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Lecture Notes in Mathematics Editors:
A. Dold, Heidelberg
B. Eckmann, Zurich
F. Takens, Groningen
1542
Jiirg Frohlich
Quantum Groups,
Quantum Categories and
Quantum Field Theory
SpringerVerlag
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London Paris Tokyo
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Authors
Jiirg Frohlich
Theoretische Physik
ETH  Honggerberg
CH8093 Zurich, Switzerland
Thomas Kerler
Department of Mathematics
Harvard University
Cambridge, MA 02138, USA
Mathematics Subject Classification (1991): 0002, 13, 18D10, 18D99, 15A36.16D60,
16W30,16W20,18E.20F36.20K01,20L17.46L,46M,81R05,81R50,81T05.81T40
ISBN3540566236SpringerVerlag Berlin Heidelberg New York
ISBN0387566236SpringerVerlag New York Berlin Heidelberg
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© SpringerVerlag Berlin Heidelberg 1993
Printed in Germany
Typesetting: Cameraready by author/editor
46/3140543210  Printed on acidfree paper
Contents
1 Introduction and Survey of Results 1
2 Local Quantum Theory with Braid Group Statistics 17
2.1 Some Aspects of LowDimensional, Local Quantum Field Theory 17
2.2 Generalities Concerning Algebraic Field Theory 24
2.3 Statistics and Fusion of Intertwiners; Statistical Dimensions 32
2.4 Unitary Representations of the Braid Groups Derived from Local Quantum
Theory; Markov Traces 41
3 Superselection Sectors and the Structure of Fusion Rule Algebras 45
3.1 Definition of and General Relations in Fusion Rule Algebras, and their
Appearance in Local Quantum Field Theories 46
3.2 Structure Theory for Fusion Rule Algebras 51
3.3 Grading Reduction with Automorphisms and Normality Constraints in
Fusion Rule Algebras 60
3.4 Fusionrules with a Generator of Dimension not Greater than Two 72
4 Hopf Algebras and Quantum Groups at Roots of Unity 102
v
5 Representation Theory of U\ed{sl2) 119
5.1 Highest Weight Representations of i7^d(s£d+1) 119
5.2 The Irreducible and Unitary Representations of jy^ed(s£2) 122
5.3 Decomposition of Tensor Product Representations 126
5.4 Fusion Rules, and qDimensions: Selecting a List of Physical Representations 135
6 Path Representations of the Braid Groups for Quantum Groups at
Roots of Unity 141
6.1 Quotients of Representation Categories : 141
6.2 Braid Group Representations and Fusion Equations 152
6.3 Unitarity of Braid Group Representations Obtained from Uq{slj.+i) .... 160
6.4 Markov Traces 171
7 Duality Theory for Local Quantum Theories, Dimensions and Balancing
in Quantum Categories 176
7.1 General Definitions, Towers of Algebras 176
7.2 Quantum Group Symmetries of Charged Fields 190
7.3 The Index and Fundamental Decompositions 197
7.4 Balancing Phases 222
7.5 Theta  Categories 245
8 The Quantum Categories with a Generator of Dimension less than Two 284
8.1 Product Categories and Induced Categories 284
8.2 The An  Categories and Main Results 360
vi
A Undirected Graphs with Norm not Larger than Two 412
A.l Bicolorable, finite graphs 413
A.2 Bicolorable, infinite graphs (corresponding to N = oo) 415
A.3 Nonbicolorable, finite graphs 416
A.4 Nonbicolorable, infinite graphs (N = oo) 416
A.5 The higher graded fusionrule algebras 417
B Fusion Rule Algebra Homomorphisms 418
B.l an : A2n * A\ 418
B.2
B.4 aDB : D16 * Es 421
Bibliography 422
Index 429
vii
Chapter 1
Introduction and Survey of Results
Our original motivation for undertaking the work presented in this book* has been to
clarify the connections between the braid (group) statistics discovered in lowdimensional
quantum field theories and the associated unitary representations of the braid groups with
representations of the braid groups obtained from the representation theory of quantum
groups  such as Uq(g), with deformation parameter q — qN := exp(iTr/N), for some N =
3,4,.... Among quantum field theories with braid statistics there are twodimensional,
chiral conformal field theories and threedimensional gauge theories with a ChernSimons
term in their action functional. These field theories play an important role in string
theory, in the theory of critical phenomena in statistical mechanics, and in a variety of
systems of condensed matter physics, such as quantum Hall systems.
An example of a field theory with braid statistics is a chiral sector of the two
dimensional WessZuminoNovikovWitten model with group 517(2) at level k which is
closely related to the representation theory of su(2)fcKacMoody algebra, with k =
1, 2,3, The braid statistics of chiral vertex operators in this theory can be understood
by analyzing the solutions of the KnizhnikZamolodchikov equations. Work of Drinfel'd
[4] has shown that, in the example of the Sf(2)WZNW model, there is a close connection
between solutions of the KnizhnikZamolodchikov equations and the representation theory
'This book is based on the Ph.D. thesis of T.K. and on results in [6, 11, 24, 28, 42, 61]
1
of ¢/,(3/3) if the level k is related to the deformation parameter q by the equation q =
exp(iir/(fc + 2)), and k is not a rational number. For an extension of these results to the
negative rationals see [62]. Unfortunately, the Sf(2)WZNW model is a unitary quantum
field theory only for the values k = 1, 2,3, • • •, not covered by the results of Drinfel'd. Our
goal was to understand the connections between the field theory and the quantum group
for the physically interesting case of positive integer levels. (This motivates much of our
analysis in Chapters 2 through 7.)
The notion of symmetry adequate to describe the structure of superselection sectors
in quantum field theories with braid statistics turns out to be quite radically different
from the notion of symmetry that is used to describe the structure of superselection
sectors in higher dimensional quantum field theories with permutation (group) statistics,
(i.e., FermiDirac or BoseEinstein statistics). While in the latter case compact groups
and their representation theory provide the correct notion of symmetry, the situation
is less clear for quantum field theories with braid statistics. One conjecture has been
that quantum groups, i.e., quasitriangular (quasi)Hopf algebras, might provide a useful
notion of symmetry (or of "quantized symmetry") describing the main structural features
of quantum field theories with braid statistics. It became clear, fairly soon, that the
quantum groups which might appear in unitary quantum field theories have a deformation
parameter q equal to a root of unity and are therefore not semisimple. This circumstance
is the source of a variety of mathematical difficulties which had to be overcome. Work
on these aspects started in 1989, and useful results, eventually leading to the material in
Chapters 4, 5 and 6, devoted to the representation theory of Uq(g), q a root of unity, and
to the socalled vertexSOS transformation, were obtained in the diploma thesis of T.K.;
see [6]. Our idea was to combine such results with the general theory of braid statistics
in lowdimensional quantum field theories, in order to develop an adequate concept of
"quantized symmetries" in such theories; see Chapter 7, Sects. 7.1 and 7.2.
In the course of our work, we encountered a variety of mathematical subtleties and
difficulties which led us to study certain abstract algebraic structures  a class of (not
necessarily Tannakian) tensor categories  which we call quantum categories. Work of
2
Doplicher and Roberts [29] and of Deligne [56] and lectures at the 1991 Borel seminar in
Bern played an important role in guiding us towards the right concepts.
These concepts and the results on quantum categories presented in this volume,
see also [61], are of some intrinsic mathematical interest, independent of their origin in
problems of quantum field theory. Although problems in theoretical physics triggered our
investigations, and in spite of the fact that in Chapters 2, 3 and 7, Sects. 7.1 through 7.4
we often use a language coming from local quantum theory (in the algebraic formulation
of Haag and collaborators [17, 18, 19, 20]), all results and proofs in this volume (after
Chapter 2) can be understood in a sense of pure mathematics: They can be read without
knowledge of local quantum theory going beyond some expressions introduced in Chapters
2 and 3, and they are mathematically rigorous.
In order to dispel possible hesitations and worries among readers, who are pure
mathematicians, we now sketch some of the physical background underlying our work,
thereby introducing some elements of the language of algebraic quantum theory in a
nontechnical way. For additional details the reader may glance through Chapter 2.
For quantum field theories on a spacetime of dimension four (or higher) the
concept of a global gauge group, or symmetry G is, roughly speaking, the following one: The
Hilbert space Ti of physical states of such a theory carries a (highly reducible) unitary
representation of the group G. Among the densely denned operators on H there are the
socalled local field operators which transform covaxiantly under the adjoint action of the
group G. The fixed point algebra, with respect to this group action in the total field
algebra, is the algebra of observables. This algebra, denoted by A, is a C*algebra obtained
as an inductive limit of a net of von Neumann algebras A(0) of observables localized in
bounded open regions O of spacetime. The von Neumann algebras A(0) are isomorphic
to the unique hyperfinite factor of type IIIi, in all examples of algebraic field theories that
one understands reasonably well. The Hilbert space 7i decomposes into a direct sum of
orthogonal subspaces, called superselection sectors, carrying inequivalent representations
of the observable algebra A. All these representations of A can be generated by composing
a standard representation, the socalled vacuum representation, with 'endomorphisms of
3
A. Each superselection sector also carries a representation of the global gauge group G
which is equivalent to a mulitple of a distinct irreducible representation of G. As shown
by Doplicher, Haag and Roberts (DHR) [19], one can introduce a notion of tensor
product, or "composition", of superselection sectors with properties analogous to those of the
tensor product of representations of a compact group. The composition of superselection
sectors can be defined even if one does not know the global gauge group G of the theory,
yet. Prom the properties of the composition of superselection sectors, in particular from
the fusion rules of this composition and from the statistics of superselection sectors, i.e.,
from certain representations of the permutation groups canonically associated with
superselection sectors, one can reconstruct important data of the global gauge group G. In
particular, one can find its character table and its 6j symbols. As proven by Doplicher
and Roberts [29], those data are sufficient to reconstruct G. The representation category
of G turns out to reproduce all properties of the composition of superselection sectors,
and one is able to reconstruct the algebra of local field operators from these data. One
says that the group G is dual to the quantum theory described by A and Ti.
The results of Doplicher and Roberts can be viewed as the answer to a purely
mathematical duality problem (see also [56]): The fusion rules and the 6j symbols obtained
from the composition of superselection sectors are nothing but the structure constants of
a symmetric tensor category with C* structure. The problem is how to reconstruct from
such an abstract category a compact group whose representation category is isomorphic
to the given tensor category. It is an old result of Tannaka and Krein that it is always
possible to reconstruct a compact group from a symmetric tensor category if the category
is Tannakian, i.e., if we know the dimensions of the representation spaces and the Clebsch
Gordan matrices, or 3j symbols, which form the basic morphism spaces. The results of
Doplicher and Roberts represent a vast generalization of the TannakaKrein results, since
the dimensions and ClebschGordan matrices are not known a priori.
Another duality theorem related to the one of Doplicher and Roberts is due to
Deligne [56] which requires integrality of certain dimensions but no C* structure on the
symmetric tensor category. (It enables one to reconstruct algebraic groups from certain
4
symmetric tensor categories.) Disregarding some subtleties in the hypotheses of these
duality theorems, they teach us that it is equivalent to talk about compact groups or
certain symmetric tensor categories.
Quantum field theories in two and three spacetime dimensions can also be
formulated within the formalism of algebraic quantum theory of DHR, involving an algebra A
of observables and superselection sectors carrying representations of A which are
compositions of a standard representation with 'endomorphisms of A. This structure enables us
to extract an abstract tensor category described in terms of an algebra of fusion rules and
6j symbols. Contrary to the categories obtained from quantum field theories in four or
more spacetime dimensions, the tensor categories associated with quantum field theories
in two and three spacetime dimensions are, in general, not symmetric but only braided.
Therefore, they cannot be representation categories of cocommutative algebras, like group
algebras. In many physically interesting examples of field theories, these categories are
not even Tannakian and, therefore, cannot be identified, naively, with the representation
category of a Hopf algebra or a quantum group; see [61]. The. complications coming from
these features motivate many of our results in Chapters 6 through 8.
The following models of two and threedimensional quantum field theories yield
nonTannakian categories:
(1) Minimal conformal models [7] and WessZuminoNovikovWitten models [8]
in two spacetime dimensions .
The basic feature of these models is that they exhibit infinitedimensional
symmetries. The example of the 5i/(n)WZW model can be understood as a Lagrangian
field theory with action functional given by
+ 5^/^((51^).
where, classically, a field configuration g is a map from the twosphere 52 to the
group G — SU(n), and g is an arbitrary extension of g from 52 = dB3 to the ball
B3; (such an extension always exists, since ir2 of a group is trivial). The second term
5
in S(g) is the socalled WessZumino term which is denned only modk'L.
Classically, the theory exhibits a symmetry which is the product of two loop groups, for
right and left movers, respectively. For Jb = 1,2,3,..., the quantum theory
associated with S(g) has conserved currents generating two commuting 5u(n)KacMoody
algebras at level fc, whose universal enveloping algebras contain Virasoro algebras;
(Sugawaxa construction). From the representation theory of the infinitedimensional
Lie algebras of symmetry generators in these models, i.e., the representation
theory of Virasoro or KacMoody algebras, one can construct algebras of socalled
chiral vertex operators which play the role of ClebschGordan operators of (a semi
simple quotient of) the representation category of the Virasoro or KacMoody
algebra. Local conformally covaxiant field operators are then constructed by taking
linear combinations of products of two such chiral vertex operators, a holomorphic
one (left movers) and an antiholomorphic one (right movers).
Of interest in relation to the main subject of our work is that the algebras of chiral
vertex operators, the holomorphic ones, say, appearing in these models provide
us with categorial data corresponding to nonTannakian braided tensor categories.
(This can be understood by studying the multivaluedness properties and operator
product expansions of chiral vertex operators. A very thorough analysis of the
5i/(2)WZW model can be found in the papers of Tsuchiya and Kanie and of
Kohno quoted in [9]; see also [8, 61].)
Zamolodchikov and others have studied "noncritical perturbations" of minimal con
formal models which are integrable field theories [10]. Their results suggest that
there are plenty of massive quantum field theories in two spacetime dimensions
with fields exhibiting nonabelian braid statistics, as originally described in [11].
(A perturbation of minimal conformal models giving rise to massive integrable field
theories is obtained from the ^(li3)field; a field with braid statistics is the field
obtained from a chiral factor of the ^(3ii)field, after the perturbation has been
turned on [12].) To such nonconformal field theories one can also associate certain
braided tensor categories. However, the general theory of superselection sectors in
twodimensional, massive quantum field theories leads to algebraic structures more
6
general than braided tensor categories, including ones with nonabelian fusion rule
algebras. A general understanding of these structures has not been accomplished,
yet.
(2) Threedimensional ChernSimons gauge theory, [13, 14, 15] .
Consider a gauge theory in three spacetime dimensions with a simply connected,
e.ff.
compact gauge group G = SU(n). Let A denote the gauge field (vector potential)
with values in g = Lie(G), the Lie algebra of the gauge group G, and let ip be a
matter field, e.g. a twocomponent spinor field in the fundamental representation of
G. There may be further matter fields, such as Higgs fields. The action functional
of the theory is given by
S[A,TJ,,i>] C# g'ftr(F2)dvol.
±Jtr(AAdA + lAAAAA) (1.1)
+ XJi>(pA+m)ipdvol. +•••,
where g, A and m axe positive constants, and I is an integer.
This class of gauge theories has been studied in [13, 14, 15]. Although the results in
these papers are not mathematically rigorous, the main properties of these theories
are believed to be as follows:
The gluon is massive, and there is no confinement of colour. Interactions
persisting over arbitrarily large distances are purely topological and are, asymptotically,
described by a pure ChernSimons theory. Thus the statistics of coloured particles
in ChernSimons gauge theory is believed to be the same as the statistics of static
colour sources in a pure ChernSimons theory which is known explicitly [16]. The
statistics of coloured asymptotic particles can be studied by analyzing the
statistics of fields creating coloured states from the vacuum sector. Such fields are the
Mandelstam string operators, ipa(yx), which are defined, heuristically, by
lM7.) = "5>hMz)J>(exp/ ^.tfWW", (12)
where a and /3 are group indices; 7X is a path contained in a spacelike surface,
starting at x and reaching out to infinity, N is some normal ordering prescription,
7
and P denotes path ordering. (Similarly, conjugate Mandelstam strings ^>a(7ir) are
denned.)
For the field theories described in (1) and (2), one observes that when the group G is
SU(2) the combinatorial data of a braided tensor category, an algebra of fusion rules and
6j symbols (braid and fusion matrices), can be reconstructed from these field theories
which is isomorphic to a braided tensor category that is obtained fr