This paper is an exploration of relationships between the Jones polynomial and quantum computing. We discuss the structure of the Jones polynomial in relation to representations of the Temperley Lieb algebra, and give an example of a unitary representation of the braid group. We discuss the evaluation of the polynomial as a generalized quantum amplitude and show how the braiding part of the evaluation can be construed as a quantum computation when the braiding representation is unitary. The question of an efficient quantum algorithm for computing the whole polynomial remains open. 1

An extension of the method and results of A. Schwarz for evaluating the partition function of a quadratic functional is presented. This enables the partition functions to be evaluated for a wide class of quadratic functionals of interest in topological quantum field theory, for which no method has previously been available. In particular it enables the partition functions appearing in the semiclassical approximation for the Witten-invariant to be evaluated in the most general case. The resulting k−dependence is precisely that conjectured by D. Freed and R. Gompf. 1

This paper is an exposition of the relationship between the heuristics of Witten's functional integral and the theory of knots and links in three-dimensional space.

These notes summarize the lectures delivered in the V Mexican School of Particle Physics, at the University of Guanajuato. We give a survey of the application of Ashtekar’s variables to the quantization of General Relativity in four dimensions with special emphasis on the application of techniques of analytic knot theory to the loop representation. We discuss the role that the Jones Polynomial plays as a generator of nondegenerate quantum states of the gravitational field. 1 Quantum Gravity: why and how? I wish to thank the organizers for inviting me to speak here. This may well be a sign of our times, that a person generally perceived as a “General Relativist ” would be invited to speak at a Particle Physics School. It just reflects the higher degree of interplay these two fields have enjoyed over the last years. In these lectures we will see more reasons for this enhanced interplay. We will see several notions from Gauge Theories, as Wilson Loops for instance, playing a central role in gravitation. An even greater interplay takes place with Topological Field Theories. We will see the important role that the Chern-Simons form, the Jones Polynomial and other notions of knot theory seem to play in General Relativity. The quantization of General Relativity is a problem that has defied resolution for the last sixty years. In spite of the long time that has been invested in trying to solve it, we believe that several people do not necessarily fully appreciate the reasons of our failure and the magnitude of the problem. It is a general perception –especially among particle physicists – that “General Relativity is nonrenormalizable ” and that is the basic problem with the theory. This statement is misleading in three ways: a) The fact that a theory is nonrenormalizable does not necessarily mean that the theory has an intrinsic problem or is “bad ” in any way. It merely says that perturbation

Chern-Simons theories, which are topological quantum field theories, provide a field theoretic framework for the study of knots and links in three dimensions. These are rare examples of quantum field theories which can be exactly and explicitly solved. Expectation values of Wilson link operators yield a class of link invariants, the simplest of them is the famous Jones polynomial. Other invariants are more powerful than that of Jones. These new invariants are sensitive to the chirality of all knots at least upto ten crossing number unlike those of Jones which are blind to the chirality of some of them. However, all these invariants are still not good enough to distinguish a class of knots called mutants. These link invariants can be alternately obtained from two dimensional vertex models. The R-matrix of such a model in a particular limit of the spectral parameter provides a representation of the braid group. This in turn is used to construct the link invariants. Exploiting theorems of Lickorish and Wallace and also those of Kirby, Fenn and Rourke which relate three-manifolds to surgeries on framed links, these link invariants in S 3 can also be used to construct three-manifold invariants.

Abstract. We study the analog of the Yang-Mills heat flow on the moduli space of framed bundles on a cut surface. Existence and convergence of the heat flow give a stratification of Morse type invariant under the action of the loop group. We use the stratification to prove versions of Kähler quantization commutes with reduction and Kirwan surjectivity. 1.

Abstract. We derive Verlinde’s formula from the fixed point formula for loop groups proved in the companion paper [FP], and extend it to compact, connected groups that are not necessarily simply-connected. 1.

This paper is a quick introduction to key relationships between the theories of knots,links, three-manifold invariants and the structure of quantum mechanics. In section 2 we review the basic ideas and principles of quantum mechanics. Section 3 shows how the idea of a quantum amplitude is applied to the construction of invariants of knots and links. Section 4 explains how the generalisation of the Feynman integral to quantum fields leads to invariants of knots, links and three-manifolds. Section 5 is a discussion of a general categorical approach to these issues. Section 6 is a brief discussion of the relationships of quantum topology to quantum computing. This paper is intended as an introduction that can serve as a springboard for working on the interface between quantum topology and quantum computing. Section 7 summarizes the paper.

Each rule f that assigns a vector f(G) to an (n+1)-graph G determines a class (or property) of n-manifold invariants. An invariant v = v(M) is in this class if, for any triangulated manifold |G | = M, one has that v(M) is a linear function of f(G). This paper defines a flag vector f(G) for i-graphs, whose associated invariants might be quantum, and which is of interest in its own right. The definition (via the concept of shelling, and a ‘disjoint pair of optional cells ’ rule for the link) seems to apply to any finite combinatorial object, and so to any compact topological object that can be triangulated. It also applies to finite groups. 1

Abstract: Algebraic properties of orbifold models on arbitrary Riemann surfaces are investigated. The action of mapping class group transformations and of standard geometric operations is given explicitly. An infinite dimensional extension The interest in studying orbifold models is twofold. First, they provide a mean to construct new conformal field theories from known ones, whose properties may in principle be determined from the knowledge of the original theory [1]. This may prove important in attempts to classify CFTs. Secondly, they provide an interesting