Marathe K. Topics in Physical Mathematics

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278 9 4-Manifold Invarian t s

SU(2)-bundle over S

. This bundle is the quaternionic Hopf ﬁbration of H

over HP

,whereH is the space of quaternions and HP

is the quaternionic

projective space, of quaternionic lines through the origin in the quaternionic

plane H

.WeidentifyH with R

by the map H → R

given by

x = x

+ ix

+ jx

+ kx

−→ (x

Then H

is isomorphic to R

and each quaternionic line intersects the 7-

sphere S

⊂ H

in S

. On the other hand, the base HP

can be identiﬁed

with S

. Thus, the quaternionic Hopf ﬁbration leads to the bundle

SU(2) = S

As indicated in the diagram, the ﬁber S

can be identiﬁed with the group

SU(2) of unit quaternions and its action on H

by right multiplication re-

stricts to S

. This makes it into a principal SU(2)-bundle over S

.This

follows from the observation that α ∈ SU(2) and (x, y) ∈ S

⊂ H

imply

that (αx, αy) ∈ S

and that the SU(2)-action is free. This principal bun-

dle is clearly non-trivial and admits a canonical connection ω

(also called

the universal connection), whose curvature Ω

is self-dual and hence satis-

ﬁes the instanton equation (and hence also the full Yang–Mills equations). It

corresponds to a BPST instanton of instanton number 1. The entire BPST

family of instantons can be generated from this solution as follows. The group

SO(5, 1) acts on S

by conformal transformations and this action induces an

action on ω

.Ifg ∈ SO(5, 1), then we denote the induced action on ω

also

by g.Thengω

is also an SU(2)-connection over S

and it has self-dual

curvature. Since the Yang–Mills action is conformally invariant, the solution

generated by gω

also has instanton number 1. The connection gω

is gauge-

equivalent to ω

(and hence determines the same point in the moduli space)

if and only if g is an isometry of S

, i.e., if and only if g ∈ SO(5) ⊂ SO(5, 1).

Thus, the space of gauge-inequivalent, self-dual, k =1,SU(2)-connections

on S

, or the moduli space M

,SU(2)), is given by

,SU(2)) = SO(5, 1)/(SO(5)).

We note that the quotient space SO(5, 1)/(SO(5)) can be identiﬁed with the

hyperbolic 5-space H

We now give an explicit local formulation of the BPST family. Consider

the chart deﬁned by the map ψ

: S

−{e

}→R

9.2 Moduli Spaces of Instantons 279

where ψ

is obtained by projecting from e

=(0, 0, 0, 0, 1) ∈ R

onto the

tangent hyperplane to S

at −e

. This chart gives conformal coordinates on

. Identifying R

with H, the metric can be written as

=4|dx|

/(1 + |x|

where |x|

= x¯x, |dx|

= dxd¯x. Identifying the Lie algebra su(2) with the

set of pure imaginary quaternions, we can write the gauge potential A

(1)

corresponding to the universal connection ω

, as follows:

(1)

=Im



xd¯x

1+|x|



It is possible to give a similar expression for the potential in the chart

obtained by projecting from −e

and to show that these expressions are

compatible, under a change of charts and hence deﬁne a global connection

with corresponding Yang–Mills ﬁeld. However, we apply the removable sin-

gularities theorem of Uhlenbeck (see [386]) to guarantee the extension of the

local connection to all of S

. This procedure is also useful in the general

construction of multi-instantons, where there is a ﬁnite number of removable

singularities. The Yang–Mills ﬁeld F

(1)

, corresponding to the gauge potential

(1)

,isgivenby

(1)

dx ∧ d¯x

(1 + |x|

)

Using the formula

vol(S

2m+1

(2m)!

and calculating the Euclidean norm |F

(1)

= 3, we can evaluate the Yang–

Mills action

8π



(1)

=3· vol(S

)/(8π

)=1.

On the other hand, A

= k for a self-dual instanton with instanton number

k. Thus we see that the above solution corresponds to k = 1. We now apply

the conformal dilatation

: R

→ R

deﬁned by x → x/λ, 0 <λ<1,

to obtain induced connections A

(λ)

= f

∗

(1)

and the corresponding Yang–

Mills ﬁelds F

(λ)

= f

∗

(1)

. Because of conformal invariance of the action

, the ﬁelds F

(λ)

have the same instanton number k =1.Thelocal

expressions for A

(λ)

and F

(λ)

are given by

280 9 4-Manifold Invarian t s

(λ)

=Im



xd¯x

+ |x|



and

(λ)

dx ∧ d¯x

(λ

+ |x|

)

If we write the above expressions in terms of the quaternionic components we

recover the formulas for the BPST instanton. The connections corresponding

to diﬀerent λ are gauge-inequivalent, since |F |

is a gauge-invariant func-

tion. Moreover, choosing an arbitrary point q ∈ S

and considering the chart

obtained by projecting from this point, we obtain gauge inequivalent con-

nections for diﬀerent choices of q. In fact, we have a 5-parameter family of

instantons parametrized by the pairs (q, λ), where q ∈ S

and λ ∈ (0, 1).

Two pairs (q

,λ

)and(q

,λ

) give gauge-equivalent connections if and only

if q

= q

and λ

= λ

. It is customary to call q the center of the instanton

and λ its size. The map deﬁned by

(q, λ) → (1 − λ)q from S

× (0, 1) to R

is an isomorphism onto the punctured open ball B

−{0}. The universal

connection A

(1)

corresponds to the origin. Thus, the moduli space of gauge

inequivalent-instantons is identiﬁed with the open unit ball B

,whichisthe

Poincar´e model of the hyperbolic 5-space H

. The connection corresponding

to (q, λ)asλ → 0 can be identiﬁed with a boundary point of the open ball

. This realizes S

as the boundary of the ball B

. Thus our base space

appears as the boundary of the moduli space. This is one of the key ideas

of Donaldson in his work on the topology of the moduli space of instantons

[106]. The moduli space of the fundamental BPST instantons or self-dual

SU(2) Yang–Mills ﬁelds with instanton number 1 over the Euclidean 4-sphere

is denoted by M

.Itcanbeshown[20] that the action of the group

SO(5, 1) of conformal diﬀeomorphisms of S

induces a transitive action of

SO(5, 1) on the moduli space M

with isotropy group SO(5). Thus M

is diﬀeomorphic to the homogeneous hyperbolic 5-space SO(5, 1)/SO(5). In

particular, the topology of M

is the same as that of R

. A more general

result of Donaldson, discussed in the next section, shows that any 1-connected

4-manifold M with positive deﬁnite intersection form, can be realized as a

boundary of a suitable moduli space.

Most of the solutions of the pure Yang–Mills equations that have been con-

structed are in fact solutions of the self-dual or anti-dual instanton equations,

i.e., instantons or anti-instantons. The instanton and anti-instanton solutions

are also called collectively pseudo-particle solutions. The ﬁrst such solu-

tions, consisting of a 5-parameter family of self-dual Yang–Mills ﬁelds on R

was constructed by Belavin, Polyakov, Schwartz, and Tyupkin ([36]) in 1975

and it is these solutions or their extension to S

that are commonly referred

to as the BPST instantons. We will show that the BPST instanton solu-

9.2 Moduli Spaces of Instantons 281

tions correspond to the gauge group SU(2) over the base manifold S

and

have instanton number k =1. In this case the instanton number k is deﬁned

k := −c

(P (S

,SU(2)))[S

where c

denotes the second Chern class of the bundle P and [S

] denotes

the fundamental cycle of the manifold S

. The BPST solution was general-

ized to the so-called multi-instanton solutions, which correspond to the

self-dual Yang–Mills ﬁelds with instanton number k. A5k-parameter family

of solutions was obtained by ’t Hooft (unpublished) and a (5k+ 4)-parameter

family was obtained by Jackiw, Nohl, and Rebbi in 1977. For a given in-

stanton number k the maximum number of parameters in the corresponding

instanton solution can be identiﬁed with the dimension of the space of gauge

inequivalent solutions (the moduli space). For SU(2)-instantons over S

this

dimension of the moduli space was computed by Atiyah, Hitchin, and Singer

([20]) by using the Atiyah–Singer index theorem and turns out to be 8k −3.

Thus, for k = 1 the moduli space has dimension 5 and the 5-parameter BPST

solutions correspond to this space. For k>1, the ’t Hooft and Jackiw, Nohl,

Rebbi solutions do not give all the possible instantons with instanton number

k. Instead of describing these solutions we will describe brieﬂy the construc-

tion of the most general (8k−3)-parameter family of solutions, which include

the above solutions as special cases.

The construction of the (8k−3)-parameter family of instantons was carried

out by Atiyah, Drinfeld, Hitchin, and Manin [19] by using the methods of

analytic and algebraic geometry. It is called the ADHM construction.It

starts by generalizing the twistor space construction of Penrose (see Wells

[399] for a discussion of twistor spaces and their applications in ﬁeld theory)

as follows.

Atiyah, Hitchin and Singer [20] studied the properties of self-dual connec-

tions over self-dual base manifolds M, which have positive scalar curvature.

In this case the bundle of projective anti-dual spinors PV

−

has a complex

structure and the The Ward correspondence:holds:

Let E be a Hermitian vector bundle with self-dual connection over a com-

pact, self-dual 4-manifold M.LetPV

−

denote the bundle of projective anti-

dual spinors on M. Then we have the following commutative diagram

−

∗

where F := h

∗

E is the pull-back of the bundle E to PV

−

.Then

1. F is holomorphic on PV

−

with holomorphically trivial ﬁbers.

282 9 4-Manifold Invarian t s

2. Let ρ : PV

−

→ PV

−

denote the real structure on PV

−

. Then there is a

holomorphic bundle isomorphism σ : ρ

∗

F → F , which induces a positive

deﬁnite Hermitian structure on the space of holomorphic sections of F on

each ﬁber.

3. Every bundle F on PV

−

satisfying the above two conditions is the pull-

back of a Hermitian vector bundle E over M, with self-dual connection.

When M = S

, the bundle PV

−

can be identiﬁed with CP

.Thisis

the original twistor space of Penrose. In this case standard techniques from

algebraic geometry can be used to study the bundle F and to obtain all self-

dual connections on G-bundles over S

. It turns out that for G = SU(2),

the general solution can be described explicitly by starting with a vector

λ =(λ

,...,λ

) ∈ H

and a k ×k symmetric matrix B over H satisfying the

following conditions:

1. (B

†

B + λ

†

λ)isarealmatrix(† is the quaternionic conjugate transpose).

2. rank



B−xI



= k, ∀x ∈ H.

The local expression for the gauge potential A

(λ,B)

determined by the pair

(λ, B),λ∈ H

,B∈ H

k×k

,isgivenby

(λ,B)

= f

∗

(A(u)),

where u =(u

,...,u

) ∈ H

, f : H → H

is deﬁned by f (x)=

[λ(B − xI)

−1

]

†

, ∀x ∈ H,andA(u)istheSU(2)-gauge potential on the space

given by

A(u)=Im



udu

†

1+|u|



The gauge ﬁeld F

(λ,B)

corresponding to the gauge potential A

(λ,B)

contains a

ﬁnite number of removable singularities and hence, by applying Uhlenbeck’s

theorem on removable singularities, the solution can be extended smoothly

to S

. The potentials determined by (λ, B)and(λ



) are gauge-equivalent

if and only if there exists α ∈ SU(2) and T ∈ O(k) such that



= αλT and B



= T

−1

BT.

We may carry out a na¨ıve counting of free real parameters as follows. The

pair (λ, B)gives

4k +4



k(k +1)



=2k

+6k (9.7)

real parameters. The reality condition (i) above involves 3k(k −1)/2 param-

eters. The condition (ii) on the rank does not restrict the number of parame-

ters. The gauge equivalence further reduces the number of parameters in (9.7)

by 3 (by the SU(2)-action) and by k(k − 1)/2(bytheO(k)-action). Thus,

we have the following count for free real parameters.

+6k −

k(k − 1) − 3 −

(k − 1) = 8k − 3.

9.2 Moduli Spaces of Instantons 283

It can be shown that this family of (8k −3)-parameter solutions exhausts all

the possible solutions up to gauge equivalence. Thus, the space of these solu-

tions may be identiﬁed as the moduli space M

,SU(2)) of k-instantons

(i.e., instantons with instanton number k)overS

. In the theorem below we

give an alternative characterization of this moduli space.

Theorem 9.1 Let k be a positive integer. Write

B(q)=A

+ A

where, q =(q

) ∈ H

,andA

∈ Hom

, H

k+1

), i =1, 2. Deﬁne the

manifold M and Lie group G by

M := {(A

) | B(q)

†

B(q) ∈ GL(k, R), ∀q =0},

G := (Sp(k +1)× GL(k, R))/Z

Then M = ∅, G acts freely and properly on M by the action

(a, b) · (A

):=(aA

−1

,aA

−1

and the quotient space M/G is a manifold of dimension 8k − 3,whichis

isomorphic to the moduli space M

,SU(2)) of instantons of instanton

number k.

9.2.1 Atiyah–Jones Conjecture

Soon after the diﬀerential geometric setting of the Yang–Mills theory was

established, Atiyah and Jones [22] took up the study of the topological aspects

of the theory over the 4-sphere S

. They proved the following theorem.

Theorem 9.2 Let B

(resp., M

) denote the moduli space of based gauge-

equivalence classes of connections (resp., instantons) on the principal bundle

,SU(2)) with instanton number k.ThenB

is homotopy equivalent to

the third loop space Ω

(SU(2)), and the natural forgetful map

: M

→B

∼

(SU(2))

induces a surjection

in homology through the range q(k); i.e., there exists

afunctionq(k) such that

: H

) → H



(SU(2))



for i ≤ q(k) << k

is a surjective homomorphism.

In its strong form, the Atiyah–Jones conjecture states that θ

is a homo-

topy equivalence through the range q(k) and that the function q(k)canbe

284 9 4-Manifold Invarian t s

explicitly determined as a function of k. The stable version of the conjecture

was proved by Taubes in [365]. In fact, he proved the following theorem for

the general case of the gauge group SU(n).

Theorem 9.3 Let M

∞

(resp., θ) be the direct limit of the moduli spaces

(resp., maps θ

). Then

θ : M

∞

→ Ω

(SU(n)),n≥ 2

is a homotopy equivalence.

Boyer et al. [63] proved the following theorem, settling the Atiyah–Jones

conjecture in the aﬃrmative.

Theorem 9.4 For al l k>0 the natural inclusion map

: M

→ Ω

(SU(2))

is a homotopy equivalence through dimension q(k)=[k/2]−2, i.e., θ

induces

an isomorphism

in homotopy

: π

) → π

(Ω

(SU(2)) = π

i+3

((SU(2)) ∀i ≤ q(k).

In particular, using the known homotopy groups of SU(2) we can conclude

that the low-dimensional homotopy groups of M

are ﬁnite.

The result of this theorem has been extended to the gauge group SU(n)

in [375](seealso[61, 62]).

The long period between the formulation of the Atiyah–Jones conjecture

and its proof is an indication of the diﬃculty involved in the topology and

geometry of the moduli spaces of gauge potentials, even for the case when

the topology and the geometry of the base manifold is well known. In fact, an

explicit description of moduli spaces of instantons on S

available through the

ADHM formalism [19] did not ease the situation. A surprising advance did

come through Donaldson’s study of the moduli space M

(M,SU(2)), where

M is a positive deﬁnite 4-manifold. He showed that M

is a 5-manifold with

singularities and used the topology of M

to obtain new obstructions to

smoothability of M.

We begin by considering Taubes’s fundamental contributions to the anal-

ysis of Yang–Mills equations. His work on the existence of self-dual and anti-

dual solutions of Yang–Mills equations on various classes of 4-manifolds paved

the way for many later contributions including Donaldson’s. A direct ap-

plication of the steepest descent method to ﬁnd the global minima of the

Yang–Mills functional A

does not work in dimension 4, although this

technique does work in dimensions 2 and 3. In fact, the problem of ﬁnding

critical points of A

in 4 dimensions is a conformally invariant variational

problem. It has some features in common with sigma models (the harmonic

map problem in 2 dimensions). Taubes used analytic techniques to study the

9.2 Moduli Spaces of Instantons 285

problem of existence of self-dual connections on a manifold. This allowed him

to remove some of the restrictions on the base manifolds that were required

in earlier work. He does impose a topological condition on the base manifold

M, namely that there be no anti-dual harmonic 2-forms on M. This condition

can be written using the second cohomology group of the de Rham complex:

−

deR

(M; R)) = 0.

This condition is equivalent to the statement that the intersection form ι

M be positive deﬁnite. It is precisely this condition that appears in Donald-

son’s fundamental theorem, which gives a new obstruction to smoothability

of a 4-manifold. We discuss this theorem in the next section.

In the rest of this section we take M to be a compact, connected, oriented,

4-dimensional, Riemannian manifold with a compact, connected, semisimple

Lie group G as gauge group. We begin by recalling the classiﬁcation of princi-

pal G-bundles over M. The isomorphism classes of principal bundles P (M, G)

are in one-to-one correspondence with the elements of the set [M; BG]ofho-

motopy classes of maps of M into the classifying space BG for G.Usingthis

isomorphism, we can consider the isomorphism class [P ]asanelementofthe

set [M; BG]. Since G is semisimple, its Lie algebra g is a direct sum of a ﬁnite

number of non-trivial simple ideals, i.e., there exists a positive integer k such

that

g = g

⊕ g

⊕···⊕g

In this case, we can write the ﬁrst Pontryagin class of the vector bundle ad(P )

as a vector

p =(p

,...,p

) ∈ Z

Each isomorphism class of P uniquely determines a set of multipliers {r

one for each simple ideal g

of g, which we write as a vector

r =(r

,...,r

) ∈ Q

wherewehavewrittenr

for r

.Thevaluesofr

when h is the Lie algebra

of the simple group H are given in Table 9.1

Tab le 9. 1 Multipliers for the Pontryagin classes of P

H SU(n) Spin(n) Sp(n) G

4n 4n − 2 4n +4 16 36 48 72 120

The classiﬁcation of principal bundles by Pontryagin classes can be de-

scribed as follows.

Proposition 9.5 Deﬁne the map

286 9 4-Manifold Invarian t s

φ :[M; BG] → Z

by [P ] → (r

−1

,...,r

−1

Then we have the following result. The map φ is a surjection and there exists

amapη

η :[M ; BG] → H

(M; π

(G))

such that φ restricted to the kernel of η is an isomorphism. In particular, if

G is simply connected, then φ is an isomorphism.

We now state two of Taubes’ theorems on the existence of irreducible,

self-dual, Yang–Mills connections on 4-manifolds.

Theorem 9.6 Let M be a compact, connected, oriented, Riemannian, 4-

manifold with p

−

deR

(M)=0and let G be a compact, semisimple Lie group.

Then there exists a principal bundle P (M,G) which admits irreducible self-

dual connections.

Moreover, we have the following theorem.

Theorem 9.7 Let M,G be as in the previous theorem and let P (M,G) be a

principal bundle with non-negative Pontryagin classes such that [P ] ∈ ker η.

Then we have the following.

1. The space A(P ) contains a self-dual connection.

2. Let Q(S

,G) be a principal bundle with the same Pontryagin classes as

P .IfQ admits an irreducible, self-dual connection B, then there exists

an irreducible connection A on P .Furthermore,suchanA has a neigh-

borhood in A(P)/G(P ) in which the moduli space of irreducible self-dual

connections is a manifold of dimension

(ad P ) −

(dim G)(χ + σ),

where p

(ad P )=



i=1

is the sum of the Pontryagin classes of ad P , χ

is the Euler characteristic of M,andσ is the signature of M.

The analytical tools developed in these and related theorems by Taubes play

a fundamental role in the construction and analysis of Yang–Mills ﬁelds and

their moduli spaces on several classes of manifolds. We now discuss a special

class of these moduli spaces, namely the moduli spaces of instantons.

We restrict ourselves to considering self-dual Yang–Mills ﬁelds on M with

gauge group G, i.e., with G-instantons. Similar considerations apply to anti-

dual ﬁelds or anti-instantons. Since both the instanton and anti-instanton

solutions are used in the literature, we shall state important results for both.

The conﬁguration space of G-instantons is denoted by C

(P ) and is deﬁned

(P ):={ω ∈A|F

= ∗F

where A is the space of gauge connections. The group G of gauge transfor-

mations acts on C

(P ) and the quotient space under this action is called the

9.2 Moduli Spaces of Instantons 287

moduli space of instantons with instanton number k. It is denoted by

(M,G), or simply by M

(M,G). Thus, we have

(M,G):=C

(P (M,G))/G.

We now brieﬂy indicate how the dimension of the moduli space M

(M,G)

can be computed by applying the Atiyah–Singer index theorem. The funda-

mental elliptic complex comes from a modiﬁcation of the generalized de Rham

sequence. For a vector bundle E over M associated to the principal bundle

P (M, G), the generalized de Rham sequence can be written as

0 −→ Λ

(M,E)

−→ Λ

(M,E)

−→ Λ

(M,E)

−→···,

where Λ

(M,E)=Γ (Λ

∗

M) ⊗ E). On the oriented Riemannian 4-

manifold M the space Λ

∗

M) splits under the action of the Hodge ∗ opera-

tor into a direct sum of self-dual and anti-dual 2-forms. This splitting extends

to Λ

(M,E)sothat

(M,E)=Λ

(M,E) ⊕ Λ

−

(M,E),

where Λ

(M,E)(resp.,Λ

−

(M,E)) is the space of self-dual (resp., anti-dual)

2-forms with values in the vector bundle E.Let

: Λ

(M,E) → Λ

(M,E)

be the canonical projections deﬁned by

(1 ±∗).

Then we have the following diagram

0 Λ

(M,E)

···

−

(M,E)

−

(M,E)