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Topological BF Theories in 3 and 4 Dimensions

Aberto S. Cattaneo, Paolo Cotta-Ramusino, Juerg Froehlich, Maurizio Martellini

Abstract

In 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.

Topological BF Theories in 3 and 4 Dimensions

Abstract

In 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.

Paper Structure

This paper contains 11 sections, 89 equations.

Table of Contents

  1. I. Introduction
  2. II. Geometry of $BF$ theories
  3. III. $BF$ observables in 3 dimensions
  4. IV. Formal relations between Chern--Simons and $BF$ theories
  5. V. Choice of gauge and v.e.v's
  6. VI. The framing of the knot and the skein relation
  7. VII. Choice of gauge and link-invariants
  8. IX. The $BF$ theory without cosmological constant and the Alexander--Conway polynomial
  9. X. Observables for the four-dimensional $BF$ theory
  10. Acknowledgements
  11. Appendix: On the coefficients of the skein-polynomials