Index of content:
Volume 36, Issue 11, November 1995

Higher‐dimensional algebra and topological quantum field theory
View Description Hide DescriptionThe study of topological quantum field theories increasingly relies upon concepts from higher‐dimensional algebra such as n‐categories and n‐vector spaces. We review progress towards a definition of n‐category suited for this purpose, and outline a program in which n‐dimensional topological quantum field theories (TQFTs) are to be described as n‐category representations. First we describe a ‘‘suspension’’ operation on n‐categories, and hypothesize that the k‐fold suspension of a weak n‐category stabilizes for k≥n+2. We give evidence for this hypothesis and describe its relation to stable homotopy theory. We then propose a description of n‐dimensional unitary extended TQFTs as weak n‐functors from the ‘‘free stable weak n‐category with duals on one object’’ to the n‐category of ‘‘n‐Hilbert spaces.’’ We conclude by describing n‐categorical generalizations of deformation quantization and the quantum double construction.

Asymptotic expansions of Witten–Reshetikhin–Turaev invariants for some simple 3‐manifolds
View Description Hide DescriptionFor any Lie algebrag and integral level k, there is defined an invariant Z _{ k } ^{*}(M, L) of embeddings of links L in 3‐manifolds M, known as the Witten–Reshetikhin–Turaev invariant. It is known that for links in S ^{3}, Z _{ k } ^{*}(S ^{3}, L) is a polynomial in q=exp (2πi/(k+c _{g} ^{ v }), namely, the generalized Jones polynomial of the link L. This paper investigates the invariant Z _{ r−2} ^{*}(M,○/) when g =sl_{2} for a simple family of rational homology 3‐spheres, obtained by integer surgery around (2, n)‐type torus knots. In particular, we find a closed formula for a formal power series Z _{∞}(M)∈Q[[h]] in h=q−1 from which Z _{ r−2} ^{*}(M,○/) may be derived for all sufficiently large primes r. We show that this formal power series may be viewed as the asymptotic expansion, around q=1, of a multivalued holomorphic function of q with 1 contained on the boundary of its domain of definition. For these particular manifolds, most of which are not Z‐homology spheres, this extends work of Ohtsuki and Murakami in which the existence of power series with rational coefficients related to Z _{ k } ^{*}(M, ○/) was demonstrated for rational homology spheres. The coefficients in the formal power series Z _{∞}(M) are expected to be identical to those obtained from a perturbative expansion of the Witten–Chern–Simons path integral formula for Z*(M, ○/). © 1995 American Institute of Physics.

Path integration in two‐dimensional topological quantum field theory
View Description Hide DescriptionA positive, diffeomorphism‐invariant generalized measure on the space of metrics of a two‐dimensional smooth manifold is constructed. We use the term generalized measure analogously with the generalized measures of Ashtekar and Lewandowski and of Baez. A family of actions is presented which, when integrated against this measure gives the two‐dimensional axiomatic topological quantum field theories, or TQFTs, in terms of which Durhuus and Jonsson decompose every two‐dimensional unitary TQFT as a direct sum.

Topological BF theories in 3 and 4 dimensions
View Description Hide DescriptionIn this paper we discuss topological BF theories in 3 and 4 dimensions. Observables are associated to ordinary knots and links (in 3 dimensions) and to 2‐knots (in 4 dimensions). The vacuum expectation values of such observables give a wide range of invariants. Here we consider mainly the 3 dimensional case, where these invariants include Alexander polynomials, HOMFLY polynomials and Kontsevich integrals.

Quantum gravity as topological quantum field theory
View Description Hide DescriptionThe physics of quantum gravity is discussed within the framework of topological quantum field theory. Some of the principles are illustrated with examples taken from theories in which space–time is three dimensional.

Clock and category: Is quantum gravity algebraic?
View Description Hide DescriptionWe investigate the possibility that the quantum theory of gravity could be constructed discretely using algebraic methods. The algebraic tools are similar to ones used in constructing topological quantum field theories. The algebraic structures are related to ideas about the reinterpretation of quantum mechanics in a general relativistic context.

Noncommutative geometry and reality
View Description Hide DescriptionWe introduce the notion of real structure in our spectral geometry. This notion is motivated by Atiyah’s KR‐theory and by Tomita’s involution J. It allows us to remove two unpleasant features of the ‘‘Connes–Lott’’ description of the standard model, namely, the use of bivector potentials and the asymmetry in the Poincaré duality and in the unimodularity condition.

Fuzzy mass relations for the Higgs
View Description Hide DescriptionThe noncommutative approach of the standard model produces a relation between the top and the Higgs masses. We show that, for a given top mass, the Higgs mass is constrained to lie in an interval. The length of this interval is of the order of m ^{2} _{τ}/m _{ t }.

The gravitational sector in the Connes–Lott formulation of the standard model
View Description Hide DescriptionWe studied the Riemannian aspect and the Hilbert–Einstein gravitational action of the noncommutative geometry underlying the Connes–Lott construction of the action functional of the standard model. This geometry involves a two‐sheeted, Euclidian space–time. We show that if we require the space of forms to be locally isotropic and the Higgs scalar to be dynamical, then the Riemannian metrics on the two sheets of Euclidian space–time must be identical. We also show that the distance function between the two sheets is determined by a single, real scalar field whose vacuum expectation value (VEV) sets the weak scale.

Simplicial quantum gravity in four dimensions and invariants of discretized manifolds
View Description Hide DescriptionThe use of Regge’s discrete formulation of classical general relativity in attempts at constructing a theory of simplicial quantum gravity is discussed briefly. Recent work on invariants of discretized three‐manifolds, which provide a regularized version of the much earlier result of Ponzano and Regge, where the state sum was related to a Feynman path integral with the Regge action, has led to the search for invariants of discretized four‐manifolds, which might be the basis for a four‐dimensional theory of quantum gravity. This search is described, and recent developments in Regge calculus arising from it are outlined. This paper is based on a talk given at the LMS Durham Symposium on Quantum Concepts in Space and Time, July 1994.

A definition of the Ponzano–Regge quantum gravity model in terms of surfaces^{a)}
View Description Hide DescriptionWe show that the partition function of the Ponzano–Regge quantum gravity model can be written as a sum over surfaces in a (2+1)‐dimensional space–time. We suggest a geometrical meaning, in terms of surfaces, for the (regulated) divergences that appear in the partition function.

Dynamical triangulations, a gateway to quantum gravity?
View Description Hide DescriptionWe show how it is possible to formulate Euclidean two‐dimensional quantum gravity as the scaling limit of an ordinary statistical system by means of dynamical triangulations, which can be viewed as a discretization in the space of equivalence classes of metrics. Scaling relations exist and the critical exponents have simple geometric interpretations. Hartle–Hawking wave functionals as well as reparametrization invariant correlation functions which depend on the geodesic distance can be calculated. The discretized approach makes sense even in higher‐dimensional space–time. Although analytic solutions are still missing in the higher‐dimensional case, numerical studies reveal an interesting structure and allow the identification of a fixed point where we can hope to define a genuine non‐perturbative theory of four‐dimensional quantum gravity.

Monte Carlo simulations of 4d simplicial quantum gravity
View Description Hide DescriptionDynamical triangulations of four‐dimensional Euclidean quantum gravity give rise to an interesting, numerically accessible model of quantum gravity. We give a simple introduction to the model and discuss two particularly important issues. One is that contrary to recent claims there is strong analytical and numerical evidence for the existence of an exponential bound that makes the partition function well defined. The other is that there may be an ambiguity in the choice of the measure of the discrete model which could even lead to the existence of different universality classes.

Holonomy and entropy estimates for dynamically triangulated manifolds
View Description Hide DescriptionWe provide an elementary proof of the exponential bound to the number of distinct dynamical triangulations of an n‐dimensional manifoldM (n≥2), of given volume and fixed topology. The resulting entropy estimates emphasize the basic role, in simplicial quantum gravity, of the moduli spaces Hom (π_{1}(M),G )/G associated with the representations of the fundamental group of the manifold, π_{1}(M), into a Lie groupG.

The world as a hologram
View Description Hide DescriptionAccording to ’t Hooft the combination of quantum mechanics and gravity requires the three‐dimensional world to be an image of data that can be stored on a two‐dimensional projection much like a holographic image. The two‐dimensional description only requires one discrete degree of freedom per Planck area and yet it is rich enough to describe all three‐dimensional phenomena. After outlining ’t Hooft’s proposal we give a preliminary informal description of how it may be implemented. One finds a basic requirement that particles must grow in size as their momenta are increased far above the Planck scale. The consequences for high‐energy particle collisions are described. The phenomenon of particle growth with momentum was previously discussed in the context of string theory and was related to information spreading near black hole horizons. The considerations of this paper indicate that the effect is much more rapid at all but the earliest times. In fact the rate of spreading is found to saturate the bound from causality. Finally we consider string theory as a possible realization of ’t Hooft’s idea. The light front lattice string model of Klebanov and Susskind is reviewed and its similarities with the holographictheory are demonstrated. The agreement between the two requires unproven but plausible assumptions about the nonperturbative behavior of string theory. Very similar ideas to those in this paper have long been held by Charles Thorn.

On the dynamics of characteristic surfaces
View Description Hide DescriptionWe formulate the vacuum Einstein equations as differential equations for two functions, one complex and one real on a six‐dimensional manifold,M×S ^{2}, with M eventually becoming the space–time and the S ^{2} becoming the sphere of null directions over M. At the start there is no other further structure available: the structure arising from the two functions. The complex function, referred to as Λ[M×S ^{2}], encodes information about a sphere’s worth of surfaces through each point of M. From knowledge of Λ one can define a second rank tensor on M which can be interpreted as a conformal metric, so that the ‘‘surfaces’’ are automatically null or characteristics of this conformal metric. The real function, Ω, plays the role of a conformal factor: it converts the conformal metric into a vacuum Einstein metric. Locally, all Einstein metrics can be obtained in this manner. In this work, we fully develop this ‘‘null surface version of general relativity (GR):’’ we display, discuss and analyze the equations, we show that many of the usual geometric quantities of GR (e.g., the Weyl and Ricci tensors, the optical parameters, etc.) can be easily expressed in terms of the Λ and Ω, we study the gauge freedom and develop a perturbation theory. To conclude, we speculate on the significance and possible classical and quantum uses of this formulation.

Linking topological quantum field theory and nonperturbative quantum gravity
View Description Hide DescriptionQuantum gravity is studied nonperturbatively in the case in which space has a boundary with finite area. A natural set of boundary conditions is studied in the Euclidean signature theory in which the pullback of the curvature to the boundary is self‐dual (with a cosmological constant). A Hilbert space which describes all the information accessible by measuring the metric and connection induced in the boundary is constructed and is found to be the direct sum of the state spaces of all SU(2) Chern–Simon theories defined by all choices of punctures and representations on the spatial boundary S. The integer level k of Chern–Simons theory is found to be given by k=6π/G ^{2}Λ+α, where Λ is the cosmological constant and α is a CP breaking phase. Using these results, expectation values of observables which are functions of fields on the boundary may be evaluated in closed form. Given these results, it is natural to make the conjecture that the quantum states of the system are completely determined by measurements made on the boundary. One consequence of this is the Bekenstein bound, which says that once the two metric of the boundary has been measured, the subspace of the physical state space that describes the further information that may be obtained about the interior has finite dimension equal to the exponent of the area of the boundary, in Planck units, times a fixed constant. Finally, these results confirm both the categorical‐theoretic ‘‘ladder of dimensions’’ picture of Crane and the holographic hypothesis of Susskind and ’t Hooft.

Quantization of diffeomorphism invariant theories of connections with local degrees of freedom
View Description Hide DescriptionQuantization of diffeomorphism invariant theories of connections is studied and the quantum diffeomorphism constraint is solved. The space of solutions is equipped with an inner product that is shown to satisfy the physical reality conditions. This provides, in particular, a quantization of the Husain–Kuchař model. The main results also pave the way to quantization of other diffeomorphism invariant theories such as general relativity. In the Riemannian case (i.e., signature ++++), the approach appears to contain all the necessary ingredients already. In the Lorentzian case, it will have to be combined in an appropriate fashion with a coherent state transform to incorporate complex connections.

Quantum aspects of 2+1 gravity
View Description Hide DescriptionWe review and systematize several recent attempts to canonically quantize general relativity in 2+1 dimensions, defined on space–times R×Σ^{ g }, where Σ^{ g } is a compact Riemann surface of genus g. The emphasis is on quantizations of the classical connection formulation, which use Wilson loops as their basic observables, but results from the ADM formulation are also summarized. We evaluate the progress and discuss the possible quantum (in)equivalence of the various approaches.