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Higherdimensional algebra and topological quantum field theory
 Jour. Math. Phys
, 1995
"... For a copy with the handdrawn figures please email ..."
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Cited by 146 (14 self)
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For a copy with the handdrawn figures please email
4Dimensional BF Theory as a Topological Quantum Field Theory
 Lett. Math. Phys
, 1996
"... Starting from a Lie group G whose Lie algebra is equipped with an invariant nondegenerate symmetric bilinear form, we show that 4dimensional BF theory with cosmological term gives rise to a TQFT satisfying a generalization of Atiyah's axioms to manifolds equipped with principal Gbundle. The c ..."
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Cited by 11 (6 self)
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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 4dimensional BF theory with cosmological term gives rise to a TQFT satisfying a generalization of Atiyah's axioms to manifolds equipped with principal Gbundle. The case G = GL(4; R) is especially interesting because every 4manifold is then naturally equipped with a principal Gbundle, namely its frame bundle. In this case, the partition function of a compact oriented 4manifold 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 4manifolds. 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....
Homologically Twisted Invariants Related to (2+1) and (3+1) Dimensional StateSum Topological Quantum Field Theories
"... 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 complexvalued functions on HD−2(M D, GrC), where GrC is the abelian group of functorial tensor automorphisms on the artinian to ..."
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Cited by 5 (1 self)
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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 complexvalued 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.
On the Classicality of Broda’s SU(2) Invariants of 4Manifolds
, 1993
"... Abstract: Recent work of Roberts [R] has shown that the surgical 4manifold invariant of Broda [B1] and (up to an unspecified normalization factor) the statesum 4manifold invariant of CraneYetter [CY] are equivalent to the signature of the 4manifold. Subsequently Broda [B2] defined another surgi ..."
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Abstract: Recent work of Roberts [R] has shown that the surgical 4manifold invariant of Broda [B1] and (up to an unspecified normalization factor) the statesum 4manifold invariant of CraneYetter [CY] are equivalent to the signature of the 4manifold. Subsequently Broda [B2] defined another surgical invariant of 4manifolds 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 CraneYetter [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 semisimple tortile category (cf. [S, Y]) in which all objects are selfdual 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 4manifolds. In what follows, we use TemperleyLieb 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 thetanet with edge labelled a,b, and c, and 15 − j denotes the evaluation of the TemperleyLieb version of the CraneYetter quantum
4Dimensional BF Theory with Cosmological Term as a Topological Quantum Field Theory
, 1995
"... Starting from a Lie group G whose Lie algebra is equipped with an invariant nondegenerate symmetric bilinear form, we show that 4dimensional BF theory with cosmological term gives rise to a TQFT satisfying a generalization of Atiyah’s axioms to manifolds equipped with principal Gbundle. The case G ..."
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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 4dimensional BF theory with cosmological term gives rise to a TQFT satisfying a generalization of Atiyah’s axioms to manifolds equipped with principal Gbundle. The case G = GL(4,R) is especially interesting because every 4manifold is then naturally equipped with a principal Gbundle, namely its frame bundle. In this case, the partition function of a compact oriented 4manifold 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 4manifolds. 1
Path integrals and lowdimensional topology
, 1997
"... The aim of our talk is to present a specific, nonperturbative, pathintegral 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 fourdimensional case in ..."
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The aim of our talk is to present a specific, nonperturbative, pathintegral 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 fourdimensional case in Sect. 4, which is rather rigorous but less
POLISH ACADEMY OF SCIENCES WARSZAWA 19** A GAUGEFIELD APPROACH TO 3 AND 4MANIFOLD INVARIANTS
, 1996
"... Abstract. An approach to construction of topological invariants of the ReshetikhinTuraevWitten type of 3 and 4dimensional manifolds in the framework of SU(2) ChernSimons gauge theory and its hidden (quantum) gauge symmetry is presented. 1. Intoduction. The issue of topological classification of ..."
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Abstract. An approach to construction of topological invariants of the ReshetikhinTuraevWitten type of 3 and 4dimensional manifolds in the framework of SU(2) ChernSimons gauge theory and its hidden (quantum) gauge symmetry is presented. 1. Intoduction. The issue of topological classification of lowdimensional 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 3dimensional manifolds took place a few years ago, when a new (numerical) topological invariant of closed orientable 3manifolds, 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 3dimensional invariant, known as the ReshetikhinTuraevWitten (RTW) invariant, is also frequently referred to as the SU(2)invariant