Low-Dimensional Topology and Quantum Field Theory

318 Pages·1993·10.482 MB·English

by  Louis H. Kauffman (auth.), Hugh Osborn (eds.)

Low-Dimensional Topology
and Quantum Field Theory
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Series B: Physics
Low-Dimensional Topology
and Quantum Field Theory
Edited by
Hugh Osborn
University of Cambridge
Cambridge, United Kingdom
Springer Science+Business Media, LLC
Proceedings of a NATO Advanced Research Workshop on
Low-Dimensional Topology and Quantum Field Theory,
held September 6-12, 1992,
in Cambridge, United Kingdom
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Library of Congress Cataloglng-ln-PublIcat1on Data
Low-dimensional topology and quantum field theory / edited by Hugh
Osborn.
p. cm. — (NATO ASI series. Series B. Physics ; v. 315)
Includes bibliographical references and index.
ISBN 978-1-4899-1614-3
1. Quantum field theory—Congresses. 2. Low-dimensional topology¬
-Congresses. 3. Mathematical physics—Congresses. I. Osborn,
Hugh. II. Series.
QC174.45.A1L68 1993
530. 1" 43—dc20 93-28686
CIP
ISBN 978-1-4899-1614-3 ISBN 978-1-4899-1612-9 (eBook)
DOI 10.1007/978-1-4899-1612-9
© Springer Science+Business Media New York 1993
Originally published by Plenum Press, New York in 1993
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Preface
The motivations, goals and general culture of theoretical physics and mathematics
are different. Most practitioners of either discipline have no necessity for most of the
time to keep abreast of the latest developments in the other. However on occasion
newly developed mathematical concepts become relevant in theoretical physics and
the less rigorous theoretical physics framework may prove valuable in understanding
and suggesting new theorems and approaches in pure mathematics. Such interdis
ciplinary successes invariably cause much rejoicing, as over a prodigal son returned.
In recent years the framework provided by quantum field theory and functional in
tegrals, developed over half a century in theoretical physics, have proved a fertile
soil for developments in low dimensional topology and especially knot theory. Given
this background it was particularly pleasing that NATO was able to generously sup
port an Advanced Research Workshop to be held in Cambridge, England from 6th to
12th September 1992 with the title Low Dimensional Topology and Quantum Field
Theory. Although independently organised this overlapped as far as some speak
ers were concerned with a longer term programme with the same title organised by
Professor M Green, Professor E Corrigan and Dr R Lickorish. The contents of this
proceedings of the workshop demonstrate the breadth of topics now of interest on the
interface between theoretical physics and mathematics as well as the sophistication
of the mathematical tools required in current theoretical physics.
As director of the workshop I would first like to thank all the participants for their en
thusiasm throughout the meeting and readiness to attend lectures from early morning
to late evening. The authors of articles to this volume also deserve much thanks for
their rapid response in providing their contributions and for extending my knowledge
of dialects of TEX. I would also like to express my thanks to Professor E Corrigan
for much assistance and smoothing over of difficulties, despite conflicting pressures,
before and during the workshop. I should also mention the unfailing help and coop
eration I received from Mrs L Troake, the warden of Wolfson Court, Girton College
where most of the participants stayed during the meeting. My thanks are also due
to Dr P Landshoff for resolving various administrative hassles in concluding the final
paperwork. Finally I would like to express my unreserved gratitude to Gerard Watts
and David McAvity for helping with their TEXpertise in ensuring that the articles in
this volume have at least approximately a uniform style. The skills of hacking should
no longer be derided.
Hugh Osborn
April 1993
v
Contents
Combinatorial recoupling theory and 3-manifold invariants 1
L.H. Kauffman
Quantum field theory and A,B,C,D IRF model invariants 19
O.J. Backofen
On combinatorial three-manifold invariants 31
G. Felder and O. Grandjean
Schwinger-Dyson equation in three dimensional
simplicial quantum gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
H.Ooguri
Observables in the Kontsevich model 73
P. Di Francesco
Matrix models in statistical mechanics and quantum field theory,
recent examples and problems ... ...................... ....... ..... 85
G.M. Cicuta
Dilogarithms and W-algebras 95
W. Nahm
Dilogarithm identities and spectra in conformal field theory 99
A.N. Kirillov
Physical states in topological coset models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 109
J. Sonnenschein
Finite W symmetry in finite dimensional integrable systems ............... , 123
T. Tjin
On the "Drinfeld-Sokolov" reduction of the
Khizhnik-Zamolodchikov equation . . . .. . . . . . . . . .. .. . . . . .. .. . . .. . . .. 131
P. Furlan, A.Ch. Ganchev, R. Paunov and V.B. Petkova
Noncritical dimensions for critical string theory:
life beyond the Calabi-Yau frontier ................................ 143
R. Schimmrigk
Woo algebra in two-dimensional black holes ................................ 159
T. Eguchi, H. Kanno and S.K. Yang
vii
Graded Lie derivatives and short distance expansions in two dimensions .... 169
R. Dick
2D Black holes and 2D gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 177
F. Ardalan
The structure of finite dimensional affine Hecke algebra quotients
and their realization in 2D lattice models . . . . . . . . . . . . . . . . . . . . . . . . .. 183
D. Levy
An exact renormalisation in a vertex model ................................ 193
B.W. Westbury
New representations of the Temperley-Lieb algebra with applications ....... 203
H.N.V. Temperley
Order-disorder quantum symmetry in G-spin models . . . . . . . . . . . . . . . . . . . . . .. 213
K. Szlachanyi
Quantum groups, quantum spacetime and Dirac equation . . . . . . . . . . . . . . . . .. 221
A. Schirrmacher
Hamiltonian structure of equations appearing in random matrices ........... 231
J. Harnad, C.A. Tracy and H. Widom
On the existence of pointlike localized fields in
conformally invariant quantum physics . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 247
M. Joerss
The phase space of the Wess-Zumino-Witten model . . . . . . . . . . . . . . . . . . . . . . . .. 261
G. Papadopoulos and B. Spence
Regularization and renormalization of Chern-Simons theory . . . . . . . . . . . . . . . .. 269
G. Giavarini, C.P. Martin and F. Ruiz Ruiz
Ray-Singer torsion, topological field theories and the
=
Riemann zeta function at s 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 279
C. Nash and D.J. O'Connor
Monstrous moonshine and the uniqueness of the moonshine module . . . . . . . .. 289
M.P. Tuite
Lie algebras and polynomial solutions of differential equations . . . . . . . . . . . . .. 297
A. Turbiner
Torus actions, moment maps, and the symplectic geometry of
the moduli space of flat connections on a two-manifold . . . . . . . . . . . . .. 307
L.C. Jeffrey and J. Weitsman
Geometric quantization and Witten's semiclassical manifold invariants . . . . .. 317
L.C. Jeffrey and J. Weitsman
Index ...................................................................... 323
viii
COMBINATORIAL RECOUPLING THEORY AND
3-MANIFOLD INVARIANTS
Louis H. Kauffman
Department of Mathematics, Statistics and Computer Science
University of Illinois at Chicago
Chicago, Illinois 60680
ABSTRACT
This paper discusses combinatorial recoupling theory, first in relation to the
vector cross product algebra and a reformulation of the Four Colour Theorem, and
secondly in relation to the Temperley-Lieb algebra, the Jones polynomial and the
SU(2) 3-Manifold invariants of Witten, Reshetikhin and Turaev.
1. INTRODUCTION
This paper discusses combinatorial recoupling theory, first in relation to the
vector cross product algebra and a reformulation of the Four Colour Theorem, and
secondly in relation to the Temperley-Lieb algebra, the Jones polynomial and the
SU(2) 3-Manifold invariants of Witten, Reshetikhin and Turaev.
Section 2 discusses a simple recoupling theory related to the vector cross product
algebra that has implications for the colouring problem for plane maps. Section 3 dis
cusses the combinatorial structure of the Temperley-Lieb algebra. We investigate an
algebra of capforms and boundaries (the boundary logic) that underlies the structure
of the Temperley-Lieb algebra. This capform algebra gives insight into the nature of
the Jones-Wenzl projectors that are the basic construction for the recoupling theory
for Temperley-Lieb algebra discussed in Section 4. Section 5 discusses the definition
of the Witten-Reshetikhin-Turaev invariant of 3-manifolds. Section 6 explains how
to translate the definition in section 5 into a partition function on a 2-cell complex
by using a reformulation of the Kirillov-Reshetikhin shadow world appropriate to the
recoupling theory of Section 4. The work described in 4 and 6 is joint work of the
author and S. Lins and will appear in [KL92J.
The foundations of the recoupling theory presented here go back to the work of
Roger Penrose on spin networks in the 1960's and 1970's. On the mathematical physics
side this has led to a number of interrelations with the 3-manifold invariants discussed
here and theories of quantum gravity in two dimensions of space and one dimension
of time. There has not been room in this paper to go into these relationships. For
the record, the reference list includes papers by Penrose and also more recent authors
on this topic (Hasslacher and Perry, Crane, Williams and Archer, Ooguri). It is
the author's belie! that the approach to the recoupling theory discussed herein will
illuminate questions about these matters of mathematical physics.
2. TREES AND FOUR COLOURS
It is amusing and instructive to begin with the subject of trees and the Four
Colour Theorem. This provides a miniature area that illustrates many subtle issues
in relation to recoupling theory.
Recall the main theorem of [KAUF90j. There the Four Colour Theorem is refor
mulated as a problem about the non-associativity of the vector cross product algebra
in three dimensional space. Specifically, let V = {i, -i,j, -j, k, -k, 1, -1, O} closed
under the vector cross product (Here the cross product of a and b is denoted in the
usual fashion by a x b.):
i x j = k, j x i = -k,
j x k = i, k x j = -i,
k xi = j, i x k = -j,
i x i = j x j = k x k = 0,
a x 0 = 0 x a = 0 for any a
-0=0
ax(-l)=-lxa=-a for any a
=
-(-a) a for any a.
V is a multiplicatively closed subset of the usual vector cross product algebra. V is
not associative, since (e.g.)
i x (i x j) = i x k = -j while (i x i) x j = 0 x j = O.
Given variables aI, a2, a3, ... , an, let L and R denote two parenthesizations of the
product al x a2 ... x an. Then we ask for solutions to the equation L = R with values
for the Ai taken from the set V· = {i,j,k} and such that the resultant values of L
and R are non-zero. In [KAUF90j such a solution is called a sharp solution to the
equation L = R, and it is proved that the existence of sharp solutions for all nand
all Land R is equivalent to the Four Colour Theorem.
The graph theory behind this algebraic reformulation of the Four Colour Theorem
comes from the fact that an associated product of a collection of variables corresponds
to a rooted planar tree. This correspondence is illustrated below. Each binary product
corresponds to a binary branching of the corresponding tree.
2