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Bundle Gerbes for ChernSimons and WESSZUMINOWITTEN THEORIES
, 2005
"... We develop the theory of ChernSimons bundle 2gerbes and multiplicative bundle gerbes associated to any principal Gbundle with connection and a class in H4 (BG, Z) for a compact semisimple Lie group G. The ChernSimons bundle 2gerbe realises differential geometrically the CheegerSimons invarian ..."
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We develop the theory of ChernSimons bundle 2gerbes and multiplicative bundle gerbes associated to any principal Gbundle with connection and a class in H4 (BG, Z) for a compact semisimple Lie group G. The ChernSimons bundle 2gerbe realises differential geometrically the CheegerSimons invariant. We apply these notions to refine the DijkgraafWitten correspondence between three dimensional ChernSimons functionals and WessZuminoWitten models associated to the group G. We do this by introducing a lifting to the level of bundle gerbes of the natural map from H 4 (BG, Z) to H3 (G, Z). The notion of a multiplicative bundle gerbe accounts geometrically for the subtleties in this correspondence for nonsimply connected Lie groups. The implications for WessZuminoWitten models are also discussed.
Parallel transport and functors
"... Parallel transport of a connection in a smooth fibre bundle yields a functor from the path groupoid of the base manifold into a category that describes the fibres of the bundle. We characterize functors obtained like this by two notions we introduce: local trivializations and smooth descent data. Th ..."
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Cited by 32 (9 self)
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Parallel transport of a connection in a smooth fibre bundle yields a functor from the path groupoid of the base manifold into a category that describes the fibres of the bundle. We characterize functors obtained like this by two notions we introduce: local trivializations and smooth descent data. This provides a way to substitute categories of functors for categories of smooth fibre bundles with connection. We indicate that this concept can be generalized to connections in categorified bundles, and how this generalization improves the understanding of higher dimensional parallel transport. Table of Contents
L∞algebra connections and applications to String and ChernSimons ntransport
, 2008
"... We give a generalization of the notion of a CartanEhresmann connection from Lie algebras to L∞algebras and use it to study the obstruction theory of lifts through higher Stringlike extensions of Lie algebras. We find (generalized) ChernSimons and BFtheory functionals this way and describe aspect ..."
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Cited by 26 (13 self)
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We give a generalization of the notion of a CartanEhresmann connection from Lie algebras to L∞algebras and use it to study the obstruction theory of lifts through higher Stringlike extensions of Lie algebras. We find (generalized) ChernSimons and BFtheory functionals this way and describe aspects of their parallel transport and quantization. It is known that over a Dbrane the KalbRamond background field of the string restricts to a 2bundle with connection (a gerbe) which can be seen as the obstruction to lifting the P U(H)bundle on the Dbrane to a U(H)bundle. We discuss how this phenomenon generalizes from the ordinary central extension U(1) → U(H) → P U(H) to higher categorical central extensions, like the Stringextension BU(1) → String(G) → G. Here the obstruction to the lift is a 3bundle with connection (a 2gerbe): the ChernSimons 3bundle classified by the first Pontrjagin class. For G = Spin(n) this obstructs the existence of a Stringstructure. We discuss how to describe this obstruction problem in terms of Lie nalgebras and their corresponding categorified CartanEhresmann connections. Generalizations even beyond Stringextensions are then straightforward. For G = Spin(n) the next step is “Fivebrane structures” whose existence is obstructed by certain generalized ChernSimons 7bundles classified by the second Pontrjagin class.
Higher symplectic geometry
"... I wish to thank my advisor John Baez for his encouragement and guidance in completing this thesis. I would also like to thank Julie Bergner and YatSun Poon for serving on my committee. I wish to acknowledge the following individuals for helpful discussions, comments, ..."
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Cited by 15 (6 self)
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I wish to thank my advisor John Baez for his encouragement and guidance in completing this thesis. I would also like to thank Julie Bergner and YatSun Poon for serving on my committee. I wish to acknowledge the following individuals for helpful discussions, comments,
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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Cited by 12 (1 self)
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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
Transgression to loop spaces and its inverse, II: Gerbes and fusion bundles with connection
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Division Algebras, Supersymmetry and Higher Gauge Theory
, 2012
"... A dissertation is the capstone to a doctoral program, and the acknowledgements provide a useful place to thank the countless people who have helped out along the way, both personally and professionally. First, of course, I thank my advisor, John Baez. It is hard to imagine a better advisor, and no o ..."
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Cited by 5 (1 self)
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A dissertation is the capstone to a doctoral program, and the acknowledgements provide a useful place to thank the countless people who have helped out along the way, both personally and professionally. First, of course, I thank my advisor, John Baez. It is hard to imagine a better advisor, and no one deserves more credit for my mathematical and professional growth during this program. “Thanks ” does not seem sufficient, but it is all I have to give. Also deserving special mention is John’s collaborator, James Dolan. I am convinced there is no subject in mathematics for which Jim does not have some deep insight, and I thank him for sharing a few of these insights with me. Together, John and Jim are an unparalleled team: there are no two better people with whom to talk about mathematics, and no two people more awake to the joy of mathematics. I would also like to thank Geoffrey Dixon, Tevian Dray, Robert Helling, Corinne Manogue, Chris Rogers, Hisham Sati, James Stasheff, and Riccardo Nicoletti for helpful conversations and correspondence. I especially thank Urs Schreiber for many discussions of higher gauge theory and L∞superalgebras, smooth ∞groups, and supergeometry.