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A COLLECTION OF EXERCISES IN
TWODIMENSIONAL PHYSICS
PART I
COSTAS J. EFTHIMIOU
Newman Laboratory of Nuclear Studies
CORNELL UNIVERSITY & FLORIDA SOUTHERN COLLEGE
DONALD A. SPECTOR
Department of Physics
HOBART & WILLIAM SMITH COLLEGES
RELEASE: 1.0
Cornell CLNS 99/1612
©All rights reserved.
This book was typeset in TeX using the LaTeX2e Document Preparation System.
A DEFENSE FOR TWO DIMENSIONS
or
AN EXTRAORDINARY PREFACE
In his classic book Flatland [Abbott], E. A. Abbott described an imaginary twodimensional world, embedded in three dimensions, and populated by twodimensional figures that think, speak, and have all the human emotions. The author, a schoolmaster with a classics background who was interested in literature and theology, wrote his science fiction fantasy in pure mathematics for entertainment. He published it under a pseudonym to avoid any negative criticism resulting from it reflecting poorly on his more formal works. Apparently, he never imagined that this work not only would entertain many generations of physicists and mathematicians, but also would contain concepts that future scientists would work on.
Almost 120 years after Flatland was written^{1}^{1}1The first edition was published in 1880, while a second corrected edition was printed in 1884. Beyond the mathematical ideas explored, the book reflects many attitudes that were normative in Western society in the late nineteenth century, but that today are rejected as offensive and illinformed., much has happened to our ideas and concepts of spaces with various dimensions. Relativity placed humans in a fourdimensional spacetime; more recently, String Theory claims that the dimensionality of our universe has to be updated to 10 or 11 dimensions. We have thus learnt to feel comfortable in a space of many dimensions.
We have also become more comfortable in spaces of few dimensions. Twodimensional science has grown to one of the most important, wellcontrolled, and welldeveloped areas of today’s physics and mathematics. Abstract mathematics and theoretical physics have established many breathtaking results, while condensed matter physics has devised many systems that behave as twodimensional, some of which have great impact on our society.
Of course, despite these twodimensional applications, many physicists have argued that ‘our universe’ could not have been twodimensional, that life could not have existed in two dimensions, and that three dimensions is the minimal feasible dimensionality for our world. In particular, S. Hawking in talks and M. Kaku in his bestseller Hyperspace have argued that the digestive system of living beings would separate them in two disjoint sets in two dimensions and, therefore, even this simple argument can rule out a universe in two dimensions. However, this argument is too quick and facile (although it does point out that evolution on Earth has selected a digestive system suitable to three dimensions, and inappropriate for Flatland). Indeed, if one puts one’s mind to it, a model for life in two spatial dimensions can be developed, as is done quite thorougly in A. K. Dewdney’s book Planiverse [Dewdney].
So, after all, life in two dimensions might be possible. And there may even be other universes out there that realize such a possibility. We may never discover such places. But we can certainly study them. And we will be glad that we did so!
CJE’S ORDINARY PREFACE
In contrast to Warren Siegel [Siegel], when I was a graduate student I used to love books that contained in the title ‘‘Introduction to ...”. This is probably because, by the time I was a graduate student, many good books delivering on this promise^{2}^{2}2This has already become better with the creation of the LANL archives by P. Ginsparg. Many beautifully written articles which include “Introduction to …” in their titles are posted there frequently in a variety of fields. Papers of all flavors are available there for virtually any taste. had been written.
Even as an undergraduate student, I recall (with some kind of nostalgia for those years) that I was lucky enough to read some great introductory books. Written on the standard core of physics, those books usually would not include in the title the previous phrase. Among these books, I especially liked Problems in Quantum Mechanics by Constantinescu and Magyari. The book is nicely written and printed. It contains a number of problems on Quantum Mechanics (QM), divided thematically into chapters. Each chapter contains a brief summary of the theory, a collection of problems, and finally their solutions. I have been influenced so greatly by this book, that the architecture of the present document is modeled after it — although I can only hope that we approach the quality of that book’s presentation.
Although the subject of the present document is distinct from the subject of its stylistic sibling, the two are not completely unrelated. What one learns in [CM], can be used here. After all, QM is 1dimensional Quantum Field Theory (QFT); the present document explores the developments in 2dimensional QFT.
During the last two decades, we have witnessed an amazing explosion in the progress of mathematical physics. String Theory^{3}^{3}3According to the standard lore of naming theories in Field Theory, the name Quantum Nematodynamics (QND) is more appropriate. Perhaps the name String Theory should be used only at the classical level. Another possible name, given the recent developments regarding branes, is Quantum Branodynamics (QBD); however, I personally would vote in favor of QND as I find Quantum Nematodynamics the most euphonic choice. has emerged as the leading candidate for the Theory of Everything (TOE), and has led to revolutions in our understanding of the principles underlying fundamental physics. In conjunction with the rise of string theory, we have witnessed the discovery of and advancements in many other areas of mathematical physics: Conformal Field Theory (CFT), Integrable Models (IMs), 2Dimensional Gravity, Quantum Groups (QGs), and Dualities in QFT, to name just a few areas. The developments in each field have influenced the developments in the other fields, at times in profound and radical ways, at other times in subtler and more controlled fashion.
Thus, I embarked on the preparation of this collection of problems, for those who wish to study mathematical physics and want to see some solved problems to whet their appetite. I hope that people who learn the subjects treated in this collection of problems will find it useful. In this sense, it might also prove useful to people who teach material related to the subjects explored herein.
In his book, Siegel also writes, “It is therefore simultaneously the best time for someone to read a book and the worst time for someone to write one.” This sentence remains true for this collection as well. Recent times have proven so fertile for mathematical physics that many results are considered ‘common knowledge’ just a few days after they are posted on the LANL archives. As a result, no review or book can cover completely such a vast terrain. The present document is certainly no exception. It covers only a fraction of the relevant subjects, and even for those covered, only a limited number of possible themes have been touched. However, depending on the interest, we hope that in a later edition, more topics will be added, and more exercises within each area.
We have partitioned the material in LABEL:CountChapters chapters. Although there are not always sharp boundaries among these chapters, this approach was taken to enhance the pedagogical value of this work.
Finally, we would like to say that for such a subject, it is impossible to give an exhaustive list of references. We primarily cite review papers and books, as those may be of great help for the reader who would like to study the material. However, we have also included many significant original papers in the bibliography that might not be cited in the text. In this spirit, we would like apologize to all people whose works are not cited; there is no way to be exhaustive, nor have we attempted to be, and in some cases such omissions are doubtless a result of our ignorance. If this document serves as an introduction to the field of twodimensional physics and its literature, it will have served its purpose.
DAS’S ORDINARY PREFACE
The very first paper I published offered new demonstrations of the integrability of some nonlinear sigma models, with and without supersymmetry, in two spacetime dimensions. While that paper predates (by just a little bit) the revolution in 2dimensional physics that this manuscript addresses, I am struck by how much the topics of that paper are echoed here.
If I trace my own personal history of interest in the topics covered here, there are two pivotal moments, beyond that first paper of mine. The first moment is the ICGTMP (International Colloquium on Group Theoretical Methods in Physics) held in Montreal in 1988. Those were heady times. The conference itself was a cornucopia of string theory, conformal field theory, and quantum groups, held in a political context that allowed attendance by an extensive collection of physicists and mathematicians from around the globe. How could one not be hooked?
The second moment was my decision to pursue a career in a liberal arts institution. These are places not wellunderstood outside the USA; they are colleges with no graduate students or postdocs, but that does not mean that they are void of research. On the contrary, my colleagues are some of the most vibrant minds I know. But it does mean that many of us do not have the benefit of spending time each day, bumping into colleagues in the hallway or seminar room, and learning things almost by osmosis. The value of such a pedagogical document as the present one of course transcends my own personal context; but this context has made clearer to me what the value of such a manuscript is.
My coauthor Costas Efthimiou has been the driving force behind this project, and the rationale he presents above is indeed the same rationale that drew me into this project, and I am glad to have been drawn in. There is no need for me to repeat the ideas Costas has expressed above. But I will express my hope that this document will find multiple uses, from students beginning their explorations of theoretical physics in graduate school, to established scientists trying to move into new areas of research, to faculty seeking inspiration for their courses.
ACKNOWLEDGEMENTS
This manuscript and our approach to the topics therein has benefited from discussions on various occasions with C. Ahn, M. Ameduri, S. Apikyan, P. Argyres, S. Chaudhuri, J. Distler, B. Gerganov, B. Greene, Z. Kakushadze, Y. Kanter, T. Klassen, A. LeClair, G. Shiu, and H. Tye. CJE thanks A. LeClair in particular, for providing an introduction to some of the subjects discussed in this document, and would especially like to thank M. Ameduri who typed some of the problems from handwritten notes, providing the momentum needed to continue on this project. Last, but not least, CJE thanks the Florida Southern College where some of the final details were written, and in particular Professor M. Jamshid who gave him the opportunity to visit the college, while DS acknowledges the support of NSF Grant PHY9970771.
Comments and criticism are welcomed and greatly encouraged.
Other mistakes may perchance…await the penetrating glance of some critical reader, to whom the joy of discovery, and the intellectual superiority which he will thus discern, in himself, to the author of this little book, will, I hope, repay to some extent the time and trouble its perusal may have cost him!
Lewis Carroll
COMMENTS

When you read this document, please keep in mind that the document is still in its infancy. Most sections are brief, and the presentation at times somewhat abbreviated. The reader is also warned that, despite our best efforts, no doubt many typos remain. We apologize, too, that different conventions may still be used in different parts of the document! This is due to the fact that several sections have their origins in projects undertaken before conceiving the plan to prepare one comprehensive pedagogical collection. (Some would say this might even be valuable, preparing the reader for the array of conventions in the published literature!) We hope that the reader nonetheless finds this work valuable and beneficial. If all proceeds according to our expectations, when the document reaches its adult stage, it will have been cured of all these childhood diseases!

On the cover page, a release number
RELEASE is given (see on the cover). It should be interpreted as follows. A higher release number of course signals a newer version. A larger means that simple typos have been corrected, wording may have been improved, conventions and notation may have been uniformized, additional references may have been added, but no essential changes have been made. Reprinting the document is in this case strongly discouraged — save the forests! A larger number means that new material has been added (e.g. new problems in previously existing sections, new sections in previously existing chapters, or even new chapters) or conceptual or other important mistakes have been corrected.

Of course, there are many topics that could be added to the present document. A list of topics that would appear as natural extensions to our work would include (see the list of abbreviations on page 1):

Background Material in QFT and in Mathematics

Supersymmetric CFT

Higher Genus CFT

CFT in Dimensions

IMs in QM

Classical IMs

Quantization of Classical IMs

Bethe Ansatz

Form Factors for IMs

Boundary IMs

Vertex Models

Applications to Condensed Matter

Knot Theory

Matrix Models

Topological Field Theories

String Theory

SeibergWitten (SW) Theory and IMs
As the release number increases, this list of missing items should shorten until it dissappears (what optimism!!!). However, even if this occurs, it would by no means imply that we had achieved a complete coverage of these topics; it should be only interpreted as a completion of our target, which is to allow the reader a foundation for indulging in the exploration of mathematical physics and physical mathematics.


Abbreviations are in general defined at the place of their first occurrence. However, especially if you do not read this document sequentially, relying on this for definitions may be somewhat cumbersome. Certain wellknown abbreviations may even not be expanded in any place in the document. Therefore, we have included a table of abbreviations (Table 1), which appears on the following page.
Abbreviation  Explanation 

BCFT  Boundary Conformal Field Theory 
CFT  Conformal Field Theory 
CBC  Conformal Boundary Condition 
CGC  Coulomb Gas Construction 
CGF  Coulomb Gas Formulation 
DSZ  DiracSchwingerZwanziger 
GUT  Grand Unified Theory 
IM  Integrable Model 
LANL  Los Alamos National Laboratory 
l.h.s.  left hand side 
MM  Minimal Model 
OPE  Operator Product Expansion 
QCD  Quantum Chromodynamics 
QED  Quantum Electrodynamics 
QFT  Quantum Field Theory 
QG  Quantum Group 
QM  Quantum Mechanics 
RCFT  Rational Conformal Field Theory 
reg  NonSingular Terms in Operator Product Expansion 
r.h.s.  right hand side 
RSOS  Restricted SolidonSolid 
SCFT  Supersymmetric Conformal Field Theory 
SG  SineGordon 
Smatrix  Scattering Matrix 
SOS  SolidonSolid 
SUSY  Supersymmetry 
SUGRA  Supergravity 
SW  SeibergWitten 
TOE  Theory of Everything 
UMM  Unitary Minimal Model 
Wmatrix  Wall (or Reflection) Matrix 
w.r.t.  with respect to 
WZWN  WessZuminoWittenNovikov 
YBE  YangBaxter Equation 
Contents
PART I
PART II
Warning! Part II has not yet
been released. The contents and page numbers for Part II are therefore
preliminary, and subject to change.