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

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precision in Z was obtained by ever-finer estimates;

presently, one knows that

EðZÞ¼E

7=3

þ C

5=3

þ oðZ

5=3

Þ½14

where the leading contribution E

7=3

is the

Thomas–Fermi energy, (1=4)Z

is the Scott correc-

tion, and C

5=3

is the Dirac–Schwinger term. The

computation of this last term requires semiclassical

analys is sketc hed in the sect ion ‘‘Quan tization and

semiclass ical limit.’’

For more details, see Cycon et al. (1987), Rauch

and Simon (1997), Thirring (1997), and Solovej

(2003). See also the article Stability of Matter in this

encyclopedia.

Scattering Theory

The study of the properties of the propagator

exp(itH ) of a self-adjoint operator H = H



,as

t !1, is the concern of scattering theory. To

obtain a well-defined mathematical object in this

limit, it is necessary to compose e xp(itH)with

the inverse of some expli citly accessible com par-

ison dynamics before passing to the limit t !1.If

V is a short-range potential, that is, V is relatively

-compact and jV( x)jCjxj



,forsome>1

and C < 1, then the comparison dynamics appro-

priate for H

is generated by H

:thewave

operators 



are defined as the strong limits





:¼ lim

t!1

itH

itH

½15

A general technique in scattering theory to prove the

existence of such limits is Cook’s argument, which

formally amounts to an application of the funda-

mental theorem of calculus. For example, for the

existence of 

, one writes



 1 ¼

itH

itH





¼i

dt fe

itH

V e

itH

g½16

and additionally proves the absolute integrability of

t 7!e

itH

itH

’, for ’ in a dense subset of H, like

dom(H

) = dom(H

Research in scattering theory in the past two

decades or so was focused around the question of

asymptotic completeness, which is a mathematically

precise formulation

Ran

¼ Ran



¼H

ðH

Þ½17

of the physical expectation that the states in H are

either bound states (eigenvectors) of H

scattering states (states in the range of 



)ofH

The intertwining property H





=



(which

easily follows from [15]) implies that the restriction

of H

to Ran



is unitarily equivalent to H

, hence

Ran



H

) H

). The difficult part of

the proof of asymptotic completeness is to show that

)  Ran



Much effort has been spent to prove asymptotic

completeness for N-body Schro¨ dinger operators on

(N)

n = 1

) of the form

ðVÞ¼

n¼1



2 m

þVðxÞ

with VðxÞ :¼

1m<nN

ðx

 x

Þ½18

where each pair potential V

obeys j@



(y)j

C(1 þjy j)

jj

,with 2 N

being a multi-index. If

>1 for all m 6¼ n then V is called a short-range

potential. Conversely, if 0 < 1thenV is a long-

range potential. Note that even though each V

decays at infinity, jxj

= x

þ x

þþx

alone does not imply that V(x) !1.Infact,physical

intuition tells us that for a cluster C of N particles,

whose dynamics is generated by H

(V), several

scenarios for the long-time asymptotic behavior of

the evolution are possible:

1. The N particles stay together in their cluster C

whose center of mass moves in space at constant

velocity.

2. The cluster breaks up into two (or even more)

subclusters, C

and C

,ofN

and N

= N  N

particles, respectively, whose centers of mass drift

apart from each other at constant velocities (in

the short-range case). For each subcluster C

and

, both scenarios may appear again, after wait-

ing sufficiently longer.

3. In the lim it t !1, possibly after going through

(1) and (2) several times, the initial cluster C is

broken up into 1  K  N subclusters

, ..., C

, w hose centers of mass drift apart

from each other at constant velocities according

to a free and independent dynamics of their

centers of mass.

In some sense, asymptotic completeness says that

nothing else than (1)–(3) can possibly happen.

(Strictly speaking, asymptotic completeness is a

statement about the limit t !1 and only

involves (3) – the actual behavior of exp [itH

]

at intermediate times in terms of (1)–(3) is beyond

the reach of current mathematics.) It is a key

insight of scattering theory that the asymptotics of

the time evolution in the sense of (3) is completely

492 Schro¨ dinger Operators

characterized by the asymptotic velocity defined

by the strong limit

:¼ lim

t!1

itH

ðVÞ

itH

ðVÞ



½19

It is a nontrivial fact that P

exists, commutes with

(V), and that bound states are precisely the states

with zero asymptotic velocity, while states with

nonzero asymptotic velocity are scattering states in

Ran



. This then implies asymptotic completeness

for short-range potentials. The proof of this dichot-

omy builds essentially upon positive commutator or

Mourre estimates. Given an interval J localized (in

energy) away from any eigenvalue of any possible

subcluster configuration C

, ..., C

(called thresh-

olds), the Mourre estimate asserts the existence of a

positive constant M > 0 and a compact operator

R 2B(H

(N)

) such that

i½H

ðVÞ; A1

 M1

 R ½20

as a quadratic form, for some suitable operator

A. This operat or A is often chosen to be the

dilation generator A = (1=2){p  x þ x  p} or a var-

iant thereof.

Again, the proof of asymptotic compl eteness for

long-range potentials is still more difficult and has

been carried out only for >

ﬃﬃﬃ

 1. The addi-

tional problem is the comparison dynamics of the

relative motion of the clusters C

and C

in (2),

which is not the free one; the clusters rather

influence each other even at large distances.

For more details, see Reed and Simon (1980c) and

Derezinski and Ge´rard (1997). See also the articles

Scattering in Relativistic Quantum Field Theory:

Fundamental Concepts and Tools, Scattering,

Asymptotic Completeness and Bound States in this

encyclopedia.

Random Schro¨ dinger Operators

Schro¨ dinger operators H(V

)onL

)or‘

)

with a random potential V

are called random

Schro¨ dinger operators. (If H(V

)actson‘

then the (continuum) Laplacian  is replaced by the

discrete Laplacian on Z

defined by [

disc

f ](x)=

 =1

{2f (x) f (x e



) f (x þe



).) More precisely,

given a probability space (,P,) and a random

variable  3!7!V

, the family {H(V

)}

!2

defines

an operator-valued random variable that we refer to

as a random Schro¨ dinger operator. Random quantum

systems are physically relevant as models for amor-

phous materials, and for solids in very heterogenous

external fields or coupled to quantized fields. Suitable

ergodicity assumptions on ! !V

ensure that the

domain of H

and even many spectral properties (in

particular, the spectrum (H(V

)) R itself) are

independent of ! P-almost surely. For example,

assuming an independent, identical distribution

(i.i.d.) of V

in the discrete case on Z

, one arrives

at the Anderson model, which has been most

thoroughly studied. Its counterpart for continuum

models is a Poisson-distributed V

. A model which

also has ergodic properties, although deterministic, is

the Hofstadter or the Mathieu problem. Most

research has been focused on localization, that is,

spatial decay properties of the resolvent {H(V

) 

1

(x,y)ofH(V

), as jx yj!1,andparticularly

the question of presence or absence of exponential

decay (localization), as this is an important indicator

for the transport properties of the material under

consideration. Exponential localization of eigenstates

has been established for d=1 or strong disorder or

sufficiently high energies E 1. Localization is also

intimately related to bounds on moments of the form

=2

kC





. The study of the asymptotic dis-

tribution of eigenvalues close to the lowest threshold

leads to the so-called Lifshitz tails.

The reader is referred to Figotin and Pastur

(1992), Cycon et al. (1987),andStollmann (2001).

(Pseudo)relativistic Schro¨ dinger

Operators

Schro¨ dinger operators of the form H(V) = p

þ V(x)

do not observe the invariance principles of (special)

relativity, as their derivation is based in classical

(Newtonian) mechanics. The free Dirac operator

D := a  p þ m (here, 



and  are self-adjoint

4  4 matrices) possesses the desired relativistic

invariance, but it is not semibounded, and the

definition of an interacting Dirac operator is

notoriously difficult (and unsolved). The replace-

ment of the kinetic energy (1=2m)p

by the Klein–

Gordon operator

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

þ m

is a step towards

relativistic invariance, which, at the same time,

yields a positive operator. This replacement may

also be viewed as the restriction of the free Dirac

operator to its positive-energy subspace. The virtue

of this replacement is that it immediately allows for

the study of interacting N-particle operators,

rel

ðZ; RÞ¼

n¼1

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ



þ m



k¼1

 R

()

1‘<nN

‘

 x

½21

much like in [12]. Since

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

þ m

jpj,asp !1,

the pseudorelativistic kinetic energy

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

þ m

can

Schro¨ dinger Operators 493

balance only less severe local singularities of the

potential V than the nonrelativistic kinetic energy

(1=2m)p

. Indeed, already the qua dratic form

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

þ m

 gjxj

1

on C

) associated to a hydro-

gen-like atom is unbounded from below if g > 2=.

Hence, the stability of matter becomes a more subtle

property of pseudorelativistic matter. The relaxation

of the restriction onto the positive subspace of the free

Dirac operator also got into the focus of research.

For more details, we refer the reader to Thirring

(1997).

See also: Deformation Quantization; Elliptic Differential

Equations: Linear Theory; h-Pseudodifferential Operators

and Applications; Localization for Quasiperiodic

Potentials; Nonlinear Schro

dinger Equations; Normal

Forms and Semiclassical Approximation; N-Particle

Quantum Scattering; Quantum Hall Effect; Quantum

Mechanical Scattering Theory; Scattering, Asymptotic

Completeness and Bound States; Stability of Matter;

Stationary Phase Approximation.

Further Reading

Amrein W, Hinz A, and Pearson D (2005) Sturm–Liouville

Theory – Past and Present. Boston: Birkha¨user.

Cycon H, Froese R, Kirsch W, and Simon B (1987) Schro¨ dinger

Operators, 1st edn. Berlin: Springer.

Derezinski J and Ge´rard C (1997) Scattering Theory of Classical

and Quantum N-Particle Systems, Text and Monographs in

Physics. Berlin: Springer-Verlag.

Dimassi M and Sjo¨strand J (1999) Spectral Asymptotics in

the Semi-Classical Limit. London Mathematical Society

Lecture Notes Series, vol. 268. Cambridge: Cambridge

University Press.

Erdo¨ s L and Solovej JP (2004) Uniform Lieb–Thirring inequality

for the three-dimensional Pauli operator with a strong

non-homogeneous magnetic field. Annales Henri Poincare´ 5:

671–741.

Figotin A and Pastur L (1992) Spectra of Random and Almost-

Periodic Operators. Grundlehren der Mathematischen

Wissenschaften, vol. 297. Berlin: Springer-Verlag.

Kato T (1976) Perturbation Theory of Linear Operators, 2 edn.,

Grundlehren der mathematischen Wissenschaften, vol. 132.

Berlin: Springer-Verlag.

Messiah A (1962) Quantum Mechanics, 1st edn., vol. 2. Amsterdam:

North-Holland.

Rauch J and Simon B (eds.) (1997) Quasiclassical Methods.IMA

Volumes in Mathematics and Its Applications, vol. 95. Berlin:

Springer-Verlag.

Reed M and Simon B (1978) Methods of Modern Mathematical

Physics IV. Analysis of Operators, 1st edn., vol. 4. San Diego:

Academic Press.

Reed M and Simon B (1980a) Methods of Modern Mathematical

Physics: I. Functional Analysis, 2nd edn., vol. 1. San Diego:

Academic Press.

Reed M and Simon B (1980b) Methods of Modern Mathematical

Physics: II. Fourier Analysis and Self-Adjointness, 2nd edn.,

vol. 2. San Diego: Academic Press.

Reed M and Simon B (1980c) Methods of Modern Mathematical

Physics: III. Scattering Theory, 2nd edn., vol. 3. San Diego:

Academic Press.

Robert D (1987) Autour de l’Approximation Semi-Classique,

1st edn. Boston: Birkha¨ user.

Schro¨ dinger E (1926) Quantisierung als Eigenwertproblem.

Annalen der Physik 79: 489.

Simon B (1979) Functional Integration and Quantum Physics,

Pure and Applied Mathematics. New York: Academic Press.

Simon B (2000) Schro¨ dinger operators in the twentieth century.

Journal of Mathematical Physics 41: 3523–3555.

Solovej JP (2003) The ionization conjecture in Hartree–Fock

theory. Annals of Mathematics 158: 509–576.

Stollmann P (2001) Caught by Disorder. Progress in Mathema-

tical Physics, vol. 20. Boston: Birkha¨ user.

Thirring W (ed.) (1997) The Stability of Matter: From Atoms to

Stars – Selecta of Elliott H. Lieb, 2 edn. Berlin: Springer-

Verlag.

Schwarz-Type Topological Quantum Field Theory

R K Kaul and T R Govindarajan, The Institute of

Mathematical Sciences, Chennai, India

P Ramadevi, Indian Institute of Technology Bombay,

Mumbai, India

Introduction

Topological quantum field theories (TQFTs) provide

powerful tools to probe topology of manifolds,

specifically in low dimensions. This is achieved by

incorporating very large gauge symmetries in the

theory which lead to gauge-invariant sectors with

only topological degrees of freedom. These theories

are of two kinds: (1) Schwarz type and (2) Witten

type.

In a Witten-type topological field theory, action is a

BRST exact form, so is the stress energy tensor T



that their functional averages are zero (Witten 1988).

The BRST charge is associated with a certain shift

symmetry. The topological observables form cohomo-

logical classes and semiclassical approximation turns

out to be exact. In four dimensions, such theories

involving Yang–Mills gauge fields provide a field-

theoretic representation for Donaldson invariants.

On the other hand, Schwarz-type TQFTs are

described by local action functionals which are not

total derivatives but are explicitly independent of

metric (Schwarz 1978, 1979, 1987, Witten 1989).

494 Schwarz-Type Topological Quantum Field Theory

The examples of such theories are topological

Chern–Simons (CS) theories and BF theories.

Metric independence of the action S of a Schwarz-

type gauge theory implies that stress–energy tensor

is zero:

 S

 g



 T



¼ 0

More generally, in the gauge-fixed version of such

theories, stress–energy can be BRST exact, where

BRST charge corresponds to gauge fixing in contrast

to Witten-type theories where corresponding BRST

charge corresponds to a combination of shift

symmetry and gauge symmetry. There are no local

propagating degrees of freedom; the only degrees of

freedom are topological. Expectation values of

metric-independent operators W are also indepen-

dent of the metric:

hWi

g



¼ 0

Three-dimensional CS theories are of particu lar

interest, for these provide a framework for the study

of knots and links in any 3-manifold. Pioneering

indications of the fact that topological invariants

can be found in such a setting came in very early

when A S Schwarz demonstrated that a particular

topological invariant, Ray–Singer analytic torsion

(which is equiva lent to combinatorial Reidemeister–

Franz torsion) can be interpreted in terms of the

partition funct ion of a quantum gauge field theory

(Schwarz 1978, 1979). In particular, in the weak-

coupling limit of CS theory of gauge group G on a

manifold M, contribution from each topologically

distinct flat connection (characterized by the equiva-

lence classes of homomorphisms: 

(M) !G) to the

partition function is given by metric-independent

Ray–Singer torsion of the flat connection up to a

phase. This phase factor is also a topological

invariant of framed 3-manifold M (Witten 1989).

It was Schwarz who first discussed CS theory as a

topological field theory and also conjectured that

the well-known Jones polynomial may be related to

it (Schwarz 1987). In his famous paper Witten

(1989) not only demonstrated this connection, but

also set up a general field-theoretic framework to

study the topological properties of knots and links in

any arbitrary 3-ma nifold. In addition, this frame-

work provides a method of obtai ning some new

manifold invariants. As discussed by A Achu´caro

and P K Townsend, CS theory also describes gravity

in three-dimensional spacetime (Carlip 2003).

BF theories in three dimensions provide another

framework for field-theoreti c description of

topological properties of knots and links. These

theories with bilinear action in fields can also be

defined in higher dimensions. In particular in D = 4,

BF theory, besides describing two-dimensional gen-

eralizations of knots and links, also provides a field-

theoretic interpretation of Donaldson invariants.

This provides a connection of these theories with

Witten-type TQFT s of Yang–Mills gauge fields. We

shall not discuss BF theories in the following and

refer to the article BF Theories in this Encyclopedia.

Witten (1995) has also formulated CS theories in

three complex dimensi ons described in terms of

holomorphic 1-forms. Such a theory on Calabi–Yau

spaces can be interpreted as a string theory in terms

of a Witten- type topological field theory of a sigma

model coupled to gravity. General topological sigma

models in Batalin–Vilkovisky formalism have been

constructed by Alexandrov et al. (1997). This is a

Schwarz-type theory. However, in its gauge-fixed

version, it can also be interpreted as a Witten-type

theory. This construction provides a general for-

mulation from which numerous topological field

theories emerge. In particular, the Witten A and B

models and also multidimensional CS theories are

special cases of this construction.

In the following, we shall survey three-dimensional

CS theory as a description of knots/links, indicate

how manifold invariants can be constructed from

invariants for framed links, and also discuss its

application to three-dimensional gravity.

Three-Dimensional CS Theory with

Gauge Group U(1)

The simplest Schwarz-type topological field theory is

the U(1) CS theory described by the action:

S ¼

8

A dA ½1

where A is a connection 1-form A = A



and M is

the 3-manifold, which we shall take to be S

for the

discussion below. The action ha s no dependence on

the metric. Besides being the U(1) gauge invariant, it

is also general coordinate invariant.

In quantum CS field theory, we are interested in

the functional averages of gauge-invariant and

metric-independent functionals W[A]:

hW½Ai ¼

½DAW½AexpfikSg

Z¼

½DAexpfikSg

½2

This theory captures some of the simple, but

interesting, topological properties of knots and links

in three dimensions. For a knot K, we associate a knot

Schwarz-Type Topological Quantum Field Theory 495

operator

A which is gauge invariant and also does

not depend on the metric of the 3-manifold. Then for

a link made of two knots K

and K

, we have the loop

correlation function h

Ai, which can be

evaluated in terms of two-point correlator



(x)A



(y)i in R

(with flat metric). This correlator

in Lorentz gauge (@



= 0) is:



ðxÞA



ðyÞi¼





ðx  yÞ



jx  yj

so that for two distinct knots K

and K



4i

LðK

; K

Þ½3

where

LðK

; K

Þ¼

4









ðx  yÞ



jx  yj

This integral is the well-known topological invariant

called ‘‘Gauss linking number’’ of two distinct

closed curves. It is an integer measuring the number

of times one knot K

goes through the other knot

. Linking number does not depend on the

location, size, or shape of the knots. In electro-

dynamics, it has the physical interpretation of work

done to move a monopole around a knot while

electric current runs through the other knot.

Abelian CS theory also provides a field-theoretic

representation for another topological quantity

called ‘‘self-linking number,’’ also known as ‘‘fram-

ing number,’’ of the knot. It is related to the

functional average of h

Ai where two loop

integrals are over the same knot. Coincidence

singularity is avoided by a topological loop-splitting

regularization. For a knot K given by x



(s) para-

metrized along the length of the knot by s,we

associate another closed curve K

given by



(s) = x



(s) þ  n



(s), where  is a small parameter

and n



(s) is a principal normal to the curve at s. The

coincidence limit is then obtained at the end by

taking the limit  !0. Such a limiting procedure is

called framing and knot K

is the ‘‘frame’’ of knot K.

Linking number of the knot K and its frame K

is the

self-linking number of the knot:

SLðK; n



Þ¼

4









ðx  yÞ



jx  yj

Hence coincidence two loop correlator is



4i

SLðK; n



Þ½4

Notice that the self-linking number of a knot is

independent of the regularization parameter ,but

does depend on the topological character of the

normal vector field n



(s). It is also related to two

geometric quantities called ‘‘twist’’ T(K) and ‘‘writhe’’

w(K) through a theorem due to Calugareanu:

SLðKÞ¼TðKÞþ!ðKÞ½5

where

TðKÞ¼

2

ds 









!ðKÞ¼

4

dt 









Here



ðs; tÞ¼



ðtÞy



ðsÞ

jyðtÞyðsÞj

is a unit map from K  K ! S

and n



(s) is a normal

unit vector field. T(K)and!(K) are not in general

integers and represent the amount of twist and coiling

of the knot. These are not topological invariants but

their sum, self-linking number, is indeed always an

integer and a topological invariant. This result has

found interesting applications in the studies of the

action of enzymes on circular DNA.

Nonabelian CS Theories

Nonabelian CS theories provide far more informa-

tion about the topological properties of the mani-

folds as well as knots and links.

Nonabelian CS theory in a 3-manifold M (which

as in last section is taken to be S

) is described by

the action functional

S ¼

4

tr A ^ dA þ

A ^ A ^ A



½6

where A is a gauge field 1-form which takes its value

in the Lie algebra LG of a compact semisimple Lie

group G. For example, we may take this group to be

SU(N) and A = A

, where T

is the fundamental

N-dimensional representation with trT

= 1=2

Under homotopically nontrivial gauge transforma-

tions this action is not invariant, but changes by an

amount 2n where integers n are the winding

numbers characterizing the gauge transformations

which fall in homotopic classes given by 

(G) = Z

for a compact semisimple group G. However, for

quantum theory what is relevant is exp[ikS] which

is invariant even under homotopically nontrivial

gauge transformations provided the coupling k

takes integer values. This quantized nature of the

coupling was pointed out by Deser et al. (1982a,b)

(and also they were first to introduce the non-

abelian CS term as a gauge-invariant topological

496 Schwarz-Type Topological Quantum Field Theory

mass term in gauge theories). So for integer k, the

quantum field theory we discuss here is gauge

invariant.

The topological operators are Wilson loop opera-

tors for an oriented knot K:

½K¼tr P exp

½7

where A

= A

with T

as the representation

matrices of a finite-dimensional representation R of

the LG. P stands for the path ordering of the

exponential. The observable Wilson link operator

for a link L =

, carrying representations R

the respective component knots, is

R

L½¼

½K

½8

Expectation values of these operators are:

R

½L¼

½DAW

R

½Le

ikS

½DAe

ikS

½9

The measure [DA] has to be metric independent.

These expectation values depend not only on the

isotopy of the link L but also on the set of the

representations {R

}. These can be evaluated in

principle nonperturbatively. For example, when

LG= su(N) and each of the component knot of the

links carries the fundamental N-dimensional repre-

sentation, the Wilson link expectati on va lues satisfy

a recursion relation involving three link diagrams

which are identical except for one cross ing where

they differ as over crossing (L

), under crossing



), and no crossing (L

) as shown in the Figure 1.

The expectation values of these links are related

as (Witten 1989):

N=2

½L

q

N=2

½L





¼ q

1=2

 q

1=2



½L

½10

where

q ¼ exp

2i

k þ N



This is precisely the well-known skein relation for

the HOMFLY polynomial. The famous Jones one-

variable polynomial (whose two-variable

generalization is the HOMFLY polynomial) corre-

sponds to the case of spin-1/2 representation of

SU(2) CS theory: V

[L] = Jones polynomial [L], up

to an overall normalization. These skein relations

are sufficient to recursively find all the expectation

values of links with only fundamental representation

on the components. To obtain invariants for any

other representation, more general methods have to

be developed. A complete and explicit solution of

the CS field theory is thus obtained. One such

method has been reviewed in Kaul (1999). The

method makes use of the following important

statement:

Proposition: CS theory on a 3-manifold M

with boundary  is described by a WZNW

(Wess–Zumino–Novikov–Witten) conformal field

theory (CFT) on the boundary (Figure 2).

Using the same identification, functional average

for Wilson lines ending at n points on the boundary

 is obtained from WZNW field theory on the

boundary with n punctures carrying repre sentations

(Figure 3):

We can represent CS functional integral as a

vector (Witten 1989) in the Hilbert space H

associated with the n-point vacuum expectation

values of primary fields in WZNW conformal field

theory on the boundary . Next, to obtain a

complete and explicit nonperturbative solution of

the CS theory, the theory of knots and links and

their connection to braids is invoked.

–

Figure 1 Skein related links.

Figure 2 Relation of CS to CFT.

Figure 3 CS functional integrals with Wilson lines and CFT on

punctured boundary.

Schwarz-Type Topological Quantum Field Theory 497

Knots/Links and Braids

Braids have an intimate connection with knots and

links which can be summarized as follows:

1. An n-braid is a collection of nonintersecting

strands connecting n points on a horizontal rod

to n points on another horizontal rod below

strictly excluding any backward traversing of the

strands. A general braid can be written as a word

in terms of elementary braid generators.

2. We associate representations R

of the group with

the strands as their colors. We also put an

orientation on each strand. When all the repre-

sentations are identical and also all strands are

unoriented, we get ordinary braids, otherwise we

get colored oriented braids.

3. The colored oriented braids form a groupoid

where product of the different braids is obtained

by joining them with both colors and orientations

matching on the joined strands. Unoriented

monochromatic braids form a grou p.

4. A knot/link can be formed from a given braid by

a process called platting. We connect adjacent

strands namely the (2i þ 1)th strand to 2ith

strand carrying the same color and opposite

orientations in both the rods of an even-strand

braid (Figure 4a).

There is a theorem due to Birman which states

that all colored oriented knots/links can be

obtained through platting. This construction is

not unique.

5. There is another construction associated with

braids which relates them to knots and links. We

obtain a closure of a braid by connecting the ends

of the first, second, third, ... strands from above

to those of the respe ctive first, second, third, ...

strands from below as shown in the Figure 4b.

There is theorem due to Alexander which states

that any knot or link can be obtained as a closure

of a braid, though again not uniquely.

Link Invariants

This connection of braids to knots and links can be

used to construct link invariants, say in S

.Todoso,

two nonintersecting 3-balls are removed from the

3-manifold S

to obtain a manifold with two S

boundaries. Then we arrange 2n Wilson lines of, say

SU(N) CS theory, as a 2n-strand oriented braid

carrying representations R

in this manifold. The CS

functional integral over this manifold is a state in

the tensor product of the Hilbert spaces H

H

associated with conformal field theory on the two

boundaries. These boundaries have 2n punctures

carrying the set of representations {R

} and {R

respectively, the two sets being permutations of each

other. This state can be expanded in terms of some

convenient basis given by the conformal blocks for

the 2n -point correlation functions of SU(N)

WZNW conformal field theory. The duality of

these correlation functions represents the transfor-

mation between different bases for the Hilbert

space. Their monodromy properties allow us to

write down representations of the braid generators.

Since an arbitrary braid is just a word in terms of

these generators, this construction provides us a

matrix representation B({R

}, {R

}) for the colored

oriented braid in the manifold with two S

bound-

aries. Then we plat this braid by gluing two balls B

and B

with Wilson lines as shown in Figure 5.

Each of the two caps again represent s a state

j ({R

})i in the Hilbert space associated with the

conformal field theory on punctured boundary (S

Platting of the braid then simply is the matrix

element of braid representation B({R

}, {R

}) with

respect to these states j ({R

})i and j ({R

})i corre-

sponding to two caps B

, B

. Thus, for a link in S

the invariant is given by the following theorem:

Theorem The vacuum expectation value of Wilson

loop operator of a link L constructed from platting

of a colored oriented 2n braid with representation

B({R

}, {R

}) is given by (Kaul 1999):

V½L¼h ðfR

gÞjBðfR

g; fR

gÞj ðfR

gÞi ½11

This theorem can be used to calculate the

invariant for any arbitrary link. For an unknot U

(a) (b)

Figure 4 (a) Platting and (b) closure of braids.

B({R

}, {R

})

′

ψ({R

})〉

′

〈ψ({R

})

⏐⏐

Figure 5 Construction of the link invariant.

498 Schwarz-Type Topological Quantum Field Theory

carrying an N-dimensional representation in an

SU(N) CS theory, the knot invariant is:

½U¼½N; where ½N¼

N=2

 q

N=2

1=2

 q

1=2

Wilson link expectation values calculated this way

depend on the regularization, that is, the definition

of framing used in defining coincident loop correla-

tors. One such regularization usually used is the

standard framing, where the frame for every knot is

so chosen that its self-linking number is zero.

The procedure outlined here has been used for

explicit computations of knot/link invariants. This

has led to answers to several questions of knot

theory. One such question relates to distinguishing

chirality of knots (Kaul 1999). In this context, newer

invariants constructed with arbitrary representations

living on the knots are more powerful than the older

polynomial invariants. For example, invariants with

spin-3/2 representation in an SU(2) CS theory are

sensitive to chirality of many knots which otherwise

is not detected by Jones, HOMFLY, and Kauffman

polynomials. However, invariants obtained from CS

theories do not distinguish all chiral knots. There is

a class of links known as ‘‘mutants’’ which are not

distinguished by CS link invariants (Kaul 1999). A

mutant link is obtained by removing a portion of

weaving pattern in a link and then gluing it back

after rotating it about any one of three orthogonal

axes by an amount .

The CS invariants of knots and links can also be

used to constru ct special 3-manifold invariants.

Hence, CS theory provides an important tool to

study these.

Manifold Invariants from CS Theory

Different 3-manifolds can be constructed through a

procedure called ‘‘surgery of framed knots and

links’’ in S

(Lickorish–Wallace theorem). This

construction is not unique. That is, there are many

framed knots and link s which give the same

manifold. However , rules of this equivalence are

known: these are called ‘‘Kirby moves.’’

Classification of 3-manifolds would involve find-

ing a method of associating a quantity with the

manifold obtained by surgery on the corresponding

framed knot/link on S

. If the Kirby moves on the

framed knot/link leave this quantity unchanged,

then it is a 3-manifold invariant. Knot/link invar-

iants of nonabelian CS theories provide a method of

finding such 3-manifold invariants. Equivalently,

this procedure gives an algebraic meaning to the

surgery construction of 3-manifolds. Detai ls of this

method for generating manifold invariants are given

in Kaul (1999) and Kaul and Ramadevi (2001).

Surgery of Framed Knots/Links and Kirby Moves

As discussed earlier, fram e of a knot K is an

associated closed curve K

going along the length

of the knot wrapping around it certain number of

times. Self-linking number (also called framing

number) is equal to the linking number of the knot

with its frame. There are several ways of fixing this

framing. The ‘‘standard’’ framing is one in which the

frame number of the knot, that is, the linking

number of the knot and its frame is zero. On the

other hand, ‘‘vertical’’ framing is obtained by

choosing the frame vertically above the knot

projected on to a plane. In such a frame, the framing

number of a knot is the same as its crossing number.

In constructing the 3-manifold invariants from CS

theories, we need vertical framing. The framing

number may be deno ted by writing the integer by

the side of knot. We denote a framed r-component

link by [L, f ] where framing f = (n(1), n(2), ..., n(r))

is a set of integers denoting the framing number of

component knots K

, K

, ..., K

in the link L.

According to the Lickorish–Wallace theorem,

surgery over links with vertical framing in S

yields

all the 3-manifolds. This surgery is performed in the

following way.

Take a framed r-component link [L, f ]inS

Thicken the component knots K

, K

, ..., K

such

that the solid tubes N

, N

, ..., N

so obtained are

nonintersecting. Then the compliment S



þ N

þþN

) will have r toral boundaries.

On the ith toral boundary, we imagine an

appropriate curve winding n(i) times around the

meridian and once along the longitude. Perform a

modular transformation so that this curve bounds

a disk. This construction is done w ith each of the

toral boundaries. The tubes N

, N

, ..., N

are

thengluedbackintotherespectivegaps.This

surgery thus yields a new 3-manifold. This

construction is not unique. The rules of equiva-

lence for surgery on framed knots/links in S

are

two independent Kirby moves.

Kirby move I Take an arbitrary r-component

framed link [L, f ]inS

and consider a curve C

with framing number þ1 going around the unlinked

strands of L as in Figure 6a. We refer to this (r þ 1)-

component link as H[X], where X represents a

weaving pattern of the strands. Kirby move I

consists of twisting the disk enclosed by C in the

clockwise direction from below by an amount 2.

This twisting thereby introduces new crossings

Schwarz-Type Topological Quantum Field Theory 499

between the curve C and the strands enclosed by it.

Then the curve C is removed giving us a new

r-component link U[X]ofFigure 6b. Framing

numbers n

(i) of the component knots in link U[X]

are related to the framing number n(i) of framed link

[L, f ]asn

(i) = n(i)  (L(K

, C))

, where L(K

, C)is

the linking number of knot K

and closed curve C.

The surgery of the framed links in Figures 6a and 6b

will give the same 3-manifold.

Inverse Kirby move I involves removal of a curve

C with framing number 1 (instead of þ1) after

making one complete anticlockwise twist from

below on the disk enclosed by C. In the process the

unlinked strands get twisted in the anticlockwise

direction leading to changed framing number s

(i) = n(i) þ (L(K

, C))

of the component knots K

Kirby move II This move consists of removing a

disjoint unknot C with framing 1 from framed link

[L, f ] without changing the rest of the link as in

Figure 7. Surgery of the two links in Figure 7 will

give the same manifold.

Inverse Kirby move II involves removal of a

disjoint unknot with framing þ1 (instead of 1)

from a framed link.

3-Manifold Invariants

Now a 3-manifold invariant can be constructed by

an appropriate combination of the invariants of

framed links in such a way that this algebraic

expression is unchanged under the Kirby moves. We

need for this purpose invariants for links in S

with

vertical framing.

Let M be the manifold obtained from surgery

of an r-component framed link [L, f ]inS

. Then

a manifold invariant

(G)

[M] is given as a linear

combination of the framed link invariants V

(G)

,..., R

[L, f ], with representations R

, R

, ..., R

living on

component knots, obtained from CS theory based

on a compact semisimple group G:

ðGÞ

½M ¼ 

½L;f 

;... R

i¼1



 V

ðGÞ

;...; R

½L; f ½12

Here [L, f ] is the signature of the linking matrix

and 

= S

,  = e

ic=4

, where c is the central

charge of the associated WZNW conformal field

theory and S

denotes the matrix element of the

modular matrix S. General S-matrix elements for

any compact group are given by

¼ðiÞ

ðdrÞ=2

=Lj

1=2

ðk þC

1=2



!2W

ð!Þexp

2i

k þC

ð!ð

þÞ; 

þÞ



where W denotes the Weyl group and its elements !

are words constructed using the generator s



–that

is, ! =



and (!) = (1)

‘(!)

with ‘(!)aslengthof

the word. Here 

’s denotes the highest weights of

the representations R

’s and  is the Weyl vector. The

action of the Weyl generator s



on a weight 



ð

Þ¼

 2

ð

;Þ

ð; Þ

and j L

=Lj is the ratio of weight and coroot lattices

(equal to the determinant of the Cartan matrix for

simply laced algebras). Also C

is quadratic Casimir

invariant for the adjoint representation.

It is important to stress that the expression

(G)

[M] is unchanged under both Kirby moves I

and II (for detailed proof, see Kaul (1999) and Kaul

and Ramadevi (2001)). Notice that for every

compact gauge group, we have a new 3-manifold

invariant.

Few examples of 3-manifolds Table 1 lists the

algebraic expressions of this invariant calculated

explicitly from the formula in eqn [12] for a few

3-manifolds. All these examples can be constructed

by surgery on an unknot U(f ) with differe nt frame

numbers f.

In Table 1 L[p, q] stands for Lens spaces of the

type (p, q) and C

is the quadratic Casimir invariant

n(i )

n ′(i )

H [X ] U [X ]

(b)(a)

Figure 6 Kirby move I.

–1

Figure 7 Kirby move II.

500 Schwarz-Type Topological Quantum Field Theory

for representation R of the Lie algebra of the gauge

group G.

Partition function of a CS theory on M is also an

invariant characterizing the 3-manifold. This has

been calculated for several manifolds by differe nt

methods. Invariant

(G)

[M] listed above for various

manifolds is related to the CS partition function

(G)

[M]:

(G)

[M] = S

1

(G)

[M]. So the method of

constructing 3-manifold invariants above can also

be used to calculate the partition function of CS

theories.

3D Gravity and CS Theory

Three-dimensional CS theory also provides a

description of gravity. The 3D gravity including

cosmological constant has been first discussed by

Deser and Jackiw (1984). The action with cosmolo-

gical constant =1=‘

is:

S ¼

16G

ﬃﬃﬃﬃﬃﬃﬃ

g

R  2ðÞ½13

G is the Newton’s constant, g



is the metric on the

3-manifold M, and R is scalar curvature. Solutions

of Einstein equations of motion have a consta nt

positive (negative) curvature if  is positive (nega-

tive). It is also well known that there are no

dynamical degrees of freedom for gravity in dimen-

sions D  3; it is indeed described by topological

field theories. The gravity action above can be

rewritten as a CS gauge theory in first-order

formulation (Carlip 2003). For triads e



and spin

connection !



of Euclidean gravi ty, we define

1-forms e = e



, ! = !



, which have

values in the Lie algebra of SU(2) whose generators

are T

= i

=2 with 

as three Pauli matrices.

In terms of these we define two gauge field 1-forms

A and



A as:

A ¼

‘

þ !



;



A ¼

‘

 !



Then the Euclidean gravity action can be written

in terms of two CS actions, S

[A] and S

[



A], as

S ¼ kS

½AkS



A½14

where the coupling constant k = ‘=(4G) for negative

cosmological constant =1=‘

. The gauge group

for this theory is SL(2, C). Infinitesimal diffeo-

morphisms are described by field-dependent gauge

transformations. The corresponding gauge group for

Minkowski gravity with negative cosmological con-

stant  is SO(2, R)  SO(2, R). For positive , one

gets SO(3, 1) and SO(4) for Minkowski and Euclidean

metrics, respectively. For =0, we have ISO(2, 1)

(ISO(3)) as the gauge group for Minkowski

(Euclidean) gravity. Hence, the sign of cosmological

constant determines the gauge group of the CS

theory.

Identification of 3D gravity with CS theory can be

used with some advantag e to find the partition

function for a black hole in 3D gravity with negative

cosmological constant. This in turn yields an

expression for entropy of the black hole.

BTZ Black Hole and Its Partition Function

Only for neg ative  we have a black hole solution of

the Einstein’s equations. This solution, known as the

BTZ black hole (Carlip 2003), in Euclidean gravity

is given by the metric

¼M þ





d

þM þ





1

þ r

d 

d



It is specified by two parameters M and J (the mass

and angular momentum). By a coordinate transfor-

mation, this metric can be rewritten as ds

)(dx

þ dy

þ dz

), with z > 0. This is the 3D

upper-half hyperbolic space and can be rewritten

using spherical polar coordinates as

sin



þ R

d

þ R

sin

d



We have the identifications (R, , )  (R exp {2r

=l},

 þ {2r



=l}, ) where r

and r



are the outer and

inner horizon radii, respectively. It is clear from this

identification that topologically the metric corre-

sponds to a solid torus. Functional integral over

this manifold represents a state in the Hilbert space

specified by the mass and angular momentum. It is

the microcanonical ensemble partition function and

its logarithm is the entropy of the black hole.

To evaluate this partition function, the connection

1-form is kept at a constant value on the toroidal

boundary through a gauge transformation. We

Table 1 Invariants for some simple manifolds

U(f ) M

(G)

[M]

U(0) S

 S

1=S

U(1) S

U(þ2) RP



1

U(þp) L[p,1] 

1

Schwarz-Type Topological Quantum Field Theory 501