[Lecture Notes in Physics New Series M, Monographs _ M36] Gerald Dunne - Self-dual Chern-Simons theories (1995, Springer)

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Gerald Dunne
Self-Dual
Chern-Simons Theories
Springer
Author
Gerald Dunne
Physics Department
University of Connecticut
Storrs, CT 06269, USA
Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Dunne, Gerald:
Self-dual Chern-Simons theories I Gerald Dunne. - Berlin;
Heidelberg; New York; Barcelona; Budapest; Hong Kong;
London; Milan; Paris; Tokyo: Springer, 1995
(Lecture notes in physics : N.s. M, Monographs ; 36)
ISBN 3-540-60257-7
NE: Lecture notes in physics / M
ISBN 3-540-60257-7 Springer-Verlag Berlin Heidelberg New York
This work is subject to copyright. Allrights are reserved, whether the whole or part of the
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For Elyse
Preface
This book is illtended to provide a pedagogical illtroduction to the
subject of self-dual Chern-Simons theories, which comprises a rela-
tively recent addition to the 'zoo' of self-dual field theories. I have
tried to give a detailed presentation of the basic features and prop-
erties of these theories. It is not possible to cover the entire subject
in an introductory text, so I have treated some of the more advanced
developments in less detail; nevertheless, I have attempted to balance
this compromise witIl an extensive 1)i1)liograpIly and all all1l0tated
overview of 1110re recent alld CtlITent "vork.
Tllis book grew fro111 various sell1illars alld lectures I Ilave given
over recent years. I thank Jilln E. I(iln and CII0011kyu Lee for the op-
portunity to presellt some of tIlis lnaterial at the X I 11th Symposium
on Theoretical Physics (Field Theory and Mathematical Physics)
held at Mt. Sorak (I(orea) in 1994. Some material is also drawn
from a series of seillinars at tIle University of COll11ecticut.
I 11ave learlled a great deal frolll tIle selnillars alld researcll l)al)ers
of colleagues worl{illg ill tIlis field, alld I \vould lil{e eSl)ecially to ac-
knowledge the piolleering contributiollS of Roman Jacki"v, Choonkyu
Lee, I<:ilnyeollg Lee, So-YOUllg Pi alld Ericl{ Weillberg.
It was a pleasure to collaborate ,vitIl Ronla,n Jacki,v, So-Young
Pi and Carlo Trugellberger 011 SOl11e of tllis material. In addi-
tiOll, I all1 grateful for discussiollS and correSl)OllClellce witIl Giovalll1i
Alnelill0-Calllelia, DOllgsu Bak, Daniel Cangell1i, i\lldrea Capl)elli,
Dall Freed111a,n, !(urt Haller, Peter HorvatI1Y, BUll1-Hoon Lee, Al-
berto Lerda and Guillerll10 Zell11)a.
Special tIlanks are due to Bo1) Ja,ffe for Ilis generous advice and
assistance cOllcerllillg tIle ptlblicatioll of tIlis book.
I anl grateful for partial support from tIle Departlnellt of Energy
and from the University of Conllecticut ResearcIl Founclation.
VII
Contents
I Introduction 1
A Self-Dual Theories . . . . . . . . 1
B Chern-Simons Theories: Basics . 9
II Abelian Nonrelativistic Model 16
A Lagrangian Formulation. 16
B Hamiltonian Formulation 19
C Static Self-DualSolutions 21
D Dynamical Symmetries 26
E Explicit Self-Dual Solutions:
The Liouville Equation . . . . . 33
F Time-Dependent Solutions .. 38
G Solutions in External Fields . . 40
H Maxwell-Chern-Simons Model 44
III Nonabelian Nonrelativistic Model 49
A Nonrelativistic Self-Dual Chern-Simons Equa-
tions: General Matter Coupling. . . . . . . . . .. 50
B Nonrelativistic Self-Dual Chern-Simons' Equa-
tions: Adjoint Matter Coupling. . . . 54
C Algebraic Ansatze and Toda Theories 59
D 'Yang' Approach. . . . . . . . . . . . 68
E Self-Duality Equations and the Chiral Model 70
F Unitons and General Solutions
IV
to the SDCS Equations ..
G Unitons and Toda Theories
Abelian Relativistic Model
A Lagrangian and Hamiltonian Formulation
B Self-Duality Equations .
IX
74
78
83
84
86
C
D
E
F
G
Self-Dual Solutions: Topological Vortices
and Nontopological ~olitons .
Nonrelativistic Limit .
Symmetry Breaking and the Cllern-Simons Higgs
Mechanism .
Self-Dual Chern-Simons Theories and Extended
Supersymmetry .
Maxwell-Chern-Simons Model
90
95
98
105
113
V Nonabelian Relativistic Model 119
A Relativistic Self-Dtlal Cllerll-Silnolls Equations:
General Matter Coupling . . . . . . . . . . . . . . 120
B Relativistic Self-Dual Cllern-Sill1011S Equatiolls:
Adjoint Matter Coupling . . . .. ..... 125
C Classificatioll of Millillla. . 131
D Vacuunl Mass Spectra . . . 135
E Mass Matrices for Real Fields . 143
VI Quantum Aspects 154
A Nonrelativistic Matter-Cllern-Silnons Field Tlleory 155
B Scale Invariance in Quantized Planar Models 166
C Aharollov-Bollm Scatterillg
and Cheril-Silllons Tlleory 170
D Quantum Aspects of Relativistic SDCS Tlleories . 180
VII Bibliography
x
188
I. INTRODUCTION
A. Self-Dual Theories
"Self-duality" is a powerful notion in classical mechanics and clas-
sical field tlleory, ill qualltum mecllallics and quantum field theory.
It refers to theories in which the interactions have particular forms
and special strengths such that the secolld order equations of motion
(in general, a set of coupled nOlllinear partial differential equations)
reduce to first order equations ,vllicll are silnpler to analyze. The
"self-dual point", at Wllicll tIle illteractiollS alld coupling strengtlls
take their special self-dual values, corresponds to tIle millilllization
of some functional, often the energy or the action. Tllis gives self-
dual tileories crucialJ)hysical sigllificallce. For exaillple, the self-dual
Yallg-Mills equatiolls 11ave lllillill1Ulll action solutiollS kll0Wll as ill-
stantons, the Bogomol'nyi equatiolls of self-dual Yallg-Mills-Higgs
theory llave nlinimum ellergy solutions knowll as ~t Hooft-Polyakov
monopoles, and tIle plallar Al)elian Higgs lllodel 11as millill1um en-
ergy self-dual solutiollS l(nown as Nielsen-Olesen vortices. Instan-
tons, monopoles and vortices 11ave become paradigms of topologi-
cal structures ill field theory and qualltuln lnechallics, with impor-
tant applications in particle pllysics, astropllysics, condensed matter
pllysics alld l11atllelllatics. III tllese Lecture Notes, I discuss a llew
class of self-dual tlleories, self-d'llal Chern-Simons theories, Wllicll in-
volve cllarged scalar fields l11illill1ally coupled to gauge fields wll0se
'dynamics' is provided l)y a Cllerll-Sill10l1S terlll ill 2 + 1 dilnell-
sions (i. e. two spatial dilnellsions). TIle pilysical COlltext in which
tilese self-dual Cllerll-Silnolls nl0dels arise is tllat of allyollic quan-
tUlll field theory, with direct a!)plicatiolls to SUCII !)lanar l110dels as
tIle quallttlill Hall effect, allyollic Stl!)erCOllductivity alld Allaronov-
Bohm scattering. In additioll, tllere are furtller illteresting COllllec-
1
tions with tIle more mathematically inspired theory of integrable
models.
A novel feature of these self-dual Chern-Simons theories is that
they permit a realization with either relativistic or nonrelativistic
dynamics for the scalar fields. Self-dual theories for instantons,
monopoles and (Nielsen-Olesen) vortices arise from relativistic field
theories. III the nonrelativistic case, the self-dual point of tIle Chern-
Simons system corresponds to a quartic scalar potential, with overall
strength deterillilled by tIle Cllerll-Silll0ns couplillg strengtll. The
nonrelativistic self-dual Cllerll-Silll0llS equatiolls lllay be solved com-
pletely for all finite charge solutions, alld the solutions exllibit many
interesting relations to two dimensional (Euclidean) integrable mod-
els. The classification of these finite cllarge solutions is also strongly
analogous to the classificatioll of instanton solutiolls ill four dimen-
sional Euclidean spacetinle. In the relativistic case, while tIle general
exact solutiolls to tIle self-duality equatiolls are llOt eXI)licitly kllOWll,
the solution~ correspond to topological and nontol)ological solitons
and vortices, mallY cllaracteristics of wllich call be deduced from al-
gebraic and asymptotic data. These self-dual Cilern-Silllons theories
also have the property that, at the self-dual point, tlley may be em-
bedded into a model witll all extellded sUI)ersylllnletry. Tllis serves
as another explicit illustratioll of tllis gelleral featllre of self-dual tlle-
ories - indeed, recent ,vork has sougllt to characterize self-duality di-
rectly ill terlllS of a related extended sUI)ersyll1l1letry structure. Tllis
approach is especially attractive, as it provides a bridge between the
classical and tIle quantum analysis of these models.
Before introducing the self-dual Chern-Simons theories, in this
Illtroductioll I briefly review SOl11e otller ill11)OI'tallt self-dual tlleories,
in part as a meallS of illustratillg tIle general idea of self-duality, but
also because various specific prOI)erties of tllese theories ,viII apI)ear
2
(1.1)
explicitly ill our analysis of tIle self-dual Chern-Simons systems.
Perhaps tIle most familiar, and in a certain sense the most funda-
mental, self-dual theory is that of four dimensional self-dual Yang-
Mills theory. The Yang-Mills action is
SYM = Jd4x tr (Fp.vFp.v)
wllere FJ-lv == oJ-lAv - ovAJ-l + [AJ-l' Av] is tIle gauge field curvature.
TIle Euler-Lagrange equations forln a complicated set of coupled
nonlinear partial differential equations:
DJ-lFJ-lV = 0 (1.2)
wllere DJ-l == oJ-l + [AJ-l' ] is tIle covariant derivative. Ho\vever, in
four dimensional Euclideall space tIle Yang-Mills actioll (1.1) is nlin-
ilnized by solutions of the self-dual (or anti-self-dual) Yang-Mills
equations:
(1.3)
wllere FJ-lv - ~EJ-lvpaFpa is tIle dual field strellgth. Note tllat tIle
self-duality equatiolls (1.3) are first order equatiolls (ill contrast to
tIle secolld order equatiolls of lnotioll (1.2)), and tlleir "illstallton"
solutiollS are kllO\Vll ill detail (for a review see [211,212,168]). Also,
solutiollS to tIle self-duality equations (1.3) are automatically solu-
tions to tIle origillal Euler-La,grallge equatiolls (1.2) since tIle dual
field strellgtll FJ-lv satisfies tIle Bianclli idelltity DJ-lFJ-lv = 0, illde!Jen-
dent of allY equations of 1110tioll. We sIlal1 see tllat tIle llollrelativistic
self-dual Cllern-Simolls equations llave nlany illterestillg connections
witll these self-dual Yallg-Mills equatiolls.
Another important class of self-dual equatiolls consists of the
"BOgOI1101'llyi equations"
(1.4)
3
where <I> is a scalar field. The Bogomol'nyi equations arise in the
theory of lTIagnetic mOll0poles in 3 + 1 dimensional space-time.
They conle from a millinlization of tIle static energy functional of a
Yang-Mills-Higgs system in a special parametric limit known as the
Bogomol'nyi-Prasad-Sommerfield (BPS) limit [210,24]. It is interest-
ing to note that these Bogomol'nyi equations can also be obtained
from the (anti-) self-dual Yang-Mills equations (1.3) by a 'dimen-
sional reduction'from four dimensions to three dimensions. In the
allti-self-dual Yallg-Mills equatiolls (1.3) consider all fields to be in-
dependent of x 4 , and identify A4 ,vitll the scalar field <I>. Then the
equations (1.3) reduce to
D 1<I> = -F23
(1.5)
wllicll are precisely tIle BOgOITIol'llyi equatiolls (1.4). We shall see
that the nonrelativistic self-dual Cllern-SilTIOnS equations nlay also
be obtained from the self-dual Yang-Mills equations by a similar di-
mensional reduction, altll0ugll in tllat case the dimellsional reduction
is from four dimensiolls to t\VO dilTIellSions. FurtllerlTIOre, the rela-
tivistic self-dual Cllern-Sinl0lls equatiolls illvolve a Sl)ecial algebraic
embedding problelll (tllat of eillbeddillg SU(2) illto tIle gauge alge-
bra) wllicll also plays a crucial role ill tIle allal)'sis of tIle BOgOlll01'llyi
equations (1.4).
The abelian Higgs model in 2 + 1 dimensions is a model of a
complex scalar field <P illteracting ,vitll a U(1) gauge field witll con-
ventional Maxwell dynamics. This model supports vortex solutions
known as Nielsen-Olesell vortices, alld is a relativistic analogue of
4
the Landau-Ginzburg pllenomenological model for superconductiv-
ity [196,24,137]. The static energy functional for this system is
Tllis static energy may be re-expressed as
E = Jd2x[I(Dj - iEjkDk) <p1 2 + ~ (B + 1<p1 2 - v2 )2 + v2B
-~ (1<p1 2 - v2) 2 +V (I <pI)]
Thus, for a special "self-dual" quartic potential,
1 (2 2)2~elf-dual = 2" 1<p1 - v
(1.6)
(1.7)
(1.8)
the static ellergy fUllctiollal is boullded below by (v2 tillles) tIle mag-
netic flux, alld this "BOgOI1101'llyi bound" is saturated by solutions
to the following set of self-duality equations:
(1.9a)
(1.9b)
TIle term "self-duality" arises fronl the appearance of the duality
operation ill (1.9a). Tllese equations llave vortex solutiollS wllich
llave been studied ill great detail [196,24,137,245]. We sllall see that
the self-duality equatiolls ill tIle self-dua.! Cllern-SilllOl1S systeills also
arise from minimizing the energy functional in a 2 + 1 dilllensional
theory, and the resulting Cllern-SilllollS self-duality equations have a
very sinlilar forl11 to the abelial1 Higgs Illodel self-duality equations
(1.9). Tllis is true of botll tIle relativistic and the nonrelativistic
self-dual Chern-Simons systems.
Yang [252] proposed all a!)prOaCII to tIle four dilnensiollal, self-
dual Yang-Mills equatiolls (1.3) ill Wllicll tlley call l)e viewed as the
5
consistency conditions for a set of first order differential operators.
This idea is fundamental to the notion of "integrability" of systems
of differential equations, a subject with many deep connections. to
self-dual theories [241-243,255,42,114]. Indeed, it has been conjec-
tured that all low-dimensional integrable systems can be obtained
from the self-dual Yang-Mills equations (1.3) by some sort of di-
mensional reduction, for some particular gauge algebra. This re-
lationship has been explicitly studied for classic integrable systems
SUCll as tIle 110lllillear Schrodillger eqllation, tIle I<dV equatioll, tIle
Sille-Gordoll eqllation and otilers. TIle general relatioll is due to tIle
zero-curvature apl)roacil to illtegral)ility, Wilicll 11as a 11atural fOflTIU-
lation in terms of gauge tlleory [206,82-,42].
If tIle self-dual Yallg-Mills equatiolls (1.3) are rewrittell in terlns
of the null coordinates u = (xl + ix2)/ V2 and v = (x3 + ix4)/ V2,
they become '
Fiiv = 0
(1.10)
Tllese equations express tIle COllsistellcy conditions for tIle first order
equatiolls
(1.11)
where ( is known as a "spectral parailleter". TIle first two equations
in (1.10) can be solved locally to give
6
(1.12)
where Hand !{ are gauge group elements. Then, defining J =
H !{-l, the third of the self-duality equations in (1.10) take the
'Yang' forln [252]
(1.13)
If we now mal(e a dimensiollal reduction ill ,vllicll tIle fields are CllO-
sen to be independent of x 2 alld x 4 , tllis equatioll becollles the two
dimensional equation
(1.14)
wllicll is known as tIle chiral model equation. TIle clliral model equa-
tion will playa very important role in our analysis of tIle nonrelativis-
tic self-dual Cllerll-SilllOllS equatiolls. Also 110te tllat if J E SU(N)
and J is furtller restricted to satisfy the condition J2 == 1, then
(1.14) is the equation of motioll for tIle CpN -1 model, which is yet
another well-knoWll self-dual systelll [211,256].
TIle fillal class of 11lodels tllat we recall in tllis Illtroductioll are
kllown as Toda tlleories. TIle origillal Toda systelll described tIle
displacements of a line of masses joilled by sprillgs witll an exponen-
tial sprillg tellsion [230]. Tllis provides a beautiful 110l1lillear, but
still integrable, gelleralizatioll of tIle stalldard lillear (Hooke's law)
model. TIle equatiolls of 1110tioll for tIle Toda lattice are
(1.15)
where the matrix Cij is tIle tridia,gonal discrete approxilllation to the
second derivative, and call be cllosen for periodic or opell boundary
conditiolls. This system is classically integrable ill tIle lilllit of an
7
infillite number of masses, in tIle sense that it possesses an infinite
number of conserved quantities in involution. The Toda lattice sys-
tem also has a deep algebraic structure due to the fact that the
matrix Cij in (1.15) is tIle Cartan matrix of the Lie algebra SU(N)
(or its affine extension). Indeed, tllis relationship allows one to ex-
tend the original Toda system to a Toda lattice based on other Lie
algebras.
The Toda system gelleralizes still further, to all illtegrable set of
nonlinear partial differential equations ill two dimensions
(1.16)
where V'2 is tIle two dimensiollal Laplacian. Tllese Toda equations
are not only illtegrable, but also solvable, ill tIle sense that tIle so-
lution may be writtell ill terms of 21' arbitrary functions, wilere r is
tIle rank of tIle classical Lie algel)ra \vhose Cartall lllatrix alJpears
ill (1.16) [158,177,192]. For SU(2) tIle classical Toda systelll reduces
to the nonlinear Liouville equation
(1.17)
which was solved by Liouville [182], and ,vllicil llas played a signifi-
callt role ill string tlleory alld 1110dels of quantuill gravity. Botll the
Liouville alld Toda equatiolls, togetller witll tlleir solutions, appear
pronlinelltly ill tIle allalysis of tIle 110nrelativistic self-dual Cllerll-
Simons models. Moreover, tIle Toda equatiolls also arise from the
Bogomol'llyi equatiolls (1.4) wIlen Olle looks for spllerically symmet-
ric monopole solutiolls [179]. Tllis reduction illvolves an algebraic
embedding problem very silnilar to Olle tilat appears ill tIle treat-
ment of the relativistic self-dual Cileril-Simons lnodels.
8
(1.18)
B. Chern-Simons Theories: Basics
The self-dual Chern-Simons theories discussed in these Lecture
Notes describe charged scalar fields in 2 + 1 dimensional space-time,
minimally coupled to a gauge field whose dynamics is given by a
Chern-Simolls Lagrangian rather than by a conventional Maxwell (or
Yang-Mills) Lagrangian. The possibility of describing gauge theories
with a Chern-Simons terlU rather tllall with a Yang-Mills term is a
special feature of odd-dimensional space-time, and the 2 + 1 dimen-
siollal case is especially distillguislled ill tIle sellse tllat the derivative
part of the Cherll-Silll0ns Lagrallgiall is quadratic ill tIle gauge fields.
To COllclude tllis Illtrodllction, I l)riefly review SOllle of tIle illlportallt
prOI)erties [46,249,54,55] of tIle Cllerll-Silll0llS Lagrallge dellsity:
£cs = fJiVPtT (OJiAvAp + ~AJiAvAp)
The gauge field AJl takes values in a finite dilllellsiollal represelltation
of tIle gauge Lie algel)ra g. TIle totally alltisylllilletric E-SYlll1)01 EJlVP
is normalized witll E012 == 1. III an abeliall tlleory, tIle gauge fields
AJl commute, alld so tIle trilillear terlll ill (1.18)vanislles due to
tIle antisylllllletry of tIle E-Syillbol. TIle Euler-Lagrallge equations of
motioll derived from tllis Lagrallge dellsity are sillll)ly
FJlV == 0
wllicll follows directly frolll tIle fact tllat
8£cs _ JiVpF
£5A - E vp
Jl
(1.19)
(1.20)
At first sigllt, tIle equa.tiolls of 1110tiol1 (1.19) Seel1l s0111ewllat trivial,
with solutions tllat are Sillll)ly l)llre gallges AJl == g-18Jlg. However,
much recent work llas revealed tllat "vitll tIle inclusioll of topological
effects and exterllal sources tllese equatiolls are far frolll trivial, as
9
they possess an extraordinarily ricll structure witll important appli-
cations to conformal field theory, quantum groups and condensed
matter physics [249,25,76,257,90].
For our purposes here, it is very important to notice that the
equations of motion (1.19) are first-order in space-time derivatives,
in contrast to the Yang-Mills equations of motion (1.2) which are
second-order. One viewpoint of self-dual theories is that the equa-
tions of motion are such that they luay be factorized into simpler
first-order equatiolls. But for a gauge tlleory witll a Cllerll-Siluons
Lagrange density, tIle gauge equatioll of lllotion is already first-order
and so it can be used directly as Olle of tIle self-duality equations.
Later, we sllall see ill detail 11ow tllis fact !)roves useflll ill tIle COlltext
of tIle self-dual Cllern-Sinlons 1110dels.
TIle equatiolls of 1110tioll (1.19) aTe gauge covariallt under tIle
gauge trallsforluation
A Ag - -IA -laJl --t Jl = 9 Jlg + 9 Jl9 (1.21 )
and so tIle Lagrange dellsity (1.18) defilles a sellsil)le gauge tlleory
(at least at the classical level) evell tllOUgll tIle Lagrange density
(1.18) itself is 110t illvariallt ullder tIle gauge trallsforll1atioll (1.21).
Illdeed, ullder a gauge trallsforll1atioll £cs trallsforllls as
(1.22)
For an abeliall Cherll-Sill1011S tlleory, tIle final terll1 ill (1.22) vallisiles
alld tIle Clla.Ilge ill £cs is a total s!)aCe-tillle derivative. Hellce tIle
action, S = Jd3x£cs, is gauge illvariallt and so ,ve expect that a sen-
sil)le qualltull1 gauge tileory 11lay l)e forll1ulated. Ho,vever, for a non-
abelian Cllern-Sill1011S tlleory tIle fillal terlll ill (1.22) is !)ro!)ortional
10
to the winding number of the group element g, alld so the Chern-
Simons action changes by a constallt under a gauge transformation
with nontrivial winding number. This has important implications
for tIle developlnent of a qualltum nonabelian Chern-Simons theory.
To ensure that the quantum amplitude exp( i S) remains invariant,
the Chern-Simons Lagrange density (1.18) must be multiplied by a
dimensionless coupling parameter K' which assumes quantized values
[46,249]
, iT~teger
K = (1.23)
47r '
witll standard normalizatiolls. Tllis argulnellt for a quantized cou-
pling parameter is remilliscent of Dirac's quantization condition for
the quantuln mecilallicalillagiletic nlOllO!)ole [50,51,100], wllicll was
developed further for a field tlleory of monopoles by Schwinger
[220-222]. In a field tlleory, Wllicll in gelleral will require renor-
malization, olle call ask wlletller tIle quantizatioll COllditioll (1.23)
a!)!)lies to tIle bare cou!)lillg or to tIle rellOI"lllalized couplillg. Fortu-
nately, Chern-Simons theories perlnit a deeper probing of this ques-
tiOll, witll tIle result tllat at olle-Ioo!) tIle bare couplillg !)arailleter ill
a llonabeliall theory receives a finite additive renormalization sllift
which is such that 47rK is sllifted by all integer. Moreover, there are
strong indications (botll computatiollal and fundamental) tllat this
result is valid to all orders. Tllese COlll!)utatiolls were illitially per-
formed for gauge tlleories involving botll a Yang-Mills alld a Chern-
Simons term in tIle gauge field Lagrangian [205], alld tllen extended
to pure Cllern-Simons tlleories [249,25,76]. More recently, tllis issue
has been explored, witll similar conclusions, witll tIle illclusion of
matter fields alld also spolltalleous sylllinetry breakillg [33].
All0tller ilnportallt feature of Cllerll-Silnolls tlleories is tllat tIle
Cllerll-Silll0l1S terln descril)es a to]Jologicalga,llge field theory [249,23]
ill tIle sellse that tllere is 110 eXI)licit depelldellce 011 tIle SI)aCe-tinle
11
metric. This follows because the Lagrange density (1.18) can be
written directly as a 3-forlll : £cs = tr(AdA + A3 ). Thus tIle action
is illdepelldent of tIle space-time metric, and so tIle Chern-Simons
Lagrange density Lcs does not cOlltribute to the energy momentum
tensor. This may also be understood by noting tilat Lcs is first
order in space-tinle derivatives
(1.24)
The time derivative part of Lcs illdicates tIle canonical structure of
tIle tlleory, witll Al alld A2 beillg callollically cOlljugate fields. Tllis
is radically differellt frolll cOllventiollal (Yallg-Mills) gallge tlleory ill
wllicll the gauge field COlllpollellts Ai 111ay be regarded as coordillate
fields, callollically conjugate to tIle electric field E i =FOi • TIle Ao
part of the Lagrange dellsity produces tIle Gauss law constraillt, and
there is no contribution to the Hailliitollian. Tllis ill1plies tilat tIle
Cllerll-SilllOl1S gauge field does not llave allY real dYllalllics of its o,,,n
- it is a 110111)rOpagating field \VllOSe dYllalllics COllles frolll tIle fields
to Wilicil it is 111illilllally COUI)led. Th~ l)recise illll)leillelltatioll of
this classical canollical strlIcture in a, qualltum tlleory leads to IllallY
illteresting features ill Cllerll-Silll0llS tlleories.
TIle Cllern-Simolls Lagrallge dellsity (1.18) lllay be COUI)led to
an exterllal nlatter currellt JI-t as
K
£, = 2"£'cs - ir (A,JJl) (1.25)
(Note tllat we include tIle factor of 1/2 in tIle Cileril-Silllons couplillg
coefficiellt for later cOllvelliellce.) Tllis leads to tIle equatiolls of
IllOtioll [gelleralizillg Equation (1.19)]
1
FJ-LI/ = --EJ-Ll/pJP (1.26)
K
involving tIle covarialltly conserved (DI-tJJ1. = 0) current. TIllIS, the
till1e COlllpollellt JO of tIle lllatter currellt is proportiollal to the mag-
lletic field
12
(1.27)
which is the Cilern-Simons Gauss law constraint, and which is im-
portant for the interpretation of Chern-Simons theories as field the-
ories for anyons [175]. The spatial components Ji of tIle current are
everywhere perpendicular to the electric field
(1.28)
With this coupling to externalillatter, the equations of motion (1.26)
are still first order in spa,ce-time derivatives a,ctillg on the fields (in
contrast to tIle correS!)Olldillg equatiolls of 11l0tio11, DJ1.FJ1.V == JV, ill
conventional Yang-Mills tlleory). Tllis fact is crucial for tIle forillu-
latiol1 of self-dual tlleories illvolving Cllerll-Silll0llS gauge fields. TIle
relatiolls (1.27-1.28) are fUl1daillel1tai to tIle a!)plicatioll of Cllern-
SilllOI1S tlleories to condensed matter systenls sucll as tIle quantum
Hall effect [209,229,86,257].
TIle last property of Cilern-Sinl0ns theories tllat we melltion ill
this Illtroduction is tllat tIle Higgs 111ecllallism bellaves very differ-
ently wIlen tIle gauge fields are Cllern-Simolls fields. In a COllven-
tional gauge tlleory, tIle Higgs 11lecllallislll produces a 111assive gauge
mode in a brol{en vacuum in \vIlicil the scalar field, to wilich tIle
gauge fields are coupled, possesses a llonvanisllillg vaCUUlll expecta-
tion value. Forillally, tI1is mechallislll is illdependellt of the dimen-
sion of spacetime, but in 2+ 1 dimensiolls tIle possibility of illcluding
a Chern-SilllollS term for tIle gauge fields leads to a ricller variety of
111ass gelleratioll effects. For exaillple, evel1 \vitl10ut any SYlllllletry
breal{illg at all, a gauge tlleory \vitl1 l)otll a Yallg-Mills alld a Cllerll-
Silllons terlll describes a 111assive dYllanlical gauge 11lode, witil IllasS
deterillilled by tIle Cllerll-Silll0llS couplillg paralneter/\', alld witll
Spill ±1 given l)y the Sigll of /\'. T11is systenl llas beell dubl)ed "to!)O-
logically massive gauge tileory" [46]. TIle 110vellnass alld Spill pro!)-
erties of fields in 2 + 1 dill1ellsioilS nlay be ullderstood ill ternlS of
13
the representations of the corresponding Poincare and Lorentz alge-
bras [22,223,18,234]. Now if such a gauge field is coupled to a scalar
field with a symmetry-breaking minimum in its potential, the 'Higgs
mechanism' leads now to two massive modes, as one new mode is
generated by combining the Goldstone boson with the longitudinal
part of the gauge field in the standard manner, and tIle mass of the
existing (in the unbroken vacuum) topologically massive photon is
also shifted [205,246,110]. Furtllerillore, if tIle Yang-Mills term is
not presellt (alld so tllere is no lllassive gauge lllode in tIle Ulll)ro-
ken vacuuIll) thell tIle 'Clleril-Silllons-Higgs mecllanislll' produces
a single massive gauge excitation ill a syillmetry breaking vacuum
[48]. However, since tIle Cherll-Simons ternl is first-order in space-
time derivatives, the mass of tllis gauge excitation is proportional
to the square of the vacuum expectation value of tIle scalar field, in
COlltrast to tIle cOllvelltiollal Higgs lllecllallism for wllicll tIle gauge
mass is proportiollal to tIle magnitude of tIle scalar field vaCUUIll
expectatioll value. The relativistic self-dual Cllerll-SilllOllS tlleories
described ill tllese Lecture Notes illvolve a self-dual scalar field po-
tential which possesses nontrivial symmetry brea,king vacua. The
quantum analysis of these lllodels tllerefore provides an illterestillg
forum for application of tIle Cllern-Simons-Higgs mechanism. We
shall see tllat tIle particular self-dual forlll of the scalar potential
leads to illtricate lllass sl)ectra in tIle brokell vacua.
TIlere is, of course, lllucll lllore tIlat could be writtell about
Chern-Simons models, but tIle above brief outlille covers tIle fea-
tures that are most inllllediately relevallt to tIle tOI)ics discussed ill
these Lecture Notes. Further details will be introduced as necessary
at the appropriate POillts ill tIle discussion tIlrougIlout tIle book. For
general field theoretical properties of Cllerll-Simons tIleories, tIle ill-
terested reader is referred to [46,249,123,208,11], witll apI)lications
14
to conformal field theory discussed ill [76,25]. For applications of
CIlern-Simons theories to allyon pIlysics see [85,90,97,118,175,247].
General applications to planar cOlldensed lnatter physics are dis-
cussed in [86,234]; with tIle quantum Hall effect in [209,229,257] and
anyonic superconductivity in [185].
15
II. ABELIAN NONRELATIVISTIC MODEL
III this Cllapter we i11troduce tIle abelian nonrelativistic self-dual
Chern-Simons model [125]. Tllis 1nodel serves as a field theoreti-
cal description of anyons - point particles in the plane with frac-
tional statistics. In this Chapter we concentrate on the symmetry
and self-dual structure of the classical field theory, deferring dis-
cussion of quantum aspects to Chapter 6. This 2 + 1 dimensional
theory has a Bogomol'llyi style lower bound for the e11ergy whe11 the
scalar pote1ltial takes a purely quartic for111 , tIle overall strengtll of
Wllicll depends 011 tIle Cllerll-Si11101lS coupli11g K. Tllis e11ergy lower
bOUlld (Wllicll is ill fact zero) is saturated l)y solutiollS to a set of
self-duality equations a1ld, as a consequence of a special dY11amical
symmetry, tllese self-dual solutiollS exllaust all static solutiollS. The
self-duality equations Inay be COllverted into the Liouville equatioll,
all integrable ll0nlillear partial differential equatioll \Vll0Se solutions
are explicitly kll0Wll. Tllis tllerefore leads to a COll11)lete cllaracteri-
zation of the vortex-like self-dual solutiollS. Otller 11lore COll11)licated
solutiollS 11lay tllell l)e gellerated fro111 these l)y sllital)le coordi11ate
tra11sfornlatio11s. Finally, we SIIOW 110W to ge11eralize tllis 1110del to
include Maxwell dynall1ics for tIle gauge field, \vllile still preserving
tIle self-dual structure.
A. Lagrangian Formulation
TIle al)elia11 11011relativistic self-dual ClleI'l1-Silll0IlS systell1 de-
scril)es a C01111)lex scalar field 1/J( X, t) \vhicll is 111i11ill1ally COUI)led to
all abelian gauge field AJ.l (x, t). Tllis 1110clel is defilled ill two dill1ell-
sio1lal sl)ace (wllicll is takell to l)e R2 , unless otller\vise Sl)ecified) a11d
the dynamical equations for tIle matter field 1/J (and its conjllgate,
16
(2.1 )
1/;*) are nonrelativistic. However, tIle gauge field AJl is most conve-
niently expressed using a "Lorentz covariant" notation AJl = (Ao, Ji)
with "Minkowski metric" gJlV = diag(-1, 1, 1) and with "c" set to
unity. The gauge field is chosen to have only a Chern-Simons term in
the Lagrange density. Tllis has the consequence that the gauge field
does not possess any dynamics of its own - in fact, tIle gauge field is
determined by the nonrelativistic matter. Thus, even thougil some
of tIle formulae involvillg tIle gauge field look relativistic, tilis system
is illdeed llollrelativistic. TIle La,grange dellsity for tllis systell1 is
K 1 I.... 12 9 4£, = 2€IlV PAIl8v Ap + i7jJ* Do'l/J - 2m D'l/J + 2I'l/JI
Here m is tIle 111ass of tIle scalar field 1/;~ K is a COUI)ling constallt
Wilicil deterl11illes tIle strellgtil of tIle Cilerll-Sill101lS terl11
(2.2)
and 9 is a couplillg COIIstailt Wllicll deterillines tIle strellgtil of tIle
11/;14 IIOlllillearity. TIle totally antisyll1111etric Syll11)01 EJlVP is cllosen
witll E012 = +1. TIle gauge covariallt derivative DJl is defilled as
(2.3)
The Euler-Lagra.llge equations of 1110tioll Wilicil follow fro111 tIle
Lagrange dellsity (2.1) are
(2.4a)
(2.41))
wilere JJl =(p, J) is a Lorelltz covariallt 110tatioll for tIle (co1Iserved)
n01lrelativistic cilarge a1ld curre1lt dellsities:
(2.5a)
17
(2.5b)
The matter equation of motion (2.4a), togetller with the field-
current relation (2.4b), is referred to as the planar gauged nonlinear
Schrodinger equation [125].
The equations of motion (2.4) are invariant under tIle abelian
gauge transformation
(2.6)
The Lagrange density (2.1) is not in,rariant under tllis gauge trans-
forlllation as the CIlerll-SilllOllS ternl cIlanges by a total derivative:
(2.7)
Nevertlleless, tIle Lagrallge dellsity (2.1) defines a sensible gauge
tlleory (at least at tIle classical level) because tIle equations of lllotion
(2.4) are gauge invariallt.
TIle gauge field equatioll of nlotioll (2.4b) may be re-ex!)ressed
as
1
B= -p
K
wIlere tIle !)lallar "lllaglletic field" strellgtll is
tIle planar "electric field" strengtIl is
(2.8a)
(2.8b)
(2.9)
(2.10)
and we use the spatial alltisymmetric symbol Eij witll E12 == +1. In
terms of tllese gauge invariant quantities, tIle equations of motion
18
(2.8) imply that the charge density p is proportional to the magnetic
field B, alld the currellt density J is perpendicular to the electric
field E. This fact is ilnportallt for pllenomellological applications
of Cllern-Simons theories as effective field tlleories for the quantum
Hall effect [257]~ Illdeed, tIle abelian model discussed in tilis Chapter
has been gelleralized to a multi-abelian [U(l)]N theory [149] (see also
[73]), which is relevant for the fractional quantum Hall effect.
As a consequence of tIle Euler-Lagrange equations of motion
(2.4), tIle cilarge alld current dellsities satisfy tIle cOlltinuity equation
(2.11)
wllicll is cOllvellielltly expressed as
(2.12)
USillg tIle "Lorelltz covariallt" llotation. TIle llet cllarge
(2.13)
is cOllserved alld, as a COllsecluellce of (2.8a), is !)ro!)oItiollal to tIle
totaIlllaglletic flux
(2.14)
with correspondence
(2.15)
B. Hamiltonian Formulation
The abelian nonrelativistic systeln descril)ed in SectionII A 1llay
also be described ill a Hall1iltolliall forillulatioll, a fact tilat will prove
19
useful in ullderstandillg tIle static solutions of the system. TIle La-
grange density (2.1) may be rewrittell as
(2.16)
where irrelevant total derivative terms have been dropped. From this
first-order form of the Lagrange density, the field Ao is recognized as
a Lagrange multiplier field wllicll enforces the Chern-Simons Gauss
law constraint (2.8a). This constraillt may be solved by expressing
the vector !)otelltial A in terlllS of tIle cllarge density p as
... 1 J 2 ,'" ..., ...,A = -;, d x G(x,x )p(x)
where the Greell's fUllctioll G(x, X') is .defined byl
G(x, X') = G(x - X')
(2.17)
(2.18a)
(2.18b)
(2.18c)
This systelll is tllerefore a constrailled Hamiltollian system, witll
hamiltoniall
(2.19)
where tIle vector potelltial A wllicll appears illside tIle covariallt
derivative D'ljJ is given l)y tIle expression (2.17). TIle hanliltonian
lSome special properties of these definitions for the Green's function
have been discussed in the literature [106,130].
20
(2.19) is therefore a nonlocal, nOlllillear expression in terms of the
lnatter field 'ljJ. The tillle evolution for the matter field 'ljJ is obtained
from the Heisenberg equation
(2.20)
where it is important to remember that in varying H with respect to
1jJ* Olle must also vary Awitll respect to 'ljJ* accordillg to the relation
(2.17). Tllis generates the Ao1/J term appearillg in tIle original matter
equation of 1110tioll (2.4) ,vitll ..40 clefined ill terlllS ~f tIle lnatter fields
as
1/ 2 ,-+ -+( -+-+(Ao = -- d x G(x, x ) · J(x )
K
(2.21 )
Note tllat tllis relatioll is consistellt ,vitll tIle 'other' gauge field equa-
tion of lllotioll (2.8b) a,lld witil tIle tilne evolutioll of tIle constraint
(2.17) defilling A in terll1S of p. Tilese equatiolls definillg tIle gauge
fields ill terlns of tIle 111atter fielcls illvolve a cll0ice of gauge (for
eXallll)le, tIle relation (2.8) deterlllines A Ollly UI) to a divergellce
terI11), all issue whicll ,ve shall cOllfrollt ill tIle clualltizatioll of tllese
systell1s ill Cllapter 6.
c. Static Self-Dual Solutions
To seek soIutiollS to tIle Euler-Lagra.Ilge eClllatiolls (2.4) ,ve sllall
filld it useful to 11lake a ·'self-clual" allsatz for tIle lllatter fields [125].
Su!)pose tile 111a,tter field ~ satisfies tIle self-dual allsatz:
Tllis Inay be written in a more cOlllpact form
D=r-~=O
21
(2.22)
(2.23)
if we introduce tile characteristic coordinates
(2.24a)
(2.24b)
(2.24c)
Wit}l tIle fields subject to tIle ansatz (2.23) (eqllivalelltly (2.22)),
the currellt dellsity (2.5b) simplifies to
(2.25)
and so the electric field equatioll of motion (2.8b) becomes:
(2.26)
It is also useful to record tIle followillg factorization identity:
(2.27)
This identity, wllich factorizes tIle covariant Laplacian (producing
an additiollal B1/J term) plays a proillinent rqle throughout these
Lectures. Using the magnetic field equation of motioll (2.8a) wilich
relates F12 to the cilarge dellsity p, we see tilat tIle nlatter field
equation (2.4a), tIle gauged 11011lillear Scilrodillger equatioll, becoilles
(2.28)
Witll tIle self-dual ansatz, tIle first terlll 011 tIle RHS of (2.28) van-
ishes, givillg
iao1/; == - ((g =F _1) 11/J1 2 - Ao) 1/J
21TIK,
22
(2.29)
By inspection, we see that tIle equations (2.26) and (2.29) with the
self-dual ansatz (2.23) are solved by static solutions
oo1/J = 0
(2.30)
witll Ao cIlosell as
(2.31)
provided tIle strengtIl 9 of tIle lloillinear coul)ling ill (2.1) is cIlosen
to take tIle Sl)ecial critical vallIe
1
g=±-
mK
(2.32)
(2.33)
wIlich we sIlal1 refer to as tIle "self-dual couplillg". TIle correspond-
illg self- dual Lagrallge dellsity is
K 1 1- 12 1£ == _EJ-LVPA ovA + i1/J* Do1/J - - D1/J ± - 11/J1 42 J-L P 2111, 2mK
We tI1US arrive at tIle "llollrelativistic self-dual Clleril-Silllons
equations" :
D~1/J == 0
1 2
F12 == -11/J1
~
(2.34a)
(2.34b)
As described above, ,vitIl tIle critical value (2.32) for tIle nonlillear
coupling g, solutiollS to tIle llollrelativistic self-dllal CIlern- Silllons
equations provide static solutiollS to tIle Euler-Lagrange equations
of motion. Also note tIlat the self-duality equations (2.34) are first-
order equations, rather tIlan the second-order equatiolls of lllotion
(2.4). TIlis is a familiar feature of self-dualillodels, as was illdicated
ill tIle Illtroductioll.
23
A clearer illSigilt illtO tIle physical nature of tllese self-duality
equations COllles frolll COllsidering the Hamiltoniall formalism. The
factorization identity (2.27) may be re-expressed as:
(2.35)
Using this identity, together with the Gauss law COllstraint (2.8a),
we filld that tIle Hanliltolliall (2.19) is
(2.36)
where we Ilave drOI)ped a Sl)atial boulldary terlll ~ JEijaiJj . Witll
the critical self-dual couplillg (2.32), tIle energy reduces to
(2.37)
whicll is lllanifestly positive, alld Wilicil is lllillilnized by cOllfigura-
tiOllS satisfyillg tIle self-duality equatiolls (2.34). Sillce H is lllin-
imized by tIle self-dual soIutiollS, tllese solutiollS llecessarily corre-
spond to static solutions of tIle Euler-Lagrange equations of motion,
as delnollstrated explicitly above. III fact, tIle relationsllil) between
tIle static SOllltiollS alld tIle self-dllality equatiolls (2.34) is deeper
tllall is illdicated 11ere. So far we Ilave sl10wn tllat tIle self-dual so-
lutions are llecessarily static. III tIle llext Section we SI10W that tIle
correspolldellce works ill tIle otller directioll also: all static solutiollS
are solutiollS of the self-dllality equations.
TIle particular seif-dlial forln of tIle potelltial ill (2.33) lnay also
be understood as a Pauli-like lllagnetic illteractioll. COllsider a two-
co111ponent spill0rx (as is al)l)rOpriate for a plallar tlleory) alld let
(2.38)
wilere tIle Pauli lllatrices a 1 alld a2 satisfy
24
Then the Pauli energy is
E = _1_ Jd2xStS
2m
= _1_ J d2x lDxI2+ _1_ Jd2xBxta3x
2m 21n
Taking X to be all eigenstate of a3 , with eigenvalue ±1,
(2.39)
(2.40)
(2.41 )
tIle Pauli energy (2.40) beCOlTIeS
(2.42)
after USillg tIle Gauss law cOllstraillt (2.8a). Tllus, tIle self-dual 11011-
linear interaction terlTI ill (2.33) lTIay be alternatively vie,ved as a
magnetic l110n1ent illteractioll ,vitl1 a magnetic field tllat is given
self-consistelltly ill terlTIS of tIle cl130rge density p by the Cl1ern- Si-
111011S equatiol1 (2.830). Tl1is 1)lal1ar Pauli systel11 is well-knowl1 to be
all eXamlJle of sUIJersyl11l11etric qualltuln l11ecll3ollics [96,38]. TIle role
of supersyl11l11etry is ill fact l11ucll deelJer, as tIle self-dualll10del witl1
La,gral1ge del1sity (2.33) l11a.y lJe el111Jedded il1tO a, tlleory IJOSsessil1g
a super-Galilea,n il1Va,ria.I1Ce [161]. Also, tIle scala.r n1atter-Cl1ern-
Simons self-dual models descrilJed 11ere lnay be generalized to spinoI'
tl1eories witl1 an analogous self-dual structure [69].
25
D. Dynamical Symmetries
As expected, the nonrelativistic field theory witll Lagrange den-
sity (2.1) possesses the kinematic symmetry of Galilean invariance.
However, we shall see below that tllere is in addition a further dynam-
ical invariance under cOllformal reparametrizations of time [133,134].
The standard Galilean transformations are:
1. time translation:
t -+ t' == t + a
(2.43)
2. space trallslatioll:
t -+ t' == t
(2.44)
3. Sl)ace rotation:
t -+ t' == t
(2.45)
wllere R ij (w) is tIle rotatioll l11atrix for a rotation tllrougll allg1e
w.
4. Galileall boost:
t -+ t' == t
x -+ X' == X+ iJt
26
(2.46)
Under tIle first three types of Galilean transformation (time
translation, space translation and space rotation) the matter field
1/J transformsas a scalar
(2.47)
but, as is well known ill llollrelativistic quantum tlleory, ullder a
Galilean boost (2.46) tIle field 1/J transforms with a l-cocycle
1/J'(t', X') = eimiJo(x+iJt/2)1/J(t, x) (2.48)
TIle llollrelativistic Cllerll-SilllOl1S systelll witll Lagrallge dellsity
(2.1) llas additiollal illvariallces, beyolld tIle Galileall symilletries
(2.43,2.44,2.45,2.46), cOrreslJondillg to cOllforlllal relJarallletrizatiolls
of tIle tillle coordillate. COllsider the special class of coordillate trall-
forillatiolls
t --+ t' = T(t)
(2.49)
ill Wllicll tIle tillle coorclinate is relJarallletrized, alld tIle SIJatial co-
ordinates are lllultiplied lJy all associated tillle delJelldellt factor.
Define tIle trallsfornled field as
(2.50)
Wllicll includes botll a Jacol)iall weigllt factor alld a cocycle factor.
Then if 1/J' (i' , X') satisfies tIle origillal gauged 110I1lillear Scllrodillger
equatioll (2.4a) ill terll1s of tIle IJrill1ed coordillates, tIle field 1/J(t, x)
satisfies
o 1 ....2 1 12 1 2) .....)zDo1/J = --D 1/J - g 1/J 1/J + -111W (t x-'ljJ2m 2
27
(2.51 )
This is the gauged nonlinear Schrodinger equation (2.4a) with a new
term corresponding to a harmonic potential with time-dependent
frequency (squared)
(2.52)
There are tIlree special trallsfornlations of tIle fornl (2.49) for wIlich
this frequency in (2.52) vanishes. These transformations therefore
correspond to symmetries of tIle system. The first is just tIle time
trallslatioll syllll11etry (2.43). TIle otIler two are
5. till1e dilatioll:
t ---t t' == at
-+ ...., r::-+
x ---t x == V ax
6. Sl)ecial cOllforlnal tillle transforll1ation:
1 1 1
----t-=-+a
t t' t
i~i'= 1 i
1 + at
(2.53)
(2.54)
It is straigIltforward to cIleck tIlat w(t) given ill (2.52) does illdeed
vallisII for eacII of tIle trallsforlnatiolls (2.43,2.53,2.54).
Ullder tIle tillle dilatioll (2.53) tIle field 1/J transforms as
1/J' (t' , i') = ~ 1/J(t, i)
va
(2.55)
wIlile ullder tIle Sl)ecial cOllforlllal transforillatioll (2.54), ,~, trallS-
forills as
(2.56)
28
Note that for eacll of these symmetry operations, the matter density
p transforills with a Jacobiall factor
p' (t' , X') == .:Jp(t, x)
( ax
i
):J = det -a.X') (2.57)
and the gauge fields All, defined ill (2.17) and (2.21), transform co-
variantly
, '-I) oxll _
..4H (t ,x == -a ~411(t, x)
r x'll (2.58)
It is therefore straiglltforward to clleck tllat tllese trallsforillations
do indeed leave the actioll correspollding to tIle Lagrange density
(2.1) illvariant, and so correS!)Olld to symmetry operatiolls. The
correspolldillg cOllserved gellerators call be obtailled fronl Noetller's
theorenl.
TIle gellerators of tIle standard Galileall trallsforillatiolls
(2.43,2.44,2.45,2.46) call also l)e fOUlld froll1 tIle energy-Ill011lelltulll
tellsor. TIle ellergy dellsity is
(2.59)
TIle mOlllelltum dellsity is
(2.60)
TIle ellerg~y alld lll01llelltulll del1sit~y satisfy tIle COlltil1Uity eqautiollS
(2.61a)
(2.61b)
29
where T is the energy flux
and Tij is tIle momentum flux (or stress tensor)
Tij = ~ ((Di1/J)* Dj1/J + (Dj1/J)* Di1/J - <5ij ID1/J1 2 )
+~ (<5ijV2 - 28i 8j ) p + <5ijE
(2.62)
(2.63)
Notice tllat tIle energy flux T i =T iO does 110t equal tIle 11l01llen-
tum dellsity pi =T Oi , as ,vould be tIle case ill a relativistic tIleory.
However, tIlis tlleory is rotatiollally illvariallt, alld so tIle stress tell-
sorTij is symmetric. Also note tllat tIle energy density £ is one-half
the trace of tIle stress tellsor
(2.64)
wllich reflects the COllformal illvariance of tllis nOllrelativistic systelll
(contrast witll a relativistic cOllforlllally illvariallt 11l0del, for wllicll
£ = L,i T ii ). Tllis relatioll l)et\Veell tIle ellergy alld tIle trace of tIle
stress tellsor will l)e illll)Ortallt ill tIle cillalltizatioll of tllese systeills
- see Cllal)ter 6.
TIle Galileall constants of ll10tion correspondillg to tIle symmetry
transforlllatiollS (2.43,2.44,2.45,2.46) are:
1. ellergy
(2.65)
2. mOlllelltulll
(2.66)
30
3. angular nl0lllelltum
(2.67)
4. Galilean boost
(2.68)
and are stalldard for any Galilean invariant theory. The conserva-
tion of these quantities follows directly from the continuity equations
(2.61) togetller witll tIle lllatter dellsity COlltilluity equation (2.11)
for p alld J, WIlicll ex!)resses tIle al)eliall pIlase illvariallce of tIle
systeln.
TIle cOllserved qualltities corres!)ondillg to tIle additiollal cOllfor-
lnal trallsforlnatiolls (2.53,2.54) are:
5. dilatioll
1 J 2 ~D = tE - 2" d x X · P
6. s!)ecial cOllforlllal
~ 2 1TlJ2 ~2k=-tE+2tD+ 2 dxpx
(2.69)
(2.70)
At tIle classical level, USillg callollical POiSSOll l)rackets for tIle
fields, one fillds tllat tIle tIlree generators E, D alld !{ satisfy tIle
50(2,1) cOll1lnutation relations, COI1Stitutillg a dynanlical symmetry
algebra of tIle systeln. It is illterestillg to llote that tIle magnetic
vortex systeln also Ilas all 50(2,1) dYllalllical syllllnetry [124]. For
a general discussioll of scale illvariallce in pIlysics alld field tIleory
see [121].
We stress tIlat tIlis dynanlical cOllforll1al sylllllletry is presellt at
the classical level for any value of the nonlinear coupling consta.nt
9 appearillg in tIle Lagrange density (2.1). Later (see CIlapter 6)
we shall consider tIle question of tIle fate of tllis classical symnletry
31
in the quantum tlleory and we sllall see that this classical conformal
symmetry is brol(en, unless 9 tal(es a critical value corresponding to
the self-dual value in (2.32).
This dYllamical sylnmetry guaralltees tllat static solutions are
necessarily self-dual. To see tllis, consider tIle dilation gellerator D
defined in (2.69). D itself is cOllserved and so is tilne illdependent.
For static solutions 13 is also tillle illdependellt, and so (2.69) implies
that E must vallisll. But from (2.37) we see tllat tIle energy vanishes
only for self-dual solutiollS. A sinlilar argument using tIle special
conformal gellerator !( in (2. 70) sllows that botll E alld D vallisll
for static solutiollS.
Furtller, froln tIle Galileall boost -gellerator Bill (2.68) we see
that for static solutions tIle 1110111elltull1 P vallislles (as it sllould!).
TIle angular 1110111elltulll for a static (self-dual) solutioll reduces to
(2.71)
whicll is proportional to tIle net lnatter density. For self-daul solu-
tions, tIle Galilea.lll)oost generator ill (2.68) reduces to all expressioll
prOI)OItiollal to tIle electric dil)ole l110nlent
.... J ?B = -111 d-x px
wllile tIle dilatioll gellerator ill (2.69) l)eCOllles
(2.72)
(2.73)
TIle self-dual special cOllforll1al gellerator ill (2. 70) is prOI)Ortiollal
to tIle electric quadrUI)ole l110l11ellt
(2.74)
To conclude this Section on tIle dynamical sylnnletries of tIle 11011-
relativistic self-dual Cherll-Sill10ns tlleory, we note tllat tllis system
32
has also been formulated in a nonrelativistic I(aluza-I(lein frame-
work [67] in wllich the nonrelativistic 2 + 1 dimensional space-time
is obtained froIn a 3+1 dimellsiollal Lorentz nlanifold by dimensional
reduction. TIle llonrelativistic conforlllal synlmetries ill tIle 2 + 1 di-
mellsional model may tilell be related to tIle relativistic symllletries
of tIle origillal Lorelltz manifold Wilicil survive tIle reductioll. Tilis
interesting alternative viewpoillt deserves furtller illvestigation, in
particular to discover the geometrical sigllificance of the self-duality
COlldition.
E. Explicit Self-Dual Solutions: The Liouville Equation
TIle llollrelativistic self-cIua'! Cllern-Silll0llS eqllatiolls (2.34) call,
ill fact, be solved COlll1)letely alld eXl)licitly. If tIle (Colllplex) field 1/J
is decomposed as
(2.75)
then the first self-duality equatioll (2.34a) yields the vector potelltial
as
(2.76)away from tIle zeros of p. Illsertillg tllis forl11 for A illtO tIle secolld
self-dualityequatioll (2.34b) l)roduces tIle followillg ec!Uatioll for tIle
cllarge dellsity p
.... 2 2V' lTtp = =f-P
K
(2.77)
Wiliclll11Ust lJe satisfied a.\vay fro111 the zeros of p. We recogllize (2.77)
as tIle falnous Liouville equatioll, ,vllicll is kno,vll to be integrable
alld illdeed conlpletel)T solval)le [182]. TIle l)llaSe w of 1/J is deterlnilled
33
by demallding A to be nonsingular ill the vicillity of the zeros of p,
as is illustrated below in detail.
To describe the general solutions to the Liouville equation (2.77)
we first re-express it in terms of tIle cllaracteristic coordinates (2.24)
(2.78)
It is a simple matter to verify tllat
(2.79)
satisfies tIle Liouville equatioll (2.78) for any fUllctiollS f alld g. Tllis
follows silnply because
p = ±,,; (1 + j(x-)g(X+))2
(where I' =0-1 and g' =o+g) so tllat
(2.80)
inp = in (±,,;) + In (!'(x-)) + In (g'(x+)) - 2In (1 + j(x-)g(x+))
(2.81 )
frolll w hiel1 (2. 78) il11111ediately follo\vs. In fact, tl1is solutio11 (2.79)
is the nl0st gelleral solutioll [182].
For real and regularsolutiolls for p, we take g(x+) = (f(x-))*, so
that
p = ±,,;V2in (1 + l!(x-)1 2 )
If'(x-)1 2
=±K-----(1 + If(x-)12)2 (2.82)
Since p = 17/71 2 is a cllarge density ,ve require it to be positive, alld
so we must cll00se tIle ± Sigll accordillg to tIle Sigl1 of K ill such a
way that
34
(2.83)
(2.84)
Thus, in order to obtain an appropriate, regular solution to tIle Li-
ouville equation (2.77) (which ill turn, yields a solution to the self-
duality equations (2.34)), we see tllat it is llecessary to correlate the
choice of a self-dual or anti-self-dual ansatz ill (2.23) with the sign
of K so that (2.83) is satisfied.2 Another way to say this is that only
the Liouville equation witil sign
~2 2V lnp == --pIKI
has real positive regular solutions.
FrOlll 110W 011, we clloose tIle Sigll of K to l)e positive ill wllich
case we obtaill self-dual solutions b)' solvillg tIle self-duality equatioll
(2.34a) witll a D_:
(2.85)
(2.86)
It is very importallt to stress tllat \vitll tIle Sigll correlatioll in (2.83),
the self-dual COUl)lillg strellgtll g in (2.33) ,vitll tIle critical value
(2.32) is always IJositive (illclel)enclellt of tIle sign of K), correSl)Olldillg
always to an attractive self- dual potelltial [125,126]
1 4
VSD = - 2m1K1 1"p1
Explicit radially SYlllllletric solutiolls nlay be ol)tailled by takillg
2Note that if we c1100se the op]Josite sign correlation, ±n: = -II\, I, it
would still be possible to obtain positive real solutions for p by choosing
g(x-) = - f(x+)*, but these solutions would not be regular. We return
to this possibility in Chapter 6, when we consider quantization.
35
(2.87)
where we have introduced tIle polar coordinates x± = ~e=fi8; {note
the factor of 1/2, whicll ellters because of the definition of the charac-
teristic coordinates in (2.24)). TIle corresponding self-dual solution
has charge density
(2.88)
As r --t 0, tIle cllarge dellsity bellaves as
wllile as T --t 00
-2-2nprvr
TIle vector potential for r --+ 0 bellaves as
(2.89)
(2.90)
(2.91 )
We can therefore avoid sillgularities in the tIle vector potelltial at
the origin if we choose tIle pllase of 'ljJ to be
W = ±{11 - 1)0
Thus the self-dual 'ljJ field is
~/. _2nM (r/ror-1 ±i(n-l)O
0/ - ? e
ro 1 + {r/ro)_n
(2.92)
(2.93)
Requirillg tllat 1/J be sillg1e-vailled \ve find tllat 11 lllust be all illteger,
alld for p to decay at illfillity \ve require tllat 11 be positive. For
n > 1 the 'ljJ solutioll llas vorticity 11 at tIle origill alld p goes to zero
at tIle origill.
36
The net matter charge Q corresponding to the solution (2.82) is
Q = 1,,;1 Jd2xV2ln (1 + 1112)
= 21f1,,;1 [r~ In (1 + 111 2)]~
For the radial solution (2.88), the net matter charge is
and tIle cOrreSIJOlldillg flux is
<I> = 41rl1,
(2.94)
(2.95)
(2.96)
whicll represellts an even llumber of flux units. Tllis qualltized cllar-
acter of tIle flux is related to a SIJecial inversion symmetry of the
Liouville equation, and is not particular to radially symmetric solu-
tions [151]. TIle allgular luonleiltuill generator (2.67) is
and the conforlual cllarge (2. 70) is
!( = 2m1rIKI11T"5 ( .1f/1; )
Sl111r 11
(2.97)
(2.98)
TIle radially sylllilletric solutioll (2.93) arose from choosillg tIle
llolomorpllic fUllction f(x-) ill (2.82) as
(2.99)
and corresponds to n solitolls superiluposed at tIle origin. A solution
correspondillg to n separated solitOllS may be obtailled by taking
n
f( -) '"'" Ca
X = L.J x- - x-
a=l a
37
(2.100)
Note that there are 4n real parameters involved in tllis solution:
2n real parameters x;; (a = 1, ... n) describing tIle locations of the
solitons, and 2n real parameters Ca (a = 1, ... n) corresponding to
the scale and phase of eacll solit011. TIle solution in (2.100) llas ill fact
beell ShOWll, by all index tlleory counting argumellt, to be tIle most
general multi-solitoll solution [150]. SolutiollS witll a periodic matter
density p may be obtained by Clloosillg tIle function fill (2.82) to
be a doubly periodic function [198]. Other properties of these self-
dual Cllerll-Sill10l1S vortices, ill relatioll to vortices ill l)lallar Maxwell
electrodynamics, llave beell discussed ill [53].
F. Time-Dependent Solutions
As discussed in Sectioll lID, solutions to tIle self-duality equa-
ti011S (2.34) l)rovide all tIle solutiollS to tIle static Etller-Lagrallge
equations of motion (2.4). \Ve can then make use of tIle Galilean
alld cOllforlllal sylllilletries of tIle tlleory to cOllstruct eXl)licitly tillle-
depelldellt solutiollS l)ased 011 tllese static solutiollS [131]. TIle ll10St
obvious of tllese is tIle Galileall boost of a static solutioll 1Pstatic(X):
TIle correspolldillg dellsity
p(t, x) = Pstatic(X + vt)
(2.101)
(2.102)
correspollds to a trallslatioll, witll ulliforln velocity V, of tIle static
density Pstatic(X). As SUCll, tllis till1e-depelldellt solutioll is 110t l)ar-
ticularly illterestillg.
More illterestillg is tIle time del)elldent solutioll ol)tained by ap-
plying the conforlnal transformation (2.54) to a static solutioll:
38
f)/,(t x) = 1 eimar2/2(!+at).I. . ( x )
tf/, 1 + at tf/statlc 1 + at
Tllis leads to a tilue-depelldellt dellsity
(2.103)
(2.104)
The conserved generators (2.65,2.66,2.67, 2.68,2.69,2.70) may be
evaluated for these time dependent solutions. For tIle Galilean
boosted solution (2.101)
E = ~m:i;2Q
M=Q(1+1JtVX <x»
D 1J[2 .... p"" 1 .... ....= -- (, X~D' - -111V' < X >2 2
wllere < x > dell0tes tIle ex!)ectatioll value
(2.105)
(2.106)
Both the Galilean boost gellerator B alld tIle cOllforlllal generator
!{ are uncllallged for this time depelldellt solutioll. The gellerators
ill (2.105) illustrate tIle fact tlla,t tIle l)oosted solitOll l)ellaves like a
particle of total mass 1nQ.
For tIle confornlally boosted SOllItiOll (2.103) one fillds
E = ~ma2Q < 5;2 >
2
j3 = 1JtaQ < x>
39
(2.107)
(2.108)
with Band !{ unchanged. Finally, one may combine the Galilean
boost and the special cOllformal transformation (tIle order is irrel-
evant since Band !{ commute as generators) to obtain even more
general tinle depelldellt solutiollS [131]. TIle existence properties of
these fornlal time dependellt solutions have recelltly beell investi-
gated in tIle COlltext of tIle illitial value problelll for tIle nonrelativis-
tic self-dual Cllern-Sill10llS systeln [19]. Based 011 tIle relation (2.70)
(Wllich is valid for all till1e)
; Jd2xpfi2 = Et2 -2Dt+I<
and tIle lllallifest l)ositivity of tIle LHS for all till1e, Olle fillds furtller
COllditiollS 011 E, D and !( 'Vllicll llltlSt be satisfied for COllsistellt time
evolutioll. III l)articular,for certaill illitial data, tIle till1e depelldent
solutiollS are fOUlld to collal)Se ill a fillite time [19].
G. Solutions in External Fields
The tilne-depelldellt solutiolls described ill Section II F were ob-
tailled by transforll1illg tIle static solutiolls accordillg to a sym-
metry operatioll of tIle systell1, ,vllicll leaves tIle actioll illvariallt.
Alternatively, Olle lllay l11ake a trallsforlllatioll of tIle action wllicll
changes tIle action to tIle actioll for a llew l110del. Tllen solutions
of tIle origillalll10del may be transforlned to solutiollS (in general,
now tilne depelldent solutions) of tIle new lnodel. Tllis is, of course,
Ollly a useful exercise if tIle transforllled systelll is all illteresting
one. Fortullately, it is a l)roperty of llollrelativistic dynalnics tllat
a llollrelativistic actioll l1lay be easily trallsforllled so as to illtro-
duce all additiollal llarllloilic l)otelltial l! rv ~w2X2. III two Sl)atial
40
dimensions this is of direct interest for two reasons. First, tIle har-
monic potential may be used as a confining potential wllich permits
an elegant and simple treatment of the thermodynamic and sta-
tistical mechanical properties of many-particle systems [191,37,43].
Second, a l11inor variallt of this trallsformation amounts to the in-
troductioll of an exterllal Ullifornl ll1agnetic field transverse to the
spatial plane, a configuration wllicll is of great pllenomenological in-
terest for applications of Chern-Simons theories to the quantum Hall
effect [78-81,175].
The idea is lll0St easily illtroduced witll tIle llarmollic potential
case, and without coupling to gauge fields. Suppose 1jJ(x) is a static
solution of the nonlinear3 Schrodinger equation witll zero energy
Tllen the til11e del)endellt fUllctioll
1jJ(W) (t, x) = _1_e- i (m./2)wx2 tanwt1jJ (~)
coswt coswt
satisfies tIle Scllrodillger equatioll
Tllis correSI)011ds to Clloosillg tIle trallsforlllatioll fUllCtioll
1
T(t) = -taTtwt
w
(2.109)
(2.110)
(2.111)
(2.112)
ill tIle coordillate trallsforll1ation (2.49). Tllell tIle tra,llsforllled wave-
fUllction acquires a cocycle and a Jacobian weigilt factor as in (2.50),
and tIle frequency (2.52) of tIle llew harmonic term generated in
(2.51) is constant
3This argument works both with and without the nOlllinear term.
41
(2.113)T 3 (T)2 2-. - --.- == w
2T 4 (T)2
Note that the solution (2.110) is periodic in time. This periodic-
ity is important for the semi-classical quantization of these classical
solutions [131,132].
This transformation gelleralizes to the gauged nonlinear
Schrodinger equation (2.4a). Tllat is, suppose 'l/;(x) satisfies
( 1 (-+ -+)2 2)- 2m, \7 + iA - gl1jJ1 1jJ(x) = 0 (2.114)
where the vector potential ~4 is givell by the ex!)ression (2.17). Tllen
tIle trallsforllled field 1/'(t, x) ill (2.110) satisfies
i (at + iA&w)) 1jJ(w) = ( - 2~1, (v + iA(W)) 2 _ gl1jJ(w) 12 + ; w2x2) 1jJ(w)
(2.115)
wllere
(2.116)
Evaluatillg tIle cOllserved energy alld allgular nl0lnelltum gellerators
011 tIle tilne-dependent solutiolls \ve filld
M(W) == 1'/1 (2.117)
where ]{ and M refer to the corresponding generators for the solu-
tions without the 11aflll0nic potelltial.
This idea call also be extended to tIle case of an external uni-
fOfll1 lnaglletic field, of lna,gllitude B perpendicular to tIle !)lalle, by
defilling
42
e-i8r2tan(8t/2m)/4rne-iQ8t/(41TK111) (Rii (Bt/2m) Xi)
7jJ(8) (t x) == 7jJ
, cos(Bt/2m) cos(Bt/2m)
(2.118)
which corresponds to a tinle-dependent dilation, a time-dependent
rotatioll tllrougll angle Bt/2111, and a gauge transformation. Here Q
is tIle net charge of the untransformed solution. Tllen if 7jJ(x) is a
solution of the original gauged nonlinear Scllrodinger equation (2.4a),
then 7jJ(B) (t, x) satisfies tIle gauged lloillinear Scllrodillger equation
witll all additiollal exterllal 111aglletic field:
wllere
(8) 1 J') (8) B .A· == -- d"xG·p + -E' ·xl1 K 1 2 l)~ (2.120)
TIle corresponding cOllserved qualltities for tllis llew systell1 (wllicll
illcludes tIle 111aglletic field B) lllay be ex!)ressecl ill terlllS of tIle
cOllserved gellerators of tIle origillal sJTstelll witllout B as follows:
(8) B·p. == p. + -E··B)
1. 1. 2111 1)
(2.121)
As a furtiler generalizatioll of tllese SOllltions, ratiler tilan transforlll-
ing tIle static (self-dual) solutiollS (2.93) of tIle origillal model witll-
out tIle 11arlllonic potential (or lllagiletic field), one 111ay trallsform
the l)oosted till1e clepenclellt solutions (2.101) or (2.103) descril)ed
43
in section II F. These time dependent solutions may be used in a
semiclassical quantization of tllese self-dual Chern-Simons systems
[131,132]. Furthermore, the external field solutions discussed in this
Section can also be found ill tIle nonrelativistic I<aluza-I<:Iein ap-
proach of Duval et al [68].
H. Maxwell-Chern-Simons Model
TIle llollrelativistic self-dual Cllern-SilllOllS systeln described ill
tllis Cllapter 111ay be gelleralized to illclude a !vlaxwell term for tIle
gauge field in tIle Lagrallge dellsity [57]. Tllis additioll 11as a dra-
matic effect 011 tIle 1110del becallse tIle gauge field 110\V acquires a
pllysical propagating nlassive lnode - whereas \vitllout the Maxwell
terlll tIle gauge field is itself 11011dyllall1ical, beillg deterll1illed by
tIle nlatter fields as in (2.17) alld (2.21). This also 11leans tilat the
gauge equatiolls of ll10tioll becolne second order equations, ill con-
trast to tIle first order equatiolls (2.4b) for tIle pure Cllerll-Silnons
case. Beillg first order already, tIle l)ure Cilerll-Sill10llseqtlations
of motion did not need to be factorized ill order to produce first-
order self-duality equations. Tllus tIle self-duality equatiolls (2.34)
COI1Sist of tIle factored lnatter equatioll alld tIle (already first order)
Gauss law COllstraillt. With a Maxwell terll1 illcillcled for tIle gauge
field the ga,uge equatiolls 111USt also l)e factored, ill additioll to tIle
(llew) factorization of tIle lllatter equatioll. It is a renlarkable sign
of robustlless tllat tIle Cilern-SilllollS systelll still adll1its a self-dual
forlllulatioll, provided tIle additioll of tIle Max\veII gauge terll1 is ac-
compallied by tIle inclusioll of a.n extra real scalar field of lllass equal
to tIle luass of tIle l)rOpagatillg gatlge luode. Tllis is demollstrated
below.
COllsider tIle Lagrallge clellsity
44
Note that e2 has dimensions of mass in 2 + 1 dimensions, and the
dimensionless Chern-SilllollS coupling parameter has been written as
flK=-
e2
(2.123)
where fl is tIle l11ass of tIle propagatillg gauge lllode. TIle l11ass of
the neutral scalar field N llas been chosell to be equal to tllis gauge
IIIass
111 N = fl = 111gauge (2.124)
TIle !Jotelltial terms ill (2.122) have beell cllosell to satisfy tIle self-
duality requiremellt, as vve sllo,v below. Notice tllat tllese self-
dual potential terms illvolve interactiollS bet,veell the llonrelativistic
scalar dellsity p = 1'¢12 and tIle lleutral scalar field N.
TIle pure Cllerll-SilllOllS li111it is achieved l)y the COl1lbilled lilllit
ill ,vllicll botl1 lnass scales e2 alld It l)eCOllle illfillite, but ill sucll a
way tllat tlleir ratio K relllains fixed:
fL' --t 00
It
K = ? = fixed
e-
III tllis linlit, tIle neutral scala.r field is forced to
N,. 1=-p
2111K
ill wllicll case the potelltial becolnes
45
(2.125)
(2.126)
(2.127)
which is the self-dual quartic potelltial in (2.33) for tIle pure Chern-
Simons self-dual model. TIle Lagrange density (2.122) may also
be obtained as a nonrelativistic lilnit [57] of a relativistic self-dual
Maxwell-Chern-Simons theory [166] and the massive neutral scalar
field N is essential for such a lilnit to be well-defined. This relativis-
tic Maxwell-CIlern-Simolls tlleory is described in Section IV G.
TIle Euler-Lagrangeequatiolls of ll10tioll for tIle systenl (2.122)
are
(2.128a)
(2.128b)
(2.128c)
wllere JV is tIle llollrelativistic l11atter currellt defilled ill (2.5). III
particular, the Gauss law COllstraillt 110W reads
-+ -+ 2
V · E == J1B - e p (2.129)
wIlere B alld E are tIle lnagnetic and electric fields defilled ill (2.9)
alld (2.10).
TIle energy dellsity is
(2.130)
For static fields we take BoN = 0 alld tIle Gauss law (2.129) ilnplies
tllat we can identify (up to surface terll1s)
46
(2.131)
Furthernl0re, USillg tIle factorizatioll identity (2.35) we can express
tIle energy density as
1 2 1 ( J-l2 ) 1 (2 2)£=-ID_1P1 +-B 1-- B+-N J-l -V N2nl 2e2 V2 2e2
e2 1 1 1
--p-p + J-lB-p + -Bp2 V'2 V'2 2m
_(1 + .!!-) N p + e2 p2
2m 8111,2
wilich may be factorized as
(2.132)
(2.133)
TIle ellergy density is bOllllded l)elo,v l)y zero, and tllis bound is
saturated by fields satisfyillg tIle follo\villg equatiolls
( 2 'J) e2 ( 'J)J-L - V'- B = 2m 2mJ-L + V'- p
(2.134a)
(2.134b)
(2.134c)
Note tllat tIle tilird equatioll (2.134c) is just tIle Euler-Lagrallge
equation of nl0tioll (2.1281)) \vhicil cleterll1illes tIle auxiliary neutral
47
scalar field N in terms of tIle scalar density p. The first equation
(2.134a) illl!)lies that
1 2B=--\11np
2
(2.135)
so that the self-duality equations (2.134) reduce to the following
nonlocal equation for p:
V2[ = _ e2 (2mJl + \12)
np 2 ~2 P
m Jl - v
(2.136)
III tIle pure Cileril-Silllons lilllit (2.125), tllis expressioll reduces to
.J 2\l-lr~p = --p
K
(2.137)
wllich we recognize as tIle Liouville equation (2.77) for the pure
Chern-Silnolls solitons. Wllile tIle gelleral Maxwell-Chern-Simons
equation (2.136) cannot be solved in closed form, it is possible to
describe tIle global and asylnptotic properties of solutiollS. Tllis is
due to the fact that the Cllerll-Simolls dynamics dOlllinates tIle long-
distallce l)llysics. SolutiollS \vith alld \vithout vorticity are clescribed
in [57]. SOll1e rigorous l11athelllatical results for tllis systelll are pre-
sellted ill [227], ,,,llere it is Sl10'''11 tllat tllere exist doubly periodic
condellsate solutiollS for Wllicll tIle llulnber of vortices ill a l)eriodic
lattice cell can be arbitrary. Tllese are analogous to tIle condensate
solutiollS studied in [198] for tIle pure Chern-Silnolls system.
48
III. NONABELIAN NONRELATIVISTIC MODEL
This Chapter deals with the nonabelian generalization of the
abelian nonrelativistic self-dual Chern-Simons models discussed in
Chapter 2. This nonabelian generalization is of interest for the un-
derstanding of nonabelian fractional statistics, a rich subject with
potential applications in various quantum Hall systems [257,31]. As
in the abelian models, there is a Bogomol 'nyi lower bound for the
energy which is saturated by static zero energy configurations which
satisfy a set of first order self-duality equations. These self-duality
equations may be reduced, by certain algebraic ansatze, to many
examples of two dimensional integrable partial differential equations
such as the classical Toda or affine Toda equations. Furthermore,
with adjoint matter coupling the self-duality equations may be con-
verted into the (Euclidean) two dimensional chiral model equation
by a judicious choice of gauge. This fact leads to a complete classifi-
cation of finite charge solutions in terms of Dhlenbeck's classification
of finite action solutions to the chiral model (or "harmonic map")
equations. This analysis is very similar in both spirit and style to
the classification of finite action ("instanton") solutions to the self-
dual Yang-Mills equations (1.3) in four dimensional Euclidean space.
In fact, the nonabelian nonrelativistic self-dual Chern-Simons equa-
tions may be obtained from the four dimensional self-dual Yang-Mills
equations by a dimensional reduction. These results lead to an inter-
esting new relationship between the Toda and chiral model systems.
49
A. Nonrelativistic Self-Dual Chern-Simons Equations:
General Matter Coupling
The nonrelativistic self-dual Chern-Simons system discussed in
Chapter 2 may be generalized from an abelian theory to a non-
abelian theory [103,58]. We consider now a multiplet IlJ of complex
scalar fields which transform under nonabelian local gauge transfor-
mations according to some definite representation n of the gauge
algebra g. (In general, 9 could be any compact simple Lie algebra,
but for ease of presentation we shall primarily focus on the 5U(N)
case. Noncompact groups, in particular 5£(2, R) and 150(2,1),
have been considered in [30].) The abelian gauge field AI-' general-
izes to the fields A~, with a = 1 ... D, where D is the dimension of
the gauge algebra. The Lagrange density (2.33) becomes
where
(3.2)
with fa being antihermitean generators of the gauge algebra, in
the representation n corresponding to the matter fields 1lJ. The
matter and gauge fields are minimally coupled through the covariant
derivative
(3.3)
The nonabelian Chern-Simons Lagrange density Les is
(3.4)
where rbc denotes the structure constants of the gauge algebra:
50
(3.5)
(3.6)
The coefficient of the nonlinear papa term in the Lagrange density
(3.1) has been chosen to take the self-dual value 2.32. Witllout loss
of generality, we have cllosen tIle Chern-Simons coupling parameter
~ to be positive, yielding an attractive self-dual potential4
1; 1 a a
VSD == ---p p2m~
While the Chern-Simolls Lagrange density (3.4) is not gauge in-
variallt (ullder a gauge trallsforl1latioll it cllanges 1))' a total derivative
term and a tOI)ological terl11 - see Equatioll (1.22)), its variatioll witll
reSI)ect to tIle gauge fields ..4~ !)roduces gauge covariant equations of
lllotiOfl. The resulting Euler-Lagrange equations are
(3.7a)
(3.7b)
wllere
(3.8)
is tIle nOlla1)eliall gauge curvature alld Ja J.l =(pa., JU) is a Lorelltz
covariallt sl10rthalld for tIle nonrelativistic ll1atter current witll pa as
ill (3.2) alld
(3.9)
41f f\, were negative., then the critical coupling in (2.32) would still yield
an attractive potential, bllt the notion of self-duality and anti-self-duality
would be interchanged~ just as in the abelian case - see Section lIE.
51
Note that JaJ.l satisfies the gauge covariant ~contilluity equation':
(3.10)
In addition to the gauge current JaJ.l there is an abelian current QJ.l,
correspondillg to tIle global U{l) symilletry of the Lagrange density
(3.1 ):
(3.11a)
(3.11b)
wllicll satisfies tIle ordillary COlltillllity equatioll
(3.12)
The Cllerll-Sill10ns eCluation of Illotion (3. 71)) Illay l)e ex!)ressed ill
terills of tIle usual Ilollabeliall Illaglletic alld electric fields sa al1d
i a as:
(3.13a)
(3.13b)
To seek SOltltiollS to tIle Euler-La,gra.nge eqtlations (3.7) \ve llla.ke a
self-dual allsatz for the fielcls, as ill (2.22,2.23),
(3.14)
TIle llo11abeliall versiOll of tIle iclelltity (2.27) is
(3.15)
Therefore, tIle nOlla1)eliall gauged IlOlllillear Scilrodillger equatioll
(3.7a) becoll1es
52
(3.16)
wllere we have used (3.15), together with the self-dual ansatz (3.14)
and the Gauss law constraint (3.13a). This can be solved by static
solutiol1S
(3.17)
(3.18)
with Ao given by
Aa = __i_q,tyaq, = L
o 2111~ 2nl~
Note tllat tllese SOltltiol1S are C011sistel1t witll tIle electric field equa-
tiOl1 (3.13b) because for self-dual fields satisfying (3.14) tIle nOllrel-
ativistic current dellsity (3.9) Sil1ll)lifies to
(3.19)
in whicll case the electric field equatioll of ll10tioll (3.13b) l)eCOllles
Tllerefore~ static solutiollS of tIle Euler-Lagrange equations of ll10tioll
(3.7) correSI)Olld to SOltltiollS of tIle first-order self-cltlality equatiol1s:
(3.21a)
(3.21b)
which are tIle natural 1101labelial1 gelleralization of tIle