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

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where

Avðjruj

; x;"Þ

:¼

jBðx;"Þ\j

Bðx;"Þ\

jruðyÞj

and f : [0, þ1) ![0, þ1) is any increasing continu-

ous function with f (0) = 0, f

(0) = 1, and f (t) !1=2

as t !þ1. Then for every sequence "

!0 the

sequence F

-converges to F.

Given g 2 L

(), for every ">0 let u

be a

solution of the minimum problem

min

u2W

1;2

ðÞ



f " Avðjruj

; x;"Þ







ju  gj



From the basic properties of -convergence it

follows that there exists a sequence "

!0 such

that u

converges in L

() to a solution u of [26],

so that (u,

) is a solution of [24].

Other approximations by nonlocal functionals use

finite differences instead of averages of gradients.

A different approximation can be obtained by

using the local functionals G

:(L

())

![0, þ1]

defined by

ðu;vÞ:¼





ðvÞjruj

jrvj

hðvÞ



if ðu; vÞ2ðW

1;2

ðÞÞ

þ1 otherwise

where g



(t):= 

þt

,0<

<< ", and h(t):=

(1 t)

for 0 t 1, while h(t):= þ1 otherwise. Let

G:(L

())

![0, þ1] be the functional defined by

Gðu; vÞ :¼

FðuÞ if v ¼ 1a:e: on 

þ1 otherwise



where F is defined [25]. Then for every sequence

! 0 the sequence G

-converges to G.

Given g 2 L

(), for every ">0 let (u

, v

)bea

solution of the minimum problem

min

ðu;vÞ2ðW

1;2

ðÞÞ





ðvÞjruj

jrvj

hðvÞþju  gj



dx ½27

From the basic properties of -convergence it

follows that there exists a sequence "

!0 such

that u

converges in L

() to a solution u of [26],

so that (u,

) is a solution of [24].

The approximation of the solutions of [24] based

on [27] has been used to construct numerical

algorithms for image segmentation.

Free discontinuity problems similar to [24] appear

in the math ematical treatment of Griffith’s model in

fracture mechanics. In this case, u is a vector-valued

function, which represents the deformation of an

elastic body, the first term in [24] is repl aced by a

more general integral functional which represents

the energy stored in the elastic region nK, while the

second term is interpreted as the energy dissipated to

produce the crack K. An approximation based on

minimum problems similar to [27] has been used to

construct numerical algorithms to study the process

of crack growth in brittle materials.

An important research line, connected with these

problems, has been developed in the last years to

derive the macroscopi c theories of fracture

mechanics from the microscopic theories of inter-

atomic interactions. Using -convergence, some

theories expressed in the language of continuum

mechanics can be obtained as limits of discrete

variational models on lattices, as the distance

between neighboring points tends to zero.

See also: Convex Analysis and Duality Methods; Elliptic

Differential Equations: Linear Theory; Free Interfaces

and Free Discontinuities: Variational Problems;

Geometric Measure Theory; Image Processing:

Mathematics; Variational Techniques for Ginzburg–

Landau Energies; Variational Techniques for

Microstructures.

Further Reading

Allaire G (2002) Shape Optimization by the Homogenization

Method. Berlin: Springer.

Ambrosio L, Fusco N, and Pallara D (2000) Functions of

Bounded Variation and Free Discontinuity Problems. Oxford:

Oxford University Press.

Bakhvalov N and Panasenko G (1989) Homogenisation: Aver-

aging Processes in Periodic Media. Mathematical Problems in

the Mechanics of Composite Materials (translated from the

Russian by D. Leıtes). Dordrecht: Kluwer.

Bensoussan A, Lions JL, and Papanicolaou GC (1978) Asymptotic

Analysis for Periodic Structures. Amsterdam: North-Holland.

Braides A (1998) Approximation of Free-Discontinuity Problems,

Lecture Notes in Mathematics, vol. 1694. Berlin: Springer.

Braides A (2002) -Convergence for Beginners. Oxford: Oxford

University Press.

Braides A and Defranceschi A (1998) Homogenization of Multi-

ple Integrals. Oxford: Oxford University Press.

Cherkaev A and Kohn RV (eds.) (1997) Topics in the Mathema-

tical Modelling of Composite Materials. Boston: Birkha¨user.

Christensen RM (1979) Mechanics of Composite Materials. New

York: Wiley.

Cioranescu D and Donato P (1999) An Introduction to Homo-

genization. New York: The Clarendon Press, Oxford Uni-

versity Press.

Dal Maso G (1993) An Introduction to -Convergence. Boston:

Birkha¨user.

Friesecke G, James RD, and Mu¨ ller S (2002) A theorem on

geometric rigidity and the derivation of nonlinear plate theory

456 G-Convergence and Homogenization

from three-dimensional elasticity. Communications on Pure

and Applied Mathematics 55: 1461–1506.

Jikov VV, Kozlov SM, and Oleınik OA (1994) Homogenization

of Differential Operators and Integral Functionals (translated

from the Russian by G. A. Yosifian). Berlin: Springer.

Milton GW (2002) The Theory of Composites. Cambridge:

Cambridge University Press.

Oleınik OA, Shamaev AS, and Yosifian GA (1992) Mathematical

Problems in Elasticity and Homogenization. Amsterdam:

North-Holland.

Panasenko G (2005) Multi-scale Modelling for Structures and

Composites. Dordrecht: Springer.

Sanchez-Palencia E (1980) Nonhomogeneous Media and Vibration

Theory, Lecture Notes in Physics, vol. 127. Berlin: Springer.

Gauge Theoretic Invariants of 4-Manifolds

S Bauer, Universita

t Bielefeld, Bielefeld, Germany

Introduction

Poincare´ duality is fundamental in the study of

manifolds. In the case of an orientable closed

manifold X, this duality ap pears as an isomorphism

: H

ðX; ZÞ!H

nk

ðX; ZÞ

between integral cohomology and homology. The

map is defined by cap product with a chosen

orientation class. This article focuses on dimension

n = 4, where Poincare´ duality induces a bilinear

form Q on H

(X; Z) by use of the Kronecker pairing

Qð; 

Þ¼h

1

ðÞ;

i2Z

One of the outstanding achievements of modern

topology, the classification of simply connected

topological 4-manifolds by Freedman (1982),can

be phrased in terms of the intersection pairing Q.

Indeed, two simply connected differentiable

4-manifolds X and X

are orientation pr eserving ly

homeomorphic if and only if the associated

pairings Q and Q

are equivalent. Freedman’s

classification scheme has been extended to also

cover a wide range of fundamental groups,

resulting in a fair understanding of topological

4-manifolds (Freedman and Quinn 1990).

When it comes to differentiable 4-manifolds, the

situation changes drastically. On the one hand, there is

an abundance of topological 4-manifolds which do not

admit a differentiable structure at all. On the other

hand, there also are topological 4-manifolds support-

ing infinitely many distinct differentiable structures.

A classification of differentiable 4-manifolds up to

differentiable equivalence seems out of reach of

current technology, even in the most simple cases.

The discrepancy between topological and differen-

tiable 4-manifolds was uncovered by gauge-theoretic

methods, applying the concepts of instantons and of

monopoles. In order to study these, one has to equip a

4-manifold both with a Riemannian metric and some

additional structure: a Hermitian rank-2 bundle in

the case of instantons and a spin

-structure in the case

of monopoles. Given such data, instantons and

monopoles arise as solutions to partial differential

equations the gauge equivalence classes of which form

finite-dimensional moduli spaces. As it turns out,

these moduli spaces encode significant information

about the differentiable structures of the underlying

4-manifolds.

A decoding of such information contained in the

instanton moduli and in the monopole moduli is

achieved through Donaldson invariants and Seiberg–

Witten invariants, respectively. This article outlines

these theories from a mathematical point of view.

Instantons and Donaldson Invariants

Let X denote a closed, connected, oriented differ-

entiable Riemannian 4-manifold. We will consider a

principal bundl e P over X with fiber a compact Lie

group G with Lie algebra g. Connections on P form

an infinite-dimensional affine space A(P) = A



(X; g

) modeled on the vector space of 1-forms

with values in the adjoint bundle

¼ P 

AdðGÞ

The curvature F

2 

(X, g

) of a connection A is a

-valued 2-form satisfying the Bianchi identity

= 0. The group G of principal bundle auto-

morphisms of P acts in a natural way on the space

of connections with quotient space

BðPÞ¼AðPÞ=G

The Yang–Mills functional

YM : AðPÞ!R

0

associates to a connection A the norm square

¼

trðF

^F

of its curvature. Here  deno tes the Hodge star

operator defined by the metric on X and the

orientation. The metric tr: g  g !R is Ad(G)-

invariant and hence YM is invariant under the

Gauge Theoretic Invariants of 4-Manifolds 457

action of G. In particular, the Yang–Mills functional

descends to a function on the space B(P) of a gauge

equivalence class of connections.

The Euler–Lagrange equations for the critical

points of YM, called Yang–Mills equations, are of

the form

ðF

Þ¼0

and can be derived easily from the formula

Aþa

¼ F

þ D

ðaÞþ½a ^a

Satisfying the equations

ðF

Þ= 0andD

ðF

Þ= 0

a Yang–Mills connection is characterized by the fact

that it is harmonic with respect to its own Laplacian.

The bundle ^



X of 2-forms on X decomposes

into (1)-eigenbundles of the Hodge operator. This

orthogonal splitting leads to a decomposition of

curvature forms

¼ F

þ F



into self-dual and anti-self-dual components. The

differential form (1=4

)tr(F

) represents a

characteristic class of the principal bundle P.In

particular, the integral

ðPÞ¼

trðF

¼kF

kF



is independent of the connection A. The Yang–Mills

functional therefore is bounded

YMðPÞjðPÞj

and attains this mi nimum at conn ections A which

satisfy the equation

F

¼F

Such connections are either self-dual, anti-self-dual

or both, that is, flat, depending on whether (P)is

negative, positive , or zero. The moduli space of

instantons on P is the subset of minima of the Yang–

Mills functional

MðPÞ¼YM

1

ðjðPÞjÞ  BðPÞ

The moduli space thus consists of gauge equivalence

classes of connections which are either self-dual or

anti-self-dual. Donaldson theory indeed considers

anti-self-dual connections on principal bundles with

structure group PU(2) = SO(3).

The Hodge  operator induces a decomposition of

the second cohomology

ðXÞ¼H

ðXÞH



ðXÞ

into (1)-eigenspaces of dim ension b

and b



Unless specified differently, coh omology groups are

meant with real coefficients. In order to simplify the

exposition, we will assume X to be simply con-

nected. The Donaldson invariants then are defined if

is odd and greater than 1.

A ‘‘homology orie ntation’’ consists of an orienta-

tion of H

(X) and an integral homology class

c 2 H

(X; Z). The Donaldson invariant D

X, c

= D

is defined after fixing such a homology orientation.

It is a linear function

: AðX Þ!R

where A(X) is the graded algebra

AðXÞ¼Sym



ðH

ðXÞH

ðXÞÞ

in which H

(X) has degree (1=2)(4  i). The sig-

nificance of D

is its functoriality

; f ðcÞ

ðf ðÞÞ ¼ D

X; c

ðÞ

under diffeomorphisms f : X !X

which preserve

both orientation and homology orientation. Switch-

ing the orientation of H

(X; R) reverses the sign of

. Similarly,

¼ð1Þ

ððcc

Þ=2Þ

if c  c

2 2H

(X, Z)  H

(X; Z).

The construction of this invariant makes use of

the following facts:

1. An SO(3) principal bundle P over X is

determined by its first Pontrjagin number p

(P) and

its Stiefel–Whitney class w

(P) 2 H

(X; Z=2). As X

is simply connected, this Stiefel–Whitney class

admits integer lifts. Let c be such a lift and let c

be shorthand for the intersection pairing Q(c, c).

A pair (p

, w

) is realized by a principal bundle

provided it satisfies the relation p

 c

modulo 4.

2. If b

is nonvanishing, then for generic metrics

on X, the moduli space M(P) is a manifold of

dimension

2p

ðPÞ3ð1 þ b

This follows from a transversality theorem whose

main ingredient in the Sard–Smale theorem. The

dimension is computed by use of the Atiyah–Singer

index theorem: to an anti-self-dual connection A on

P there is an associated elliptic complex

0 ! 

ðX; g

Þ!



ðX; g

!



ðX; g

Þ!0

where 

(X; g

) denotes g

-valued i-forms on X.

This complex describes the tangential structure of

458 Gauge Theoretic Invariants of 4-Manifolds

the moduli space at the equivalence class of A. The

space 

(X; g

) is the tangent space of A(P)atA,



(X; g

) is the tangent space of the group G at the

identity, and D

is the differential of the orbit map.

The differential operator D

is the linearization of

the anti-self-duality map

a 7! F

Aþa

¼ D

ðaÞþ½a ^ a

3. The moduli space M (P) can be oriented if it is

a manifold. The orientation depends on an orienta-

tion of H

(X) and on a U(2)-pr incipal bundle which

has P as its PU(2)-quotient bundle. It is determined

by an integer lift of w

(P). The elliptic complex

above then can be compared with a corresponding

elliptic complex where the differentials are given by

a complex Dirac operator. This leads to an almost-

complex structure on the tangent space for each

point in the moduli space and in particul ar to an

orientation on the moduli space itself.

4. Over the product M(P) X there is a universal

PU(2)-bundle P with first Pontrjagin class p

(P).

Taking slant product with the class (1=4)p

(P)

results in a homomorphism

 : H

ðXÞ!H

4i

ðMðPÞÞ

5. The moduli space M(P) in general is noncom-

pact. There is an Uhlenbeck compactification

M(P)

describing ‘‘ideal instantons.’’ Such an ideal instanton

consists of an element (x

, ..., x

) 2 Sym

(X)andan

anti-self- dual connection A

on the principal bundle

on X with w

) = w

(P) for which the equality

ðP

Þp

ðPÞ¼4n

of Pontrjagin numbers holds. Uhlenbeck’s compact-

ness theorem describes what happens if a sequence

of anti-self-dual connections has no convergent

subsequence: afte r passing to a subsequence, the

sequence converges to an anti-self-dual connection

on the restriction of P to Xn{x

, ..., x

}. This limit

connection extends to a connection A

on the

principal bundle P

. The functions j F

on X

converge to the measure

i¼1

8



The compactification M(P) is a stratified space and

not usually a manifold. If w

(P) 6¼0, then the

singular set of codimension at least 2 and thus the

space

M(X) carries a fundamental class. In the case

(P) = 0, such a fundamental class in general can

only be defined if p

(P) > 4 þ 3b

. In practice,

this problem can be circumvented by blow ing up X

and considering bundles with w

(P) 6¼ 0 over the

connected sum X#

. Note that the complex

projective plane CP

as a complex manifold carries

a natural orientation. The notation CP

indicates a

reversed orientation.

6. The classes () 2 H

(M(P)) for  2 H

(X)

extend over the compactification. The same holds for

the class (x), where x 2 H

(X; Z) is the generator

corresponding to the orientation, as long as w

(P) 6¼ 0.

Otherwise, there are certain dimension restrictions.

However, the same blow-up trick as mentioned above

allows to handle the case w

(P) = 0 as well.

Now fix an element c 2 H

(X; Z) and let

¼t

d0

MðP

c;d

denote the disjoint union of all moduli spaces

of anti-self-dual connections on principal PU(2)-

bundles P

c, d

whose second Stiefel–Whitney class is

Poincare´-dual to c modulo 2 and whose Pontrjagin

number equals d  (3=2)(b

þ 1).

Our assumption of b

being odd corresponds

to the fact that the dimension 2d of the moduli

space M(c, d) is even and congruent to c

(1=2)(1 þ b

) modulo 4. Neglecting the difficulties

in the case w

(P) = 0 mentioned above, we may use

the cup product on H



) to extend  to an

algebra homomorphism

 : AðXÞ!H



ðM

The Donaldson invariant D

is nonzero only on

elements z of A(X) whose total degree d is congruent

to c

þ (1=2)(1 þ b

) modulo 4. For such an

element it is defined by

ðzÞ¼hðzÞ; MðP

c;d

Þi¼

MðP

c;d

ðzÞ

The Donaldson series D

is defined as a formal

power series

ðÞ¼D

ðexpð

ÞÞ ¼

d¼0



for  2 H

(X)and

 = (1 þ (x=2)).

Computations and Structure Theorems

The first results about these invariants are due to

S Donaldson. He proved both a vanishing and a

nonvanishing theorem (Donaldson and Kronheimer

1990):

Theorem 1 If both b

(X) > 0 and b

(Y) > 0, then

all Don aldson invariants vanish for the connected

sum X#Y.

Gauge Theoretic Invariants of 4-Manifolds 459

Theorem 2 If c represents a divisor on a complex

algebraic surface X and  represents an ample

divisor, then

ð

Þ 6¼ 0 for r  0

The second theorem is a consequence of the fact that in

thecaseofanalgebraicsurface the instanton moduli

can be describ ed in algebraic geometric terms: the

moduli space

M(P

c, d

) associated to the metric induced

from the Fubini–Study metric on CP

by an embedding

X ,!CP

carries the structure of a project ive variety.

This variety is reduced and of complex dimension d,as

soon as d is large enough. Furthermore, (d) is the first

Chern class of an ample line bundle.

The translation of instanton moduli into algebraic

geometry uses two steps: suppose the first Chern class

of a U(r)-principal bundle P on a Ka¨hler surface is also

the first Chern class of a holomorphic line bundle. Then

the absolute minima of the Yang–Mills functional are

achieved by Hermite–Einstein connections. These are

connections for which the Ricci curvature is a constant

multiple of the identity. The second step, the transla-

tion from differential geometry into algebraic geome-

try, is called the Kobayashi–Hitchin correspondence,

which again was proved by Donaldson.

The Donaldson invariants have been computed

for a number of 4-manifolds. A simply connected

4-manifold is said to have simple type, if the relation

ðx

zÞ¼4D

ðzÞ

is satisfied by its Donaldson invariant for all z 2 A(X)

and c 2 H

(X; Z). It is known that this simple type

condition holds for many 4-manifolds. Indeed, it is an

open question whether there are 4-manifolds which

are not of simple type. For manifolds of simple type

the Donaldson series D

completely determines the

Donaldson invariant D

. A main result is due to

Kronheimer and Mrowka (1995):

Theorem 3 Let X be a simply connected 4-manifold

of simple type. Then, there exist finitely many basic

classes 

, ..., 

2 H

(X; Z) such that

¼ expðQ=2Þ

i¼1

ð1Þ

ðc



cÞ=2

expð

as analytic functions on H

(X). The numbers a

are

rational and each basic class 

is characteristic, that

is, it satisfies 

 Q(, 

) modulo 2 for all

 2 H

(X; Z). The homology class 

in this formula

acts on an arbitrary homology class by intersection.

The geometric significance of the basic classes is

underlined by the following theorem (Kronheimer

and Mrowka 1995):

Theorem 4 If  2 H

(X : Z) is represented by an

embedded surface of genus g with self-intersection



 2, then for each basic class  the following

adjunction inequality is satisfied:

2g  2  

þjQð; Þj

There are many 4-manifolds for which the Donaldson

series have been computed (Friedman and Morgan

1997). The basic classes for complete intersections, for

example, are the canonical divisor and its negative.

Another example is given by elliptic surfaces. Let

E(n; p, q) be a minimal elliptic surface, that is, a

holomorphic surface admitting a holomorphic map to

with generic fiber f an elliptic curve. For any

numbers n, p,andq with p < q coprime, there exists

such a simply connected elliptic surface with Euler

characteristic 12n and two multiple fibers of multi-

plicity p and q, respectively. The Donaldson series of

E(n; p, q)forc = 0thenisgivenby

D ¼ exp



sinh

ðf Þ

sinhðf =pÞsinhðf =qÞ

Another important formula relates the Donaldson

series D amanifoldX of simple type and the

Donaldson series

D of the blow-up X#

¼ D

 expðe

=2ÞcoshðeÞ

cþe

¼D

 expðe

=2ÞsinhðeÞ

Here e 2 H

(CP

; Z) denotes a generator. Indeed, a

more general blow-up formula is known which

relates the Donaldson invariants for X and its

blow-up even in case X is not of simple type. This

formula, due to Fintushel and Stern (199 6), involves

Weierstraß sigma-functions.

The instanton moduli space carries nontrivial

information about 4-manifolds even in the case

(X)  1. However, one has to deal with singula-

rities in the moduli space. Let us first consider the

case b

(X) = 0. If the intersection form on X is

negative definite, the instanton moduli spaces in

general are bound to have singularities. Indeed,

Donaldson examined the case with the Pontrjagin

number p

(P) = 4andw

(P) = 0. In this case, the

moduli space for a generic metric on X will be an

orientable smooth manifold except at isolated

singular points. The singularities are cones over

and they correspond to reducible connections,

that is, reductions of the structure group of P to

U(1). These reductions are in bijective correspon-

dence to pairs  2 H

(X; Z) with 

= 1. The

Uhlenbeck compactification of the moduli space

thus leads to an oriented cobordism between X and

the disjoint union t



over all pairs  in

(X; Z) of square 1. As the signature of a

460 Gauge Theoretic Invariants of 4-Manifolds

manifold is an invariant of oriented cobordism,

there have to be b



many pairs  of square (1) in

(X; Z) and, in particul ar, the intersection form Q

is represented by the negative of the identity matrix

(Donaldson 1983):

Theorem 5 The intersection form on a differenti-

able manifol d with negative-definite intersection

form is diagonal.

Indeed, from rank 8 on there are lots of definite

unimodular forms which are not diagonal. By

Freedman’s (1982) classification, any unimodular

form is realized as the intersection form of a simply

connected topological manifold. This theore m

shows that most of these manifolds do not support

differentiable structures.

The case b

(X) = 1 is also interesting. Here, the

moduli space is a smooth manifold for a generic

metric, giving rise to Donaldson invariants. How-

ever, over a smooth path of metrics, there is in

general no smooth cobordism of moduli spaces. So

the invariants depend on the chosen metric. The

singularities in the cobordisms again correspond to

classes in H

(X; Z) with negative square. An

analysis of these singularities leads to wall-crossing

formulas describing how different choices of the

metric do affect Donaldson invariants. The case of

is special, as there are no elements of negative

square in H

(CP

; Z). The Donaldson invariants for

as well as the wall-crossing formulas turn out

to be closely related to modular forms ( Go¨ ttsche

2000).

Monopoles and Seiberg–Witten Invariants

A spin

-structure on an oriented Riemannian

4-manifold is a Spin

(4)-principal bundle P projecting

to the orthonormal tangent frame bundle

P over X

through the group homomorphism Spin

(4) !SO(4)

with kernel U(1). The group H

(X; Z) acts freely

and transitively on the set of all spin

-structures.

A spin

-connection is a lift to P of the Levi-Civita

connection on

P. Fixing a background spin

connection A

, the monopole m ap

 : ðA;Þ7! D

; F

 





; d





is defined (Witten 1994) for spin

-connections A 2

þ 

(X;iR) and positive spinors . Here, D

denotes the complex Dirac operator associated to A

and d



a for a 2 

(X;iR) is the adjoint of the de

Rham differential on forms. The section 





of the

traceless endomorphism bundle of positive spinors is

viewed as a self-dual 2-form on X.

In case the first Betti number vanishes, this map –

after suitable Sobolev completion – becomes a map

between Hilbert spaces  : A!C which is a compact

deformation of a linear Fredholm map. The

Weitzenbo¨ ck formula can be used to show that

preimages under  of bounded sets in C are bounded

in A. Furthermore,  is U(1)-equivarant, where U(1)

acts by complex multiplication on spinors and

trivially on forms. If b

(X) > 0, the monopole map

is a map between Hilbert space bundles over the

torus H

(X)=H

(X; Z). These properties of the

monopole map allow for an interpretation in terms

of stable homotopy (Bauer 2004):

Theorem 6 If the first Betti number of X vanishes,

then  defines an element

½2

Uð1Þ

ðS

in an equivariant stable homotopy group of spheres.

The index i = ind D

 H

(X) as an element of the

real representation ring RO(U(1)) is determind by

the analytic index of the linearization of .

In the case b

(X) > 1, these equivariant

stable homotopy groups can be identified with

nonequivariant stable cohomotopy groups



1

(CP

d1

). Here, d denotes the index of the

complex Dirac operator ind D

. Fixing an orienta-

tion of H

(X) results in a Hurewicz homomorphism

h : 

1

ðCP

d1

Þ!H

1

ðCP

d1

; ZÞ

If b

(X) is odd, the image

hð½Þ ¼ SWðXÞt

ðb

1Þ=2

is an integer multiple of a power of the generator

t 2 H

(CP

d1

; Z). This integer SW(X) is known as

the Seiberg–Witten invariant (Witten 1994).

This invariant alternatively can be defined by

considering the moduli space M(a) = 

1

(a). Assum-

ing b

> 0, this is a smooth oriented manifold with a

free U(1)-action for generic a 2 

(X; iR). The

Seiberg–Witten invariant is the characteristic

number obtainable by these data. In general, the

stable homotopy invariant [] encodes global inform-

ation about the monopole map, which cannot be

recovered by only considering the moduli space. In

case the spin

-structure is associated to an almost-

complex structure, however, there is a fortunate

coincidence: the Hurewicz homomorphism in this

case is an isomorphism. So for almost-complex

spin

-structures, the invariants [] and SW carry

the same information.

The Seiberg–Witten invariants turn out to be directly

computable for Ka¨hler manifolds and to some degree

also for symplectic manifolds (Taubes 1994). Indeed,

Gauge Theoretic Invariants of 4-Manifolds 461

the following theorem follows from arguments of

Witten and of Taubes:

Theorem 7 Let X be a 4-manifold with b

> 1 and

= 0 which can be equipped with a Ka¨ hler or a

symplectic structure. If [] is nonvanishing for a

spin

-structure on X, then the spin

-structure is

associated to an almost- complex structure. For the

canonical spin

-structure on X the Seiberg–Witten

invariant is 1.

Seiberg–Witten invariants and Donaldson invar-

iants are closely related: Witten gave physical

arguments that an equality of the form

D ¼ 2

expðQ=2Þ



SWðÞexp ðÞ

should hold for the Donaldson series for c = 0of

a simply connected manifold of simple type. Here,

 2 H

(X; Z) denotes the first Chern class of the

complex determinant line bundle. This first Chern

class characterizes spin

-structures in the simply

connected case. The number k is related to the

signature  and the Euler characteristic  of the

manifold X by the formula

4k ¼ 11 þ 7 þ 2

A mat hematical proof of this formula is known in

special cases (Feehan and Leness 2003).

As is the case for Donaldson invariants, the Seiberg–

Witten invariants vanish for connected sums X#Y if

both b

(X) > 0andb

(Y) > 0 holds. This is not the

case for the stable homotopy refinement as follows

from the following theorem (Bauer 2004).

Theorem 8 For a connected sum X#Yof

4-manifolds the stable equivariant homotopy

invariants are related by smash prod uct

½

X#Y

¼½

^½



As an example application, consider connected sums of

elliptic surfaces of the form E(2n; p, q). Now suppose X

and X

are each connected sums of at most four copies of

such elliptic surfaces. Then X and X

are diffeomorphic

if and only if the summands were already diffeomorphic.

This contrasts to the fact that the connected sum

E(2n; p, q)#CP

is diffeomorphic to a connected sum

of 4n  1 copies of CP

and 20n  1copiesofCP

independently of p and q.

As a final applicati on, we consider the case of spin

manifolds. If the manifold X is spin, then the

intersection form Q is even, that is, Q(, ) = 0mod

2for 2 H

(X, Z). According to Rochlin’s theorem,

the signature of a spin 4-manifold is divisible by 16.

The monopole map  for the spin structure admits

additional symmetry. It is Pin(2)-equivariant. The

nonabelian group Pin(2) appears as the normalizer

of the maximal torus SU(2). Methods from equivar-

iant K-theory lead to Furuta’s (2001) theorem:

Theorem 9 Let X be a spin 4-manifol d. Then

ðXÞ>

jðXÞj

Manifolds with Boundary

Both Donaldson invariants and Seiberg–Witten invar-

iants to some extent satisfy formal properties which

fit into a general conceptual framework known as

‘‘topological quantum field theories (TQFTs).’’ Such

aTQFTin3þ 1 dimensions is a functor on the

cobordism category of oriented 3-manifolds to the

category of, say, vector spaces over a ground field: it

assigns to an oriented 3-manifold Y avectorspace

h(Y). To a disjoint union it assigns

hðY

t Y

Þ¼hðY

ÞhðY

Reversing orientation corresponds to dualizing

hð

YÞ¼hðYÞ



Viewing a four-dimensional manifold X with

boundary @X =

t Y

formally as a morphism

from Y

to Y

, this functor associates to X a

homomorphism

HðXÞ : hðY

Þ!hðY

that is, an element H(X) 2 h(

t Y

). The most

important feature is the composition law

HðX

[

Þ¼HðX

ÞHðX

So if a cobordism X from Y

to Y

can be decomposed

as a cobordism X

from Y

to an intermediate

submanifold Y and a cobordism X

from Y to Y

then the homomorphism H(X) can be computed from

H(X

)andH(X

) as their composition.

Donaldson invariants and Seiberg–Witten invar-

iants fit neatly into the framework of a TQFT if one

restricts to 3-manifolds which are disjoint unions of

homology 3-spheres. In both the instanton and

the mono pole case, the vector spaces h(Y) are

Floer homology group s. The construction of Floer

homology carries the Morse theory description of

the homology of a finite-dimensional manifold over

to an infinite-dimensional setting. In the instanton

case, one considers the Chern–Simons function

CSðaÞ¼

8

tr a ^ da þ

a ^ a ^ a



This function is defined on the space of gauge

equivalence classes of SU(2)-connections on Y. Note

that for a homology 3-sphere, any SU(2) or PU(2)

462 Gauge Theoretic Invariants of 4-Manifolds

principal bundle over Y is trival. Choosing a

trivialization, a connection becomes identified with

a Lie-algebra-valued 1-form a. Critical points for the

Chern–Simons functional lead to generaters in a

chain complex the homology of which then gives the

Floer groups. Such critical points correspond to flat

connections on Y. The Floer homology groups HF



(Y)

are Z=8-graded in the SU(2) case and Z=4-graded in

the SO(3) case. If X is a 4-manifold with b

(X) = 0

and b

(X) > 1 and such that the boundary @X is a

disjoint union of homology 3-spheres, then the

Donaldson invariants are linear maps

: AðXÞ!HF



ð@XÞ

These invariants satisfy a composition law on the

subring of A(X) generated by two-dimensional

homology classes (Donaldson 2002).

In the monopole case, one considers a Chern–

Simons–Dirac functional

CSDða; Þ¼

h ; D

idvol 

a ^ da



and obtains integer graded Floer homology groups.

Details and proofs of the relevant composition laws

are announced.

See also: Floer Homology; Four-Manifold Invariants and

Physics; Gauge Theory: Mathematical Applications;

Instantons: Topological Aspects; Moduli Spaces: An

Introduction; Several Complex Variables: Basic

Geometric Theory; Topological Quantum Field Theory:

Overview.

Further Reading

Bauer S (2004) Refined Seiberg–Witten invariants. In: Donaldson,

et al. (eds.) Different Faces of Geometry. New York, NY:

Kluwer Academic/Plenum Publishers.

Donaldson SK (1983) An application of gauge theory to four-

dimensional topology. Journal of Differential Geometry 18:

279–315.

Donaldson SK (2002) Floer Homology Groups in Yang–Mills

Theory. Cambridge: Cambridge University Press.

Donaldson SK and Kronheimer PB (1990) The Geometry of Four-

Manifolds. Oxford: Oxford University Press.

Feehan PMN and Leness TG (2003) On Donaldson and Seiberg–

Witten invariants. In: Topology and Geometry of Manifolds,

Proc. Sympos. Pure Math. (Athens, GA, 2001), vol. 71.

pp. 237–248. Providence, RI: American Mathematical Society.

Fintushel R and Stern R (1996) The blow up formula for Donaldson

invariants. Annals of Mathematics (2) 143: 529–546.

Freedman MH (1982) The topology of 4-dimensional manifolds.

Journal of Differential Geometry 17: 357–453.

Freedman MH and Quinn F (1990) Topology of 4-Manifolds,

Princeton Math. Ser., vol. 39. Princeton, NJ: Princeton University

Press.

Friedman R and Morgan JW (eds.) (1997) Gauge Theory and the

Topology of Four-Manifolds, IAS/Park City Math. Ser., vol. 4.

Providence, RI: American Mathematical Society.

Furuta M (2001) Monopole equation and the 11/8-conjecture.

Mathematical Research Letter 8: 363–400.

Go¨ ttsche L (2000) Donaldson invariants in algebraic geometry.

School on Algebraic Geometry (Trieste 1999), 101–134. ICTP

Lecture Notes 1, Trieste, 2000.

Kronheimer PB and Mrowka T (1995) Embedded surfaces and

the structure of Donaldson’s polynomial invariants. Journal of

Differential Geometry 41: 573–735.

Taubes CH (1994) The Seiberg–Witten invariants and sympletic

forms. Mathematical Research Letter 1: 809–822.

Witten E (1994) Monopoles and four-manifolds. Mathematics

Research Letter 1: 769–796.

Gauge Theories from Strings

P Di Vecchia, Nordita, Copenhagen, Denmark

Introduction

One of the most exciting properties of string theory,

which led ten years ago to the formulation of the M

theory as the unique theory unifying all interactions,

has been the discovery that type II theories, besides a

perturbative spectrum consisting of closed-string

excitations, contain also a nonperturbative one

consisting of ‘‘solitonic’’ p-dimensional objects

called Dp branes. They are characterized by two

important properties. They are coupled to closed-

string states as the graviton, the dilaton, and the

R–R (p þ 1)-form potential, and are described by a

classical solut ion of the low-energy string effective

action. Their dynamics is, on the other hand,

described by open strings having the endpoints

attached to their world volume and therefore

satisfying Dirichlet boundary conditions in the

directions transverse to their world volume. This is

the reason why they are called D (Dirichlet) branes.

Since the lightest open-string excitation corresponds

to a gauge field, they have a gauge theory living on

their world volume. This twofold description of

D-branes has opened the way to study both the

perturbative and nonperturbative properties of the

gauge theory living on their world volume from

their dynamics in terms of closed strings. With the

addition of the decoupling limit, these two proper-

ties have led to the Maldacena (1998) conjecture of

the equivalence between the maximally supersym-

metric and conformal N = 4 super Yang–Mills and

type IIB string theory on AdS

 S

They have also been successfully applied to less

supersymmetric and nonconformal gauge theories

Gauge Theories from Strings 463

that live on the world volume of fractional and

wrapped branes. For general reviews of various

approaches see Bertolini et al. (2000), Herzog et al.

(2001), Bertolini (2003), Bigazzi et al. (2002), and

Di Vecchia and Liccardo (2003). Also in these cases,

one has constructed a classical solution of the

supergravity equations of motion corresponding to

these more sophisticated branes. These equations

contain not only the supergravity fields present in

the bulk ten-dimensional action but also boundary

terms corresponding to the location of the branes. It

turns out that in general the classical solution

develops a naked singularity of the repulson type

at short distances from the branes. This means that

at short distances, it does not provide a reliable

description of the branes. In the case of N = 2

supersymmetry, this can be explicitly seen because

of the appearance of an enhanc˛on located at

distances slightly higher than the naked singularity

(Johnson et al. 2000). The enhanc˛on radius corre-

sponds, in supergravity, to the distance where a

brane probe becomes tensionless, and, in the gauge

theory living on the branes, to the dynamically

generated scale 

QCD

. Then, since short distances

in supergravity correspond to large distances in

the gauge theory, as implied by holography, the

presence of the enhanc˛on and of the naked

singularity does not allow to get any information

on the nonperturbative large-distance behavior of

the gauge theory living on the D-branes. Above the

radius of the enhanc˛on, instead, the classical solu-

tion provides a good description of the branes and

therefore it can be used to get information on the

perturbative behavior of the gauge theory. This

shows that, if we want to use the D-branes for

studying the nonperturbative properties of the gauge

theory living on their world volume, we must

construct a classical solution that has no naked

singularity at short distances in supergravity. We

will see in a specific example that it will be possible

to deform the classical solution, eliminating the

naked singularity, and use it to describe nonpertur-

bative properties as the gaugino condensate.

In this article, we review some of the results obtained

by using fractional D3 branes of some orbifold and D5

branes wrapped on 2-cycles of some Calabi–Yau

manifold. The analysis of the supersymmetric gauge

theories living on the world volume of these D-branes

will be based on the gauge/gravity relations that relate

the gauge coupling constant and the -angle to the

supergravity fields (see, e.g., reference Di Vecchia et al.

(2005) for a derivation of them):

4

ð2

ﬃﬃﬃﬃﬃ



e



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

detðG

þ B

½1

and



2

ðC

þ C

Þ½2

where C

is the 2-cycle where the branes are

wrapped.

In the next section, we will describe the case of

the fractional D3 branes of the orbifold C

and

show that the classical solution corresponding to a

system of N D3 and M D7 branes reproduces the

perturbative behavior of N = 2 super-QCD.

Then, we will consider D5 branes wrapped on 2-

cycles of a Calabi–Yau manifold described by the

Maldacena–Nu´n˜ ez classical solution (Maldacena

and Nu´n˜ ez 2001, Chamseddine and Volkov 1997)

and show that in this case we are able to reproduce

the phenomenon of gaugino condensate and to

construct the complete -function of N = 1 super

Yang–Mills.

Fractional D3 Branes of the Orbifold

and N = 2 Super-QCD

In this section, we consider fractional D3 and D7

branes of the noncompact orbifold C

in order

to study the properties of N = 2 super-QCD. We

group the coordinates of the directions (x

, ..., x

)

transverse to the world volume of the D3

brane where the gauge theory lives, into three

complex quantities: z

= x

þ ix

, z

= x

þ ix

= x

þ ix

. The nontrivial generator h of Z

acts as z

!z

, z

!z

, leaving z

invariant.

This orbifold has one fixed point, located at

= z

= 0 and corresponding to a vanishing

2-cycle. Fractional D3 branes are D5 branes

wrapped on the vanishing 2-cycle and therefore

are, unlike bulk branes, stuck at the orbifold fixed

point. By considering N fractional D3 and M

fractional D7 branes of the orbifold C

,weare

able to study N = 2 super-QCD with M hyper-

multiplets. In order to do that, we need to

determine the classical solution corresponding to

the previous brane configuration. For the case of

the orbifold C

, the complete classical solution

is found in Bertolini et al. (2002b);seealso

references therein and Bertolini et al. (2000) for a

review on fractional branes. In the following, we

write it explicitly for a system of N fractional D3

branes with their world volume along the direc-

tions x

, x

,andx

and M fractional D7 branes

containing the D3 branes in their world volume

and having the remaining four world-volume

directions along the orbifolded ones. The metric,

464 Gauge Theories from Strings

the 5-form field strength, the axion, and the

dilaton are given by

¼H

1=2









þ H

1=2



‘m

‘

þ e







½3

ð5Þ

¼dH

1

^^dx





1

^^dx



½4

  C

þ ie



¼ i1

2

log





z  x

þ ix

¼ ye

i

½5

where the self-dual field strength

(5)

is given in

terms of the NS–NS and R–R 2-forms B

and C

and of the 4-form potential C

(5)

= dC

þ C

. The warp factor H is a function of the

coordinates (x

, ..., x

) and  is an infrared cutoff.

We denote by  and  the four directions corre-

sponding to the world volume of the fractional D3

brane, by ‘ and m those along the four orbifolded

directions x

, x

, and x

, and by i and j the

directions x

and x

that are transverse to both the

D3 and the D7 branes. The twisted fields are instead

given by B

= !

b, C

= !

c where !

is the volume

form of the vanishing 2-cycle and



ð2

ﬃﬃﬃﬃﬃ



1 þ

2N  M



log





c þ C

b ¼2

g

ð2N  MÞ

½6

The expression of H (Kirsch and Vaman 2005)shows

that the previous solution has a naked singularity of

the repulson type at short distances. On the other

hand, if we use a brane probe approaching from

infinity the stack of branes, described by the previous

classical solution, it can also be seen that the tension

of the probe vanishes at a distance that is larger than

that of the naked singularity. The point where the

probe brane becomes tensionless is called ‘‘enhanc˛on’’

(Johnson et al. 2000) and at this point the classical

solution does not describe anymore the stack of

fractional branes.

Let us now use the gauge/gravity relations given in

the introduction, to determine the coupling con-

stants of the world-volume theory from the super-

gravity solution. In the case of fractional D3 branes

of the orbifold C

, that is characterized by one

single vanishing 2-cycle C

, the gauge coupling

constant given in eqn [1] reduces to

4g

ð2

ﬃﬃﬃﬃﬃ





½7

By inserting the classical solution in eqns [7] and [2],

we get the following expressions for the gauge coupling

constant and the 

angle (Bertolini et al. 2002b):

8g

2N  M

16

log





¼ð2N  MÞ

½8

Notice that the gauge coupling constant appearing

in the previous equation is the ‘‘bare’’ gauge

coupling constant computed at the scale m  y=

while the square of the bare gauge coupling constant

computed at the cutoff   =

is equal to 8g

In the case of an N = 2 supersymmetric gauge

theory, the gauge multiplet contains a complex

scalar field  that corresponds to the complex

coordinate z transverse to both the world volume

of the D3 brane and the four orbifolded directions:

  z=2

. This is another example of holographic

identification between a quantity, , peculiar of the

gauge theory living on the fractional D3 branes and

another one, the coordinate z, peculiar of super-

gravity. It allows one to obtain the gauge theory

anomalies from the supergravity background. In

fact, since we know how the scale and U(1)

transformations act on , from the previous gauge/

gravity relation we can deduce how they act on z,

namely

 ! se

2i

 () z ! se

2i

z ¼) y ! sy

 !  þ 2

½9

Those transformations do not leave invariant the

supergravity background in eqn [6] and when

we use them in eqns [7] and [2], they generate the

anomalies of the gauge theory living on the

fractional D3 branes. In fact, by acting with

those transformations in eqns [8], we get

2N  M

8

log s



! 

 2ð2N  MÞ

½10

The first equation generates the -function of N = 2

super-QCD with M hypermultiplets given by

ðg

Þ¼

2N  M

16

½11

while the second one reproduces the chiral U(1)

anomaly (Klebanov et al. 2002, Bertolini et al. 2002a).

In particular, if we choose  = 2=(2(2N  M)),

then 

is shifted by a factor 2. But since 

periodic of 2, this means that the subgroup Z

2(2NM)

not anomalous in perfect agreement with the gauge

theory results.

Gauge Theories from Strings 465