Self-dual Chern-Simons theories

226 Pages·1995·7.97 MB·English

by  Gerald Dunne

Lecture Notes in Physics
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Gerald Dunne
Self-Dual
Chern-Simons Theories
Springer
Author
GeraldDunne
PhysicsDepartment
UniversityofConnecticut
Storrs, CT06269, USA
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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
ISBN3-540-60257-7Springer-Verlag Berlin Heidelberg NewYork
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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 ofthe more advanced
developmentsin lessdetail; nevertheless, Ihave 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 oftIlis lnaterial at the X I11th 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 ofseillinars at tIle University of COll11ecticut.
I11ave 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 piolleeringcontributiollSofRoman 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-Dual Solutions 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
to the SDCS Equations .. 74
G Unitons and Toda Theories 78
IV Abelian Relativistic Model 83
A Lagrangian and Hamiltonian Formulation 84
B Self-Duality Equations . 86
IX
C Self-DualSolutions: TopologicalVortices
and Nontopological ~olitons . 90
D Nonrelativistic Limit . 95
E Symmetry Breaking and the Cllern-Simons Higgs
Mechanism . 98
F Self-Dual Chern-Simons Theories and Extended
Supersymmetry . 105
G Maxwell-Chern-Simons Model 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 188
x
I. INTRODUCTION
A. Self-Dual Theories
"Self-duality" is a powerful notion inclassicalmechanicsand 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 ofmotion
(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)hysicalsigllificallce. 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 ofself-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