Volume 35, Issue 10, October 1994
Index of content:

Supersymmetric Yang–Mills theory on a four‐manifold
View Description Hide DescriptionBy exploiting standard facts about N=1 and N=2 supersymmetric Yang–Mills theory, the Donaldson invariants of four‐manifolds that admit a Kahler metric can be computed. The results are in agreement with available mathematical computations, and provide a powerful check on the standard claims about supersymmetric Yang–Mills theory.

Four‐dimensional topological quantum field theory, Hopf categories, and the canonical bases
View Description Hide DescriptionA new combinatorial method of constructing four‐dimensional topological quantum field theories is proposed. The method uses a new type of algebraic structure called a Hopf category. The construction of a family of Hopf categories related to the quantum groups and their canonical bases is also outlined.

η‐invariants and determinant lines
View Description Hide DescriptionThe η‐invariant of an odd dimensional manifold with boundary is investigated. The natural boundary condition for this problem requires a trivialization of the kernel of the Dirac operator on the boundary. The dependence of the η‐invariant on this trivialization is best encoded by the statement that the exponential of the η‐invariant lives in the determinant line of the boundary. Our main results are a variational formula and a gluing law for this invariant. These results are applied to reprove the formula for the holonomy of the natural connection on the determinant line bundle of a family of Dirac operators, also known as the ‘‘global anomaly formula.’’ The ideas developed here fit naturally with recent work in topological quantum field theory, in which gluing (which is a characteristic formal property of the path integral and the classical action) is used to compute global invariants on closed manifolds from local invariants on manifolds with boundary.

The Chern–Simons action in noncommutative geometry
View Description Hide DescriptionA general definition of Chern–Simons actions in noncommutative geometry is proposed and illustrated in several examples. These examples are based on ‘‘space–times’’ which are products of even‐dimensional, Riemannian spin manifolds by a discrete (two‐point) set. If the * algebras of operators describing the noncommutative spaces are generated by functions over such ‘‘space–times’’ with values in certain Clifford algebras the Chern–Simons actions turn out to be the actions of topological gravity on the even‐dimensional spin manifolds. By constraining the space of field configurations in these examples in an appropriate manner one is able to extract dynamical actions from Chern–Simons actions.

Reshetikhin’s formula for the Jones polynomial of a link: Feynman diagrams and Milnor’s linking numbers
View Description Hide DescriptionFeynman diagrams are used to prove a formula for the Jones polynomial of a link derived recently by Reshetikhin. This formula presents the colored Jones polynomial as an integral over the coadjoint orbits corresponding to the representations assigned to the link components. The large k limit of the integral can be calculated with the help of the stationary phase approximation. The Feynman rules allow one to express the phase in terms of integrals over the manifold and the link components. Its stationary points correspond to flat connections in the link complement. A relation between the dominant part of the phase and Milnor’s linking numbers is conjectured. It is checked explicitly for the triple and quartic numbers by comparing their expression through the Massey product with Feynman diagram integrals.

On the self‐linking of knots
View Description Hide DescriptionThis note describes a subcomplex F of the de Rham complex of parametrized knot space, which is combinatorial over a number of universal ‘‘Anomaly Integrals.’’ The self‐linking integrals of Guadaguini, Martellini, and Mintchev [‘‘Perturbative aspects of Chern–Simons field theory,’’ Phys. Lett. B 227, 111 (1989)] and Bar‐Natan [‘‘Perturbative aspects of the Chern–Simons topological quantum field theory,’’ Ph.D. thesis, Princeton University, 1991; also ‘‘On the Vassiliev Knot Invariants’’ (to appear in Topology)] are seen to represent the first nontrivial element in H ^{0}(F)—occurring at level 4, and are anomaly free. However, already at the next level an anomalous term is possible.

Extended reflection equation algebras, the braid group on a handlebody, and associated link polynomials
View Description Hide DescriptionThe correspondence of the braid group on a handlebody of arbitrary genus to the algebra of Yang–Baxter and extended reflection equation operators is shown. Representations of the infinite‐dimensional extended reflection equationalgebra in terms of direct products of quantum algebra generators are derived; they lead to a representation of this braid group in terms of R‐matrices. Restriction to the reflection equation operators only gives the colored braid group. The reflection equation operators, describing the effect of handles attached to a three‐ball, satisfy characteristic equations that give rise to additional skein relations and thereby invariants of links on handlebodies. The origin of the skein relations is explained and they are derived from an adequately adapted handlebody version of the Jones polynomial. Relevance of these results to the construction of link polynomials on closed three‐manifolds via Heegard splitting and surgery is indicated.

Classification and construction of unitary topological field theories in two dimensions
View Description Hide DescriptionIt is proven that unitary two‐dimensional topological field theories are uniquely characterized by n positive real numbers λ_{1},...,λ_{ n }, which can be regarded as the eigenvalues of a Hermitian handle creation operator. The number n is the dimension of the Hilbert space associated with the circle, and the partition functions for closed surfaces have the form Z _{ g }=∑^{ n } _{ i=1}λ^{ g−1} _{ i }, where g is the genus. The eigenvalues can be arbitrary positive numbers. It is shown how such a theory can be constructed on triangulated surfaces.

A path‐integral approach to polynomial invariants of links
View Description Hide DescriptionA self‐contained, genuinely three‐dimensional, monodromy‐matrix based, nonperturbative, covariant path‐integral approach to polynomial invariants of knots and links in the framework of (topological) quantum Chern–Simons field theory is proposed. The idea of ‘‘physical’’ observables represented by an auxiliary topological quantum‐mechanical model in an external gauge field is introduced as a replacement for the limited notion of the Wilson loop. The possibility of using various generalizations of the Chern–Simons action (also higher‐dimensional ones) as well as a purely functional language thereby arises. The theory is quantized in the framework of the antibracket‐antifield formalism of Batalin and Vilkovisky, which is well adapted to this case. Using Stokes’s theorem and formal translational invariance of the path‐integral measure we derive a monodromy matrix corresponding to an arbitrary pair of irreducible representations of an arbitrary semisimple Lie group.

Space–time topology change and stringy geometry^{a)}
View Description Hide DescriptionRecent work which has significantly honed the geometric understanding and interpretation of the moduli space of certain N=2 superconformal field theories is reviewed. This has resolved some important issues in mirror symmetry and has also established that string theory admits physically smooth processes which can result in a change in topology of the spatial universe. Recent work which illuminates some properties of physically related theories associated with singular spaces such as orbifolds is described.

An explicit description of the symplectic structure of moduli spaces of flat connections
View Description Hide DescriptionThe moduli space of flat connections on a principal G‐bundle over a compact oriented surface of genus g≥1 is considered herein. Using the holonomies around noncontractible loops, the moduli space is described as a quotient of a submanifold of G ^{2g }. An explicit expression is obtained for the symplectic form on the smooth part of moduli space, and several properties of this form are established.

The semiclassical limit of the two‐dimensional quantum Yang–Mills model
View Description Hide DescriptionThe semiclassical limit of the quantum Yang–Mills partition function on a compact oriented surface is related to the symplectic volume of the moduli space of flat connections, by using an explicit expression for the symplectic form. This gives an independent proof of some recent results of Witten and Forman.

Topological interpretations of quantum Hall conductance
View Description Hide DescriptionThe high precision of the quantum Hall effect is cited as evidence that the Hall conductance is a topological quantum number invariant under reasonably small perturbations. In this article a survey is made of the Hall conductance as a topological quantum number, of relations between the various interpretations of the integer quantum Hall effect, and of their generalization to the fractional quantum Hall effect.

The noncommutative geometry of the quantum Hall effect
View Description Hide DescriptionAn overview of the integer quantum Hall effect is given. A mathematical framework using nonommutative geometry as defined by Connes is prepared. Within this framework, it is proved that the Hall conductivity is quantized and that plateaux occur when the Fermi energy varies in a region of localized states.

Quantum temporal logic and decoherence functionals in the histories approach to generalized quantum theory
View Description Hide DescriptionThe recent suggestion that a temporal form of quantum logic provides the natural mathematical framework within which to discuss the proposal by Gell‐Mann and Hartle for a generalized form of quantum theory based on the ideas of histories and decoherence functionals is analyzed and developed herein. Particular stress is placed on properties of the space of decoherence functionals, including one way in which certain global and topological properties of a classical system are reflected in a quantum history theory.

Topology change in (2+1)‐dimensional gravity
View Description Hide DescriptionIn (2+1)‐dimensional general relativity, the path integral for a manifoldM can be expressed in terms of a topological invariant, the Ray–Singer torsion of a flat bundle over M. For some manifolds, this makes an explicit computation of transition amplitudes possible. In this paper, the amplitude for a simple topology‐changing process is evaluated. It is shown that certain amplitudes for spatial topology change are nonvanishing—in fact, they can be infrared divergent—but that they are infinitely suppressed relative to similar topology‐preserving amplitudes.

Exotic smoothness and physics
View Description Hide DescriptionThe essential role played by differentiable structures in physics is reviewed in light of recent mathematical discoveries that topologically trivial space–time models, especially the simplest one, R ^{4}, possess a rich multiplicity of such structures, no two of which are diffeomorphic to each other and thus to the standard one. This means that physics has available to it a new panoply of structures for space–time models. These can be thought of as a source of new global, but not properly topological, features. This article reviews some background differential topology together with a discussion of the role which a differentiable structure necessarily plays in the statement of any physical theory, recalling that diffeomorphisms are at the heart of the principle of general relativity. Some of the history of the discovery of exotic, i.e., nonstandard, differentiable structures is reviewed. Some new results suggesting the spatial localization of such exotic structures are described and speculations are made on the possible opportunities that such structures present for the further development of physical theories.

A gravitational lens need not produce an odd number of images
View Description Hide DescriptionGiven any space–time M without singularities and any event O, there is a natural continuous mapping f of a two‐dimensional sphere into any spacelike slice T not containing O. The set of future null geodesics (or the set of past null geodesics) form a two‐sphere S ^{2} and the map f sends a point in S ^{2} to the point in T which is the intersection of the corresponding geodesic with T. Considering the f for each point of a world‐line W gives us a map F:S ^{2}×W→T. The local degree of F at a regular value y in T has the same parity as the number of null geodesics from W to y.

Multiple Hamiltonian structures for Toda‐type systems
View Description Hide DescriptionResults on the finite nonperiodic Toda lattice are extended to some generalizations of the system: The relativistic Toda lattice, the generalized Toda lattice associated with simple Lie groups and the full Kostant–Toda lattice. The areas investigated include master symmetries, recursion operators, higher Poisson brackets, invariants, and group symmetries for the systems. A