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218 7 Theory of Fields, II: Quan t um and Topological

det Q := exp(−ζ



(0)). (7.26)

We note that equation (7.26) reduces to the usual deﬁnition of determinant

in the ﬁnite-dimensional case.

Example 7.3 Consider a real scalar ﬁeld φ on a compact Riemannian man-

ifold (M, g) with Lagr angian density

L =

∂

φ∂

φ −m

) −V (φ), (7.27)

where m denotes the mass and V the potential. Then the vacuum to vacuum

amplitude Z in the absence of sources is given by

Z =



exp



−



<φ,Aφ>dv



dμ(φ), (7.28)

where μ is a measure on the space H of all sc a lar ﬁelds, dv

is the volume

form on M determined by the metric g,andA is an operator on H deﬁned

A := −g

∂

+ m

∂V

∂φ

. (7.29)

Under certain conditions H is a separable Hilbert space, A is a positive, sym-

metric operator with a discrete set of eigenvalues λ

,andH has an orthonor-

mal basis ψ

consisting of eigenvectors of A. Then the zeta function re gular-

ization of the det A allows us to evaluate the inte grals deﬁning Z to obtain

Z =exp





(0)



. (7.30)

The discussion given above is implicit in most evaluations of such amplitudes.

We note that the zeta function regularization can be applied also in the

fermionic case. This requires an extension of usual calculus to Grassmann or

anti-commuting variables. An introduction to this subject may be found in

Berezin [38].

7.3.2 Heat Kernel Regularization

Let E be a vector bundle over a compact Riemannian manifold M of di-

mension m =2n.LetD : Γ (E) → Γ (E) be a self-adjoint, non-negative

elliptic operator on the space Γ (E) of the sections of E over M.Fromthe

standard theory of elliptic operators we know that D extends to the Hilbert

space completion L

(E)ofΓ (E) as a self-adjoint, non-negative elliptic op-

erator and that L

(E) has a complete orthonormal basis of eigenvectors

{φ

| Dφ

= λ

}. Furthermore, the spectrum {λ

} is discrete and non-

7.3 Semiclassical Approximation 219

negative and each eigenspace is ﬁnite-dimensional. In physics literature ker D

with basis {φ

| Dφ

=0} is called the space of zero modes. One often fac-

tors out the zero modes to obtain an operator D

with positive spectrum. In

what follows we shall consider that this has been done and denote D

just

by D.

The heat family associated with D is the unique family of operators H

on L

(E) satisfying the heat equation

∂

∂t

)=−DH

, (7.31)

with H

= id. It is customary to write formally H

= e

−tD

. We note that for

z ∈ C,Re(z) >> 0, the operator D

−z

is well-deﬁned and is of trace class.

This complex power of the operator D, its zeta function ζ

,andtheheat

family are related by

(z)=Tr(D

−z

Γ (z)



∞

z−1

Tr(e

−tD

)dt, (7.32)

where Γ (z) is the usual gamma function. The operators H

are themselves

of trace class and are given by convolution with the heat kernel K

.Heat

equation was used by Patodi in his study of the index theorem. It can also be

used to study harmonic forms and the Hodge decomposition theorem, (see,

for example, [214]). It is the detailed study of the heat kernel that enters

into the computation of the semiclassical measure when applied to suitable

Laplace operators on Λ

(M,ad(P )).

The calculations of the semiclassical expectation are quite involved. We

state the result for the expectations of |k| and λ in the case that M = S

and G = SU(2) and refer to [174] for further details.

Theorem 7.1 Let M = S

and G = SU(2).LetM

denote the moduli space

of k =1instantons over M . Then for any smooth, bounded, gauge-invariant

function Φ,thesemiclassical partition function

(Φ)=



Φ(ω)e

−S

(ω)

dν (7.33)

is ﬁnite. Moreover, one has an explicit formula for the semiclassical measure

dν on M

). In particular, the semiclassical expectations of |k| and λ,

where k is the instanton number and λ the instanton scale size, are given by

|k|

≈ 0.05958Cμ

−8π

/μ

(7.34)

and

λ

≈ 0.03693Cμ

−8π

/μ

, (7.35)

where C =2

−5/2

23/2

220 7 Theory of Fields, II: Quan t um and Topological

We note that, even though both expectations decrease exponentially as

μ → 0, their ratio is non-zero and we have

|k|

λ

≈ 0.6198.

On the other hand the ratio of classical expectations is given by

|k|

λ

≈ 0.3112.

These ratios can be interpreted as conditional expectations: the expected

scale size of a gauge potential, given that it has instanton number ±1. The

diﬀerence in the semiclassical and the classical values is due to the diﬀer-

ence between the semiclassical measure and the classical measure used in

computing these expectations. It remains to be seen how these results can

be extended to non-compact Lorentz manifolds to compute quantum ﬁeld

theoretic expectation values of physically signiﬁcant quantum variables.

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

topological ﬁeld theories. The earliest example of a TFT result occurs in the

well-known formula for the linking number by Gauss. In the next section we

discuss this as an example of TCFT.

7.4 Topological Classical Field Theories (TCFTs)

We discuss the invariants of knots and links in 3-manifolds in Chapter 11.

Here we consider one of the earliest investigations in combinatorial knot the-

ory, contained in several unpublished notes written by Gauss between 1825

and 1844 and published posthumously as part of his Nachlass

.Theydeal

mostly with his attempts to classify “Tractﬁguren,” or closed curves in the

plane with a ﬁnite number of transverse self-intersections. As we shall see in

Chapter 11, such ﬁgures arise as regular plane projections of knots in R

However, one fragment of Gauss’s notes deals with a pair of linked knots:

Es seien die Koordinaten eines unbestimmten Punkts der ersten

Linie x, y, z; der zweiten x



und



[(x



− x)

+(y



− y)

+(z



− z)

]

−3/2

[(x



− x)(dydz



− dzdy



)

+(y



− y)(dzdx



− dxdz



)+(z



− z)(dxdy



− dydx



)] = V

Estate.

Let the coordinates of an arbitrary point on the ﬁrst curve be x, y, z; of the second



and let

7.4 Topological Classical Field Theories (TCFTs) 221

dann ist dies Integr al durch beide Linien ausgedehnt

=4πm

und m die Anzahl der Umschlingungen.

Der Werth ist ge genseitig, d.h. er bleibt derselbe, wenn beide Linien

gegen einander umgetauscht werden,

1833. Jan. 22.

In this fragment, Gauss had given an analytic formula for the linking

number of a pair of knots. This number is a combinatorial topological in-

variant. As is quite common in Gauss’s work, there is no indication of how

he obtained this formula. The title of the note “Zur Elektrodynamik” (“On

Electrodynamics”) and his continuing work with Weber on the properties of

electric and magnetic ﬁelds lead us to guess that it originated in the study of

the magnetic ﬁeld generated by an electric current ﬂowing in a curved wire.

Recall that the magnetic ﬁeld due to a unit current ﬂowing along a wire C

generates the magnetic ﬁeld B(r



)atapointr



∈ C



given by

B(r



4π





− r) × dr



− r|

where we have used the vector notation r =(x, y, z)andr



=(x



)and

× is the vector product. The work W done by this magnetic ﬁeld in moving

a unit magnetic pole around the wire C



is given by

W =



B(r



) ·dr = V/4π.

Maxwell knew Gauss’s formula for the linking number and its topological

signiﬁcance and its origin in electromagnetic theory. In fact, in commenting

on this formula, he wrote:

It was the discovery by Gauss of this very inte gral expressing the work

done on a magnetic p o le while describing a closed curve in the presence

of a closed electric current and indicating the geometric connection be-

tween the two closed curves, that led him to lament the small progress

made in the Geometry of Position since the time of Leibnitz, Euler and

Vandermonde. We now have some progress to report, chieﬂy due to

Riemann, Helmholtz and Listing.

In obtaining a topological invariant by using a physical ﬁeld theory, Gauss

had anticipated topological ﬁeld theory by almost 150 years. Even the term

“topology” was not used then. It was introduced in 1847 by J. B. Listing, a

then this integral taken along both curves is = 4πm and m is the number of intertwinings

(linking number in modern terminology). The value (of the integral) is common (to the

two curves), i.e., it remains the same if the curves are interchanged,

222 7 Theory of Fields, II: Quan t um and Topological

student and proteg´e of Gauss, in his essay “Vorstudien zur Topologie” (“Pre-

liminary Studies on Topology”). Gauss’s linking number formula can also be

interpreted as the equality of topological and analytic degree of a suitable

function. Starting with this a far-reaching generalization of the Gauss inte-

gral to higher self-linking integrals can be obtained. We discuss this work

in Chapter 11. This result forms a small part of the program initiated by

Kontsevich [235] to relate topology of low-dimensional manifolds, homotopi-

cal algebras, and non-commutative geometry with topological ﬁeld theories

and Feynman diagrams in physics.

7.4.1 Donaldson Invariants

Electromagnetic theory is the prototype of gauge theories with Abelian gauge

group U(1). Its generalization to non-Abelian gauge group SU(2) was ob-

tained by Yang and Mills in 1954. A spectacular application of Yang–Mills

theory as TCFT came thirty years later in Donaldson theory. We discuss

this theory in detail in Chapter 9. Here we simply indicate its interpreta-

tion as a topological ﬁeld theory based on the classical instanton solutions

of Yang–Mills equations. Donaldson’s theorem on the topology of smooth,

closed, 1-connected 4-manifolds provides a new obstruction to smoothability

of these topological manifolds. A surprising ingredient in his proof of this

theorem was the moduli space I

of SU(2)-instantons on a manifold M .This

theorem has been applied to obtain a number of new results in topology and

geometry and has been extended to other manifolds. The space I

is a sub-

space of the moduli space M

of Yang–Mills ﬁelds with instanton number

1. The space M

in turn is a subspace of the moduli space of A/G of all

Yang–Mills ﬁelds on M.Infact,wehave

A/G =



Donaldson shows that the space I

(or a suitable perturbation of it) is

a 5-dimensional manifold with singularities and with one boundary compo-

nent homeomorphic to the original base space M. By careful study of the

remaining boundary components Donaldson obtained the following theorem.

Theorem 7.2 (Donaldson) Let M be a smooth, closed, 1-connected, oriented

manifold of dimension 4 with positive deﬁnite intersection form ι

.Then

∼

(1), the identity form of rank b

, the second Betti number of M .

This theorem is the genesis of what can be called gauge-theoretic topol-

ogy. In his later work, Donaldson used the homology of spaces M

,forsuf-

ﬁciently large k, to obtain a family of new invariants of a smooth 4-manifold

M, satisfying a certain condition on its intersection form. We now describe

these invariants, which are known as Donaldson’s polynomial invariants,

7.4 Topological Classical Field Theories (TCFTs) 223

or simply Donaldson polynomials. The Donaldson polynomials are deﬁned

by polarization of a family q

of symmetric, multilinear maps

: H

(M) ×···×H

(M)



 

d times

→ Q,

where k is the instanton number of P(M,SU(2)) and d is a certain function

of k. We shall also refer to the maps q

as Donaldson polynomials. A basic

tool for the construction of Donaldson polynomials is a map that transfers

the homology of M to the cohomology of the orbit space O

of irreducible

connections on the SU(2)-bundle P . The details of this construction are given

in Chapter 9. Donaldson polynomials are examples of TCFT invariants. It

is interesting to note that in [404] Witten has given a TQFT interpretation

for them. We comment on this in the next section, where an introduction to

TQFT is given. Before that we give two more examples of TCFT.

7.4.2 Topological Gravity

Einstein’s theory of gravity is the most extensively studied and experimen-

tally supported theory of gravity at this time. However, Einstein was not

content with it. We have already considered his introduction of the cosmo-

logical constant and indicated a generalized theory of gravity, which replaces

the cosmological constant by the cosmological function. Over the years many

alternative theories of gravitation have been proposed. Several of these start

with a variational principle with some geometric function as the Lagrangian.

Lanczos [242] in his study of Lagrangians for generalized gravitational ﬁeld

equations observed that one of his Lagrangians led to the integral that was

invariant under the action of the group of diﬀeomorphisms of the space-time

manifold and so did not contain any dynamics. Lanczos had obtained the ex-

pression for the Euler class of a 4-manifold as an integral of a polynomial in

the Riemann curvature, but he did not recognize its topological signiﬁcance.

The general formula for the Euler class of an arbitrary oriented manifold

was obtained by Chern, generalizing the earlier work for hypersurfaces by

Weil and Allendoerfer. It was in studying this generalization that Chern was

led to his famous characteristic classes, now called the Chern classes. All

Chern classes and not just the Euler class are topological invariants and can

be expressed as integrals of polynomials in the Riemann curvature. We can,

therefore, consider them topological gravity invariants if we think of these

polynomials as Lagrangians of some generalized gravitational ﬁeld in arbi-

trary dimension.

224 7 Theory of Fields, II: Quan t um and Topological

7.4.3 Chern–Simons (CS) Theory

Chern–Simons theory is a classical gauge ﬁeld theory formulated on an odd-

dimensional manifold. We discuss it in detail in Chapter 10 and apply it to

consider Witten’s QFT interpretation of the Jones polynomial. Chern–Simons

theory became widely known after Witten’s paper was published. It gives new

invariants as TCFT also. We discuss this after reviewing the 3-dimensional

CS theory. Let M be a compact, connected, oriented 3- manifold and let

P (M ) be a principal bundle over M with structure group SU(n). Then the

Chern–Simons action A

is deﬁned by

4π



tr(A ∧ F +

A ∧ A ∧A), (7.36)

where k ∈ R is a coupling constant, A denotes the pull-back to M of the

gauge potential ω by a section of P ,andF = F

= d

A is the gauge ﬁeld

on M corresponding to the gauge potential A. We note that the bundle P

always admits a section over a 3-manifold. The action is manifestly covariant

since the integral involved in its deﬁnition is independent of the metric on

M. It is in this sense that the Chern–Simons theory is a topological ﬁeld

theory. The ﬁeld equations obtained by variation of the action turn out to be

ﬂat connections. A detailed calculation showing this is given in Chapter 10.

Under a gauge transformation g the action transforms as follows:

)=A

(A)+2πkA

, (7.37)

where

24π





αβγ

tr(θ

) (7.38)

is the Wess–Zumino action functional. It can be shown that the Wess–Zumino

functional is integer-valued and hence if the Chern–Simons coupling constant

k istakentobeaninteger,thenwehave

)

= e

(A)

It follows that the path integral quantization of the Chern–Simons model

is gauge-invariant. This conclusion holds more generally for any compact

simple group. In the next subsection we discuss brieﬂy the development of

topological QFT. Chern–Simons theory played a fundamental role in this

development. It also gives a new way of looking at the Casson invariant.

This is an example of TCFT. It rests on the observation that the mod-

uli space M

(N,H)ofﬂatH-bundles over a manifold N can be identiﬁed

with the set hom(π

(N),H)/H. The moduli space M

(N,H)andtheset

hom(π

(N),H) have a rich mathematical structure, which has been exten-

sively studied. These spaces appear in the deﬁnition of the Casson invariant

for a homology 3-sphere and its gauge theoretic interpretation was given by

7.5 Topological Quantum Field Theories (TQFTs) 225

Taubes [366]. Recall that the Casson invariant of an oriented homology 3-

sphere Y is deﬁned in terms of the number of irreducible representations of

(Y )intoSU(2). As indicated above, this space can be identiﬁed with the

moduli space M

(Y,SU(2)) of ﬂat connections in the trivial SU(2)-bundle

over Y .ThemapF : ω →∗F

deﬁnes a natural 1-form and its dual vector

ﬁeld V

on A/G. Thus, the zeros of this vector ﬁeld V

are just the ﬂat

connections. We note that since A/G is inﬁnite-dimensional, it is necessary to

use suitable Fredholm perturbations to get simple zeros and to count the in-

dex of the vector ﬁeld with appropriate signs. Taubes showed that this index

equals the Casson invariant of Y . In classical geometric topology of a compact

manifold M,thePoincar´e–Hopf theorem tells us that the index of a vector

ﬁeld equals the Euler characteristic of M. This theorem does not apply to

the inﬁnite-dimensional case considered here. Is there some homology theory

associated to the above situation whose Euler characteristic would be equal

to the Casson invariant? The surprising aﬃrmative answer to this question

is provided by Floer homology. We discuss it brieﬂy in the next section and

in detail in Chapter 10.

7.5 Topological Quantum Field Theories (TQFTs)

In recent developments in low-dimensional topology of manifolds, geometric

analysis of partial diﬀerential equations plays an important role. These equa-

tions have their origins in physical ﬁeld theories. A striking example of this

is provided by Donaldson’s work on smoothability of 4-manifolds, where the

moduli space of instantons is used to obtain a cobordism between the base

manifold M and a certain manifold N whose topology is well understood,

thereby enabling one to study the topology of M. In physics the moduli

spaces of these classical, non-Abelian gauge ﬁelds are used to study the prob-

lem of quantization of ﬁelds. Even though a precise mathematical formulation

of QFT is not yet available, the methods of QFT have been applied success-

fully by Witten for studying the topology and geometry of low-dimensional

manifolds. It seems reasonable to say that his work has played a fundamental

role in the creation of topological QFT (see [41] and references therein for a

review of this fast developing ﬁeld).

Let Σ

be a compact Riemann surface of genus g. Then the moduli space

(Σ

,H) of ﬂat connections has a canonical symplectic structure ι.We

now discuss an interesting physical interpretation of the symplectic manifold

(Σ

,H),ι). Consider a Chern–Simons theory on the principal bundle

P (M, H) over the (2 + 1)-dimensional space-time manifold (M = Σ

× R)

with gauge group H and with time-independent gauge potentials and gauge

transformations. Let A (resp., H) denote the space (resp., group) of these

gauge connections (resp. transformations). It can be shown that the curvature

deﬁnes an H-equivariant moment map

226 7 Theory of Fields, II: Quan t um and Topological

μ : A→LH

∼

(M,ad P ), by ω →∗F

where LH is the Lie algebra of H. The zero set μ

−1

(0) of this map is precisely

the set of ﬂat connections and hence

∼

−1

(0)/H

is the reduced phase space of the theory, in the sense of the Marsden–

Weinstein reduction. We denote this reduced phase space by A//H and call

it the symplectic quotient of A by H. Marsden–Weinstein reduction and

symplectic quotient are fundamental constructions in geometrical mechanics

and geometric quantization. They also arise in many other mathematical and

physical applications.

A situation similar to that described above also arises in the geometric for-

mulation of canonical quantization of ﬁeld theories. One proceeds by analogy

with the geometric quantization of ﬁnite-dimensional systems. For example,

Q = A/H can be taken as the conﬁguration space and T

∗

Q as the corre-

sponding phase space. The associated Hilbert space is obtained as the space

of L

sections of a complex line bundle over Q. For physical reasons this bun-

dle is taken to be ﬂat. Inequivalent ﬂat U (1)-bundles are said to correspond

to distinct sectors of the theory. Thus we see that at least formally these

sectors are parametrized by the moduli space

(Q, U(1))

∼

hom(π

(Q),U(1))/U(1)

∼

hom(π

(Q),U(1)),

since U(1) acts trivially on hom(π

(Q),U(1)).

Now let Y be a homology 3-sphere. By considering a one-parameter family

{ω

}

t∈I

of connections of Y deﬁnes a connection on Y × I and the corre-

sponding Chern–Simons action A

. This is invariant under the connected

component of the identity in G but changes by the Wess–Zumino action under

the full group G.In[403], Witten showed how the standard Morse theory (see

Morse and Cairns [292] and Milnor [284]) can be modiﬁed by consideration

of the gradient ﬂow of the Morse function f between pairs of critical points

of f . One may think of this as a sort of relative Morse theory. Witten was

motivated by the phenomenon of the quantum mechanical tunneling eﬀect

between the states represented by the critical points. The resulting Wit-

ten complex (C

∗

,δ) can be used to deﬁne the Floer homology groups

(Y ),n∈ Z

(see [130, 131]). This is an example of TQFT. The Euler

characteristic of Floer homology equals the Casson invariant. Thus TQFT

methods give a resolution or categoriﬁcation of the Casson invariant in

terms of a new homology theory. Floer homology is discussed in detail in

Chapter 11.

Consideration of the Heegaard splitting of Y leads to interesting connec-

tions with conformal ﬁeld theory. On the other hand given a 4-manifold M

we can always decompose it along a homology 3-sphere Y into a pair of 4-

manifolds M

and M

−

. Then one can study instantons on M by studying a

7.5 Topological Quantum Field Theories (TQFTs) 227

pair of instantons on M

and M

−

and matching them along the boundary Y .

This procedure can be used to relate the Floer homology to the polynomial

invariants of M deﬁned by Donaldson.

A geometrical interpretation of the Jones polynomial invariant of a link was

provided by Witten [405], who applied ideas from QFT to the Chern–Simons

Lagrangian. In fact, Witten’s model allows us to consider the knot and link

invariants in any compact 3-manifold M .LetP (M,G) be a principal bundle

over M , with compact semisimple Lie group G. The state space is taken to

be the space of gauge potentials A

. The partition function Z of the theory

is deﬁned by

Z(Φ):=



−iA

(ω)

Φ(ω)DA,

where Φ : A

→ R is a quantum observable of the theory and A

is the

Chern–Simons action. The expectation value Φ of the observable Φ is given

Φ =



−iA

(ω)

Φ(ω)DA



−iA(ω)

Taking for Φ the Wilson loop functional W

ρ,κ

,whereρ is a suitably chosen

representation of G and κ is the link under consideration, one is led to the

following interpretation of the Jones polynomial:

Φ = V

(q), where q = e

2πi/(k+2)

Witten’s ideas have been used by several authors to obtain a geometrical in-

terpretation of various knot and link invariants and to discover new invariants

of knots, links, and 3-manifolds (see, for example, [87,178]).

In [404] it is shown that the Donaldson polynomial invariants of a 4-

manifold M appear as expectation values of certain observables in a topo-

logical QFT. The Lagrangian of this topological QFT has also been ob-

tained through consideration of an inﬁnite-dimensional version of the classical

Chern–Gauss–Bonnet theorem, in [21]. The general program of computing

the curvature of connections on inﬁnite-dimensional bundles and of deﬁning

appropriate generalizations of characteristic classes was initiated by I. Singer

in his fundamental paper [351] on Gribov ambiguity. We proceed by analogy

with the ﬁnite-dimensional case. To simplify considerations, let us suppose

that the group of gauge transformations G acts freely on the space of gauge

connections A. Then we can consider A as a principle G-bundle over the space

B = A/G of gauge equivalence classes of connections. Deﬁne the map

−

: A→Λ

−

(M,ad(P )) by ω → F

−

. (7.39)

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