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a meaning to its unknown quantities in terms of the

new Seiberg–Witten invariants. Equation [50] is a

rather remarkable prediction that has been tested in

many cases, and for which a general pro of has been

recently proposed. For a review of the subject, see

Labastida and Lozano (1998).

The situation for manifolds with b

= 1 involves a

metric dependence and has been worked out in

detail (Moore and Witten 1998). The formulation of

Donaldson invariants in field-theoretical terms has

also provided a generalization of these invariants.

This generalization has been carried out in several

directions: (1) the con sideration of higher-rank

groups, (2) the coupling to matter fields after

twisting N = 2 hypermultiplets, and (3) the twist of

theories involving N = 4 supersymm etry.

We will now look at Chern–Simons gauge theory

from the perspective that emerges from its treatment

in the context of the large-N expansion. We will

restrict the discussion to the case of knots on S

with

gauge group SU(N). Gauge theories with gauge group

SU(N) admit, besides the perturbative expansion, a

large-N expansion. In this expansion correlators are

expanded in powers of 1/N while keeping the

’t Hooft coupling t = Nx fixed, x being the coupling

constant of the gauge theory. For example, for the

free energy of the theory, one has the general form

F ¼

g0

h1

g;h

22g

2g2þh

½51

In the case of Chern–Simons gauge theory, the coupling

constant is x = 2i=(k þ N) after taking into account

the shift in k.Thelarge-N expansion [51] resembles a

string-theory expansion and indeed the quantities C

g, h

can be identified with the partition function of a

topological open string with g handles and h bound-

aries, with N D-branes on S

in an ambient six-

dimensional target space T



. This was pointed out by

Witten in 1992. The result makes a connection between

a topological three-dimensional field theory and the

topological strings described in the previous section.

In 1998 an important breakthrough took place

which provided a new approach to compute quan-

tities such as [51]. Using arguments inspired by the

AdS/CFT correspondence, Gopakumar and Vafa

(1999) provided a closed-string-theory interpretation

of the partition function [51]. They conjectured that

the free energy F can be expressed as

F ¼

g0

22g

ðtÞ½52

where F

(t) corresponds to the partition function of a

topological closed-string theory on the noncompact

Calabi–Yau manifold X called the resolved conifold,

O(1) O(1) ! P

, t being the flux of the B-field

through P

. The quantities F

(t) have been computed

using both physical and mathematical arguments,

thus proving the conjecture.

Once a new picture for the partition function of

Chern–Simons gauge theory is available, one should

ask about the form that the expectation values of

Wilson loops could take in the new context. The

question was faced by Ooguri and Vafa and they

provided the answer, later refined by Labastida,

Marin˜ o, and Vafa. The outcome is an entirely new

point of view in the theory of knot and link

invariants. The new picture provides a geometrical

interpretation of the integer coefficients of the

quantum group invariants, an issue that has been

investigated during many years. To present an

account of these developments, one needs to review

first some basic facts of large-N expansions.

To consider the presence of Wilson loops, it is

convenient to introduce a particular generating

functional. First, one performs a change of basis

from representations R to conjugacy classes C(k)of

the symmetric group, labeled by vectors

k = (k

, k

, ...) with k

 0, and jkj=

> 0.

The change of basis is W



(C(k))W

where 

are characters of the permutation group

‘

of ‘ =

elements ( ‘ is also the number of

boxes of the Young tableau associated to R).

Second, one introduces the gen erating functional:

FðVÞ¼log ZðVÞ¼

jCðkÞj

‘!

ðcÞ



ðVÞ½53

where

ZðVÞ¼

jCðkÞj

‘!



ðVÞ



ðVÞ¼

ðtr V

In these expressions jC(k)j denotes the number of

elements of the class C( k)inS

‘

. The reason behind

the introduction of this generating functional is that

the large-N structure of the connected Wilson loops,

(c)

, turns out to be very simple:

jCðkÞj

‘!

ðcÞ

g¼0

2g2þjkj

g; k

ðÞ½54

where  = e

and t = Nx is the ’t Hooft coupling.

Writing x = t=N, it corresponds to a power series

expansion in 1/N. As before, the expansion looks

like a perturbative series in string theory where g is

the genus and jkj is the number of holes. Ooguri and

Vafa conjectured in 1999 the appropriate string-

theory description of [54]. It corresponds to an open

topological string theory (notice that the ones

288 Topological Quantum Field Theory: Overview

described in the previous section were closed),

whose target space is the resolved conifold X. The

contribution from this theory will lead to open-

string analogs of Gromov–Witten invariants.

In order to describe in more detail the fact that one

is dealing with open strings, some new data need to

be introduced. Here is where the knot description

intrinsic to the Wilson loop enters. Given a knot K on

, let us associate to it a Lagrangian submanifold C

with b

= 1 in the resolved conifold X and consider a

topological open string on it. The contributions in

this open topological string are localized on holo-

morphic maps f : 

g, h

! X with h = jkj which satisfy



[

g, h

] = Q,andf



[C] = j[]fork

oriented circles

C. In these expressions  2 H

, Z), and Q2

(X, C

, Z), that is, the map is such that k

boundaries of 

g, h

wrap the knot j times, and 

g, h

itself gets mapped to a relative two-homology class

characterized by the Lagrangian submanifold C

The number of such maps (in the sense described in

the previous section) is the open-string analog of

Gromov–Witten invariants. They will be denoted by

g, k

. Comparing to the situation that led to [47] in

the closed-string case, one concludes that in this case

the quantities F

g, k

()in[54] must take the form

g; k

ðÞ¼

g; k

; t ¼

! ½55

where ! is the Ka

hler class of the Calabi–Yau

manifold X and  = e

. For any Q, one can always

write

! = Qt, where Q is in genera l a half-integer

number. Therefore, F

g, k

() is a polynomial in 

1=2

with rational coefficients.

The result [55] is very impressive but still does not

provide a representation where one can assign a

geometrical interpretation to the integer coefficients

of the quantum-group invariants. Notice that to

match a polynomial invariant to [55], after obtain-

ing its connected part, one must expand it in x after

setting q = e

keeping  fixed. One would like to

have a refined version of [55], in the spirit of what

was described in the previous section leading from

the Gromov–Witten invariants N



of [47] to the

new integer invariants n



of [48]. It turns out that,

indeed, F(V) can be expressed in terms of integer

invariants in complete analogy with the description

presented in the previous section for topological

strings. A good review on the subject can be found

in Marin˜ o (2005).

Concluding Remarks

In this overview we have introduced key features of

TQFTs and we have described some of the most

relevant results emerged from them. We have

described how the many faces of TQFT provide a

variety of important insights in a selected set of

problems in topology. Among these outstand the

reformulation of Donaldson theory and the discovery

of the Seiberg–Witten invariants, and the string-theory

description of the large-N expansion of Chern–Simons

gauge theory, which provides an entirely new point of

view in the study of knot and link invariants and points

to an underlying fascinating interplay between string

theory, knot theory, and enumerative geometry which

opens new fields of study.

In addition to their intrinsic mathematical inter-

est, TQFTs have been found relevant to important

questions in physics as well. This is so because, in a

sense, TQFTs are easier to solve than conventional

quantum field theories. For example, topological

sigma models are relevant to the computation of

certain couplings in string theory. Also, Witten-type

gauge TQFTs such as Donaldson–Wi tten theories

and its generalizations play a role in string theory as

effective world-volume theories of extended string

states (branes) wrapping curved spaces, and TQFTs

arising from N = 4 gauge theories in four dimen-

sions have shed light on field- (and string-) theory

dualities.

Most of these developments, and others that we

have not touched upon or only mentioned in passing

have their own entries in the encyclopedia, to which

we refer the interested reader for further details.

See also: Axiomatic Approach to Topological Quantum

Field Theory; BF Theories; Chern–Simons Models:

Rigorous Results; Donaldson–Witten Theory; Gauge

Theoretic Invariants of 4-Manifolds; Gauge Theory:

Mathematical Applications; Hamiltonian Fluid Dynamics;

The Jones Polynomial; Knot Theory and Physics;

Mathai–Quillen Formalism; Mathematical Knot Theory;

Schwarz-Type Topological Quantum Field Theory;

Seiberg–Witten Theory; Stationary Phase Approximation;

Topological Sigma Models.

Further Reading

Atiyah MF (1988) New invariants of three and four dimensional

manifolds. In: The Mathematical Heritage of Herman Weyl,Proc.

Symp. Pure Math., vol. 48. American Math. Soc. pp. 285–299.

Birmingham D, Blau M, Rakowski M, and Thompson G (1991)

Topological field theory. Physics Reports 209: 129–340.

Cordes S, Moore G, and Rangoolam S (1996) Lectures on 2D

Yang–Mills theory, equivariant cohomology and topological

field theories. In: David F, Ginsparg P, and Zinn-Justin J (eds.)

Fluctuating Geometries in Statistical Mechanics and Field

Theory, Les Houches Sesion LXII, p. 505 (hep-th/9411210).

Elsevier.

Donaldson SK (1990) Polynomial invariants for smooth four-

manifolds. Topology 29: 257–315.

Floer A (1987) Morse theory for fixed points of symplectic

diffeomorphisms. Bulletin of the American Mathematical

Society 16: 279.

Topological Quantum Field Theory: Overview 289

Freyd P, Yetter D, Hoste J, Lickorish WBR, Millet K, and

Ocneanu A (1985) A new polynomial invariant of knots and

links. Bulletin of the American Mathematical Society 12(2):

239–246.

Gopakumar R and Vafa C (1999) On the gauge theory/geometry

correspondence. Advances in Theoretical and Mathematical

Physics 3: 1415 (hep-th/9811131).

Jones VFR (1985) A polynomial invariant for knots via von

Neumann algebras. Bulletin of the American Mathematical

Society 12: 103–112.

Jones VFR (1987) Hecke algebra representations of braid groups

and link polynomials. Annals of Mathematics 126(2): 335–388.

Kauffman LH (1990) An invariant of regular isotopy. Transac-

tions of American Mathematical Society 318(2): 417–471.

Labastida JMF (1999) Chern-Simons gauge theory: ten years

after. In: Falomir H, Gamboa R, and Schaposnik F (eds.)

Trends in Theoretical Physics II, ch. 484, 1. New York: AIP

(hep-th/9905057).

Labastida JMF and Lozano C (1998) Lectures in topological

quantum field theory. In: Falomir H, Gamboa R, and

Schaposnik F (eds.) Trends in Theoretical Physics, ch. 419,

54. New York: AIP (hep-th/9709192).

Labastida JMF and Marin˜o M (2005) Topological Quantum Field

Theory and Four Manifolds. Dordrecht: Elsevier; Norwell, MA:

Springer.

Marin˜ o M (2005) Chern–Simons theory and topological strings.

Reviews of Modern Physics 77: 675–720.

Moore G and Witten E (1998) Integrating over the u-plane in

Donaldson theory. Advances in Theoretical and Mathematic

Physics 1: 298–387.

Vassiliev VA (1990) Cohomology of knot spaces. In: Theory of

Singularities and Its Applications, Advances in Soviet Mathe-

matics, vol. 1, pp. 23–69. American Mathematical Society.

Witten E (1982) Supersymmetry and Morse theory. Journal of

Differential Geometry 17: 661–692.

Witten E (1988a) Topological quantum field theory. Commu-

nications in Mathematical Physics 117: 353.

Witten E (1988b) Topological sigma models. Communications in

Mathematical Physics 118: 411.

Witten E (1989) Quantum field theory and the Jones polynomial.

Communications in Mathematical Physics 121: 351.

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

Research Letters 1: 769–796.

Topological Sigma Models

D Birmingham, University of the Pacific,

Stockton, CA, USA

Introduction

Topological sigma models govern the quantum

mechanics of maps from a Riemann surface  to a

target space M. In contrast to the standard super-

symmetric sigma model, the topological version has a

special local shift symmetry. This symmetry takes the

form u

= 

,where

is an arbitrary local function of

the coordinates on the base manifold . In essence, this

topological shift symmetry ensures that all local

degrees of freedom of the model can be gauged away.

As a result, the dynamics of such a model resides in a

finite number of global topological degrees of freedom.

This feature is generic to all topological field theories

of Witten type, also known as cohomological field

theories (see Topological Quantum Field Theory:

Overview). The topological shift symmetry is respon-

sible for the special topological nature of the model,

which is seen most readily by BRST quantizing the

local shift symmetry. This gives rise to a nilpotent

BRST operator Q. The properties of this BRST

operator are crucial for establishing the topological

nature of the model. The key point in the construction

of any cohomological field theory is the fact that the

full quantum action S

can be written as a BRST

commutator S

= {Q, V}, where V is a function of the

fields needed to define the path integral. In particular,

one can show that the partition function and all

correlation functions are independent of the metric on

both the base manifold  and the target space M.For

example, let us define the path integral by

Z ¼

d e

fQ;Vg

½1

where  denotes the full set of fields required at the

quantum level. In general, the function V depends

on geometric data of both  and M. Nevertheless,

one can easily establish that the partition function is

independent of this data by noting the following.

Variation of Z with respect to the metric of the

target space g (for example) gives



Z ¼

d e

fQ;Vg

fQ;

Vg½2

The right-hand side of this equation is nothing but

the vacuum expectation value of a BRST commu-

tator, and this vanishes by BRST invariance of the

vacuum. It is important to note here that the BRST

operator Q can be constructed to be independent of

g. Apart from the necessity of introducing the metric

tensor, these models also require additional geo-

metric data for their construction. The complex

structure of , and at least an almost-complex

structure on M, is required. By a similar argument,

one can show that the partition funct ion and

correlation functions are independent of this extra

290 Topological Sigma Models

geometric data. As mentioned above, these models

possess no local degrees of freedom. One can then

show that the path-integral expression for the

correlation funct ions can be localized to a finite-

dimensional moduli space of instanton configura-

tions which minimize the classical action.

We will first show how the full quantum action of

the theory can be obtained as a BRST quantization of a

classical action with a local gauge symmetry. How-

ever, we shall then highlight the fact that the gauge

algebra for this topological shift symmetry only closes

on-shell. In order to proceed with a BRST quantization

of the model, and obtain the complete quantum

action, one must take recourse to the Batalin–

Vilkovisky quantization scheme. This machinery is

ideally tailored for such a problem, with the end result

that quartic ghost terms are present in the action.

However, the presence of such terms does not affect

the arguments presented above, since the quantum is

still obtained as a BRST commutator. Following this,

we construct all observables of the theory and

demonstrate their connection to the de Rham coho-

mology of the target space. The special topological

properties of the observables are then discussed, and it

is shown how their computation is localized to the

moduli space M of holomorphic maps from  to M.

As a particular example, we show how the computa-

tion of a certain class of observables determines the

intersection numbers of the moduli space M.We

present a brief discussion of the connection between

topological sigma models with Calabi–Yau target

space M, and the mirror symmetry of M.

Construction of the Model

We begin with the following classical action:





ﬃﬃﬃ



i

j

½3

where

i

¼ G

i





þ 







½4

The fields G

i

and K

i

both sati sfy the self-duality

constraint

i

¼ P

i

j

i

¼ P

i

j

½5

where the self-dual and anti-self-dual projection

operators are defined as

i













 







½6

The above action describes a theory of maps u

()

from a Riemann surface  to an almost complex

manifold M. The coordinates on  are denoted by





( = 1, 2), while those on the target manifold M

are denoted by u

(i = 1, ..., dim M). The metric and

complex structure of  are denoted by h



and 





respectively; they obey the relations 











= 





and 



= h









. The metric tensor g

and almost-

complex structure J

of M obey analogous relations

to the above. In the general model, the target space

need only be an almost-complex manifold. This

requires the existence of a globally defined tensor

field J

such that J

= 

The action [3] is invariant under the topological

shift symmetry

u

¼ 

½7

where 

is an arbitrary local function of the

coordinates on the base manifold . Already, at

this level, we see the distinction with the standard

sigma model. The presence of this shift symmetry

means that all local degrees of freedom can be

gauged away, leaving only a finite number of global

topological degrees of freedom. It requires some

work to determine the corresponding transformati on

for G

i

, the key point being the preservation of the

self-duality constraint. We find

G

i

¼P

i

j



























j

 



l

½8

where the covariant derivative is defined by





= @





þ 



)

Having determined the classical symmetries of the

model, we can now proceed with the BRST quantized

form of the quantum action. As a topological field

theory of Witten type, one can show that the quantum

action can be written as a BRST commutator, that is,

= {Q, V}, where the gauge fermion V is defined by

V ¼



ﬃﬃﬃ



i







i



½9

where  is an arbitrary gauge-fixing parameter. The

BRST operator Q is nilpotent Q

= 0, off-shell. It is

defined by ={Q, }, and takes the form

u

¼C

C

¼0





i

¼ B

i











j

þ



k



B

i



þR

klrs





t











j







t

þ

k

½10

Topological Sigma Models 291

In the abo ve, the ghost field is denoted by C

, while

the anti-ghost field



i

and the multiplier field B

i

obey the self-dua lity constraint [5]. The key point to

note in the above transformations is the fact that the

ghost field C

is BRST invariant. Again, this is a

feature which is generic to all cohomological field

theories. The exis tence of such a field allows the

construction of an entire set of topological correla-

tion functions, as we shall see in the following

section.

While the gauge-fixing parameter  is arbitrary, a

conventional choice is to take  = 1, and then

integrate out the multiplier field B. This yields the

action in the form



ﬃﬃﬃ



















i

















k

mkjr

i

k

ðD

ÞðD

ÞC



½11

It should be stressed that the classical gauge algebra

[7] and [8] only closes on-shell. Quantization of the

model is therefore more subtle, and requires use of

the Batalin–Vilkovisky formalism. The on-shell

closure problem automatically results in the pre-

sence of quartic ghost coupling terms in the action

and consequently cubic terms in the BRST transfor-

mations. Despite this, we have established that the

full quantum action can be written as a BRST

commutator.

The form of the action simplifies when the

complex structure of the target manifold is

covariantly constant, D

= 0. In thi s c ase, the

target ma nifold M is Ka¨hler and we denote the

complex coordinates as u

, with their complex

conjugates denoted by u



. The nonzero c ompo-

nents of the metric tensor are then g



. Similarl y,

the coordinates of  are denoted 



, with nonzero

metric components h

þ

. The nonzero components

of the ghost and anti-ghost are then given by

, C



I

. The action can be writt en in t he

form



ﬃﬃﬃ

þ







ðD





Þh

þ











ðD

Þh

þ



þ









½12

Construction of Observables

Having defined the quantum action, it is now of

interest to consider the correlation functions of the

model. In the functional integral, we integrate over

all maps  !M in a fixed homotopy class. Let us

consider a correlation function

hOi ¼



i

tS

O½13

where t > 0 is a parameter, and the observable O is

BRST invariant {Q, O} = 0. From the BRST invar-

iance of the vacuum, it follows immediately that the

vacuum expectation value of a BRST commutator is

zero, h{Q, O}i= 0. An operator which is a BRST

commutator is said to be Q-exact. Hence, our

interest is in the Q-cohomology classes of operators,

that is, BRST invariant operators modulo BRST

exact operators. It is for this reason that such a

model is called a cohomological field theory.

One can now show that the variation of [13] with

respect to t is a BRST commutator, namely



hOi ¼ t



i

tS

fQ; VOg ¼ 0 ½14

As a result, one can evalu ate the correlation function

in the large-t (weak-coupling) limit. In this limit, the

path integral is dominated by fluctuations around

the classical minima. For the sigma model under

study, the classical action is minimized by the

instanton configurations



þ 





¼ 0 ½15

Indeed, this localization of the path integral to the

moduli space of instantons can also be seen by

choosing the  = 0 gauge in [9]. Integration over the

multiplier field then imposes a delta function

constraint to the instanton configurations. The key

point in the above derivation is the fact that the

quantum action is a BRST commutator, S

= {Q, V}.

By a similar argument, one can show that variations

of hOi with respect to the metric and complex

structure of  and M are also zero.

Our aim now is to construct the Q-cohomology

classes of operators in the theory. Let us first associate

an operator O

(0)

to each p-form A = A

i

^^

on the target space M,givenby

ð0Þ

¼ A

i

C

½16

where C

is the ghost field. Under a BRST

transformation, we see that

fQ; O

ð0Þ

g¼@

i

C

¼O

ð0Þ

½17

292 Topological Sigma Models

since the ghost fields are BRST invariant by [10].

Hence, O

(0)

is BRST invariant if and only if A is a

closed p-form. Similarly, if A is an exact p-form,

then the corresponding operator is Q-exact. Hence,

the BRST cohomology classes of these operators are

in one to one correspondence with the de Rham

cohomology classes on M. The reason for assigning

the peculiar superscript to the operator O

(0)

will

become clear at the end of this construction. Notice

also that operators of the form O

(0)

can be used as

building blocks for constructing new observables. If

we consider a set of closed forms A

, ..., A

, then

the product of the associated operators O

(0)

O

(0)

is clearly Q-invariant as well.

When considering the vacuum expectation values

of operators which are polynomials in the fields,

there is an implicit dependence on the points where

the operators are located. In the case at hand

however, the operator O

(0)

() at the point  has a

vacuum expectation value which is a topological

invariant, and thus cannot depend on the chosen

point. To see this explicitly, we consider all fields

defined over , and differentiate the operator with

respect to some local coordinates 



@



i

C

¼ð@

i

@



C

þ pA

i

ð@

@



C

½18

In terms of exterior derivatives, this takes the form,

ð0Þ

¼ @

i

C

þpA

i

C

¼fQ; O

ð1Þ

g½19

where O

(1)

= pA

i

C

, and we have

used the fact that A is a closed p-form. If we let 

represent any path between two arbitrary points P

and P

, then this expression has the integral form,

ð0Þ

ðPÞO

ð0Þ

ðP

Þ¼ Q;



ð1Þ



½20

and we see that the vacuum expectation value of

(0)

is point independent by the BRST invariance of

the vacuum. The same remark applies to any

product of operators of the form we are considering.

To continue our construction, consider a one-

dimensional homology cycle (@ = 0), and define

ð1Þ

ðÞ¼



ð1Þ

½21

This new operator W

(1)

() is BRST invariant by

inspection,

Q; W

ð1Þ

ðÞ



Q; O

ð1Þ



ð0Þ

¼ 0 ½22

Moreover, if  happens to be the boundary of a two-

dimensional surface ( = @), so that  is trivial in

homology, then this new operator is likewise trivial

in Q cohomology:

ð1Þ

ðÞ¼



ð1Þ



ð1Þ

¼ Q;



ð2Þ



½23

where

ð2Þ

¼

pðp  1Þ

i

^ du

C

As before, let us now associate to each homology

2-cycle (@ = 0), another BRST invariant operator

(2)

defined by

ð2Þ

ðÞ¼



ð2Þ

½24

The BRST invariance follows trivially as in [23].

In summary, we have produced three operators

(0)

, O

(1)

, and O

(2)

from any given closed form A,

which satisfy the relations:

0 ¼ Q; O

ð0Þ

; dO

ð0Þ

¼ Q; O

ð1Þ

¼ Q; O

ð2Þ

; dO

ð2Þ

¼ 0

½25

The BRST observables are then given by arbitrary

products of the integrated operators W

(i)

() =



(i)

, where  is any i-cycle in homology.

Observables and Intersection Theory

Let us consider the computation of the correlation

function hOi in the background field method. We

first pick a background instanton configuration [15],

and then integrate over the quantum fluctuations

around that instanton. The relevant part of the

quantum action is quadratic in the quantum fields,

and localization of the model then ensures that such

a computation is exact. The quantum fields are

expanded into eigenfunctions of the operators that

appear in the quadratic part of the action, and the

functional integral is replaced by an integral over the

eigenmodes. However, if there are fermionic zero

modes, then those modes do not enter in the action.

As a result, the fermionic integrals (

d = 0) over

those modes will cause hOi to vanish unless it has

the correct fermion content; the zero modes must be

absorbed. In our case, a glance at the quantum

action indicates that we should concern ourselves

with the zero modes of the ghost C

and anti-ghost

Topological Sigma Models 293



i

.AC

zero mode is clearly in the kernel of the

operator



j

¼ D





þ 





þ 







½26

and a



i

zero mode is a zero eigenfunction of its

adjoint





. In the BRST quantization of the model,

the ghost fields C

are assigned ghost number þ1,

while the anti-ghost fields



i

have gho st number

1. It is therefore apparent that the vacuum

expectation value of any observable will vanish

unless that observable has a ghost number equal to

the number of



D zero modes, a, minus the number





zero modes, b. This difference, ! = a  b,is

called the index of the operator



There is a direct link between this index and the

dimension of the moduli space of instantons. Recall

that we are considering the space of maps  !M in a

specified homotopy class, which satisfy equation [15].

It is then of interest to determine the dimension of the

space of such solutions. To this aim, we examine the

constraint that arises by considering an instanton u

and another neighboring solution u

,where

an infinitesimal deformation. To first order in

,we

see that

must be a zero mode of the operator



This is no coincidence, and we can thus interpret the

ghost fields C

as cotangent vectors to instanton

moduli space M. In particular, if M is a smooth

manifold, then dim M= a. The index of the operator



D is called the virtual dimension of the moduli space.

In generic situations, the virtual dimension is equal to

the actual dimension dim M.

It is possible to interpret some of the obs ervables

that we have described in terms of intersection

theory applied to the moduli space of instantons. In

particular, one can show that all correlation func-

tions of the form

ð0Þ

O

ð0Þ

½27

are intersection numbers of certa in submanifolds of

moduli space. In order to see this in a simple

example, we first recall the notion of Poincare´

duality and the relationship between cohomology

and homology.

Poincare´ duality can be formulated as a relation-

ship between de Rham cohomology (defined in

terms of closed differential forms) and homology

(defined in terms of subspaces of M). For our

purposes here, it is sufficient to state that we can

associate to each boundaryless submanifold N of

codimension k, a cohomology class [] 2 H

(M),

such that

 ^ ¼

½28

for all [ ] 2 H

nk

(M). By on the right-hand side of

this equation , we mean the pullback i



under the

inclusion i : N !M. Conversely, to each closed

k-form  on M, we can associate an (n  k)-cycle

N (it is in general a chain of subspaces), unique up

to homology, such that the previous relation is

satisfied. Furthermore, one can show that the

Poincare´ dual to N can be chosen in such a way

that its support is localized within any given open

neighborhood of N in M (essentially delta funct ion

support on N).

Let us now define the notion of transversal

intersection. For simplicity, we will first consider

the intersection of two submanifolds M

and M

contained in M. We will say that these two

submanifolds have transversal intersection if the

tangent spaces satisfy

ðM

ÞþT

ðM

Þ¼T

ðMÞ½29

for all x 2 M

\ M

. It is a theorem that a submanifold

of codimension k can be locally ‘‘cu t-out’’ by k smooth

functions, that is, the submanifold is locally specified by

the zeros of this set of functions. It is a worthwhile

exercise to convince oneself that the definition of

transversal intersection is equivalent to the statement

that the functions which cut-out M

are independent

from those which cut-out M

. Thus, we can write

codimðM

\ M

Þ¼codimðM

ÞþcodimðM

Þ½30

More generally, we say that the intersection M

\\

of s submanifolds is transversal if the intersection of

every pair of them is transversal. It then follows

trivially by the previous argument that the codimen-

sions must satisfy

codimðM

\\M

Þ¼

i¼1

codimðM

Þ½31

The special case which will be important for us

occurs when the intersection of submanifolds is a

collection of points, that is, when the codimension

of the intersection is equal to the dimension of M.

Since these points are isolated, the compactness of

M guarantees that they are finite in num ber.

We are now in a position to describe in what sense

correlation functions of the form O

(0)

O

(0)

determine intersection numbers in the moduli space

M of instantons. By definition, this moduli space is

the set of maps from  to M which satisfy [15].Let

us consider the generic situation, where the virtual

dimension of M (i.e., the index of



D) is equal to

dim M. For convenience, let us begin by choosing the

forms A

which represent de Rham cohomology

classes on M, together with their Poincare´dualsM

such that the forms have essentially delta function

294 Topological Sigma Models

support on their respective submanifolds. Since each

of the operators in the correlation function depends

on some fixed point 

,itismeaningfultodefinethe

submanifolds L

 {u 2Mju(

) 2 M

} M.Now,

the correlation function represents a functional

integral over the space of maps Map(, M), and we

have argued that this integral only receives contribu-

tions from the instanton configurations. Since the

operators A

(u(

)) vanish unless u 2 L

by our choice

of the Poincare´ duals, we see that the only contribu-

tion to the functional integral can be from those maps

which lie in the intersection L

\\L

.Byghost

number considerations, this correlation function must

vanish unless the codimension of the intersection

equals the virtual dimension of M. In the generic

case where the virtual dimension is equal to dim M,

this means that the intersection is simply a finite

number of points. Intersection numbers 1 can then

be assigned to each point in the intersection L

\L

, by considering the relative orientation of the

submanifolds L

at the intersection points. From the

functional integral point of view, the computation

reduces to an evaluation of the ratio of the bosonic

determinant (integration over u

) to the fermionic

determinant (integration over C

and



i

). In the

Ka¨ hler case, for example, the intersection number

assigned to each point in the intersection is always þ1.

This is due to the fact that the C







determinant is

the complex conjugate of the C



determinant.

A and B Models and Mirror Symmetry

The topological sigma model for a Ka¨ hler target

space [12] is also known as the topological A model.

In this case, the action can be recovered by twisting

the standard N = 2 supersymmetric sigma model.

This twisting procedure amount s to a reassignment

of the spins of the fields in the theory. However,

there is an alternative twisting which can be done,

and this leads to another model known as the

topological B model. The usefulness of this observa-

tion lies in the fact that the topolog ical A model on a

Calabi–Yau target space M is related to the

topological B model on the mirror of M. This

relationship and the computation of correlation

functions in the A and B model s thus sheds light

on the nature of mirror symmetry.

See also: Batalin–Vilkovisky Quantization;

BRST Quantization; Functional Integration in Quantum

Physics; Graded Poisson Algebras; Mathai–Quillen

Formalism; Mirror Symmetry: A Geometric Survey;

Several Complex Variables: Compact Manifolds;

Singularities of the Ricci Flow; Topological Gravity, Two-

Dimensional; Topological Quantum Field Theory:

Overview; WDVV Equations and Frobenius Manifolds.

Further Reading

Baulieu L and Singer I (1989) The topological sigma model.

Communications in Mathematical Physics 125: 227.

Birmingham D, Blau M, Rakowski M, and Thompson G (1991)

Topological field theory. Physics Reports 209: 129.

Birmingham D, Rakowski M, and Thompson G (1989) BRST

quantization of topological field theory. Nuclear Physics B

315: 577.

Eguchi T and Yang SK (1990) N = 2 superconformal models as

topological field theories. Modern Physics Letters A 5: 1693.

Floer A (1987) Morse theory for fixed points of symplectic

diffeomorphisms. Bulletin of the American Mathematical

Society 16: 279.

Floer A (1988) An instanton invariant for three manifolds.

Communications in Mathematical Physics 118: 215.

Gromov M (1985) Pseudo holomorphic curves in symplectic

manifolds. Inventiones Mathematicae 82: 307.

Gromov M (1986) In: Proceedings of the International Congress

of Mathematicians, p. 81. Berkeley.

Vafa C (1991) Topological Landau–Ginzburg models. Modern

Physics Letters A 6: 337.

Witten E (1988) Topological sigma models. Communications in

Mathematical Physics 118: 411.

Witten E (1991) Mirror manifolds and topological field theory.

In: Yau ST (ed.) Mirror Symmetry I, p. 121. American

Mathematical Society.

Turbulence Theories

R M S Rosa, Universidade Federal do Rio de Janeiro,

Rio de Janeiro, Brazil

Introduction

Turbulence has initially been defined as an irregular

motion in fluids. The cloud formations in the

atmosphere and the motion of water in rivers make

this point clear. These are but a few readily available

examples of a multitude of flows which display

turbulent regimes: from the blood that flows in our

veins and arterie s to the motion of air within our

lungs and around us; from the flow of water in

creeks to the atm ospheric and oceanic currents;

from the flows past submarines, ships, automobiles,

and aircraft to the combustion processes propelling

them; and in the flow of gas, oil, and water, from

the prospecting end to the entrails of the cities. The

great majority of flows in nature and in engineering

applications are somehow turbulent.

Turbulence Theories 295

But turbulent flows are much more than simply

irregular. More refined definitions were desirable

and were later coined. A definitive and precise one,

however, may only come when the phenomenon is

fully understood. Nevertheless, several characteristic

properties of a turbulent flow can be listed:

Irregularity and unpredictability A turbulent flow

is irregular both in space and time, displaying

unpredictable, random patterns.

Statistical order From the irregularity of a turbu-

lent motion there emerges a certain statistical

order. Mean quantities and corre lation are regular

and predictable (Figure 1).

Wide range of active scales A wide range of scales of

motion are active and display an irregular motion,

yielding a large number of degrees of freedom.

Mixing and enhanced diffusivity The fluid particles

undergo complicated and convoluted paths, caus-

ing a large mixing of different parts of fluid. This

mixing significantly enhances diffusion, increasing

the transport of momentum, energy, heat, and

other advected quantities.

Vortex stretching When a moving portion of fluid

also rotates transversally to its motion an increase

in speed causes it to rotate faster, a phenomenon

called vortex stretching. This causes that portion

of fluid to become thinner and elongated, and fold

and intertwine with other such portions. This is

an intrinsically three-dimensional mechanism

which plays a fundamental role in turbulence

and is associated with large fluctuat ions in the

vorticity field.

Turbulent Regimes

Turbulence is studi ed from many perspectives. The

subject of ‘‘transition to turbulence’’ attempts to

describe the initial mechanisms responsible for the

generation of turbulence starting from a laminar

motion in particular geometries . This transition can

be followed with respect to position in space (e.g.,

the flow becomes more complicated as we look

further downstream on a flow past an obstacle or

over a flat plate) or to parameters (e.g., as we

increase the angle of attack of a wing or the pressure

gradient in a pipe). This subject is divided into two

cases: wall-bounded and free-shear flows. In the

former, the viscosity, which causes the fluid to

adhere to the surface of the wall, is the primary

cause of the instability in the transition process. In

the latter, inviscid mechanisms such as mixing layers

and jets are the main factors. The tools for studying

the transition to turbulence include linearization of

the equations of motion around the laminar solu-

tion, nonlin ear amplitude equations, and bifurcation

theory.

‘‘Fully developed turbulence,’’ on the other hand,

concerns turbulence which evolves without imposed

constraints, such as boundaries and external forces.

This can be thought of turbulence in its ‘‘pure’’

form, and it is somewhat a theoretical framework

for research due to its idealized nature. Hypotheses

of homogeneity (when the mean quantities asso-

ciated with the statistical order characterizing a

turbulent flow are independent in space), stationar-

ity (idem in time), and isotropy (idem with respect

to rotations in space) concern fully developed

turbulent flows. The Kolmogorov theory was devel-

oped in this context and it is the most fundamental

theory of turbulence. Current research is dedicated

in great part to unveil the mechanisms behind a

phenomenon called intermittency and how it affects

the laws obtained from the conventional theory.

Research is also dedicated to derive such laws as

much from first principles as possible, minimizing

the use of phenomenological and dimensional

analysis.

Real turbulent flows involve various regimes at

once. A typical flow past a blunt object, for

instance, displays laminar motion at its upstream

edge, a turbulent boundary layer further down-

stream, and the formation of a turbulent wake

(Figure 2). The subject of turbulent boundary layer

is a world in itself with current research aiming to

determine mean properties of flows over rough

surfaces and varied topography. Convective turbu-

lence involves coupling with active scalars such as

Figure 1 Illustration of the irregular motion of a turbulent flow

over a flat plate (thin lines), and of the well-defined velocity

profile of the mean flow (thick lines).

Figure 2 Illustration of a flow past an object, with a laminar

boundary layer (light gray), a turbulent boundary layer (medium

gray), and a turbulent wake (dark gray).

296 Turbulence Theories

large heat gradients, occurring in the atmosphere,

and large salinity gradients, in the ocean. Geophy-

sical turbulence involves also stratification and the

anisotropy generated by Earth’s rotation. Anisotro-

pic turbulence is also crucial in astrophysics and

plasma theory. Multiphase and multicomponent

turbulence appear in flows with suspended particles

or bubbles and in mixtures such as gas, water, and

oil. Transonic and supersonic flows are also of great

importance and fall into the category of compres-

sible turbulence, much less explored than the

incompressible case.

In all those real situations one would like, from the

engineering point of view, to compute mean proper-

ties of the flow, such as drag and lift for more

efficient designs of aircraft, ships, and other vehicles.

Knowledge of the drag coefficient is also of funda-

mental importance in the design of pipes and pumps,

from pipelines to artificial human organs. Mean

turbulent diffusion coefficients of heat and other

passive scalars – quantities advected by the flow

without interfering on it, such as chemical products,

nutrients, moisture, and pollutants – are also of

major importance in industry, ecology, meteorology,

and climatology, for instance. And in most of those

cases a large amount of research is dedicated to the

‘‘control of turbulence,’’ either to increase mixing

or reduce drag, for instance. From a theoretical

point of view, one would like to fully understand

and characterize the mechanisms involved in

turbulent flows, clarif yin g this fascinating phe-

nomenon. This could also improve practical appli-

cations and lead to a better control of turbulence.

The concept of ‘‘two-dimensional turbulence’’ is

controversial. A two-dimensional flow may be

irregular and display mixing, statistical order, and

a wide range of active scales but definitely it does

not involve vortex stretching since the velocity field

is always perpendicular to the vorticity field. For this

reason many researchers discard two-dimensional

turbulence altogethe r. It is also argued that real

two-dimensional flows are unstable at complicated

regimes and soon develop into a three-dimensional

flow. Nevertheless, many believe that two-dimensional

turbulence, even lacking vortex stretching, is of

fundamental theoretical importance. It may shed

some light into the three-dimensional theory and

modeling, and it can serve as an approximation to

some situations such as the motion of the atmos-

phere and oceans in the large and meso scales and

some magnetohydrodynamic flows. The relative

shallowness of the atmosphere and oceans or the

imposition of a strong uniform magnetic field may

force the flow into two-dimensionality, at least for

a certain range of scales.

‘‘Chaos’’ serves as a paradigm for turbulence, in

the sense that it is now accepted that turbulence is a

dynamic processes in a sensitive deterministic

system. But not all chaotic motions in fluids are

termed turbulent for they may not display mixing

and vortex stretching or involve a wide range of

scales. An important such example appears in the

dispersive, nonlinear interactions of waves.

The Equations of Motion

It is usually stressed that turbulence is a continuum

phenomenon, in the sense that the active scales are

much larger than the collision mean free path

between molecules. For this reason, turbulence is

believed to be fully account ed for by the Navier–

Stokes equations.

In the case of incompressible homogeneous flows,

the Navier–Stokes equations in the Eulerian form

and in vector notation read

 u þðu rÞu þrp ¼ f ½1a

ru ¼0: ½1b

Here, u = u(x, t) = (u

, u

) denotes the velocity

vector of an idealized fluid particle located at

position x = (x

, x

), at time t . The mass density

in a homogeneous flow is constant, denoted . The

constant  denotes the kinematic viscosity of the

fluid, which is the molecular viscosity  divid ed by

. The variable p = p(x, t) is the kinemati c pressure,

and f = f (x, t ) = (f

, f

) denotes the mass density

of volume forces.

Equation [1a] expresses the conservation of linear

momentum. The term u accounts for the dissipa-

tion of energy due to molecular viscosity, and the

nonlinear term (u r)u, also called the inertial term,

accounts for the redistribution of energy among

different structures and scales of motion. Equation

[1b] represents the incompressibility condition. In

Einstein’s summation convention, these equations

can be written as

þ 

þ u

¼ f

;

¼ 0

The Reynolds Number

The transition to turbulence was carefully studied by

Reynolds in the late nineteenth century in a series of

experiments in which water at rest in a tank was

allowed to flow through a glass pipe. Starting with

dimensional analysis, Reynolds argued that a critical

Turbulence Theories 297