Abelian homotopy Dijkgraaf-Witten theory

Abstract

We construct a version of Dijkgraaf-Witten theory based on a compact abelian Lie group within the formalism of Turaev's homotopy quantum field theory. As an application we show that the 2+1-dimensional theory based on U (1) classifies lens spaces up to homotopy type. e-print archive: http://lanl.arXiv.org/abs/math.QA/0410179 322

Functional integration and abelian link invariants

“…where N L denotes the number of components of L and σ(L) represents the so-called signature of the linking matrix associated with L, i.e. σ(L) = n + − n − where n ± is the number of positive/negative eigenvalues of the linking matrix which is defined by the framed link L. Some properties of I k (M L ) (and of its generalizations) have been studied, for instance, in [33,34,35]. If M 0 is a homology 3-sphere, then [32] one finds I k (M 0 ) = 1.…”

Abelian link invariants and homology

“…A field theory interpretation of the Reshetikhin-Turaev invariant (4.1) -as a ratio of Chern-Simons partition functions-has been proposed in [14] and detailed discussions on the properties of the invariant (4.1) can be found in [27,28,29]. Proof.…”

“…The manifolds L p/r and L p/r ′ are homeomorphic iff ±r ′ ≡ r ±1 (mod p). The manifold L p admit a surgery presentation given by the unknot with surgery coefficient equal to the integer p. Special cases are L 0 ≃ S 2 × S 1 , L 1 ≃ S 3 ; equation for some integer m. Hansen, Slingerland and Turner have shown [29] that, when rr ′ ≡ −m 2 (mod p), one finds I k (L p/r ) = I k (L p/r ′ ); one example of this kind is shown in equation (4.10). Whereas, when the product rr ′ is equivalent to a quadratic residue, rr ′ ≡ m 2 (mod p), one has I k (L p/r ) = I k (L p/r ′ ), for istance The equivalence relation under orientation-preserving homotopy extends to the manifolds which are connected sum of equivalent spaces [33].…”