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

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Theorem 2 Let  be a Banach space and A be a

compact set such that d

(A) < N for some N 2 N.

Then, for ‘‘almost all’’ (2N þ 1)-dimensional planes

Lin, the corresponding projector 

:  ! L

restricted to the set A is a Ho¨ lder continuous

homeomorphism.

Thus, if the finite fractal dimensionality of the

attractor is established, then, fixing a hyperplane L

satisfying the assumptions of the Mane´ theorem

and projecting the attractor A and the DS S(t)

restricted to A onto this hyperplane (



A:=

and

S(t):=

 S(t)  

1

), we obtain, indeed, a

reduced DS (

S(t),



A) which is defined on a finite-

dimensional set



AL  R

2Nþ1

. Moreover, this DS

will be Ho¨ lder continuous with respect to the initial

data.

Estimates on the Fractal Dimension

Obviously, good estimates on the dimension of the

attractors in terms of the physical parameters are

crucial for the finite-dimensional reduction

described above, and (consequently) there exists a

highly developed machinery for obtaining such

estimates. The best-known upper estimates are

usually obtained by the so-called volume contraction

method, whi ch is based on the study of the evolution

of infinitesimal k-dimensional volumes in the neigh-

borhood of the attractor (and, if the DS considered

contracts the k-dimensional volumes, then the

fractal dimension of the attractor is less than k).

Lower bounds on the dimension are usually based

on the observation that the global attract or always

contains the unstable manifolds of the (hyperbolic)

equilibria. Thus, the instability index of a properly

constructed equ ilibrium gives a lower bound on the

dimension of the attractor.

In the following, several estimates for the classes

of equations given in the introduction are formu-

lated, beginning with the most-studied case of the

reaction–diffusion system [8]. For this system, sharp

upper and lower bounds are known, namely

volðÞd

ðAÞ  C

volðÞ½13

where the constants C

and C

depend on a and f

(and, possibly, on the shape of ), but are indepen-

dent of its size. The same types of estimates also hold

for the hyperbolic equation [7]. Concerning the

Navier–Stokes system [6] in general two-dimensional

domains , the asymptotics of the fractal dimension

as  ! 0 is not known. The best-known upper bound

has the form d

(A)  C

2

and was obtained by

Foias–Temam by using the so-called Lieb–Thirring

inequalities. Nevertheless, for periodic boundary

conditions, Constantin–Foias–Temam and Liu

obtained upper and lower bounds of the same order

(up to a logarithmic correction):



4=3

 d

ðAÞ  C



4=3

ð1 þ lnð

1

ÞÞ

1=3

½14

Global Lyapunov Functions and the Structure

of Global Attractors

Although the global attractor has usually a very

complicated geometric structure, there exists one

exceptional class of DS for which the global attractor

has a relatively simple structure which is completely

understood, namely the DS having a global Lyapunov

function. We recall that a continuous function

L:  ! R is a global Lypanov funct ion if

1. L is nonincreasing along the trajectories, that is,

L(S(t)u

) L(u

), for all t  0;

2. L is strictly decreasing along all nonequilibrium

solutions, that is, L(S(t)u

) = L(u

) for some t > 0

and u

implies that u

is an equilibrium of S(t).

For instance, in the scalar case N = 1, the

reaction–diffusion equations [8] posses s the global

Lyapunov function L(u

):=



[ajr

(x)j

þ F(u

(x))]dx, where F(v):=

f (u)du. Indeed, multiply-

ing eqn [8] by @

u and integrating over , we have

Lðuðt ÞÞ ¼ 2k@

uðtÞk

ðÞ

 0 ½15

Analogously, in the scalar case N = 1, multiplying

the hyperbolic equation [7] by @

u(t) and integrating

over , we obtain the standard global Lyapunov

function for this equation.

It is well known that, if a DS posseses a global

Lyapunov function, then, at least under the generic

assumption that the set R of equilibria is finite, every

trajectory u(t) stabilizes to one of these equilibria as

t !þ1. Moreover, every complete bounded trajec-

tory u(t), t 2 R, belonging to the attractor is a

heteroclinic orbit joining two equilibria. Thus, the

global attractor A can be described as follows:

A¼

[

ðu

Þ½16

where M

) is the so-called unstable set of the

equilibrium u

(which is generated by all heteroclinic

orbits of the DS which start from the given equilibrium

2A). It is also known that, if the equilibrium u

hyperbolic (generic assumption), then the set M

)

is a -dimensional submanifold of ,where is the

instability index of u

. Thus, under the generic

hyperbolicity assumption on the equilibria, the

106 Dissipative Dynamical Systems of Infinite Dimension

attractor A of a DS having a global Lyapunov function

is a finite union of smooth finite-dimensional sub-

manifolds of the phase space . These attractors are

called regular (following Babin–Vishik).

It is also worth emphasizing that, in contrast to

general global attractors, regular attractors are

robust under perturbations. Moreover, in some

cases, it is also possible to verify the so-called

transversality conditions (for the intersection of

stable and unstable manifolds of the equilibria)

and, thus, verify that the DS considered is a

Morse–Smale system. In particular, this means that

the dynamics restricted to the regular attractor A is

also preserved (up to homeomorphisms) under

perturbations.

A disadvantage of the approach of using a regular

attractor is the fact that, except for scalar parabolic

equations in one space dimension, it is usually

extremely difficult to verify the ‘‘generic’’ hyperbo-

licity and transversality assumptions for concrete

values of the physical parameters and the associated

hyperbolicity constants, as a rule, cannot be

expressed in terms of these parameters.

Inertial Manifolds

It should be noted that the scheme for the finite-

dimensional reduction described above has essential

drawbacks. Indeed, the reduced system (

S(t),



A)is

only Ho¨ lder continuous and, consequently, cannot

be realized as a DS generated by a system of ODEs

(and reasonable conditions on the attractor A which

guarantee the Lipschitz continuity of the Mane´

projections are not known). On the other hand, the

complicated geometric structure of the attractor

A (or



A) makes the use of this finite-dimensional

reduction in computations hazardous (in fact, only

the heuristic information on the number of

unknowns which are necessary to capture all the

dynamical effects in approximations can be

extracted).

In order to overcome these problems, the concept

of an inertial manifold (which allows one to embed

the global attractor into a smooth manifold) has

been suggested by Foias–Sell–Temam. To be more

precise, a Lipschitz finite-dimensional manifold M  

is an inertial manifold for the DS (S(t), )if

1. M is semiinva riant, that is, S(t)M  M, for all

t  0;

2. M satisfies the following asymptotic completeness

property: for every u

2 , there exists v

2 M

such that

kSðtÞ u

 SðtÞv



 Qðku



Þe

t

½17

where the positive constant  and the monotonic

function Q are independent of u

We can see that an inertial manifold, if it

exists, confirms in a perfect way the heuristic

conjecture on the finite dimensionality formulated

in the introduction. Indeed, the dynamics of S(t)

restricted to an inertial manifold can be, obviously,

described by a system of ODEs (which is called the

inertial form of the initial PDE). On the other hand,

the asymptotic completeness gives (in a very strong

form) the equivalence of the initial DS (S(t), ) with

its inertial form (S(t), M). Moreover, in turbulence,

the existence of an inertial manifold would yield an

exact interaction law between the small and large

structures of the flow.

Unfortunately, all the known constructions of

inertial manifolds are based on a very restrictive

condition, the so-called spectral gap condition,

which requires arbitrarily large gaps in the spectrum

of the linearization of the initial PDE and which can

usually be verified only in one space dimension. So,

the existence of an inertial manifold is still an

open problem for many important equations of

mathematical physics (including in particular the

two-dimensional Navier–Stokes equations; some

nonexistence results have also been proven by

Mallet–Paret).

Exponential Attractors

We first recall that Definition 1 of a global

attractor only guarantees that the images S(t)B of

all the bounded subsets converge to the attractor,

without saying anything on the rate of convergence

(in contrast to inertial manifolds, for which this

rate of convergence can be controlled). Further-

more, as elementary examples show, this conver-

gence can be arbitrarily slow, so that, until now,

we have no effective way for estim ating this rate of

convergence in terms of the phy sical parameters of

the system (an exception is given by the regular

attractors described earlier for which the rate of

convergence can be estimated in terms of the

hyperbolicity constants of the equilibria. However,

even in this situation, it is usually very difficult to

estimate these constants for concrete equations).

Furthermore, there exist many physically relevant

systems (e.g., the so-called slightly dissipative

gradient systems) which have trivial global attrac-

tors, but very rich and physically relevant transient

dynamics which are automatically forgotten under

the global-attractor approach. Another important

problem is the robustness of the global attractor

under perturbations. In fact, global attractors are

usually only upper semicontinuous under

Dissipative Dynamical Systems of Infinite Dimension 107

perturbations (which means that they cannot

explode) and the lower semicontinuity (which

means that they cannot also implode) is much

more delicate to prove and requires some hyperbo-

licity assumptions (which are usually impossible to

verify for concrete equations).

In order to overcome these difficulties, Eden–

Foias–Nicolaenko–Temam have introduced an inter-

mediate object (between inertial manifolds and

global attractors), namely an exponential attractor

(also called an inertial set).

Definition 3 A compact set M is an exponen-

tial attractor for the DS (S(t), )if

(i) M has finite fractal dimension: d

(M) < 1;

(ii) M is semi-invariant: S(t)MM, for all t  0;

(iii) M attracts exponentially the images of all the

bounded sets B  :

dist

ðSðtÞB; MÞ  QðkBk



Þe

t

½18

where the positive constant  and the monotonic

function Q are independent of B.

Thus, on the one hand, an exponential attractor

remains finite dimensional (like the global attractor)

and, on the other hand, estimate [18] allows one to

control the rate of attraction (like an inertial

manifold). We note, however, that the relaxation

of strict invariance to semi-invariance allows this

object to be nonunique. So, we have here the

problem of the ‘‘best choice’’ of the exponential

attractor. We also mention that an exponential

attractor, if it exists, always contains the global

attractor.

Although the initial construction of ex ponential

attractors is ba sed on the so-called squeezing

property (and requires Zorn’s lemma), we formulate

below a simpler construction, due to Efendiev–

Miranville–Zelik, which is similar to the method

proposed by Ladyzhenskaya to verify the finite

dimensionality of global attractors. This is done for

discrete times and for a DS generated by iterations

of some map S :  ! , since the passage from

discrete to continuous times usually arises no

difficulty (without loss of generality, the reader

may think that S = S(1) and (S(t), ) is one of the DS

mentioned in the introduction).

Theorem 3 Let the phase space 

be a closed

bounded subset of some Banach space H and let H

be another Banach space compactly embedded into

H. Assume also that the map S : 

! 

satisfies

the following ‘‘smoothing’’ property:

kSu

 Su

 Kku

 u

; u

2 

½19

for some constant K independent of u

. Then, the DS

(S, 

) possesses an exponential attractor.

In applications, 

is usually a bounded absorb-

ing/attracting set whose existence is guaranteed by

the dissipative estimate [10], H := L

() and

:= H

(). Furthermore, estimate [19] simply

follows from the classical parabolic smoothing

property, but now applied to the equ ation of

variations (as in [11], hyperbolic equations require

a slightly more complicated analogue of [19]). These

simple arguments show that exponential attractors

are as general as global attractors and, to the

best of our knowledge, exponential attractors exist

indeed for all the equations of mathematical physics

for which the finite dimensionality of the global

attractor can be established. Moreover, since A

M, this scheme can also be used to prove the finite

dimensionality of global attractors.

It is finally worth emphasizing that the control on

the rate of convergence provided by [18] makes

exponential attractors much more robust than global

attractors. In particular, they are upper and lower

semicontinuous under perturbations (of course, up to

the ‘‘best choice,’’ since they are not unique), as

shown by Efendiev–Miranville–Zelik.

Essentially Infinite-Dimensional

Dynamical Systems – The Case of

Unbounded Domains

As already mentioned in the introduction, the theory

of dissipative DS in unbounded domains is develop-

ing only now and the results given here are not as

complete as for bounded domains. Nevertheless, we

indicate below several of the most interesting (from

our point of view) results concerning the general

description of the dynamics genera ted by such

problems by considering a system of reaction–

diffusion equations [8] in R

with phase space

=L

) as a model example (although all the

results formulated below are general and depend

weakly on the choice of equation).

Generalization of the Global Attractor and

Kolmogorov’s e-Entropy

We first note that Definition 1 of a global attractor

is too strong for equations in unbounded domains.

Indeed, as seen earlier, the compactness of the

attractor is usually based on the compactness of

the embedding H

()  L

(), which does not hold

in unbounde d domains. Furthermore, an attractor,

in the sense of Definition 1, does not ex ist for most

of the interesting examples of eqns [8] in R

108 Dissipative Dynamical Systems of Infinite Dimension

It is natural to use instead the concept of the

so-called locally compact global attractor which is

well adapted to unbounded domains. This attrac-

tor A is only bounded in the phase space

=L

), but its restrictions Aj



to all bounded

domains  are compact in L

(). Moreover, the

attraction property should also be understood in

the sense of a local topology in L

). It is

known that this generalized global attractor A

exists indeed for problem [8] in R

(of course,

under some ‘‘natural’’ assumptions on the non-

linearity f and the diffusion matrix a). As for

bounded domains, its existence is based on the

dissipative e stimate [10], the smoothing property

[11], and the compactness of the embedding

loc

)  L

loc

) (we need to use the local

topology only to have this compactness).

The next natural question that arises here is how

to control the ‘‘size’’ of the attractor A if its fractal

dimension is infinite (which is usually the case in

unbounded domains). One of the most natural

ways to handle this problem (which was first

suggested by Chepyzhov–Vishik in the different

context of uniform attractors associated with

nonautonomous equations in bounded domains

and appears as extremely fruitful for the theory of

dissipative PDE in unbounded domains) is to study

the asymptotics of Kolmogorov’s "-entropy of the

attractor. Actually, since the attractor A is compact

only in a local topology, it is natural to study the

entropy of its restrictions, say, to balls B

of R

radius R cente red at x

with respect to the three

parameters R, x

, and ". A more or less complete

answer to this question is given by the following

estimate:

ðAj

ÞCðR þ log

1="Þ

log

1=" ½20

where the constant C is independent of "  1, R,

and x

. Moreover, it can be shown that this estimate

is sharp for all R and " under the very weak

additional assumption that eqn [8] possesses at least

one exponentially unstable spatially homogeneous

equilibrium.

Thus, formula [20] (whose proof is also based on

a smoothing property for the equation of variations)

can be interpreted as a natural generalization of the

heuristic principle of finite dimensionality of global

attractors to unbounded domains. It is also worth

recalling that the entropy of the embedding of a ball

of the space C

) into C(B

) has the

asymptotic H

(B)  C

(1=")

n=k

, which is essen tially

worse than [20]. So, [20] is not based on the

smoothness of the attractor A and, therefore,

reflects deeper properties of the equation.

Spatial Dynamics and Spatial Chaos

The next main difference with bounded domains is

the existence of unbounded spatial directions which

can generate the so-called spatial chaos (in addition

to the ‘‘usual’’ temporal chaos arising under the

evolution). In order to describe this phenomenon, it

is natural to consider the group {T

, h 2 R

}of

spatial translations acting on the attractor A:

ðT

ÞðxÞ :¼ u

ðx þ hÞ; T

: A!A ½21

as a DS (with multidimensi onal ‘‘times’’ if n > 1)

acting on the phase space A and to study its

dynamical properties.

In particular, it is worth noting that the lower

bounds on the "-entro py that one can derive imply

that the topological entropy of this spatial DS is

infinite and, consequently, the classical symbolic

dynamics with a finite number of symbols is not

adequate to clarify the nature of chaos in [21].

In order to overcome this difficulty, it was suggested

by Zelik to use Bernoulli shifts with an infinite

number of symbols, belonging to the whole interval

! 2 [0, 1]. To be more precise, let us consider the

Cartesian product M

:= [0, 1]

endowed with the

Tikhonov topology. Then, this set can be interpreted

as the space of all the functions v : Z

! [0, 1],

endowed with the standard local topology. We define

aDS{T

, l 2 Z

}onM

ðT

vÞðmÞ :¼ vðm þ lÞ; v 2 M

; l; m 2 Z

½22

Based on this model, the following description of

spatial chaos was obtained.

Theorem 4 Let eqn [8] in =R

possess at least

one exponentially unstable spatially homogeneous

equilibrium. Then, there exist >0 and a home-

omorphic embedding  : M

!Asuch that

l

 ðvÞ¼ T

ðvÞ; 8l 2 Z

; v 2 M

½23

Thus, the spatial dynamics, restricted to the set

(M

), is conjugated to the symbolic dynamics on

. Moreover, there exists a dynamical invariant

(the so-called mean toplogical dimension) which is

always finite for the spatial DS [22] and strictly

positive for the Bernoulli scheme M

. So, the

embedding [23] clarifies, indeed, the nature of

chaos arising in the spatial DS [21].

Spatio-Temporal Chaos

To conclude, we briefly discuss an extension of

Theorem 4, which takes into account the temporal

modes and, thus, gives a description of the spatio-

temporal chaos. In order to do so, we first note

Dissipative Dynamical Systems of Infinite Dimension 109

that the spatial DS [21] commutes obviously

with the temporal evolution operators S(t) and,

consequently, an extended (n þ 1)-parametri c semi-

group { S (t, h), (t, h) 2 R

 R

} acts on the attractor:

Sðt; hÞ :¼ SðtÞT

; S ðt; hÞ : A!A

t 2 R

; h 2 R

½24

Then, this semigroup (interpreted as a DS with

multidimensional times) is responsible for all the

spatio-temporal dynamical phenomena in the initial

PDE [8] and, consequently, the question of finding

adequate dynamical characteristics is of a great

interest. Moreover, it is also natural to consider the

subsemigroups S

(t, h)associatedwiththek-dimen-

sional planes V

of the spacetime R

 R

, k < n þ 1.

Although finding an adequate description of the

dynamics of [24] seems to be an extremely difficult

task, some particular results in this direction have

already been obtained. Thus, it has been proved by

Zelik that the semigroup [24 ] has finite topological

entropy and the entropy of its subsemigroups

(t, h) is usually infinite if k < n þ 1. Moreover

(adding a natural transport term of the form

(L, r

)u to eqn [8]), it was proved that the analog

of Theorem 4 holds for the subsemigroups S

(t, h)

associated with the n-dimensional hyperplanes V

the spacet ime. Thus, the infinite-dimensional Ber-

noulli shifts introduced in the previous subsection

can be used to describe the temporal evolution in

unbounded domains as well.

In particular, as a consequence of this embed ding,

the topological entropy of the initial purely temporal

evolution semigroup S(t) is also infinite, which

indicates that (even without considering the spatial

directions) we have indeed here essential new levels

of dynamical complexity which are not observed in

the classical DS theory of ODEs.

See also: Dynamical Systems in Mathematical Physics:

An Illustration from Water Waves; Ergodic Theory;

Evolution Equations: Linear and Nonlinear; Fractal

Dimensions in Dynamics; Inviscid flows; Lyapunov

Exponents and Strange Attractors.

Further Reading

Babin AV and Vishik MI (1992) Attractors of Evolution

Equations. Amsterdam: North-Holland.

Chepyzhov VV and Vishik MI (2002) Attractors for Equations of

Mathematical Physics. American Mathematical Society Collo-

quium Publications, vol. 49. Providence, RI: American

Mathematical Society.

Faddeev LD and Takhtajan LA (1987) Hamiltonian Methods in

the Theory of Solitons. Springer Series in Soviet Mathematics.

Berlin: Springer.

Hale JK (1988) Asymptotic Behavior of Dissipative Systems.

Mathematical Surveys and Monographs, vol. 25. Providence,

RI: American Mathematical Society.

Katok A and Hasselblatt B (1995) Introduction to the Modern

Theory of Dynamical Systems. Encyclopedia of Mathematics

and its Applications, vol. 54. Cambridge: Cambridge Uni-

versity Press.

Ladyzhenskaya OA (1991) Attractors for Semigroups and Evolu-

tion Equations. Cambridge: Cambridge University Press.

Temam R (1997) Infinite-Dimensional Dynamical Systems in

Mechanics and Physics. Applied Mathematical Sciences, 2nd

edn., vol. 68. pp. New York: Springer.

Zelik S (2004) Multiparametrical semigroups and attractors of

reaction–diffusion equations in R

. Proceedings of the

Moscow Mathematical Society 65: 69–130.

Donaldson Invariants see Gauge Theoretic Invariants of 4-Manifolds

Donaldson–Witten Theory

M Marin˜o, CERN, Geneva, Switzerland

Introduction

Since they were introduced by

Witten in 1988,

topological quantum field theories (TQFTs) have

had a tremendous impact in mathematical physics

(see

Birmingham et al. (1991) and

Cordes et al. for a

review). These quantum field theories are

constructed in such a way that the correlation

functions of certain operators provide topological

invariants of the spacetime manifold where the

theory is defined. This means that one can use the

methods and insights of quantum field theory in

order to obtain information about topological

invariants of low-dimensional manifolds.

Historically, the first TQFT was Donaldson–Witten

theory, also called topological Yang–Mills theory.

This theory was constructed by Witten (1998) starting

from N = 2 super Yang–Mills by a procedure called

110 Donaldson–Witten Theory

‘‘topological twisting.’’ The resulting model is topolo-

gical and the famous Donaldson invariants of

4-manifolds are then recovered as certain correlation

functions in the topological theory. The analysis of

Witten (1998) did not indicate any new method to

compute the invariants, but in 1994 the progress in

understanding the nonperturbative dynamics of N = 2

theories (

Seiberg and Witten 1994 a, b) led to an

alternative way of computing correlation functions in

Donaldson–Witten theory. As Witten (1994) showed,

Donaldson–Witten theory can be r educed to

another, simpler topological theory consisting of

a twisted abelian gauge theory coupled to spinor

fields. This theory leads to a different set of

4-manifold invariants, the so-called ‘‘Seiberg–

Witten invariants,’’ and Donaldson invariants can

be expressed in terms of these invariants through

Witten’s ‘‘magic formula.’’ The connection

between Seiberg–Witten and Donaldson invar-

iants was streamlined and extended by Moore

andWittenbyusingtheso-calledu-plane integral

(

Moore and Witten 1998). This has led to a rather

complete understanding of Donaldson–Witten the-

ory from a physical point of view.

In this article we provide a brief review of

Donaldson–Witten theory. First, we describe the

construction of the model, from both a mathematical

and a physical point of view, and state the main

results for the Donaldson–Witten generating func-

tional. In the next section, we present the basic results

of the u-plane integral of Moore and Witten and

sketch how it can be used to solve Donaldson–Witten

theory. In the final section, we mention some

generalizations of the basic framework. For a

complete exposition of Donaldson–Witten theory,

the reader is referred to the book by

Labastida

and Marin˜ o (2005). A short review of the u-plane

integral can be found in

Marin˜ o and Moore (1998a).

Donaldson–Witten Theory: Basic

Construction and Results

Donaldson–Witten Theory According to Donaldson

Donaldson theory as formulated in

Donaldson (1990),

Donaldson and Kronheimer (1990),and

Friedman and

Morgan (1991) starts with a principal G = SO(3)

bundle V ! X over a compact, oriented, Riemannian

4-manifold X, with fixed instanton number k

and Stiefel–Whitney class w

(V) (SO(3) bundles on a

4-manifold are classified up to isomorphism by these

topological data). The moduli space of anti-self-dual

(ASD) connections is then defined as

ASD

¼fA : F

ðAÞ¼0g=G½1

where F

(A) is the self-dual part of the curvature,

and G is the group of gauge transformations. To

construct the Donaldson polynomials, one considers

the universal bundle

P ¼ðV A



Þ=ðG  GÞ½2

where A



is the space of irreducible G-connections

on V. This is a G-bundle over B



 X, where



= A



=G is the space of irreducible connections

modulo gauge transformations, and as such has a

Pontrjagin class

ðPÞ2H



ðB



ÞH



ðXÞ½3

One can then obtain differential forms on B



taking the slant product of p

(P) with homology

classes in X. In this way we obtain the Donald son

map:

 : H

ðXÞ!H

4i

ðB



Þ½4

After restriction to M

ASD

, we obtain the following

differential forms on the moduli space of ASD

connections:

x 2 H

ðXÞ!OðxÞ2H

ðM

ASD

S 2 H

ðXÞ!I

ðSÞ2H

ðM

ASD

½5

If the manifold X has b

(X) 6¼ 0, there are also

cohomology classes associated to 1-cycles and

3-cycles, but we will not consider them here.

We can now formally define the Donaldson

invariants as follows. Consider the space

AðXÞ¼SymðH

ðXÞH

ðXÞÞ ½6

with a typical element written as x

‘

S

. The

Donaldson invariant corresponding to this element

of A(X) is the following intersection number:

ðVÞ;k

ðx

‘

S

ASD

‘

^ I

ðS

Þ^^I

ðS

Þ½7

where M

ASD

is the moduli space of ASD connec-

tions with second Stiefel–Whitney class w

(V) and

instanton number k. The integral in [7] will be

different from zero only if the degrees of the forms

add up to dim(M

ASD

It is very convenient to pack all Donaldson

invariants in a generating functional. Let

}

i = 1,...,b

be a basis of 2-cycles. We introduce the

formal sum

S ¼

i¼1

½8

Donaldson–Witten Theory 111

where v

are complex numbers. We then define the

Donaldson–Witten generating functional as

ðVÞ

ðp; v

Þ¼

k¼0

ðVÞ;k

ðe

pxþS

Þ½9

where on the right-hand side we are summing over

all instanton numbers, that is, we are summing over

all topological configurations of the SO(3) gauge

field with a fixed w

(V). This gives a formal power

series in p and v

For b

(X) > 1, the g enerating functional [9]

is a diffeomorphism invariant of X;therefore,it

is potentially a powerful tool in four-dimensional

topology. When b

(X) = 1, Donaldson invariants

are metric dependent. The metric dependence

can be described in more detail as follows. Define

the period point as the harmonic 2 -form

satisfying

! ¼ !; !

¼ 1 ½10

which depends on the conformal class of the metric.

As the conformal class of the metric varies, !

describes a curve in the cone

¼f! 2 H

ðX; R Þ : !

> 0g½11

Let  2 H

(X) satisfy

  w

ðVÞmod 2;

< 0; ð;!Þ¼0 ½12

Such an element  defines a ‘‘wall’’ in V



¼f! : ð; !Þ¼0g½13

The complements of these walls are called ‘‘cham-

bers,’’ and the cone V

is then divided in chambers

separated by walls. A class  satisfying [12] is the

first Chern class associated to a reducible solution of

the ASD equations, and it causes a singularity in

moduli space: the Donaldson invariants jump when

we pass throu gh such a wall. Therefore, when

(X) = 1, Donaldson invariants are metric inde-

pendent in each chamber. A basic problem in

Donaldson–Witten theory is to determine the jump

in the generating function as we cross a wall,



ðp; SÞZ





ðp; SÞ¼WC



ðp; SÞ½14

The jump term WC



(p, S, ) is usually called the

‘‘wall-crossing’’ term.

The basic goal of Donaldson theory is to study

the properties of the generating functional [9] and

to compute it for different 4-manifolds X.On

the mathematical side, many results have been

obtained on Z

, and some of them can be found

Donaldson and Kronheimer (1990),

Friedman

and Morgan (1991),

Stern (1998), and

Go¨ ttsche

(1996). On the other hand, Donaldson theory can be

formulated as a topological field theory, and many

of these results can be obtained by using quantum

field theory techniques. This will be our main focus

for the rest of the article.

Donaldson–Witten Theory According to Witten

Witten (1988) constructed a twisted version of

N = 2 super Yang–Mills theory which has a nilpo-

tent Becchi–Rouet–Stora–Tyutin (BRST) charge

(modulo gauge transformations)

Q¼



½15

where Q

_

are the supersymmetric (SUSY) charges.

Here

 is a chiral spinor index and A has its origin in

the SU(2) R-symmetry. The field content of the

theory is the standard twisted N = 2 vector multiplet:





;; D



;



_



; ;  ¼



½16

where (1=2)D







is a self-dual 2-form derived

from the auxiliary fields, etc. All fields are valued in

the adjoint representation of the gauge group. After

twisting, the theory is well defined on any Rieman-

nian 4-manifold, since the fields are naturally

interpreted as differential forms and the

Q charge

is a scalar (

Witten 1988).

The observables of the theory are

Q cohomology

classes of operators, and they can be constructed

from 0-form observables O

(0)

using the descent

procedure. This amounts to solving the equations

ðiÞ

¼fQ; O

ðiþ1Þ

g; i ¼ 0; ...; 3 ½17

The integration over i-cycles 

(i)

in X of the

operators O

(i)

is then an observable. These descent

equations have a canonical solution: the 1-form-

valued operator K

 _

= i



_

=4 verifies

d ¼f

Q; Kg½18

as a consequence of the supersymmetry algebra. The

operators O

(i)

= K

(0)

solve the descent equations

[17] and are canonical representatives. When the

gauge group is SU(2), the observables are obtained

by the descent procedure from the operator

O¼trð

Þ½19

The topological descendant O

(2)

is given by

ð2Þ

¼

ﬃﬃ

ðF





þD



Þ









^dx



½20

and the resulting observable is

ðSÞ¼

ð2Þ

½21

112 Donaldson–Witten Theory

O and I

(S) correspond to the cohomology classes in

[5]. One of the main results of

Witten (1988) is that

the semiclassical approximation in the twisted

N = 2 Yang–Mills theory is exact. The semiclassical

evaluation of correlation functions of the observa-

bles above leads directly to the definition of

Donaldson invariants, and the generating functional

[9] can be written as a correlation function of the

twisted theory. One then has

ðVÞ

ðp; SÞ¼

exp



pOþI

ðSÞ



½22

Results for the Donaldson–Witten

Generating Function

The basic results that have emerged from the

physical approach to Donaldson–Witten theory are

the following.

1. The Donaldson–Witten generating functional

is in general the sum of the two terms,

¼ Z

þ Z

½23

(We have omitted the Stiefel–Whitney class for

convenience.) The first term, Z

, is called the

‘‘u-plane integral.’’ It is given by a complicated

integral over C which can be written, in turn, as an

integral over a fundamental domain of the con-

gruence subgroup 

(4) of SL(2, Z). Z

depends

only on the cohomology ring of X, and therefore

does not contain any information beyond the one

provided by classical topology. Finally, Z

vanishes

if b

(X) > 1, and it is responsible for the wall-

crossing behavior of Z

when b

(X) = 1.

2. The second term of [23], Z

, is called the

Seiberg–Witten contribution. This contribution

involves the Seiberg–Witten invariants of X, which

are obtained by considering the moduli problem

defined by the Seiberg–Witten monopole equations

(

Witten 1994b):





þ 4i





Þ

¼ 0



¼ 0

½24

In these equations, M



is a section of the spinor

bundle S

 L

1=2

, L is the determinant line bundle

of a Spin

structure on X, F





=









is the

self-dual part of the curvature of a U(1) connection

on L,andD

is the Dirac operator for the bundle

 L

1=2

. The so lutions of these equations modulo

gauge equivalence form the moduli space M

and the Seiberg–Witten invariants are defined by

integrating suitable differential forms on this moduli

space. We will label Spin

structures by the class

 = c

1=2

) 2 H

(X, Z) þ w

(X)=2. We say that  is

a Seiberg–Witten basic class if the corresponding

Seiberg–Witten invariants are not all zero. If M

is zero dimensional, the Seiberg–Witten invariant

depends only on the Spin

structure associated to

 = c

1=2

), and is denoted by SW().

3. A manifold X is said to be of Seiberg–

Witten simple type if all the Seiberg–Witten basic

classes have a zero-dimensional moduli space. For

simply connected 4-manifolds of Seiberg–Witten

simple type and with b

(X) > 1, Witten determined

the Seiberg–Witten contribution and proposed the

following ‘ ‘magic formula’ ’ for Z

(

Witten 1994b):

¼2

1þ7=4þ11=4



2ið

þ

2pþS

2ðS;Þ

þi



w

ðVÞ

2pS

2iðS;Þ

SWðÞ½25

In this equation, ,  are the Euler characteristic and

signature of X, respectively, 

= ( þ ) =4 is the

holomorphic Euler characteristic of X, and 

is an integer lifting of w

(V). This formula gen-

eralizes previous results by

Witten (1994a) for

Ka¨ hler manifolds. It also follows from this formula

that the Donaldson–Witten generating function of

simply connected 4-manifolds of Seiberg–Witten

simple type and with b

(X) > 1 satisfies

 4



¼ 0

which is the Donaldson simple type condition

introduced by

Kronheimer and Mrowka (1994).

4. Using the u-plane integral, one can find explicit

expressions for Z

in more general situations (like

non-simply-connected manifolds or manifolds which

are not of Seiberg–Witten simple type).

In the next section we explain the formalism of

the u-plane integral introduced by

Moore and

Witten (1998), which makes possible a detailed

derivation of the above results.

The u -Plane Integral

Definition of the u -Plane Integral

The evaluation of the Donaldson–Witten generating

function can be made by using the results of

Seiberg

and Witten (1994 a, b) on the low-energy dynamics

of SU(2), N = 2 Yang–Mills theory. In their work,

Seiberg and Witten determined the exact low-energy

effective action of the model up to two derivatives.

Donaldson–Witten Theory 113

From a physical point of view, there are cert ainly

corrections to this effective action which are difficult

to evaluate. Fortunately, the computation in the

twisted version of the theory can be done by just

considering the Seiberg–Witten effective action. This

is because the correlation functions in the twisted

theory are invariant under rescalings of the metric,

so we can evaluate them in the limit of large

distances or equivalently of very low energies. The

effective action up to two derivatives is sufficient for

that purpose.

One way of describing the main result of the work

of Seiberg and Witten is that the moduli space of



Q-fixed points of the twisted SO(3) N = 2 theory on

a compact 4-manifold has two branches, which we

refer to as the Coulomb and Seiberg–Witten

branches. On the Coulomb branch the expectation

value

u ¼

htr 

16

breaks SO(3) ! U(1) via the standard Higgs

mechanism. The Coulomb branch is simply a copy

of the complex u-plane. The low-energy effective

theory on this branch is simply the abelian N = 2

gauge theory. However, at two points, u = 1, there

is a singularity where the moduli space meets a

second branch, the Seiberg–Witten branch. At these

points, the effective action is given by the magnetic

dual of the U(1), N = 2 gauge theory coupled to a

monopole matter hypermultiplet. Therefore, this

branch consists of solutions to the Seiberg–Witten

equations [24].

Since the manifold X is compact, the partition

function of the twisted theory is a sum over ‘‘all’’

vacuum states. Equation [23] then follow s. In this

equation, Z

comes from ‘‘integrating over the

u-plane,’’ while Z

corresponds to the points

u = 1. As we stated before, Z

vanishes for

manifolds of b

(X) > 1, but once this piece has

been determined an argument originally presented

Moore and Witten (19 98) allows one to derive

the form of Z

as well for arbitrary b

(X)  1.

The computation of Z

ispresentedindetailin

Moore and Witten (1998). The starting point of

the computation is the untwisted low-energy

theory, which has been described i n detail in

Seiberg and Witten (1994 a, b) and

Witten

(1995).ItisanN = 2 theory characterized by a

prepotential F which depends on an N = 2 vector

multiplet. The effective gauge coupling is given by

(a) = F

(a), where a is the scalar component of

the vector multiplet. The Euclidean Lagrange

density for the u-plane theory can be obtained

simply by twisting the physical theory. It can be

written as

6

FðaÞþ

16

Q; F

ðD þ F



ﬃﬃﬃ

32

Q; F

d 



ﬃﬃﬃ

3  2





000















ﬃﬃﬃ

þAðuÞtrR ^ R þBðuÞtrR ^

R ½26

where A(u), B(u) describe the coupling to gravity,

and after integration of the corresponding differen-

tial forms we obtain terms proportional to the

signature  and Euler characteristic  of X. The

data of the low-energy effective action can be

encoded in an elliptic curve of the form

¼ x

 ux

x ½27

and  is the modulus of the curve. The monodromy

group of this curve is 

(4). All the quantities

involved in the action can be obtained by integrating

a certain meromorphic differential on the curve, and

they can be expressed in terms of modul ar forms.

As for the operators, we have u = O(P )by

definition. We may then obtain the 2-observables

from the descent procedure. The result is that I(S) !

I(S) =

u =

(du=da)(D

þ F



) þ. Here D

is the auxiliary field. Although one has I(S) !

I(S)in

going from the microscopic theory to the effective

theory, it does not necessarily follow that

I(S

)I(S

) !

I(S

)

I(S

) because there can be contact

terms. If S

and S

intersect, then in passing to the

low-energy theory we integrate out massive modes.

This can induce delta function corrections to the

operator product expansion modifying the mapping

to the low-energy theory as follows:

exp IðSÞðÞ!exp

IðSÞþS

TðuÞ



½28

where T(u) is the contact term. Such contact terms

were observed in

Witten (1994a) and studied in

detail in

Losev et al. (1998). It can be shown that

TðuÞ¼

ðÞ



u ½29

where E

() is Eisenstein’s series and da=du is one of

the periods of the elliptic curve [27].

The final result of Moore and Witten is the

following expression:

ðp; SÞ¼

du d



1=2

ðÞe

2puþS

TðuÞ

 ½30

114 Donaldson–Witten Theory

Here,

ðÞ¼

d





1ð1=2Þ



=8

TðuÞ¼TðuÞþ

ðdu=daÞ

8y

½31

where y = Im  and  is the discriminant of the

curve [27]. The quantity  is essentially a Narain–

Siegel theta function associated to the lattice

(X, Z). Notice that this lattice is Lorentzian and

has signature (1, (1)



(X)

) (since b

(X) = 1). The

self-dual projection of a 2-form  can be done with

the period point ! as 

= (, !)!. The lattice is

shifted by half the second Sti efel–Whitney class of

the bundle, w

(V), that is,

 ¼ H

ðX; Z Þþ

ðVÞ

and

¼exp 

8y





2i

2

ð1Þ

ð

Þw

ðXÞ

ð; !Þþ

4y

ðS;!Þ



exp ið

 ið



 i

ðS;





½32

Here, w

(X) is the second Stiefel–Whitney class

of X,and

is a choice of lifting of w

(V)to

(X, Z). This expression can be extended to the

non-simply-connected case (see

Marin˜o and

Moore (1999) and

Moore an d Witten (1998)).

The study of the u-plane integral leads to a

systematic derivation of many important results

in Donaldson–Witten theory. We will discuss in

detail two such applications, Go¨ ttsche’s wall-

crossing formula and Witten’s ‘‘magic formula.’’

Wall-Crossing Formula

As shown by Moore and Witten, the u-plane integral

is well defined and does not depend on the period

point (hence on the metric on X) except for discontin-

uous behavior at walls. There are two kinds of walls,

associated, respectively, to the singularities at u = 1

(the semiclassical region of the underlying Yang–Mills

theory) and at u = 1, given by

u ¼1: 

¼ 0; 2 H

ðX; ZÞþ

ðVÞ

u ¼1: 

¼ 0; 2 H

ðX; ZÞþ

ðXÞ

½33

The first type of walls is precisely the one that

appears in Donaldson theory on manifolds of

(X) = 1. The discontinuity of the u-plane inte-

gral at these walls can be easily computed from

eqn [33]:

¼2

ðp; SÞ

¼

ð1Þ

ð

ðXÞÞ

2i



ðÞ

2



1

 exp 2pu

þ S

 ið; SÞ=h



½34

This expression involves the modular forms

, f

, u

, and T

(the subscript 1 refers to the

fact that they are computed at the ‘‘electric’’ frame

which is appropriate for the Seiberg–Witten curve at

u !1). They can be written in terms of Jacobi

theta functions #

(q), with q = e

2i

, and their

explicit expression is

ðqÞ¼

ðqÞ#

ðqÞ

ðqÞ¼

ðqÞ#

ðqÞ

ðqÞ¼

ðqÞþ#

ðqÞ

2ð#

ðqÞ#

ðqÞÞ

ðqÞ¼

ðqÞ

½35

The subinde x q

means that in the expansion in q of

the modular forms, we pick the constant term. The

formula [34] agrees with the formula of

Go¨ ttsche

(1996) for the wall crossing of the Donaldson–

Witten generating functional.

The Seiberg–Witten Contribution and Witten’s

Magic Formula

At u = 1, Z

jumps at the second type of

walls [33], which are called Seiberg–Witten (SW)

walls. In fact, these walls are labeled by classes

 2 H

(X; Z) þ (1 =2)w

(X), which correspond to

Spin

structures on X. At these walls, the Seiberg–

Witten invariants have wall-crossing behavior. S ince

the Donaldson polynomials do not jump at SW

walls, it must happen that the change of Z

at u = 1

is canceled by the change of Z

. As shown by

Moore and Witten, this actually allows one

to obtain a precise expression for Z

for general

4-manifolds of b

(X)  1.

On general grounds, Z

is given by the sum of

the generating functionals at u = 1. These involve

a magnetic U(1), N = 2 vector multiplet coupled to a

hypermultiplet (the monopole field). The twisted

Donaldson–Witten Theory 115