Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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Wrapped D5 Branes and N = 1 Super

Yang–Mills

In this section, we will consider the classical solution

corresponding to N D5 branes wrapped on a 2-cycle

of a noncompact Calabi–Yau space and we use it to

study the properties of the gauge theory living on

their world volume that can be shown to be N = 1

super Yang–Mills.

We start by writing the classical solution found in

Maldacena and Nu´n˜ ez (2001) and Chamseddine and

Volkov (1997). It has a nontrivial metric:

¼e



1;3





þ sin

 d

’











d

a¼1



 A

ðÞ



½12

a 2-form R–R potential

ð2Þ

4

ð þ

Þ sin 

d

^ d  sin

 d

 ^ d

’

h

cos 

cos

 d ^ d

’

2

 ^ 

 sin

 d

’ ^ 

½13

and a dilaton

2

sinh 2

½14

where

¼  coth 2 



sinh

2



¼ e

sinh 2

a ¼

2

sinh 2

½15

and

¼

2

aðrÞd



2

aðrÞsin

 d

’

¼

2

cos

 d

’

½16

with   r and 

2

= Ng



. The left-invariant

1-forms of S

are



cos d

þ sin 

sin d



¼

sin d

 sin 

cos d



d þ cos 

d

½17

with 0  

 ,0    2,and0  4.The

variables

 and

’ describe a two-dimensional sphere

and vary in the range 0 

   and 0 

’  2.

Before proceeding, here we want to stress the fact that

the presence of the function a() 6¼ 0 makes the

solution regular everywhere. This will allow us to use

it later on to describe the nonperturbative gaugino

condensate property of N = 1 super Yang–Mills.

We can now use the previous solution for comput-

ing the running coupling constant and the  parameter

of N = 1 super Yang–Mills (see Di Vecchia et al.

(2002), Bertolini and Merlatti (2003),andMu¨ck

(2003) reviewed in Bertolini (2003), Di Vecchia and

Liccardo (2003),andImeroni (2003)). In order to do

that, we have to fix the cycle on which to perform the

integrals in eqns [1] and [2]. It turns out that this

2-cycle is specified by

 ¼ 



’ ¼; ¼ 0 ½18

keeping  fixed. If we now compute the gauge couplings

on the previous cycle with B

= C

= 0, we get

4

¼  coth 2 þ

aðÞcos ½19

and



2g



¼N þaðÞsin þ

ðÞ½20

where we have kept 6¼ 0 for reasons that will

become clear in a moment. Equation [19] shows

that the coupling constant is running as a function

of the distance  from the branes. In order to obtain

the correct running of the gauge theory, we have to

find a relation between  and the renormalization

group scale . This can be obtained with the

following considerations. If we look at the previous

solution, it is easy to see that the metric in eqn [12]

is invariant under the following transformations:

! þ 2 if a 6¼ 0

! þ 2 if a ¼ 0

½21

where  is an arbitrary constant. On the other hand,

is not invariant under the previous transforma-

tions, but its flux, that is exactly equal to 

in eqn

[20], changes by an integer multiple of 2:



2

! 

2N; if a 6¼ 0

2N; if a ¼ 0;¼

k

(

½22

But since the physics does not change when



!

þ 2, one gets that the transformation in

eqn [22] is an invariance. Notice that also eqn [19] for

466 Gauge Theories from Strings

the gauge coupling constant is invariant under the

transformation in eqn [21]. The previous considera-

tions show that the classical solution and also the

gauge couplings are invariant under the Z

transfor-

mation if a 6¼ 0, while this symmetry becomes Z

if a

is taken to be zero. As a consequence, since in the

ultraviolet a() is exponentially small, we can neglect

it and we have a Z

symmetry, while in the infrared

we cannot neglect a() anymore and we have only a

symmetry left. This fits very well with the fact that

N = 1 super Yang–Mills has a nonzero gaugino

condensate <> that is responsible for the break-

ing of Z

into Z

. Therefore, it is natural to identify

the gaugino condensate precisely with the function

a() 6¼ 0 that makes the classical solution regular also

at short distances in supergravity (Di Vecchia et al.

2002, Apreda et al. 2002):

<>

¼ 

aðÞ½23

This provides the relation between the renormaliza-

tion group scale  and the supergravity spacetime

parameter . In the ultraviolet (large ) a()is

exponentially suppressed and in eqns [19] and [20]

we can neglect it obtaining

4

¼ coth 2



¼N þ

ðÞ

½24

The chiral anomaly can be obtained by performing

the transformation ! þ 2 and getting



! 

 2N ½25

This implies that the Z

transformations corre-

sponding to  = k=N are symmetries because they

shift 

by multiples of 2.

In general, however, eqns [19] and [20] are only

invariant under the Z

subgroup of Z

correspond-

ing to the transformation

! þ 2 ½26

that changes 

in eqn [20] as follows:



! 

 2N ½27

leaving invariant the gaugino condensate:

<

> ¼ 

16

3Ng

8

=Ng

i

½28

Therefore, the chiral anomaly and the breaking of

to Z

are encoded in eqns [19] and [20].

Finally, if we put = 0ineqn [19], we get

4

¼  coth 2 

aðÞ¼ tanh  ½29

This equation taken together with eqn [23] allows us to

determine the running coupling constant as a function of

.Fromit,weget(Di Vecchia et al. 2002, Di Vecchia

andLiccardo2003) the Novikov–Shifman–Vainshtein–

Zacharov (NSVZ) -function plus nonperturbative

corrections due to fractional instantons:

ðg

Þ¼

3 Ng

16

1þ

4

sinh

2

4

1

8

sinh

2

4

½30

where in the ultraviolet we have approximated

 with 4

=(Ng

) coth 4

=(Ng

See also: AdS/CFT Correspondence; Anomalies;

BF Theories; Brane Construction of Gauge Theories;

Gauge Theory: Mathematical Applications;

Noncommutative Geometry from Strings;

Nonperturbative and Topological Aspects of Gauge

Theory; Perturbation Theory and its Techniques;

Seiberg–Witten Theory; Superstring Theories.

Further Reading

Apreda R, Bigazzi F, Cotrone AL, Petrini M, and Zaffaroni A

(2002) Some comments on N = 1 gauge theories from wrapped

branes. Physics Letters B 536: 161–168 (hep-th/0112236).

Bertolini M (2003) Four lectures on the gauge/gravity correspon-

dence. International Journal of Modern Physics A18:

5647–5712 (hep-th/0303160).

Bertolini M, Di Vecchia P, Frau M, Lerda A, and Marotta R

(2002a) More anomalies from fractional branes. Physics

Letters B 540: 104–110 (hep-th/0202195).

Bertolini M, Di Vecchia P, Frau M, Lerda A, and Marotta R

(2002b) N = 2 gauge theories on systems of fractional D3/D7

branes. Nuclear Physics B 621: 157–178 (hep-th/0107057).

Bertolini M, Di Vecchia P, and Marotta R (2000) N = 2 four-

dimensional gauge theories from fractional branes. In:

Olshanetsky M and Vainshtein A (eds.) Multiple Facets of

Quantization and Supersymmetry, pp. 730–773. (hep-th/

0112195). Singapore: World Scientific.

Bertolini M and Merlatti P (2003) A note on the dual of N = 1

super Yang–Mills theory. Physics Letters B 556: 80–86 (hep-th/

0211142).

Bigazzi F, Cotrone AL, Petrini M, and Zaffaroni A (2002) Super-

gravity duals of supersymmetric four dimensional gauge theories.

Rivista del Nuovo Cimento 25N12: 1–70 (hep-th/0303191).

Chamseddine AH and Volkov MS (1997) Non-abelian BPS

monopoles in N = 4 gauged supergravity. Physical Review

Letters 79: 3343–3346 (hep-th/9707176).

Di Vecchia P, Lerda A, and Merlatti P (2002) N = 1 and N = 2

super Yang–Mills theories from wrapped branes. Nuclear

Physics B 646: 43–68 (hep-th/0205204).

Di Vecchia P and Liccardo A (2003) Gauge theories from

D branes, Lectures given at the School ‘‘Frontiers in Number

Theory, Physics and Geometry,’’ Les Houches, hep-th/0307104.

Di Vecchia P, Liccardo A, Marotta R, and Pezzella F (2005) On

the gauge/gravity correspondence and the open/closed string

duality. International Journal of Modern Physics A 20:

4699–4796. hep-th/0503156.

Gauge Theories from Strings 467

Herzog CP, Klebanov IR, and Ouyang P (2001) D-branes on the

conifold and N = 1 gauge/gravity dualities. Based on I.R.K.’s

Lectures at the Les Houches Summer School Session 76,

‘‘Gravity, Gauge Theories, and Strings,’’ hep-th/0205100.

Imeroni E (2003) Studies of the Gauge/String Theory Correspon-

dence. Ph.D. thesis, hep-th/0312070.

Johnson CV, Peet AW, and Polchinski J (2000) Gauge theory and

the excision of repulson singularities. Physical Review D 61:

086001 (hep-th/9911161).

Kirsch I and Vaman D (2005) The D3/D7 background and

flavour dependence of Regge trajectories, hep-th/0505164.

Klebanov IR, Ouyang P, and Witten E (2002) A gravity dual of the

chiral anomaly. Physical Review D 65: 105007 (hep-th/0202056).

Maldacena J (1998) The large N limit of superconformal field

theories and supergravity. Advances in Theoretical and

Mathematical Physics 2: 231–252 (hep-th/9711200).

Maldacena J and Nu´n˜ ez C (2001) Towards the large N limit of

N = 1 super Yang Mills. Physical Review Letters 86: 588–591

(hep-th/0008001).

Mu¨ ck W (2003) Perturbative and nonperturbative aspects of pure

N = 1 super Yang–Mills theory from wrapped branes. Journal

of High Energy Physics 0302: 013 (hep-th/0301171).

Gauge Theory: Mathematical Applications

S K Donaldson, Imperial College, London, UK

Introduction

This article surveys some developments in pure mathe-

matics which have, to varying degrees, grown out of the

ideas of gauge theory in mathematical physics. The

realization that the gauge fields of particle physics and

the connections of differential geometry are one and the

same has had wide-ranging consequences, at different

levels. Most directly, it has led mathematicians to work

on new kinds of questions, often shedding light later on

well-established problems. Less directly, various funda-

mental ideas and techniques, notably the need to work

with the infinite-dimensional gauge symmetry group,

have found a place in the general world-view of many

mathematicians, influencing developments in other

fields. Still less directly, the work in this area – between

geometry and mathematical physics – has been a prime

example of the interaction between these fields which

has been so fruitful since the 1970s.

The body of this article is divided into three

sections: roughly corresponding to analysis, geome-

try, and topology. However, the different topics

come together in many different ways: indeed the

existence of these links between the topics is one of

the most attractive features of the area.

Gauge Transformations

For a review of the usual foundational material

on connections, curvature, and related differential

geometric constructions, the reader is referred to

standard texts. We will, however, briefly recall

the notions of gauge transformations and gauge

fixing. The simplest case is that of abelian gauge

theory – connections on a U(1)-bundle, say over

. In that case the connection form, representing

the connection in a local trivialization, is a pure

imaginary 1-form A, which can also be identified

with a vector field A. The curvature of the

connection is the 2-form dA. Changing the local

trivialization by a U(1)-valued function g = e

i

changes the connection form to

A ¼ A  dgg

1

¼ A  id

The forms A,

A are two representations of the same

geometric object: just as the same metric can be

represented by different expressions in different

coordinate systems. One may want to fix this choice

of representation, usually by choosing A to satisfy

the Coulomb gauge condition d



A = 0 (equivalently

div A = 0), supplemented by appropriate boundary

conditions. Here we are using the standard Eucli-

dean metric on R

. (Throughout this article we will

work with positive-definite metrics, regardless of the

fact that – at least at the classical level – the

Lorentzian signature may have more obvious bear-

ing on physics.) Arranging this choice of gauge

involves solving a linear partial differential equation

(PDE) for .

The case of a general structure group G is not

much different. The connection form A now takes

values in the Lie algebra of G and the curvature is

given by the expression

F ¼ dA þ

½A; A

The change of bundle trivialization is given by a

G-valued function and the resulting change in the

connection form is

A ¼ gAg

1

 dgg

1

(Our notation here assumes that G is a matrix

group, but this is not important.) Again, we can seek

to impose the Coulomb gauge condition d



A = 0,

but now we cannot linearize this equation as before.

We can carry the same ideas over to a global

problem, working on a G-bundle P over a general

468 Gauge Theory: Mathematical Applications

Riemannian manifold M. The space of connections on

P is an affine space A: any two connections differ by a

bundle-valued 1-form. Now the gauge group G of

automorphisms of P acts on A and, again, two

connections in the same orbit of this action represent

essentially the same geometric object. Thus, in a sense

we would really like to work on the quotient space

A=G. Working locally in the space of connections, near

to some A

, this is quite straightforward. We represent

the nearby connections as A

þ a, where a satisfies the

analog of the coulomb condition



a ¼ 0

Under suitable hypotheses, this condition picks out a

unique representative of each nearby orbit. However,

this gauge-fixing condition need not single out a

unique representative if we are far away from A

indeed, the space A=G typically has, unlike A,a

complicated topology which means that it is impos-

sible to find any such global gauge-fixing condition.

As noted above, this is one of the distinctive features

of gauge theory. The gauge group G is an infinite-

dimensional group, but one of a comparatively

straightforward kind – much less complicated than

the diffeomorphism groups relevant in Riemannian

geometry for example. One could argue that one of

the most important influences of gauge theory has

been to accustom mathematicians to working with

infinite-dimensional symmetry groups in a compara-

tively simple setting.

Analysis and Variational Methods

The Yang–Mills Functional

A primary object brought to mathematicians atten-

tion by physics is the Yang–Mills funct ional

YMðAÞ¼

d

Clearly, YM(A) is non-negative and vanishes if and

only if the connection is flat: it is broadly analogous

to functionals such as the area functional in minimal

submanifold theory, or the energy functional for

maps. As such, one can fit into a general framework

associated with such functionals. The Euler–

Lagrange equations are the Yang–Mills equations



¼ 0

For any solution (a Yang–Mills connection), there is

a ‘‘Jacobi operator’’ H

such that the second

variation is given by

YMðA þ taÞ¼YMðAÞþt

a; aiþOð t

The omnipresent phenomenon of gauge invar-

iance means that Yang–Mills connections are never

isolated, since we can always generate an infinite-

dimensional family by gauge transformations. Thus,

as explained in the last section, one imposes the

gauge-fixing condition d



a = 0. Then the operator

can be written as

a ¼ 

a þ½F

; a

where 

is the bundle-valued ‘‘ Hodge Laplacian’’



þ d



and the expression [F

, a] c ombines

the bracket in the Lie algebra with the action of



on 

. This is a self-adjoint elliptic operator

and, if M is compact, the span of the negative

eigenspaces is finite dimensional, the dimension

being defined to be the index of the Yang–Mills

connection A.

In this general setting, a natural aspiration is to

construct a ‘‘Morse theory’’ for the functional. Such

a theory should relate the topology of the ambient

space to the critical points and their indices. In the

simplest case, one could hope to show that for any

bundle P there is a Yang–Mills connection with

index 0, giving a minimum of the functional. More

generally, the relevant ambient space here is the

quotient A=G and one might hope that the rich

topology of this is reflected in the solutions to the

Yang–Mills equations.

Uhlenbeck’s Theorem

The essential foundation needed to underpin such a

‘‘direct method’’ in the calculus of variations is an

appropriate compactness theorem. Here the dimen-

sion of the base manifold M enters in a crucial way.

Very roughly, when a connection is represented

locally in a Coulomb gauge, the Yang–Mills action

combines the L

-norm of the derivative of the

connection form A with the L

-norm of the

quadratic term [A, A]. The latter can be estimated

by the L

-norm of A. If dim M  4, then the

Sobolev inequalities allow the L

-norm of A to be

controlled by the L

-norm of its derivative, but this

is definitely not true in higher dimensions. Thus,

dim M = 4 is the ‘‘critical dimension’’ for this

variational problem. This is related to the fact that

the Yang–Mills equations (and Yang–Mills func-

tional) are conformally invariant in four dimensions.

For any nontrivial Yang–Mills connection over the

4-sphere, one generates a one-parameter family of

Yang–Mills connections, on which the functional

takes the same value, by applying conformal

transformations corresponding to dilations of R

In such a family of connections the integrand jF

–

the ‘‘curvature density’’ – converges to a -function

Gauge Theory: Mathematical Applications 469

at the origin. More generally, one can encounter

sequences of connections over 4-manifolds for

which YM is bounded but which do not converge,

the Yang–Mills density converging to -functions.

There is a detailed analogy with the theory of

the harmonic maps energy functional, where the

relevant critical dimension (for the domain of the

map) is 2.

The result of Uhlenbeck (1982), which makes

these ideas precise, considers connections over a ball

 R

. If the exponent p  2n, then there are

positive co nstants (p, n), C(p, n) > 0 such that any

connection with kFk

)

  can be represented in

Coulomb gauge over the ball, by a connection form

which satisfies the condition d



A = 0, together with

certain boundary conditions, and

kAk

 CkFk

In this Coulomb gauge, the Yang–Mills equations

are elliptic and it follows readily that, in this setting,

if the connection A is Yang–Mills one can obtain

estimates on all derivatives of A.

Instantons in Four Dimensions

This result of Uhlenbeck gives the analytical basis

for the dire ct method of the calculus of variations

for t he Yang–Mills functional over base manifolds

M of dimension 3. For example, any bundle over

such a manifold must admit a Yang–Mills connec-

tion, minimizing the functional. Such a statement

is definitely false in dimensions 5. For example,

an early result of Bourguignon and Lawson (1981)

and Simons asserts that there is no minimizing

connection on any bundle over S

for n  5. The

proof exploits the action of the conformal trans-

formations of the sphere. In the critical dimension

4, the s ituation is much m ore complicated. In four

dimensions, there are the renowned ‘‘instanton’’

solutions of the Yang–Mills equation. Recall that

if M is an oriented 4-manifold the Hodge

-operation is an involution of 



M which

decomposes the two forms into self-dual and

anti-self-dual parts, 



M =

 



.Thecurva-

ture of a connection can then be written as

¼ F

þ F



and a connection is a self-dual (respectively

anti-self-dual) instanton if F



(respectively F

)is0.

The Yang–Mills functional is

YMðAÞ¼kF

þkF



while the difference kF

kF



is a topological

invariant (P) of the bundle P, obtained by

evaluating a four-dimensional characteristic class

on [M]. Depending on the sign of (P), the self-

dual or anti-self-dual connections (if any exist)

minimize the Yang–Mills functional among all

connections on P. These instanton sol utions of

the Yang–Mills equations are a nalogous to the

holomorphic maps from a Riemann surface to a

Ka¨ hler manifold, which minimize the harmonic

maps energy functional in their homotopy class.

Moduli Spaces

The instanton solutions typically occur in ‘‘moduli

spaces.’’ To fix ideas, let us consider bundles with

structure group SU(2), in which case (P) =

8

(P). For each k > 0, we have a moduli space

of anti-self-dual instantons on a bundle P

with c

) = k. It is a manifold of dimension 8k  3.

The general goal of the calculus of variations in this

setting is to relate three things:

1. the topology of the space A=G of equivalence

classes of connections on P

;

2. the topology of the moduli space M

instantons; and

3. the existence and indices of other, nonminimal,

solutions to the Yang–Mills equations on P

In this direction, a very influential conjecture was

made by Atiyah and Jon es (1978). They considered

the case when M = S

and, to avoid certain

technicalities, work with spaces of ‘‘framed’’ con-

nections, dividing by the restricted group G

gauge transformations equal to the identity at

infinity. Then, for any k, the quotient A=G

homotopy equivalent to the third loop space 

of based maps from the 3-sphere to itself. The

corresponding ‘‘framed’’ moduli space

is a

manifold of dimension 8k (a bundle over M

with

fiber SO(3)). Atiyah and Jones conje ctured that

the inclusion

!A=G

induces an isomorphism

of homotopy groups 

in a range of dimensions

l  l(k), wher e l(k) increases with k. This would be

consistent with what one might hope to prove by the

calculus of variations if there were no other Yang–

Mills solutions, or if the indices of such solutions

increased with k.

The first result along these lines was due to

Bourguignon and Lawson (1981), who showed that

the instanton solutions are the only local minima of

the Yang–Mills functional over the 4-sphere. Subse-

quently, Taubes (1983) showed that the index of an

non-instanton Yang–Mills connection P

is at least

k þ 1. Taubes’ proof used ideas related to the action

of the quaternions and the hyper-Ka¨ hler structure on

the

(see the section on hyper-Ka¨ hler quotients).

Contrary to some expectations, it was shown by

470 Gauge Theory: Mathematical Applications

Sibner et al. (1989) that nonminimal solutions do

exist; some later constructions wer e very explicit

(Sadun and Segert 1992). Taubes’ index bound gave

ground for hope that an analytical proof of the

Atiyah–Jones conjecture might be possible, but this

is not at all straightforward. The problem is that in

the criti cal dimension 4 a mini–max sequence for the

Yang–Mills functional in a given homotopy class

may diver ge, with curvature densities converging to

sums of -functions as outlined above. This is

related to the fact that the

are not compact. In

a series of papers culminating in a framework for

Morse theory for Yang–Mills functional, Taubes

(1998) succeeded in proving a partial version of the

Atiyah–Jones conjecture, together with similar

results for general base manifolds M

. Taubes

showed that, if the homotopy groups of the moduli

spaces stabilize as k !1, then the limit must be

that predicted by Atiyah and Jones. Related analy-

tical techniques were developed for other variational

problems at the critical dimension involving ‘‘critical

points at infinity.’’ The full Atiyah–Jones conjecture

was established by Boyer et al. but using geometrical

techniques: the ‘‘explicit’’ description of the moduli

spaces obtained from the Atiyah–Drinfeld–Hitchin–

Manin (ADHM) construction (see below). A differ-

ent geometrical proof was given by Kirwan (1994),

together with generalizations to other gauge groups.

There was a parallel story for the solutions of the

Bogomolony equation over R

, which we will not

recount in detail. Here the base dimension is below

the critical case but the analytical difficulty arises

from the noncompactness of R

. Taubes succeeded

in overcoming this difficulty and obtained relations

between the topology of the moduli space, the

appropriate configuration space and the higher

critical points. Again, these higher critical points

exist but their index grows with the numerical

parameter corresponding to k. At about the same

time, Donaldson (1984) showed that the moduli

spaces could be identified with spaces of rational

maps (subsequently extended to other gauge

groups). The analog of the Atiyah–Jones conjecture

is a result on the topology of spaces of rational maps

proved earlier by Segal, which had been one of the

motivations for Atiyah and Jones.

Higher Dimensions

While the scope for variational met hods in Yang–

Mills theory in higher dimensions is very limited,

there are useful analytical results about solutions of

the Yang–Mills equations. An important monotoni-

city result was obtained by Price (1983). For

simplicity, consider a Yang–Mills connection over

the unit ball B

 R

. Then Price showed that the

normalized energy

EðA; Bð rÞÞ ¼

n4

jxjr

jFj

d

decreases with r. Nakajima (1988) and Uhlenbeck used

this monotonicity to show that for each n there is an

 such that if A is a Yang–Mills connection over a ball

with E(A, B(r))   then all derivatives of A,ina

suitable gauge, can be controlled by E(A, B(r)). Tian

(2000) showed that if A

is a sequence of Yang–Mills

connections over a compact manifold M with bounded

Yang–Mills functional, then there is a subsequence

which converges away from a set Z of Haussdorf

codimension at least 4 (extending the case of points in a

4-manifold). Moreover, the singular set Z is a minimal

subvariety, in a suitably generalized sense.

In higher dimensions, important examples of

Yang–Mills connections arise within the framework

of ‘‘calibrated geometry.’’ Here, we consider a

Riemannian n-manifold M with a covariant constant

calibrating form  2 

n4

. There is then an analog

of the instanton equation

¼ð ^ F

whose solutions minimize the Yang–Mills functional.

This includes the Hermitian Yang–Mills equation

over a Ka¨hler manifold (see the section on moment

maps) and also certain equations over manifolds with

special holonomy groups (Donaldson and Thomas

1998). For these ‘‘higher-dimensional instantons,’’

Tian shows that the singular sets Z that arise are

calibrated varieties.

Gluing Techniques

Another set of ideas from PDEs and analysis which

has had great impact in gauge theory involves the

construction of solutions to appropriate equations

by the following general scheme:

1. constructing an ‘‘approximate solution,’’ formed

from some standard models using cutoff

functions;

2. showing that the approximate solution can be

deformed to a true solution by means of an

implicit function theorem.

The heart of the second step usually consists of

estimates for the relevant linear differential opera-

tor. Of course, the success of this stra tegy depends

on the particular features of the problem. This

approach, due largely to Taubes, has been particu-

larly effective in finding solutions to the first-order

instanton equations and their relatives. (The applic-

ability of the approach is connected to the fact that

Gauge Theory: Mathematical Applications 471

such solutions typically occur in moduli spaces and

one can often ‘‘see’’ local coordinates in the moduli

space by varying the parameters in the approximate

solution.) Taubes applied this approach to the

Bogomolny monopole equation over R

(Jaffe and

Taubes 1980) and to construct instantons over

general 4-manifolds (Taubes 1982). In the latter

case, the approximate solutions are obtained by

transplanting standard solutions over R

– with

curvature density concentrated in a small ball – to

small balls on the 4-manifold, glued to the trivial

flat connection over the remainder of the manifold.

These types of techniques have now become a fairly

standard part of the armory of many differential

geometers, working both within gauge theory and

other fields. An example of a problem where similar

ideas have been used is Joyce’s construction of

constant of manifolds with exceptional holonomy

groups (Joyce 1996). (Of course, it is likely that

similar techniques have been developed over the

years in many other areas, but Taubes’ work in

gauge theory has done a great deal to bring them

into prominence.)

Geometry: Integrability and Moduli

Spaces

The Ward Correspondence

Suppose that S is a complex surface and ! is the

2-form corresponding to a Hermitian metric on S.

Then S is an oriented Riemannian 4-manifold and !

is a self-dual form. The orthogonal complement of !

in 

can be identified with the real parts of forms

of type (0, 2). Hence, if A is an anti-self-dual

instanton connection on a principle U(r)-bundle over

S the (0, 2) part of the curvature of A vanishes. This

is the integrability condition for the



@-operator

defined by the connection, acting on sections of the

associated vector bundle E !S. Thus, in the

presence of the connection, the bundle E is naturally

a holomorphic bundle over S.

The W ard correspondence (Ward 1877) builds

on this idea to give a complete translation of the

instanton equations over certain Riemannian

4-manifolds into holomorphic geometry. In the

simplest case, let A be an instanton on a bundle

over R

. Then, for any choice of a linear complex

structure on R

, compatible with t he metric, A

defines a holomorphic structure. The choices of

such a complex structure are parametrized by a

2-sphere; in f act, the unit sphere in 

). So, for

any  2 S

we have a complex surface S



and a

holomorphic bundle over S



. These data can be

viewed in the following way. We consider the

projection  : R

 S

and the pull-back





(E)toR

 S

. This pullback bundle has a

connection which defines a holomorphic structure

along each fiber S



 R

 S

of the other projec-

tion. T he product R

 S

is the t wistor space of R

and it is in a natural way a three-dimensional

co mplex manifold. It can be identified with the

complement of a line L

in CP

where the projection

 S

becomes the fibration of CP

by the complex planes through L

. One can see

then that 



(E) is naturally a holomorphic bundle

over CP

n L

. The construction extends to the

conformal compactification S

of R

.IfS

is viewed

as the quaternionic projective line HP

and we

identify H

with C

in the standard way, we get a

natural map  : CP

!HP

. Then CP

is the

twistor space of S

and an anti-self-dual instanton

on a bundle E over S

induces a holomorphic

structure on the bundle 



(E) over CP

In general, the twistor space Z of an oriented

Riemannian 4-manifold M is defined to be the unit

sphere bundle in 

. This has a natural almost-

complex structure which is integrable if and only if

the self-dual part of the Weyl curvat ure of M

vanishes (Atiyah et al. 1978). The antipodal map

on the 2-sphere induces an antiholomorphic involu-

tion of Z. In such a case, an anti-self-dual instanton

over M lifts to a holomorphic bundle over Z.

Conversely, a holomorphic bundle over Z which is

holomorphically trivial over the fibers of the fibra-

tion Z !M (projective lines in Z), and which

satisfies a certain reality condition with respect to

the an tipodal map, arises from a unitary instanton

over M. This is the Ward correspon dence, part of

Penrose’s twistor theory.

The ADHM Construction

The problem of describing all solutions to the

Yang–Mills instanton equation over S

is thus

reduced to a problem in algebraic geometry, of

classifying certain holomo rphic vector bundles. This

was solved by Atiyah et al. (1978). The resul ting

ADHM construction reduces the problem to certain

matrix equations. The equations can be reduced to

the following form. For a bundle Chern class k and

rank r, we require a pair of k  k matrices 

, 

k  r matrix a, and an r  k matrix b. Then the

equations are



;

½¼ab





;



þ 



;



¼ aa



 b



½1

We also require certain open, nondegeneracy condi-

tions. Given such matrix data, a holomorphic

472 Gauge Theory: Mathematical Applications

bundle over CP

is constructed via a ‘‘monad’’: a

pair of bundle maps over CP

 Oð1Þ@ > D

>> C

 C

@ > D

>> C

Oð1Þ

with D

= 0. That is, the rank-r holomorphic

bundle we construct is Ker D

=Im D

. The bundle

maps D

, D

are obtained from the matrix data in a

straightforward way, in suitable coordinates. It is

this matrix description which was used by Boyer

et al. to prove the Atiyah–Jones conjecture on the

topology of the moduli spaces of instantons. The

only other case when the twistor space of a compact

4-manifold is an algebraic variety is the compl ex

projective plane, with the nonstandard orientation.

An analog of the ADHM de scription in this case was

given by Buchdahl (1986).

Integrable Systems

The Ward correspondence can be viewed in the general

framework of integrable systems. Working with the

standard complex structure on R

, the integrability

condition for the



@-operator takes the shape

½r

þ ir

; r

þ ir

¼0

where r

are the components of the covariant

derivative in the coordinate directions. So, the

instanton equation can be viewed as a family of

such commutator equations parametrized by  2 S

One obtains many reductions of the instanton

equation by imposing suitable symmetries. Solutions

invariant under translation in one variable corre-

spond to the Bogomolny ‘‘monopole equa tion’’

(Jaffe and Taubes 1980). Solutions invariant under

three translations correspond to solut ions of Nahm’s

equations,

¼ 

ijk

½T

; T



for matrix-valued functions T

, T

of one

variable t. Nahm (1982) and Hitchin (1983)

developed an analog of the ADHM construction

relating these two equations. This is now seen as a

part of a general ‘‘Fourier–Mukai–Nahm trans-

form’’ (Donaldson and Kronheimer 1990). The

instanton equations for connections invariant

under two translations, Hitchin’s equations

(Hitchin 1983), are locally equivalent to the

harmonic map equation for a surface into the

symmetric space dual t o the structure group.

Changing the signature of the metric on R

to (2, 2),

one gets the harmonic mapping equations into Lie

groups (Hitchin 1990). More complicated reduc-

tions yield almost all the known examples of

integrable PDEs as special forms of the instanton

equations (Mason and Woodhouse 1996).

Moment Maps: the Kobayashi–Hitchin Conjecture

Let  be a compact Riemann surface. The Jacobian

of  is the complex torus H

(, O)=H

(, Z): it

parametrizes holomorphic line bundles of degree 0

over . The Hodge theory (which was, of course,

developed long before Hodge in this case) shows

that the Jacobian can also be identified with the

torus H

(, R)=H

(, Z) which parametrizes flat

U(1)-connections. That is, any holomorphic line

bundle of degree 0 admits a unique compatible flat

unitary connection.

The generalization of these ideas to bundles of

higher rank began with Weil. He observed that any

holomorphic vector bundle of degree 0 admits a flat

connection, not necessarily unita ry. Narasimhan and

Seshadri (19 65) showed that (in the case of degree 0)

the existence of a flat, irreducible, unitary connec-

tion was equivalent to an algebro-geometric condi-

tion of stability which had been introduced shortly

before by Mumford, for quite different purposes.

Mumford introduced the stability condition in order

to construct separated moduli spaces of holo-

morphic bundles – generalizing the Jacobian – as

part of his general geometric invariant theory. For

bundles of nonzero degree, the discussion is slightly

modified by the use of projectively flat unitary

connections. The result of Narasimhan and Seshadri

asserts that there are two different descriptions of

the same moduli space M

d, r

(Sigma): either as

parametrizing certain irreducible proje ctively flat

unitary connections (representations of 

()), or

parametrizing stable holomorphic bundles of degree

d and rank r. While Narasimhan and Seshadri

probably did not view the ideas in these terms,

another formulation of their result is that a certain

nonlinear PDE for a Her mitian metric on a

holomorphic bundle – analogous to the Laplace

equation in the abelian case – has a solution when

the bundle is stable.

Atiyah and Bott (1982) cast these results in the

framework of gauge theory. (The Yang–Mills

equations in two dimensio ns essentially reduce to

the condition that the connection be flat, so they are

rather trivial locally but have interesting global

structure.) They made the important observation

that the curvature of a connection furnishes a map

F : A!LieðGÞ



which is an equivariant moment map for the action

of the gauge group on A. Here the symplectic form

on the affine space A and the map from the adjoint

Gauge Theory: Mathematical Applications 473

bundle-valued 2-forms to the dual of the Lie algebra

of G are both given by inte gration of products of

forms. From this point of view, the Narasimhan–

Seshadri result is an infinite-dimensional example of

a general principle relating symplectic and complex

quotients. At about the same time, Hitchin and

Kobayashi independently pro posed an extension of

these ideas to higher dimensions. Let E be a

holomorphic bundle over a complex manifold V.

Any compatible unitary connection on E has

curvature F of type (1,1). Let ! be the (1,1)-form

corresponding to a fixed Hermitian metric on V.

The Hermitian Yang–Mills equation is the equation

F! ¼ 1

where  is a constant (det ermined by the topological

invariant c

(E)). The Kobayashi–Hitchin conjecture

is that, when ! is Ka¨ hler, this equation has an

irreducible solution if and only if E is a stable

bundle in the sense of Mumford. Just as in the

Riemann surface case, this equation can be viewed

as a nonlinear second-order PDE of Laplace type for

a metric on E. The moment map picture of Atiyah

and Bott also extends to this higher-dimensional

version. In the case when V has complex dimension

2 (and  is zero), the Hermitian Yang–Mills

connections are exactly the anti-self-dual instantons,

so the conjecture asserts that the moduli spaces of

instantons can be identified with certain moduli

spaces of stable holomorphic bundles.

The Kobayashi–Hitchin conjecture was proved in

the most general form by Uhlenb eck and Yau

(1986), and in the case of algebraic manifolds in

Donaldson (1987). The proofs in Donaldson (1985,

1987) developed some extra structure surrounding

these equations, connected with the moment map

point of view. The equations can be obtained as the

Euler–Lagrange equations for a nonlocal functional,

related to the renormalized determinants of Quillen

and Bismut. The results have been extended to non-

Ka¨ hler manifolds and certain noncompact mani-

folds. There are also many extensions to equations

for systems of data comprising a bundle with

additional structure such as a holomorphic section

or Hi ggs’ field (Bradlow et al. 1995), or a parabolic

structure along a divisor. Hitchin’s equations

(Hitchin 1987) are a particularly rich example.

Topology of Moduli Spaces

The moduli spaces M

r, d

() of stable holomorphic

bundles/projectively flat unitary connections over

Riemann surfaces  have been studied intensively

from many points of view. They have natural Ka¨ hler

structures: the complex structure being visible in the

holomorphic bundles guise and the symplectic form

as the ‘‘Marsden–Weinstein quotient’’ in the unitary

connections guise. In the case when r and d are

coprime, they are compact manifolds with compli-

cated topologies. There is an important basic

construction for producing cohomology classes

over these (and other) moduli spaces . One takes a

universal bundle U over the product M with

Chern classes

ðUÞ2H

ðM  Þ

Then, for any class  2 H

(), we get a cohomology

class c

(U)= 2 H

2ip

(M). Thus, if R



is the graded

ring freely generated by such classes, we have a

homomorphism  : R





(M). The questions

about the topology of the moduli spaces which

have been studied include:

1. finding the Betti numbers of the moduli space M;

2. identifying the kernel of ;

3. giving an explicit system of generators and

relations for the ring H



(M);

4. identifying the Pontrayagin and Chern classes of

M within H



(M); and

5. evaluating the pairings

ðWÞ

for elements W of the appropriate degree in R.

All of these questions have now been solved quite

satisfactorily. In early work, Newstead (1967) found

the Betti numbers in the rank-2 case. The main aim

of Atiyah and Bott was to apply the ideas of Morse

theory to the Yang–Mills functional over a Riemann

surface and they were able to reproduce Newstead’s

results in this way and extend them to higher rank.

They also showed that the map  is a surjection, so

the universal bundle construction gives a system of

generators for the cohomology. Newstead made

conjectures on the vanishing of the Pontrayagin

and Chern classes above a certain range which were

established by Kirwan and extended to higher rank

by Earl and Kirwan (1999). Knowing that R



maps

on to H



(M), a full set of relations can (by Poincare´

duality) be deduced in principle from a knowledge

of the integral pairings in (5) above, but this is not

very explicit. A solution to (5) in the case of rank 2

was found by Thaddeus (1992). He used results

from the Verlinde theory (see section on 3-manifolds

below) and the Riemann–Roch formula. Another

point of view was developed by Witten (1991), who

showed that the volume of the moduli space was

related to the theory of torsion in algebraic topology

and satisfied simple gluing axioms. These different

474 Gauge Theory: Mathematical Applications

points of view are compared in Donaldson (1993).

Using a nonrigorous localization principle in infinite

dimensions, Witten (1992) wrote down a general

formula for the pairings (5) in any rank, and this

was established rigorously by Jeffrey and Kirwan,

using a finite-dimensional version of the same

localization method. A very simple and explicit set

of generators and relations for the cohomology (in

the rank-2 case) was given by King and Newstead

(1998). Finally, the quantum cohomology of the

moduli space, in the rank-2 case, was identified

explicitly by Munoz (1999).

Hyper-Ka¨ hler Quotients

Much of this story about the structure of moduli

spaces extends to higher dimensions and to the

moduli spaces of connections and Higgs fields.

A particularly notable extension of the ideas

involves hyper-Ka¨ hler structures. Let M be a hyper-

Ka¨ hler 4-manifold, so there are three covariant-

constant self-dual forms !

, !

on M. These

correspond to three complex structures I

, I

obeying the algebra of the quaternions. If we single

out one structure, say I

, the instantons on M can be

viewed as holomorphic bundles with respect to I

satisfying the moment map condition (Hermitian

Yang–Mills equation) defined by the form !

Taking a different complex structure interchanges

the role of the moment map and integrability

conditions. This can be put in a general framework

of hyper-Ka¨ hler quotients due to Hitchin et al.

(1987). Suppose initially that M is compact

(so either a K3 surface or a torus). Then the !

components of the curvature define three maps

: A!LieðGÞ



The structures on M make A into a flat hyper-

Ka¨ hler manifold and the three maps F

are the

moment maps for the gauge group action with

respect to the three symplectic forms on A. In this

situation, it is a general fact that the hyper-Ka¨ hler

quotient – the quotient by G of the common zero set

of the three moment maps – has a natural hyper-

Ka¨ hler structure. This hyper-Ka¨ hler quotient is just

the moduli space of instantons over M. In the case

when M is the noncompact manifold R

, the same

ideas apply except that one has to work with

the based gauge group G

. The conclusion is that

the framed mod uli spaces

M of instantons over R

are naturally hyper-Ka¨ hler manifolds. One can also

see this hyper-Ka¨hler structure through the ADHM

matrix description. A variant of these matrix

equations was used by Kronheimer to construct

‘‘gravitational instantons.’’ The same ideas also

apply to the moduli spaces of monopoles, where

the hyper-Ka¨ hler metric, in the simplest case, was

studied by Atiyah and Hitchin (1989).

Low-Dimensional Topology

Instantons and 4-Manifolds

Gauge theory has had unexpected applications in

low-dimensional topology, particularly the topology

of smooth 4-manifolds. The first work in this

direction, in the early 1980s, involved the Yang–

Mills instantons. The main issue in 4-manifold

theory at that time was the correspondence between

the diffeomorphism classification of simply con-

nected 4-manifolds and the classification up to

homotopy. The latter is determined by the intersec-

tion form, a unimo dular quadratic form on the

second integral homology group (i.e., a symmetric

matrix with integral entries and determinant 1,

determined up to integral change of basis). The only

known restriction was that Rohlin’s theorem, which

asserts that if the form is even the signature must be

divisible by 16. The achievement of the first phase of

the theory was to show that

1. There are unimodular forms which satisfy the

hypotheses of Rohlin’s theorem but which do not

appear as the intersection forms of smooth

4-manifolds. In fact, no nonstandard definite

form, such as a sum of copies of the E

matrix,

can arise in this way.

2. There are simply connected smooth 4-manifolds

which have isomorphic intersection forms, and

hence are homotopy equivalent, but which are

not diffeomorphic.

These results stand in contrast to the homeomorph-

ism classification which was obtained by Freedman

shortly before and which is almost the same as the

homotopy classification.

The original proof of item (1) above argued with

the moduli space M of anti-self-dual instantons SU(2)

instantons on a bundle with c

= 1 over a simply

connected Riemannian 4-manifold M with a negative-

definite intersection form (Donaldson 1983). In the

model case when M is the 4-sphere the moduli space

M can be identified explicitly with the open 5-ball.

Thus the 4-sphere arises as the natural boundary of

the moduli space. A sequence of points in the moduli

space converging to a boundary point corresponds to

a sequence of connections with curvature densities

converging to a -function, as described earlier. One

shows that in the general case (under our hypotheses

on the 4-manifold M) the moduli space M has a

similar behavior, it contains a collar M  (0, )

Gauge Theory: Mathematical Applications 475