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Starting from a Lie group G whose Lie algebra is equipped with an invariant nondegenerate symmetric bilinear form, we show that 4-dimensional BF theory with cosmological term gives rise to a TQFT satisfying a generalization of Atiyah's axioms to manifolds equipped with principal G-bundle. The case G = GL(4; R) is especially interesting because every 4-manifold is then naturally equipped with a principal G-bundle, namely its frame bundle. In this case, the partition function of a compact oriented 4-manifold is the exponential of its signature, and the resulting TQFT is isomorphic to that constructed by Crane and Yetter using a state sum model, or by Broda using a surgery presentation of 4-manifolds. 1 Introduction In comparison to the situation in 3 dimensions, topological quantum field theories (TQFTs) in 4 dimensions are poorly understood. This is ironic, because the subject was initiated by an attempt to understand Donaldson theory in terms of a quantum field theory in 4 dimensions....

Abstract: We outline a general construction applicable to the Turaev/Viro [TV], Crane/Yetter [CY] and generalized Turaev/Viro invariants (cf. [Y1]) of invariants valued in complex-valued functions on HD−2(M D, GrC), where GrC is the abelian group of functorial tensor automorphisms on the artinian tortile category used to construct the TQFT.

Abstract: Recent work of Roberts [R] has shown that the surgical 4-manifold invariant of Broda [B1] and (up to an unspecified normalization factor) the state-sum 4-manifold invariant of Crane-Yetter [CY] are equivalent to the signature of the 4-manifold. Subsequently Broda [B2] defined another surgical invariant of 4-manifolds in which the 1- and 2- handles are treated differently. We use a refinement of Roberts ’ techniques developped in [CKY] to identify the normalization factor to show that the “improved ” surgical invariant of Broda [B2] also depends only on the signature and Euler character. As a starting point, let us first observe that the construction of Crane-Yetter [CY] does not really depend on the use of labels chosen from the irreps of Uq(sl2) at the principal rth root of unity: the simple objects of any artinian semi-simple tortile category (cf. [S, Y]) in which all objects are self-dual and the fusion rules are multiplicity free will suffice. In particular, if we restrict to the integer spin (bosonic) 3 irreps, we obtain a construction of a different invariant of 4-manifolds. In what follows, we use Temperley-Lieb recoupling theory (cf. [KL,L,R]). In particular, arcs are labelled with elements of {0,1,...r − 2} (twice the spin), A = e2πi/4r, q = A2, ∆(n) = (−1) n qn+1−q−n−1 q−q−1, θ(a,b,c) denoted the evaluation of the theta-net with edge labelled a,b, and c, and 15 − j denotes the evaluation of the Temperley-Lieb version of the Crane-Yetter quantum

Starting from a Lie group G whose Lie algebra is equipped with an invariant nondegenerate symmetric bilinear form, we show that 4-dimensional BF theory with cosmological term gives rise to a TQFT satisfying a generalization of Atiyah’s axioms to manifolds equipped with principal G-bundle. The case G = GL(4,R) is especially interesting because every 4-manifold is then naturally equipped with a principal G-bundle, namely its frame bundle. In this case, the partition function of a compact oriented 4-manifold is the exponential of its signature, and the resulting TQFT is isomorphic to that constructed by Crane and Yetter using a state sum model, or by Broda using a surgery presentation of 4-manifolds. 1

The aim of our talk is to present a specific, non-perturbative, path-integral approach to topological invariants of knots/links and manifolds of dimension three and four. The technique is not rigorous but very intuitive and strongly motivated by physics. An exception is the four-dimensional case in Sect. 4, which is rather rigorous but less

Abstract. An approach to construction of topological invariants of the Reshetikhin-Turaev-Witten type of 3- and 4-dimensional manifolds in the framework of SU(2) Chern-Simons gauge theory and its hidden (quantum) gauge symmetry is presented. 1. Intoduction. The issue of topological classification of low-dimensional manifolds, especially of dimensions 3 and 4 (the most difficult and interesting ones), is a challenging problem in modern mathematics. One of the most spectacular events in topology of 3-dimensional manifolds took place a few years ago, when a new (numerical) topological invariant of closed orientable 3-manifolds, parametrized by the integer k, defined via surgery on a framed link, was discovered. The idea is due to a physicist, Edward Witten, who proposed the invariant in his famous paper on quantum field theory and the Jones polynomial [Wit1]. The first explicit and rigorous construction is due to mathematicians, Reshetikhin and Turaev [RT]. Their approach is combinatorial, whereas noncombinatorial possibilities, very straightforward though mathematically less rigorous, are offered by topological quantum field theory. The 3-dimensional invariant, known as the Reshetikhin-Turaev-Witten (RTW) invariant, is also frequently referred to as the SU(2)invariant