Relativistic spin networks are defined by considering the spin covering of the group SO(4), SU(2) times SU(2). Relativistic quantum spins are related to the geometry of the 2-dimensional faces of a 4-simplex. This extends the idea of Ponzano and Regge that SU(2) spins are related to the geometry of the edges of a 3-simplex. This leads us to suggest that there may be a 4-dimensional state sum model for quantum gravity based on relativistic spin networks which parallels the construction of 3-dimensional quantum gravity from ordinary spin networks.Comment: 10 pages, amstex, some errors corrected, more reference

We give a relativistic spin network model for quantum gravity based on the Lorentz group and its q-deformation, the Quantum Lorentz Algebra. We propose a combinatorial model for the path integral given by an integral over suitable representations of this algebra. This generalises the state sum models for the case of the four-dimensional rotation group previously studied in [1], gr-qc/9709028.As a technical tool, formulae for the evaluation of relativistic spin networks for the Lorentz group are developed, with some simple examples which show that the evaluation is finite in interesting cases. We conjecture that the '10J' symbol needed in our model has a finite value.1 is that the area of a timelike surface is quantized but the area of a spacelike surface can take any value in a continuous range.After removing a trivial infinite factor of the volume of the hyperbolic space, we obtain a expression for the evaluation of the relativistic spin network associated to a 4-simplex as a multiple integral on hyperbolic space. We conjecture that this integral is absolutely convergent.The study of the q-deformation of the Lorentzian case is not so far advanced, but we give some observations on this and the possible implications for Lorentzian signature state sum models.Before explaining our results, we give a brief review of the Euclidean case.1.1. Review of the Euclidean model. The steps of the construction of the model in [1] were as follows:

We propose a new combinatorial method of constructing 4D-TQFTs. The method uses a new type of algebraic structure called a Hopf category. We also outline the construction of a family of Hopf categories related to the quantum groups, using the canonical bases.INTRODUCTION Since the seminal papers of Witten [1], and Atiyah [2], the attention of many mathematicians and physicists has been drawn to the problem of constructing topological quantum field theories (TQFTs). The mathematical form of the definition of a TQFT uses very categorical language, and is strongly influenced by the definition of a conformal field theory due to Segal [3].The subject of TQFT has a dual character, as a new development in algebraic topology on one hand, and as a branch of quantum field theory on the other. While physicists have suggested that various TQFTs exist in various dimensions by non-rigorous and ill-defined arguments involving path integrals, many rigorous constructions have been discovered using various techniques involving abstract algebra [4,5,6,7,8,9]. There has been a close interaction between the mathematical and physical approaches, which has led to the rapid development of the subject.The most exciting theories whose existence has been suggested are the cousins of Donaldson-Floer (DF) theory [1], which should exist in 4 dimensions, and be sensitive to smooth structure. So far, neither the analytic [10] nor the path integral nor any algebraic approach has succeeded in constructing Donaldson-Floer theory, except for special cases.The only 4D-TQFTs which have been rigorously constructed so far are Dijkgraaf-Witten theory [11,12,13], which counts representations of π 1 into a finite group, and Crane-Yetter theory [14], which can be used to find the signature, Euler character, second Chern class and cup product on the second cohomology of the 4 manifold. Both of these theories, in short, produce classical invariants (although by new combinatorial formulae), and hence, are not obviously related to Donaldson-Floer theory.The purpose of this paper is to propose a new algebraic construction of a 4D-TQFT, which uses a novel algebraic structure, called a Hopf category. This would not be very interesting, except that we also propose a construction of a family of Hopf categories, intimately related to the canonical bases for the quantum groups. We conjecture that the 4D-TQFTs which arise by our new method from our Hopf categories may be sensitive to smooth structure, and in fact may be related to DF theory.As far as the current status of DF theory is concerned, let us make a hypothetical historical analogy. Suppose somebody had used the implicit function theorem to show that the dimension of the space of harmonic forms on a Riemannian manifold was independent of the metric. One would then have a set of "analytic" invariants of closed manifolds. It would then take some imagining to find a combinatorial algebraic formula for these invariants from a triangulation. Of course, once that invariant had been understood as rational...

We provide, with proofs, a complete description of the authors' construction of state-sum invariants announced in [CY], and its generalization to an arbitrary (artinian) semisimple tortile category. We also discuss the relationship of these invariants to generalizations of Broda's surgery invariants [Br1,Br2] using techniques developed in the case of the semi-simple sub-quotient of Rep(Uq(sl2)) (q a principal 4rth root of unity) by Roberts [Ro1]. We briefly discuss the generalizations to invariants of 4-manifolds equipped with 2-dimensional (co)homology classes introduced by Yetter [Y6] and Roberts [Ro2].

IntroductionIn recent years, it has been discovered that invariants of three dimensional topological objects, such as links and three dimensional manifolds, can be constructed from various tools of mathematical physics, such as Von Neumann algebras [1], Quantum Groups [2], and Rational Conformal Field Theories [3].Since these different structures lead to the same 3D invariants, it is natural to wonder how they are related. A fundamental connection is that they all give rise to the same special tensor categories, which act as expressions of "quantum symmetry" in the very different physical settings [4]. It is therefore very important that the 3D invariants can all be reconstructed from a tensor category with the appropriate properties, called a "modular tensor category (MTC) [3,5]." The invariants have the property that they factorize nicely if the manifold or link is cut along a surface, which we express by saying that they form a Topological Quantum Field Theory, or TQFT.Thus the theorem proven in [3] states that a modular tensor category gives rise to a 3D TQFT. This represents a remarkable convergence with [6], in which it was realized that a suitable categorical structure would give rise to 3D topological information, although without the examples from physics.The original suggestion to look for TQFT's appears in (not mathematically rigorous) work of Atiyah [7] and Witten [8]. Atiyah devotes more attention to the 4D than the 3D situation, and poses the question whether the two are related; or more concretely, whether the invariants of Donaldson and Jones are related. Witten actually works in the 4D situation, and formally suggests that Donaldson's invariant can be fitted into a 4D TQFT.At this point, there is no mathematical construction of the 4D TQFT envisioned in [7] and [8]. Donaldson-Floer theory has so far eluded the efforts of the strongest of analysts [9].It is therefore clear that a 4D construction of a TQFT paralleling the 3D categorical one would have very great implications for mathematics. As we shall mention in the conclusion section of this paper, there may very well be important physical implications also.The construction we describe in this paper is a considerable step in this direction. Following a suggestion of Ooguri [10], we offer an expression which