghost ﬁelds.

The Lagrangian in the action term is then replaced by a new Lagrangian

containing these ghost and anti-ghost ﬁelds. This new Lagrangian is called

the eﬀective Lagrangian. The eﬀective Lagrangian is not gauge-invariant,

but it is invariant under a special group of transformations involving the ghost

and anti-ghost ﬁelds. These transformations are called the BRST (Becchi–

Rouet–Stora–Tyutin) transformations. On the inﬁnitesimal level the BRST

transformations correspond to cohomology operators and deﬁne what may be

called the BRST cohomology. The non-zero elements of the BRST cohomol-

ogy are called anomalies in the physics literature. At present there are several

interesting proposals for studying these questions, proposals that make use

of equivariant cohomology in the inﬁnite-dimensional setting and which are

closely related to the various interpretations of BRST cohomology (see, for

example, [21,193,236,404]). A detailed discussion of the material of this sec-

tion from a physical point of view may be found in the books on quantum

ﬁeld theory referred in the introduction. A geometrical interpretation of some

of these concepts may be found in [27,86].

The general program of computing the curvature of connections on inﬁnite

dimensional bundles and of deﬁning appropriate generalizations of character-

istic classes was initiated by Isadore Singer in his fundamental paper [351]on

Gribov ambiguity. Today this is an active area of research with strong links

to quantum ﬁeld theory (see, for example, [23,61,174,233,234,352,404]). We

now give a brief description of some aspects of this program.

We proceed by analogy with the ﬁnite-dimensional case. To simplify con-

siderations let us suppose that the group of gauge transformations G acts

freely on the space of gauge connections A. Then we can consider A to be a

principal G-bundle over the space B = A/G of gauge equivalence classes of

connections. Deﬁne the map

7.2 Non-perturbative Methods 213

−

: A→Λ

−

(M,ad(P )) by ω → F

−

. (7.13)

Then F

−

is equivariant with respect to the standard action of G on A and the

linear action of G on the vector space Λ

−

(M,ad(P )). Hence, we may think

of F

−

as deﬁning a section of the vector bundle E associated to A with ﬁber

−

(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

thereby 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

−

(see Theorem 6.2) is the moduli

space M

⊂B.Thus

=(F

−

)

−1

({0})/G. (7.14)

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

−

and

to 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.

In the case of Abelian gauge ﬁelds we have the following argument (see

[280] for details). 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.Weconsider

the case of quantum electrodynamics where the gauge group G is U(1). In

this case the moduli space of instantons is 0-dimensional and the computation

of the index can be carried out as in the ﬁnite-dimensional case. The index

turns out to be related to an important new invariant due to Donaldson (see

[109]). Even in this case there does not seem to be any version of the Thom

class with compact support. Instead, we have an equivariant version with

Gaussian asymptotics, which may be regularized. Locally, the action of U(1)

on R

is given by the vector ﬁeld

X = −x

∂

∂x

+ x

∂

∂x

If ω is a gauge connection form on P (M,U(1)), then a representative of an

equivariant Thom class α is given by

214 7 Theory of Fields, II: Quan t um and Topological

α =

−|x|

(u +

dω +(dx

+ ωx

) ∧ (dx

− ωx

)), (7.15)

where u is a degree-2 indeterminate, which enters in the formulation of equiv-

ariant cohomology. The precise relation of equation (7.15) to the physical

computation of invariants in quantum electrodynamics or its role in topolog-

ical quantum ﬁeld theory are yet to be understood.

On the principal G-bundle A = P (B, G), we can deﬁne a natural connection

as follows. For ω ∈A, the tangent space T

A is identiﬁed with Λ

(M,ad(P ))

and hence carries the inner product as deﬁned in Deﬁnition 6.1 with p =1.

With respect to this inner product we have the orthogonal splitting

A = V

⊕ H

, (7.16)

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 (see Section 6.3), the connection form ˆω : T A→LGof this

connection is given by

ˆω(X)=G

(δ

X),X∈ T

A, (7.17)

where G

is the Green operator, which inverts the Laplacian

= δ

: Λ

(M,ad(P )) → Λ

(M,ad(P )).

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

]), (7.18)

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. We now explain the relation of the natural

connection ˆω deﬁned above to a connection deﬁned in [23]. Regarding G as

a subgroup of Aut(P ), we have a natural action of G on P (M,G) and hence

on P (M,G) ×A. The quotient of this action can be regarded as a principal

bundle P over M ×Bcalled the Poincar´e bundle. Pulling back the natural

connection ˆω to the Poincar´e bundle P, we get a connection that coincides

with the one deﬁned in [23]. The Poincar´e bundle with this connection can

7.2 Non-perturbative Methods 215

be regarded as a universal bundle with respect to deformations of the bundle

P (M, G).

In addition to the gauge connection on the bundle P (M, G) and associated

diﬀerential operators, there are other bundles and operators which enter nat-

urally into many mathematical and physical applications. For example, on

M there is the frame bundle L(M) and the Levi-Civita connection on iti,

which are fundamental in gravitation. Another important example is that

of the Dirac operator deﬁned on spinor bundles over M ,whenM is a spin

manifold. More generally, we can consider an elliptic operator or a sequence

of elliptic operators on sections of real or complex vector bundles over M.

For deﬁniteness, we consider a ﬁxed elliptic operator D : Γ (V ) → Γ (V ),

where V is a vector bundle over M .IfE is a vector bundle associated to

P (M, G), then each connection ω ∈Ainduces a connection and correspond-

ing covariant derivative operator on sections of E. By coupling the ellip-

tic operator D with this gauge connection ω, we obtain an elliptic operator

: Γ (V ⊗ E) → Γ (V ⊗ E). Then ω →D

deﬁnes a family

D of elliptic

operators indexed by ω ∈A. From ellipticity of D

it follows that ker D

and coker D

are ﬁnite dimensional. By the Atiyah–Singer index theorem, it

follows that the numerical index

n(ω)=dimkerD

− dim coker D

(7.19)

is independent of ω. The family

D can be used to deﬁne two bundles over

A, whose ﬁbers over ω ∈Aare ker D

and coker D

,andavirtual bundle

(in the sense of K-theory), whose ﬁber over ω is ker D

− coker D

.This

virtual bundle is called the index bundle Ind

D of

D.Infact,sinceG acts

equivariantly on the operators D

, the index bundle descends to the quotient

and can be regarded as an element of the Grothendieck group K(B). In

particular, its restriction to the ﬁnite-dimensional instanton moduli space

M⊂Bdeﬁnes an element of K(M). In this general set up, we cannot apply

the index theorem for families to B. However, in the special case when D is

the Dirac operator, we have the following formula

ch(Ind

D)=



ˆa(M) ch(E), (7.20)

where ˆa(M ) is the characteristic class of the spinor bundle whose value on

the fundamental cycle of M is the

A-genus and E is a certain vector bundle

associated to the Poincar´e bundle P (see [23] for further details). The natural

connection ˆω on the principal bundle A = P(B, G) restricts to deﬁne a con-

nection on the index bundle in the case that it corresponds to a real vector

bundle. For example, this happens if coker D

=0, ∀ω ∈A(see [204]). In

this case there is also a well-deﬁned second fundamental form and one can

apply the Gauss–Codazzi equation to this situation to derive information on

the geometry of B.

216 7 Theory of Fields, II: Quan t um and Topological

Several of the topics mentioned above are now considered to be parts of

topological ﬁeld theory, which may be viewed as the study of topological

structures related to inﬁnite-dimensional manifolds and inﬁnite-dimensional

bundles over them. The mathematical foundations of this theory are not yet

precisely formulated. In the next section, we discuss one of the most widely

used perturbative techniques, namely, the semiclassical approximation to the

partition functions for Yang–Mills theories.

7.3 Semiclassical Approximation

It is well-known that an asymptotic description of physical ﬁeld theories has

become an indispensable tool in deciding the validity and usefulness of these

theories, which is done by our comparing their predictions with experimen-

tal measurements. In fact, the actual calculations carried out in quantum

electrodynamics as well as in other areas of physics have been veriﬁed to a

high degree of accuracy, even though fundamental theoretical justiﬁcation

for them is not always evident. These calculations involve various methods

of perturbation theory. For Hamiltonian systems a geometric description of

perturbation theory suitable for computer simulation and symbol manipula-

tion techniques is given in Omohundro [307]. The mathematical formulation

suitable for quantum ﬁeld theories is currently under development.

A widely used method of approximation in quantum ﬁeld theories is the

so-called semiclassical,ortheone loop, approximation to the Feynman

path integrals. We give a brief account of the semiclassical approximation

for Yang–Mills theory based on [174]. It is well-known that the partition

functions of quantum Yang–Mills theory have an expansion in powers of the

coupling constant. The leading order term in this expansion is called the

semiclassical approximation. As we observed in the previous section we have

a one parameter family of quantum expectations indexed by the coupling

constant μ in the Yang–Mills action. As μ → 0 the integrals deﬁning the

quantum expectation <Φ>of the observable Φ have an asymptotic expan-

sion in powers of μ. The leading order term in the resulting expansion of

[<Φ>] is called the semiclassical expectation of Φ and is denoted by

<Φ>

. The calculation of [<Φ>

] requires a detailed study of some

elliptic operators. Two important techniques that are required for this study

are (i) the zeta function regularization and (ii) study of the associated heat

family. These techniques are also useful in other applications. We therefore,

give a slightly more general discussion of these techniques than is necessary

for the statement of results.

7.3 Semiclassical Approximation 217

7.3.1 Zeta Function Regularization

Let Q be a symmetric, positive operator on a ﬁnite-dimensional real Hilbert

space H.WeidentifyH with R

and deﬁne the positive deﬁnite quadratic

form

Q on R

associated to Q by

Q(x)=<Q(x),x >, ∀x ∈ R

. Then the

Gaussian integral associated to Q is given by



exp(−π

Q(x)) dx =(detQ)

−1/2

, (7.21)

where Q(x)=



,detQ is the determinant of the symmetric matrix

), and dx is the standard volume form on R

. We are interested in gen-

eralizing this Gaussian integral to the case when Q is a symmetric, positive

operator on an inﬁnite-dimensional Hilbert space. We could deﬁne the inte-

gral by the same formula as formula (7.21)provideddetQ is well-deﬁned. We

now consider the method of zeta function regularization, which allows us to

deﬁne det Q for a large class of operators. This class includes the families of

Laplace operators which enter in the computation of semiclassical approxi-

mations in Yang–Mills theory. Recall ﬁrst that in the ﬁnite-dimensional case

we can write

det Q =



i=1

=exp



i=1

ln(λ

), (7.22)

where λ

,1≤ i ≤ n, is the complete set of eigenvalues of Q.NowletQ

be an operator on an arbitrary separable Hilbert space such that Q has a

discrete set of positive eigenvalues λ

,i∈ N. We deﬁne the generalized

zeta function ζ

(z):=

∞



i=1

(λ

)

−z

. (7.23)

It can be shown (see [345]) that under certain conditions on Q,thesumin

formula (7.23) converges for Re(z) >> 0 (i.e., for suﬃciently large Re(z))

and has a meromorphic continuation to the entire complex z-plane with only

simple poles. Furthermore, zero is a regular point of ζ

(z) and hence ζ



(0)

is well-deﬁned. From equation (7.23) it follows that



(z):=−

∞



i=1

ln(λ

)(λ

)

−z

. (7.24)

Thus formally one can write



(0) := −

∞



i=1

ln(λ

). (7.25)

In view of this equation we deﬁne det Q by