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15
Constructions with Bundle Gerbes
"... This thesis develops the theory of bundle gerbes and examines a number of useful constructions in this theory. These allow us to gain a greater insight into the structure of bundle gerbes and related objects. Furthermore they naturally lead to some interesting applications in physics. i iiStatement ..."
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This thesis develops the theory of bundle gerbes and examines a number of useful constructions in this theory. These allow us to gain a greater insight into the structure of bundle gerbes and related objects. Furthermore they naturally lead to some interesting applications in physics. i iiStatement of Originality This thesis contains no material which has been accepted for the award of any other degree or diploma at any other university or other tertiary institution and, to the best of my knowledge and belief, contains no material previously published or written by another person, except where due reference has been made in the text. I give consent to this copy of my thesis, when deposited in the University Library, being made available for loan and photocopying. Stuart Johnson
TQFT’s and gerbes
 Algebr. Geom. Topol
"... We generalise the notions of holonomy and parallel transport for abelian bundles and gerbes using an embedded TQFT construction, and obtain statesumlike formulae for the parallel transport along paths and surfaces. This approach has been greatly influenced by the work of Graeme Segal, in particula ..."
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We generalise the notions of holonomy and parallel transport for abelian bundles and gerbes using an embedded TQFT construction, and obtain statesumlike 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
Formal Homotopy Quantum Field Theories
 II : Simplicial Formal Maps
"... Homotopy Quantum Field Theories (HQFTs) were introduced by the second author to extend the ideas and methods of Topological Quantum Field Theories to closed dmanifolds endowed with extra structure in the form of homotopy classes of maps into a given ‘target ’ space. For d = 1, classifications of HQ ..."
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Homotopy Quantum Field Theories (HQFTs) were introduced by the second author to extend the ideas and methods of Topological Quantum Field Theories to closed dmanifolds 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 2type, giving a common generalisation of the classifying algebraic structures for the two cases previously known. The algebraic models for 2types that we use are crossed modules, C, and we introduce a notion of formal Cmap, which extends the usual latticetype constructions to this setting. This leads to a classification of ‘formal ’ 2dimensional HQFTs with target C,
Homological Quantum Field Theory
 in M. Levy (Ed.), Mathematical Physics Research Developments, Nova Publishers
"... 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 dalgebra. We construct a structure of transversal 1category on the space of chains of maps from a suspension space S(Y), with certain boundary restrictions, ..."
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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 dalgebra. We construct a structure of transversal 1category 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
Relative differential Kcharacters
 SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2008
"... We define a group of relative differential Kcharacters associated with a smooth map between two smooth compact manifolds. We show that this group fits into a short exact sequence as in the nonrelative case. Some secondary geometric invariants are expressed in this theory. ..."
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We define a group of relative differential Kcharacters associated with a smooth map between two smooth compact manifolds. We show that this group fits into a short exact sequence as in the nonrelative case. Some secondary geometric invariants are expressed in this theory.
2Representations and Equivariant 2D Topological Field Theories Contents
, 2008
"... 2 Frobenius algebras with twisted Gaction 4 ..."
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A FUNCTORIAL APPROACH TO nGERBES
, 2002
"... Abstract. We provide a characterisation of ngerbes with connection in terms of a variant of Turaev’s homotopy quantum field theories based on chains in a smooth manifold X. ..."
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Abstract. We provide a characterisation of ngerbes with connection in terms of a variant of Turaev’s homotopy quantum field theories based on chains in a smooth manifold X.