Abstract. For manifolds with freely generated first homology, we characterize gerbes with connection as functors on a certain surface cobordism category. This allows us to relate gerbes with connection to Turaev’s 1+1-dimensional homotopy quantum field theories, and we show that flat gerbes on such spaces as above are the same as a specific class of rank one homotopy quantum field theories.

Homotopy Quantum Field Theories (HQFTs) were introduced by the second author to extend the ideas and methods of Topological Quantum Field Theories to closed d-manifolds endowed with extra structure in the form of homotopy classes of maps into a given ‘target ’ space. For d = 1, classifications of HQFTs in terms of algebraic structures are known when B is a K(G,1) and also when it is simply connected. Here we study general HQFTs with d = 1 and target a general 2-type, giving a common generalisation of the classifying algebraic structures for the two cases previously known. The algebraic models for 2-types that we use are crossed modules, C, and we introduce a notion of formal C-map, which extends the usual lattice-type constructions to this setting. This leads to a classification of ‘formal ’ 2-dimensional HQFTs with target C,

We generalise the notions of holonomy and parallel transport for abelian bundles and gerbes using an embedded TQFT construction, and obtain state-sum-like formulae for the parallel transport along paths and surfaces. This approach has been greatly influenced by the work of Graeme Segal, in particular his axiomatic approach to Conformal Field Theory. 1

We show that the space of chains of smooth maps from spheres into a fixed compact oriented manifold has a natural structure of a transversal d-algebra. We construct a structure of transversal 1-category on the space of chains of maps from a suspension space S(Y), with certain boundary restrictions, into a fixed compact oriented manifold. We define homological quantum field theories HLQFT and construct several examples of such structures. Our definition is based on the notions of string topology of Chas and Sullivan, and homotopy quantum field theories of Turaev. 1

The motivation for this paper was to construct approximations to a conformal version of homotopy quantum field theory using 2-categories. A homotopy quantum field theory, as defined by Turaev in [9], is a variant of a topological quantum field theory in which manifolds come equipped with a map to some auxiliary space X.

Abstract For smooth finite dimensional manifolds, we characterise gerbes with connection as functors on a certain surface cobordism category. This allows us to relate gerbes with connection to Turaev’s 1+1-dimensional homotopy quantum field theories, and we show that flat gerbes are related to a specific class of rank one homotopy quantum field theories. AMS Classification 55P48; 57R56, 81T70

We present a general framework for TQFT and related constructions using the language of monoidal categories. We construct a topological category C and an algebraic category D, both monoidal, and a TQFT functor is This work was supported by the programme “Programa Operacional Ciência, Tecnologia, Inovação ” (POCTI) of the Fundação para a Ciência e Tecnologia (FCT), cofinanced by the European Community fund FEDER. then defined as a certain type of monoidal functor from C to D. In contrast with the cobordism approach, this formulation of TQFT is closer in spirit to the classical functors of algebraic topology, like homology. The fundamental operation of gluing is incorporated at the level of the morphisms in the topological category through the notion of a gluing morphism, which we define. It allows not only the gluing together of two separate objects, but also the self-gluing of a single object to be treated in the same fashion. As an example of our framework we describe TQFT’s for oriented 2D-manifolds, and classify a family of them in terms of a pair of tensors satisfying some relations.

Abstract. We provide a characterisation of n-gerbes with connection in terms of a variant of Turaev’s homotopy quantum field theories based on chains in a smooth manifold X.

A functorial approach to differential characters

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. 1.