Abelian BF Theories and Knot Invariants

In the context of the Batalin–Vilkovisky formalism, a new observable for the Abelian BF theory is proposed whose vacuum expectation value is related to the Alexander–Conway polynomial. The three-dimensional case is analyzed explicitly, and it is proved to be anomaly free. Moreover, at the second order in perturbation theory, a new formula for the second coefficient of the Alexander–Conway polynomial is obtained. An account on the higher-dimensional generalizations is also given. Received: 2 October 1996 / Accepted: 21 March 1997     

Wilson Surfaces and Higher Dimensional Knot Invariants

An observable for nonabelian, higher-dimensional forms is introduced, its properties are discussed and its expectation value in BF theory is described. This is shown to produce potential and genuine invariants of higher-dimensional knots.A.S.C. acknowledges partial support of SNF Grant No. 20-63821.00     

Vassiliev Knot Invariants and Chern-Simons Perturbation Theory to All Orders

At any order, the perturbative expansion of the expectation values of Wilson lines in Chern-Simons theory gives certain integral expressions. We show that they all lead to knot invariants. Moreover these are finite type invariants whose order coincides with the order in the perturbative expansion. Together they combine to give a universal Vassiliev invariant. Received: 26 March 1996 / Accepted: 7 November 1996     

Torus Knots and Links from Eikonal Equations and Knot Invariants for Classification of Atoms

The history of knot theory and physics has a deep roots. It started by Lord Kelvin, in 1867, when he conjectured that atoms were knotted vortex tubes of ether. In 1997, Faddeev and Niemi suggested that knots might exist as stable soliton solution in a simple three dimensional classical field theory. That opening up a wide range of possible applications in physics. In this work we consider the Eikonal equation, which is a partial differential equation describing the traveltime propagation, which is an important part of seismic imaging algorithms. We will follow the work of Wereszczynski of solving the Eikonal equation in cylindrical coordinates. We show that only torus knots and links do occur, so figure eight knot does not occur. We show that these solutions are not unique, which means the possible occurrence of the same knot type for different configurations. Using the idea of framed knots, it is shown that two Eikonal knots are equivalent if and only if they are ambient isotopic as a framed knots, i.e. if and only if they are of the same knot type and of the same twisting number.     

Invariants of Quantum Algebras and Their Applications in Physics

We summarize recent results^[1] on a functor that maps tangles to commutants of a quasitriangular Hopf algebra. In certain cases, images of (1, 1)-tangles are new Casimir operators of the algebraLwhose eigenvalues are link invariants. Some applications to physics are briefly discussed.     

Letter: Differential Invariants of the Kerr Vacuum

The norms associated with the gradients of the two non-differential invariants of the Kerr vacuum are examined. Whereas both locally single out the horizons, their global behavior is more interesting. Both reflect the background angular momentum as the volume of space allowing a timelike gradient decreases with increasing angular momentum becoming zero in the degenerate and naked cases. These results extend directly to the Kerr-Newman geometry.     

Three-Manifold Invariants and Their Relation with the Fundamental Group

We consider the 3-manifold invariant I(M) which is defined by means of the Chern–Simons quantum field theory and which coincides with the Reshetikhin–Turaev invariant. We present some arguments and numerical results supporting the conjecture that for nonvanishing I(M), the absolute value |I(M)| only depends on the fundamental group π₁ (M) of the manifold M. For lens spaces, the conjecture is proved when the gauge group is SU(2). In the case in which the gauge group is SU(3), we present numerical computations confirming the conjecture. Received: 15 November 1996 / Accepted: 17 June 1997     

Differential Invariants of Immersions of Manifolds with Metric Fields

The problem of finding all r^th order differential invariants of immersions of manifolds with metric fields, with values in a left (G¹_m×G¹_n)-manifold is formulated. For obtaining the basis of higher order differential invariants the orbit reduction method is used. As a new result it appears that r^th order differential invariants depending on an immersion f:M N of smooth manifolds M and N and metric fields on them can be factorized through metrics, curvature tensors and their covariant differentials up to the order (r–2), and covariant differentials of the tangent mapping Tf up to the order r. The concept of a covariant differential of Tf is also introduced in this paper. The obtained results are geometrically interpreted as well.This research is supported by grants GAR 201/03/0512 and MSM 143100006.     

Hojman Exact Invariants and Adiabatic Invariants of Hamilton System

The perturbation to Lie symmetry and adiabatic invariants are studied. Based on the concept of higher-order adiabatic invariants of mechanical systems with action of a small perturbation, the perturbation to Lie symmetry is studied, and Hojman adiabatic invariants of Hamilton system are obtained. An example is given to illustrate the application of the results.     

Solutions of Nonlinear Differential and Difference Equations with Superposition Formulas

Matrix Riccati equations and other nonlinear ordinary differential equations with superposition formulas are, in the case of constant coefficients, shown to have the same exact solutions as their group theoretical discretizations. Explicit solutions of certain classes of scalar and matrix Riccati equations are presented as an illustration of the general results.     

Invariants and chaotic maps

A one-dimensional mapf(x) is called an invariant of a two-dimensional mapg(x, y) ifg(x, f(x))=f(f(x)). The logistic map is an invariant of a class of two-dimensional maps. We construct a class of two-dimensional maps which admit the logistic maps as their invariant. Moreover, we calculate their Lyapunov exponents. We show that the two-dimensional map can show hyperchaotic behavior.     

Chiral Observables and Modular Invariants

Various definitions of chiral observables in a given M?bius covariant two-dimensional (2D) theory are shown to be equivalent. Their representation theory in the vacuum Hilbert space of the 2D theory is studied. It shares the general characteristics of modular invariant partition functions, although SL(2, ℤ) transformation properties are not assumed. First steps towards a classification are made. Received: 19 April 1999 / Accepted: 20 July 1999     

Invariants and nonequilibrium density matrices

A new method of calculating nonequilibrium density matrices with the aid of the quantum integrals of motion is proposed. The method is shown to be very effective in the case of systems described by means of quadratic Hamiltonians. The possibility of constructing phenomenological nonstationary Hamiltonians for a wide class of dissipative systems is discussed. The exact formulas for nonequilibrium density matrices of arbitrary quadratic systems are obtained. The quantum problem of the motion of a charged particle in uniform electric and magnetic fields in the presence of a frictional force proportional to the velocity is solved exactly by means of introducing the new phenomenological Hamiltonian.     

Subfactor Realisation of Modular Invariants

 We study the problem of realising modular invariants by braided subfactors and the related problem of classifying nimreps. We develop the fusion rule structure of these modular invariants. This structure is a useful tool in the analysis of modular data from quantum double subfactors, particularly those of the double of cyclic groups, the symmetric group on 3 letters and the double of the subfactors with principal graph the extended Dynkin diagram D ₅ ⁽¹⁾. In particular for the double of S ₃, 14 of the 48 modular modular invariants are nimless, and only 28 of the remaining 34 nimble invariants can be realised by subfactors. Received: 14 February 2003 / Accepted: 3 April 2003 Published online: 19 May 2003 Communicated by H. Araki, D. Buchholz and K. Fredenhagen     

Abelian Quiver Invariants and Marginal Wall-Crossing

We prove the equivalence of (a slightly modified version of) the wall-crossing formula of Manschot, Pioline and Sen and the wall-crossing formula of Kontsevich and Soibelman. The former involves abelian analogues of the motivic Donaldson–Thomas type invariants of quivers with stability introduced by Kontsevich and Soibelman, for which we derive positivity and geometricity properties.