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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 15 (5 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
pForm Electromagnetism on Discrete Spacetimes
, 2006
"... We investigate pform electromagnetism—with the Maxwell and KalbRamond fields as lowestorder cases—on discrete spacetimes, including the regular lattices commonly used in lattice gauge theory, but also more general examples. After constructing a maximally general model of discrete spacetime suitab ..."
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Cited by 6 (2 self)
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We investigate pform electromagnetism—with the Maxwell and KalbRamond fields as lowestorder cases—on discrete spacetimes, including the regular lattices commonly used in lattice gauge theory, but also more general examples. After constructing a maximally general model of discrete spacetime suitable for our purpose—a chain complex equipped with an inner product on (p + 1)cochains—we study both the classical and quantum versions of the theory, with either R or U(1) as gauge group. We find results—such as a ‘pform Bohm–Aharonov effect’—that depend in interesting ways on the cohomology of spacetime. We quantize the theory via the Euclidean path integral formalism, where the natural kernels in the U(1) theory are not Gaussians but theta functions. As a special case of the general theory, we show pform electromagnetism in p + 1 dimensions has an exact solution which reduces when p = 1 to the abelian case of 2d YangMills theory as studied by Migdal and Witten. Our main result describes pform electromagnetism as a ‘chain field theory’—a theory analogous to a topological quantum field theory, but with chain complexes replacing manifolds. This makes precise a notion of time evolution in the context of discrete spacetimes of arbitrary topology.
Lattice pform electromagnetism and chain field theory
 LOOPS ’05, ALBERT EINSTEIN INSTITUT, MAX PLANCK GESELLSCHAFT, GOLM
, 2005
"... Since Wilson’s work on lattice gauge theory in the 1970s, discrete versions of field theories have played a vital role in fundamental physics. But there is recent interest in certain higher dimensional analogues of gauge theory, such as pform electromagnetism, including the KalbRamond field in str ..."
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Cited by 4 (2 self)
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Since Wilson’s work on lattice gauge theory in the 1970s, discrete versions of field theories have played a vital role in fundamental physics. But there is recent interest in certain higher dimensional analogues of gauge theory, such as pform electromagnetism, including the KalbRamond field in string theory, and its nonabelian generalizations. It is desirable to discretize such ‘higher gauge theories ’ in a way analogous to lattice gauge theory, but with the fundamental geometric structures in the discretization boosted in dimension. As a step toward studying discrete versions of more general higher gauge theories, we consider the case of pform electromagnetism. We show that discrete pform electromagnetism admits a simple algebraic description in terms of chain complexes of abelian groups. Moreover, the model allows discrete spacetimes with quite general geometry, in contrast to the regular cubical lattices usually associated with lattice gauge theory. After constructing a suitable model of discrete spacetime for pform electromagnetism, we quantize the theory using the Euclidean path integral formalism. The main result is a description of pform electromagnetism as a ‘chain field theory’ — a theory analogous to topological quantum field theory, but with chain complexes replacing
Crossed module bundle gerbes; classification, string group and differential geometry. Available as arXiv:math/0510078v2
"... We discuss nonabelian bundle gerbes and their differential geometry using simplicial methods. Associated to any crossed module (H → D) there is a simplicial group NC(H→D), the nerve of the 1category defined by the crossed module and its geometric realization NC(H→D). Equivalence classes of princi ..."
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Cited by 3 (0 self)
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We discuss nonabelian bundle gerbes and their differential geometry using simplicial methods. Associated to any crossed module (H → D) there is a simplicial group NC(H→D), the nerve of the 1category defined by the crossed module and its geometric realization NC(H→D). Equivalence classes of principal bundles with structure group NC(H→D)  are shown to be onetoone with stable equivalence classes of what we call crossed module bundle gerbes. We can also associate to a crossed module a 2category ˜ C(H→D). Then there are two equivalent ways how to view classifying spaces of NC(H→D)bundles and hence of NC(H→D)bundles and crossed module bundle gerbes. We can either apply the Wconstruction to NC(H→D) or take the nerve of the 2category ˜ C(H→D). We discuss the string group and string structures from this point of view. Also a simplicial principal bundle can be equipped with a simplicial connection and a Bfield. It is shown how in the case of a simplicial principal NC(H→D)bundle these simplicial objects give the bundle gerbe connection and the bundle gerbe Bfield.
Overview
"... Wheeler’s dream of “matter without matter ” is realized in 3d gravity, where particles show up naturally as punctures in space, or “topological defects”. My Ph.D. thesis research has been motivated by the desire to extend this idea to 4 spacetime dimensions. The starting point for this line of resea ..."
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Wheeler’s dream of “matter without matter ” is realized in 3d gravity, where particles show up naturally as punctures in space, or “topological defects”. My Ph.D. thesis research has been motivated by the desire to extend this idea to 4 spacetime dimensions. The starting point for this line of research lies in two key facts: 1. General relativity in 3 dimensions has a generalization known as BF theory, which retains its topological character in any dimension. One thus expects matter to show up in 4d BF theory in much the same way as in 3d gravity. 2. General relativity in 4 dimensions may be viewed as a perturbation around 4d BF theory, thanks to recent work of Freidel and Starodubtsev, based on the MacDowell–Mansouri formalism. The ultimate goal is to combine these ideas to see matter emerging in a purely geometric way in the MacDowell–Mansouri approach to gravity. In pursuit of this goal, I have studied matter in BF theory [7] as well as the geometric foundations of MacDowell–Mansouri gravity [24]. Earlier, my research focussed on lattice approaches to ‘higher gauge theory’, a generalization of gauge theory based on the mathematics of higher categories [10]. The abelian case of higher gauge theory is called pform electromagnetism, since it involves replacing the gauge connection by a pform. I made a
Centro de Análise Matemática,
, 2008
"... As a natural and canonical extension of Kumjian’s Fell bundles over groupoids [17], we give a definition for a double Fell bundle (a double category) over a double groupoid. We show that finite dimensional double category Fell line bundles tensored with their dual with S oreality satisfy the finite ..."
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As a natural and canonical extension of Kumjian’s Fell bundles over groupoids [17], we give a definition for a double Fell bundle (a double category) over a double groupoid. We show that finite dimensional double category Fell line bundles tensored with their dual with S oreality satisfy the finite real spectral triples axioms but not necessarily orientability. This means that these product bundles with noncommutative algebras can be regarded as noncommutative compact manifolds more general than real spectral triples as they are not necessarily orientable. By construction, they unify the noncommutative geometry axioms and hence provide an algebraic enveloping structure for finite spectral triples to give the Dirac operator D new algebraic and geometric structures that are otherwise missing in the transition from Fredholm operator to Dirac operator. The Dirac operator in physical applications as a result becomes less ad hoc. The new noncommutative space we present is a complex line bundle over a double groupoid. Its algebra is not directly analogous to the algebra of a spectral triple. As a result of its interpretation as a 2morphism in the double category, the new structures include that the space of Ds forms part of the C ∗algebra of the double Fell bundle inheriting a hermitian structure and as a 2morphism in the double functor from double groupoid to double Fell bundle, has the role of 2transport or 2connection. This study automatically sets spectral triples in the context of higher category theory providing a possible arena for quantum gravity.
Topos Mediated Gravity: Toward the Categorical Resolution of the Cosmological Constant
, 2009
"... According to Döring and Isham the spectral topos corresponds to any quantum system. The description of a system in the topos becomes similar to this given by classical theory. However, the logic of the emergent theory is rather intuitionistic than classical. According to the recent proposition by th ..."
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According to Döring and Isham the spectral topos corresponds to any quantum system. The description of a system in the topos becomes similar to this given by classical theory. However, the logic of the emergent theory is rather intuitionistic than classical. According to the recent proposition by the author, topoi can modify local smooth spacetime structure. A way how to add gravity into the spectral topos of a system is presented. Supposing that a quantum system modifies the local spacetime structure and interacts with a gravitational field via the spectral topos, a natural pattern for nongravitating quantum lowest energy modes of the system, appears. Moreover, a theory of gravity and systems should be symmetric with respect to the 2group of automorphisms of the category of systems. Thus, any quantum system modifies the spacetime structure locally, and their lowest modes have vanishing contributions to the cosmological constant. Nullifying the contributions to the cosmological constant is deeply related with a fundamental higher symmetry group of gravity. This is the preliminary version of the paper submitted to FP.
CONNECTIONS ON NONABELIAN GERBES AND THEIR HOLONOMY
"... Abstract. We introduce an axiomatic framework for the parallel transport of connections on gerbes. It incorporates parallel transport along curves and along surfaces, and is formulated in terms of gluing axioms and smoothness conditions. The smoothness conditions are imposed with respect to a strict ..."
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Abstract. We introduce an axiomatic framework for the parallel transport of connections on gerbes. It incorporates parallel transport along curves and along surfaces, and is formulated in terms of gluing axioms and smoothness conditions. The smoothness conditions are imposed with respect to a strict Lie 2group, which plays the role of a band, or structure 2group. Upon choosing certain examples of Lie 2groups, our axiomatic framework reproduces in a systematical way several known concepts of gerbes with connection: nonabelian differential cocycles, BreenMessing gerbes, abelian and nonabelian bundle gerbes. These relationships convey a welldefined notion of surface holonomy from our axiomatic framework to each of these concrete models. Till now, holonomy was only known for abelian gerbes; our approach reproduces that known concept and extends it to nonabelian gerbes. Several new features of surface holonomy are exposed under its extension to nonabelian gerbes; for example, it carries an action of