Quantum Invariants of Knots and 3-Manifolds by Vladimir G. Turaev (De Gruyter Studies in Mathematics: De Gruyter) Due to the strong appeal and wide use of this monograph, it is now available in its second revised edition. The monograph gives a systematic treatment of 3-dimensional topological quantum field theories (TQFTs) based on the work of the author with N. Reshetikhin and O. Viro. This subject was inspired by the discovery of the Jones polynomial of knots and the Witten-Chern-Simons field theory. On the algebraic side, the study of 3-dimensional TQFTs has been influenced by the theory of braided categories and the theory of quantum groups.
The book is divided into three parts. Part I presents a construction of 3-dimensional TQFTs and 2-dimensional modular functors from so-called modular categories. This gives a vast class of knot invariants and 3-manifold invariants as well as a class of linear representations of the mapping class groups of surfaces. In Part II the technique of 6j-symbols is used to define state sum invariants of 3-manifolds. Their relation to the TQFTs constructed in Part I is established via the theory of shadows. Part III provides constructions of modular categories, based on quantum groups and skein modules of tangles in the 3-space.
This fundamental contribution to topological quantum field theory is accessible to graduate students in mathematics and physics with knowledge of basic algebra and topology. It is an indispensable source for everyone who wishes to enter the forefront of this fascinating area at the borderline of mathematics and physics.
This second edition does not essentially differ from the first one (1994). A few misprints were corrected and several references were added. The problems listed at the end of the first edition have become outdated, and are deleted here. It should be stressed that the notes at the end of the chapters reflect the author's viewpoint at the moment of the first edition.
Since 1994, the theory of quantum invariants of knots and 3-manifolds has expanded in a number of directions and has achieved new significant results. I enumerate here some of them without any pretense of being exhaustive (the reader will find the relevant references in the bibliography at the end of the book).
In the 1980s we have witnessed the birth of a fascinating new mathematical theory. It is often called by algebraists the theory of quantum groups and by topologists quantum topology. These terms, however, seem to be too restrictive and do not convey the breadth of this new domain which is closely related to the theory of von Neumann algebras, the theory of Hopf algebras, the theory of representations of semisimple Lie algebras, the topology of knots, etc. The most spectacular achievements in this theory are centered around quantum groups and invariants of knots and 3-dimensional manifolds.
The whole theory has been, to a great extent, inspired by ideas that arose in theoretical physics. Among the relevant areas of physics are the theory of exactly solvable models of statistical mechanics, the quantum inverse scattering method, the quantum theory of angular momentum, 2-dimensional conformal field theory, etc. The development of this subject shows once more that physics and mathematics intercommunicate and influence each other to the profit of both disciplines.
Three major events have marked the history of this theory. A powerful original impetus was the introduction of a new polynomial invariant of classical knots and links by V. Jones (1984). This discovery drastically changed the scenery of knot theory. The Jones polynomial paved the way for an intervention of von Neumann algebras, Lie algebras, and physics into the world of knots and 3-manifolds.
The second event was the introduction by V. Drinfel'd and M. Jimbo (1985) of quantum groups which may roughly be described as 1-parameter deformations of semisimple complex Lie algebras. Quantum groups and their representation theory form the algebraic basis and environment for this subject. Note that quantum groups emerged as an algebraic formalism for physicists' ideas, specifically, from the work of the Leningrad school of mathematical physics directed by L. Faddeev.
In 1988 E. Witten invented the notion of a topological quantum field theory and outlined a fascinating picture of such a theory in three dimensions. This picture includes an interpretation of the Jones polynomial as a path integral and relates the Jones polynomial to a 2-dimensional modular functor arising in conformal field theory. It seems that at the moment of writing (beginning of 1994), Witten' s approach based on path integrals has not yet been justified mathematically. Wit-ten's conjecture on the existence of non-trivial 3-dimensional TQFT's has served as a major source of inspiration for the research in this area. From the historical perspective it is important to note the precursory work of A. S. Schwarz (1978) who first observed that metric-independent action functionals may give rise to topological invariants generalizing the Reidemeister-Ray-Singer torsion.
The development of the subject (in its topological part) has been strongly influenced by the works of M. Atiyah, A. Joyal and R. Street, L. Kauffman, A. Kirillov and N. Reshetikhin, G. Moore and N. Seiberg, N. Reshetikhin and V. Turaev, G. Segal, V. Turaev and 0. Viro, and others (see References). Although this theory is very young, the number of relevant papers is overwhelming. We do not attempt to give a comprehensive history of the subject and confine ourselves to sketchy historical remarks in the chapter notes.
In this monograph we focus our attention on the topological aspects of the theory. Our goal is the construction and study of invariants of knots and 3-manifolds. There are several possible approaches to these invariants, based on ChernSimons field theory, 2-dimensional conformal field theory, and quantum groups. We shall follow the last approach. The fundamental idea is to derive invariants of knots and 3-manifolds from algebraic objects which formalize the properties of modules over quantum groups at roots of unity. This approach allows a rigorous mathematical treatment of a number of ideas considered in theoretical physics.
This monograph is addressed to mathematicians and physicists with a knowledge of basic algebra and topology. We do not assume that the reader is acquainted with the theory of quantum groups or with the relevant chapters of mathematical physics.
Besides an exposition of the material available in published papers, this monograph presents new results of the author, which appear here for the first time. Indications to this effect and priority references are given in the chapter notes.
The fundamental notions discussed in the monograph are those of modular category, modular functor, and topological quantum field theory (TQFT). The mathematical content of these notions may be outlined as follows.
Modular categories are tensor categories with certain additional algebraic structures (braiding, twist) and properties of semisimplicity and finiteness. The notions of braiding and twist arise naturally from the study of the commutativity of the tensor product. Semisimplicity means that all objects of the category may be decomposed into "simple" objects which play the role of irreducible modules in representation theory. Finiteness means that such a decomposition can be performed using only a finite stock of simple objects.
The use of categories should not frighten the reader unaccustomed to the abstract theory of categories. Modular categories are defined in algebraic terms and have a purely algebraic nature. Still, if the reader wants to avoid the language of categories, he may think of the objects of a modular category as finite dimensional modules over a Hopf algebra.
Modular functors relate topology to algebra and are reminiscent of homology. A modular functor associates projective modules over a fixed commutative ring K to certain "nice" topological spaces. When we speak of an n-dimensional modular functor, the role of "nice" spaces is played by closed n-dimensional manifolds.
The book consists of three parts. Part I (Chapters I—V) is concerned with the construction of a 2-dimensional modular functor and 3-dimensional TQFT from a modular category. Part II (Chapters VI—X) deals with 6j -symbols, shadows, and state sums on shadows and 3-manifolds. Part III (Chapters XI, XII) is concerned with constructions of modular categories.
It is possible but not at all necessary to read the chapters in their linear order. The reader may start with Chapter III or with Chapters VIII, IX which are independent of the previous material. It is also possible to start with Part III which is almost independent of Parts I and II, one needs only to be acquainted with the definitions of ribbon, modular, semisimple, Hermitian, and unitary categories given in Section 1.1 (i.e., Section 1 of Chapter I) and Sections II.1, 11.4, 11.5.
The interdependence of the chapters is presented in the following diagram. An arrow from A to B indicates that the definitions and results of Chapter A are essential for Chapter B. Weak dependence of chapters is indicated by dotted arrows.
The content of the chapters should be clear from the headings. The following remarks give more directions to the reader.
Chapter I starts off with ribbon categories and invariants of colored framed graphs and links in Euclidean 3-space. The relevant definitions and results, given in the first two sections of Chapter I, will be used throughout the book. They contain the seeds of main ideas of the theory. Sections 1.3 and 1.4 are concerned with the proof of Theorem 1.2.5 and may be skipped without much loss.
Chapter II starts with two fundamental sections. In Section II.1 we introduce modular categories which are the main algebraic objects of the monograph. In Section 11.2 we introduce the invariant 7 of closed oriented 3-manifolds. In Section 11.3 we prove that 7 is well defined. The ideas of the proof are used in the same section to construct a projective linear action of the group SL(2, Z). This action does not play an important role in the book, rather it serves as a precursor for the actions of modular groups of surfaces on the modules of states introduced in Chapter IV. In Section 11.4 we define semisimple ribbon categories and establish an analogue of the Verlinde-Moore-Seiberg formula known in conformal field theory. Section 11.5 is concerned with Hermitian and unitary modular categories.
Chapter III deals with axiomatic foundations of topological quantum field theory. It is remarkable that even in a completely abstract set up, we can establish meaningful theorems which prove to be useful in the context of 3-dimensional TQFT's. The most important part of Chapter III is the first section where we give an axiomatic definition of modular functors and TQFT's. The language introduced in Section III.1 will be used systematically in Chapter IV. In Section 111.2 we establish a few fundamental properties of TQFT's. In Section 111.3 we introduce the important notion of a non-degenerate TQFT and establish sufficient conditions for isomorphism of non-degenerate anomaly-free TQFT's. Section 111.5 deals with Hermitian and unitary TQFT's, this study will be continued in the 3-dimensional setting at the end of Chapter IV. Sections 111.4 and 111.6 are more or less isolated from the rest of the book; they deal with the abstract notion of a quantum invariant of topological spaces and a general method of killing the gluing anomalies of a TQFT.
In Chapter IV we construct the 3-dimensional TQFT associated to a modular category. It is crucial for the reader to get through Section IV.1, where we define the 3-dimensional TQFT for 3-cobordisms with parametrized boundary. Section IV.2 provides the proofs for Section IV.1; the geometric technique of Section IV.2 is probably one of the most difficult in the book. However, this technique is used only a few times in the remaining part of Chapter IV and in Chapter V. Section IV.3 is purely algebraic and independent of all previous sections. It provides generalities on Lagrangian relations and Maslov indices. In Sections IV.4—IV.6 we show how to renormalize the TQFT introduced in Section IV.1 in order to replace parametrizations of surfaces with Lagrangian spaces in 1-homologies. The 3-dimensional TQFT (Pie, re), constructed in Section IV.6 and further studied in Section IV.7, is quite suitable for computations and applications. This TQFT has anomalies which are killed in Sections IV.8 and IV.9 in two different ways. The anomaly-free TQFT constructed in Section IV.9 is the final product of Chapter IV. In Sections IV.1 0 and IV.1 1 we show that the TQFT's derived from Hermitian (resp. unitary) modular categories are themselves Hermitian (resp. unitary). In the purely algebraic Section IV.1 2 we introduce the Verlinde algebra of a modular category and use it to compute the dimension of the module of states of a surface.
The results of Chapter IV shall be used in Sections V.4, V.5,
VII.4, and X.8.
Chapter V is devoted to a detailed analysis of the 2-dimensional modular
functors (2-DMF's) arising from modular categories. In Section V.1 we give
an axiomatic definition of 2-DMF's and rational 2-DMF's independent of all
previous material. In Section V.2 we show that each (rational) 2-DMF gives rise to a (modular) ribbon category. In Section V.3 we introduce the more subtle notion of a weak rational 2-DMF. In Sections V.4 and V.5 we show that the constructions of Sections IV.1—IV.6, being properly reformulated, yield a weak rational 2-DMF.
Chapter VI deals with 6j -symbols associated to a modular category. The most important part of this chapter is Section VI.5, where we use the invariants of ribbon graphs introduced in Chapter Ito define so-called normalized 6j -symbols. They should be contrasted with the more simple-minded 6j -symbols defined in Section VI.1 in a direct algebraic way. The approach of Section VI.1 generalizes the standard definition of 6j -symbols but does not go far enough. The fundamental advantage of normalized 6j -symbols is their tetrahedral symmetry. Three intermediate sections (Sections VI.2—VI.4) prepare different kinds of preliminary material necessary to define the normalized 6j -symbols.
In the first section of Chapter VII we use 6j -symbols to define state sums on triangulated 3-manifolds. Independence on the choice of triangulation is shown in Section VII.2. Simplicial 3-dimensional TQFT is introduced in Section VII.3. Finally, in Section VII.4 we state the main theorems of Part II; they relate the theory developed in Part I to the state sum invariants of closed 3-manifolds and simplicial TQFT's.
Chapters VIII and IX are purely topological. In Chapter VIII we discuss the general theory of shadows. In Chapter IX we consider shadows of 4-manifolds, 3-manifolds, and links in 3-manifolds. The most important sections of these two chapters are Sections VIII.1 and IX.1 where we define (abstract) shadows and shadows of 4-manifolds. The reader willing to simplify his way towards Chapter X may read Sections VIII.1, VIII.2.1, VIII.2.2, VIII.6, IX.1 and then proceed to Chapter X coming back to Chapters VIII and IX when necessary.
In Chapter X we combine all the ideas of the previous chapters. We start with state sums on shadowed 2-polyhedra based on normalized 6j -symbols (Section X.1) and show their invariance under shadow moves (Section X.2). In Section X.3 we interpret the invariants of closed 3-manifolds 7(M) and !MI introduced in Chapters II and VII in terms of state sums on shadows. These results allow us to show that 1M I = 7(M) 7(—M). Sections X.4—X.6 are devoted to the proof of a theorem used in Section X.3. Note the key role of Section X.5 where we compute the invariant F of links in R3 in terms of 6j -symbols. In Sections X.7 and X.8 we relate the TQFT's constructed in Chapters IV and VII. Finally, in Section X.9 we use the technique of shadows to compute the invariant 7 for graph 3-manifolds.
In Chapter XI we explain how quantum groups give rise to modular categories. We begin with a general discussion of quasitriangular Hopf algebras, ribbon Hopf algebras, and modular Hopf algebras (Sections XI.1—XI.3 and XI.5). In order to derive modular categories from quantum groups we use more general quasimodular categories (Section XI.4). In Section XI.6 we outline relevant results from the theory of quantum groups at roots of unity and explain how to obtain modular categories. For completeness, we also discuss quantum groups with generic parameter; they give rise to semisimple ribbon categories (Section XI.7).
In Chapter XII we give a geometric construction of the modular categories determined by the quantum group Uq(s/2((C)) at roots of unity. The corner-stone of this approach is the skein theory of tangle diagrams (Sections XII.1 and XII.2) and a study of idempotents in the Temperley-Lieb algebras (Sections XII.3 and XII.4). After some preliminaries in Sections XII.5 and XII.6 we construct modular skein categories in Section XII.7. These categories are studied in the next two sections where we compute multiplicity modules and discuss when these categories are unitary.
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