Topological BF theories in 3 and 4 dimensions

Abstract

May 3, 1995 hep-th/9505027, IFUM 503/FT prepared for the special issue of Journal of Math. Phys. on Quantum geometry and diffeomorphism-invariant quantum field theory P.A.C.S. 02.40, 11.15, 04.60 AbstractIn 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.

On Chern-Simons theory with an inhomogeneous gauge group and BF theory knot invariants

“…As a gauge theory, CS theory with inhomogeneous gauge group is equivalent to pure BF theory and should therefore "see" knots at least for ISU(2) gauge group, as was shown by Cattaneo et al [9,10,11]. Therefore, we need a procedure that will extract some non trivial information regarding the knot in our setting.…”

“…In the mid-90's, Cattaneo et al [9,10,11,12] showed that while BF theory with cosmological constant produces the same invariants of knots as the Chern-Simons (CS) theory, the BF theory with no cosmological constant (pure BF theory) and SU(2) gauge group produces invariants that lie in the algebra generated by the coefficients of the Alexander-Conway polynomial. The BF TQFT is completely equivalent to CS theory, however while the equivalence with non-zero cosmological constant maintains the semi-simplicity property of the gauge group, the equivalence when the cosmological constant is set to zero shifts us to a CS theory with a non-semi-simple gauge group.…”

Higher-Dimensional BF Theories¶in the Batalin-Vilkovisky Formalism:¶The BV Action and Generalized Wilson Loops

“…The generalized Wilson loop in the BV superformalism. We want to define an object that generalizes the observable introduced in [11] for the 3-dimensional BF theory with cosmological term. We shall realize this proposal by introducing the new superform…”

“…Now we briefly comment on the observables defined in this paper and in [16]. First, they are defined on loops (observables related to higher-dimensional submanifolds are of course of great interest and will be discussed in a forthcoming paper; we refer to [20,11,15] for previous attempts in this direction). Second, the quantum BV formalism requires considering the so-called BV Laplacian (see subsection 4.4) and this forces one to restrict to imbeddings (more precisely, to framed imbeddings).…”

“…Topological BF theories [32,26,4] are in three dimensions just another way of writing Chern-Simons theory [36] (at least at the perturbative level and disregarding anomalies) [11]. In particular, they produce 3-manifold and knot invariants.…”