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Sep 20, 2013
09/13

by
Danny Stevenson

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In their study of the representation theory of loop groups, Pressley and Segal introduced a determinant line bundle over an infinite dimensional Grassmann manifold. Mickelsson and Rajeev subsequently generalized the work of Pressley and Segal and in the process introduced for any p >=1 another infinite dimensional Grassmann manifold and a determinant line bundle defined over it. The construction of this determinant line bundle required the notion of a regularized determinant for bounded...

Source: http://arxiv.org/abs/0802.3608v1

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Sep 22, 2013
09/13

by
Danny Stevenson

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This thesis reviews the theory of bundle gerbes and then examines the higher dimensional notion of a bundle 2-gerbe. The notion of a bundle 2-gerbe connection and 2-curving are introduced and it is shown that there is a class in $H^{4}(M;\Z)$ associated to any bundle 2-gerbe.

Source: http://arxiv.org/abs/math/0004117v1

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Jul 20, 2013
07/13

by
Danny Stevenson

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In this paper we describe a classifying theory for families of simplicial topological groups. If $B$ is a topological space and $G$ is a simplicial topological group, then we can consider the non-abelian cohomology $H(B,G)$ of $B$ with coefficients in $G$. If $G$ is a topological group, thought of as a constant simplicial group, then the set $H(B,G)$ is the set of isomorphism classes of principal $G$ bundles, or $G$ torsors, on $B$. For more general simplicial groups $G$, the set $H(B,G)$...

Source: http://arxiv.org/abs/1203.2461v1

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3.0

Jun 28, 2018
06/18

by
Danny Stevenson

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In this paper we prove that for any simplicial set $B$, there is a Quillen equivalence between the covariant model structure on $\mathbf{S}/B$ and a certain localization of the projective model structure on the category of simplicial presheaves on the simplex category $\Delta/B$ of $B$. We extend this result to give a new Quillen equivalence between this covariant model structure and the projective model structure on the category of simplicial presheaves on the simplicial category...

Topics: Algebraic Topology, Mathematics

Source: http://arxiv.org/abs/1512.04815

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Sep 24, 2013
09/13

by
Danny Stevenson

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Given a bisimplicial set, there are two ways to extract from it a simplicial set: the diagonal simplicial set and the less well known total simplicial set of Artin and Mazur. There is a natural comparison map between these two simplicial sets, and it is a theorem due to Cegarra and Remedios and independently Joyal and Tierney, that this comparison map is a weak equivalence for any bisimplicial set. In this paper we will give a new, elementary proof of this result. As an application, we will...

Source: http://arxiv.org/abs/1112.0474v2

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7.0

Jun 29, 2018
06/18

by
Danny Stevenson

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In this paper we prove a stability result for inner fibrations in terms of the wide, or fat join operation on simplicial sets. We also prove some additional results on inner anodyne morphisms that may be of independent interest.

Topics: Algebraic Topology, Mathematics

Source: http://arxiv.org/abs/1608.07699

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Sep 22, 2013
09/13

by
Danny Stevenson

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We make the category BGrb_M of bundle gerbes on a manifold M into a 2-category by providing 2-cells in the form of transformations of bundle gerbe morphisms. This description of BGrb_M as a 2-category is used to define the notion of a bundle 2-gerbe. To every bundle 2-gerbe on M is associated a class in H^4(M;Z). We define the notion of a bundle 2-gerbe connection and show how this leads to a closed, integral differential 4-form on M which represents the image in real cohomology of the class in...

Source: http://arxiv.org/abs/math/0106018v1

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10.0

Mar 22, 2022
03/22

by
The Danny Stevenson Trio

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Tracklist: 1. Maple Leaf Rag 2. Girl From Ipanema 3. Danny Boy 4. Watermelon Man 5. Rainy Day 6. Theme From The Sirloin 7. Joey 8. Exactly Like You 9. Lights Are Low

Topic: Jazz

Source: Vinyl LP

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Jul 20, 2013
07/13

by
Michael Murray; Danny Stevenson

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Let $(P, Y)$ be a bundle gerbe over a fibre bundle $Y \to M$. We show that if $M$ is simply-connected and the fibres of $Y \to M$ are connected and finite-dimensional then the Dixmier-Douady class of $(P, Y)$ is torsion. This corrects and extends an earlier result of the first author.

Source: http://arxiv.org/abs/1007.4922v1

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Sep 20, 2013
09/13

by
Varghese Mathai; Danny Stevenson

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The central result here is an explicit computation of the Hochschild and cyclic homologies of a natural smooth subalgebra of stable continuous trace algebras having smooth manifolds X as their spectrum. More precisely, the Hochschild homology is identified with the space of differential forms on X, and the periodic cyclic homology with the twisted de Rham cohomology of X, thereby generalizing some fundamental results of Connes and Hochschild-Kostant-Rosenberg. The Connes-Chern character is also...

Source: http://arxiv.org/abs/math/0404329v2

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Sep 24, 2013
09/13

by
Varghese Mathai; Danny Stevenson

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A central result here is the computation of the entire cyclic homology of canonical smooth subalgebras of stable continuous trace C*-algebras having smooth manifolds M as their spectrum. More precisely, the entire cyclic homology is shown to be canonically isomorphic to the continuous periodic cyclic homology for these algebras. By an earlier result of the authors, one concludes that the entire cyclic homology of the algebra is canonically isomorphic to the twisted de Rham cohomology of M.

Source: http://arxiv.org/abs/math/0412485v2

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Sep 22, 2013
09/13

by
Varghese Mathai; Danny Stevenson

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It has been argued by Witten and others that in the presence of a nontrivial B-field, D-brane charges in type IIB string theories are measured by twisted K-theory. In joint work with Bouwknegt, Carey and Murray it was proved that twisted K-theory is canonically isomorphic to bundle gerbe K-theory, whose elements are ordinary vector bundles on a principal projective unitary bundle, with an action of the bundle gerbe determined by the principal projective unitary bundle. The principal projective...

Source: http://arxiv.org/abs/hep-th/0201010v5

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Jul 20, 2013
07/13

by
Michael K Murray; Danny Stevenson

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We consider the construction of the basic bundle gerbe on SU(n) introduced by Meinrenken and show that it extends to a range of groups with unitary actions on a Hilbert space including U(n), diagonal tori and the Banach Lie group of unitary operators differing from the identity by an element of a Schatten ideal. In all these cases we give an explicit connection and curving on the basic bundle gerbe and calculate the real Dixmier-Douady class. Extensive use is made of the holomorphic functional...

Source: http://arxiv.org/abs/0804.3464v1

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Jul 20, 2013
07/13

by
David M Roberts; Danny Stevenson

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In this paper, motivated by recent interest in higher gauge theory, we prove that the fiberwise geometric realization functor takes a certain class of simplicial principal bundles in a suitable category of spaces over a fixed space $B$ to fiberwise principal bundles. As an application we show that the fiberwise geometric realization of the universal simplicial principal bundle for a simplicial group $G$ in the category of spaces over $B$ gives rise to a fiberwise principal bundle with structure...

Source: http://arxiv.org/abs/1203.2460v1

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Sep 23, 2013
09/13

by
John C. Baez; Danny Stevenson

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Categorifying the concept of topological group, one obtains the notion of a 'topological 2-group'. This in turn allows a theory of 'principal 2-bundles' generalizing the usual theory of principal bundles. It is well-known that under mild conditions on a topological group G and a space M, principal G-bundles over M are classified by either the first Cech cohomology of M with coefficients in G, or the set of homotopy classes [M,BG], where BG is the classifying space of G. Here we review work by...

Source: http://arxiv.org/abs/0801.3843v2

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Sep 20, 2013
09/13

by
Thomas Nikolaus; Urs Schreiber; Danny Stevenson

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We discuss two aspects of the presentation of the theory of principal infinity-bundles in an infinity-topos, introduced in [NSSa], in terms of categories of simplicial (pre)sheaves. First we show that over a cohesive site C and for G a presheaf of simplicial groups which is C-acyclic, G-principal infinity-bundles over any object in the infinity-topos over C are classified by hyper-Cech-cohomology with coefficients in G. Then we show that over a site C with enough points, principal...

Source: http://arxiv.org/abs/1207.0249v1

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Sep 20, 2013
09/13

by
Thomas Nikolaus; Urs Schreiber; Danny Stevenson

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The theory of principal bundles makes sense in any infinity-topos, such as that of topological, of smooth, or of otherwise geometric infinity-groupoids/infinity-stacks, and more generally in slices of these. It provides a natural geometric model for structured higher nonabelian cohomology and controls general fiber bundles in terms of associated bundles. For suitable choices of structure infinity-group G these G-principal infinity-bundles reproduce the theories of ordinary principal bundles, of...

Source: http://arxiv.org/abs/1207.0248v1

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Sep 18, 2013
09/13

by
Michael Murray; David Michael Roberts; Danny Stevenson

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We consider the existence of bibundles, in other words locally trivial principal $G$ spaces with commuting left and right $G$ actions. We show that their existence is closely related to the structure of the group $\Out(G)$ of outer automorphisms of $G$. We also develop a classifying theory for bibundles. The theory is developed in full generality for $(H, G)$ bibundles for a crossed-module $(H, G)$ and we show with examples the close links with loop group bundles.

Source: http://arxiv.org/abs/1102.4388v2

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Sep 18, 2013
09/13

by
John C. Baez; Alissa S. Crans; Danny Stevenson; Urs Schreiber

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We describe an interesting relation between Lie 2-algebras, the Kac-Moody central extensions of loop groups, and the group String(n). A Lie 2-algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the "Jacobiator". Similarly, a Lie 2-group is a categorified version of a Lie group. If G is a simply-connected compact simple Lie group, there is a 1-parameter family of Lie 2-algebras g_k each having Lie(G) as its Lie algebra...

Source: http://arxiv.org/abs/math/0504123v2

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Sep 24, 2013
09/13

by
Pedram Hekmati; Michael K. Murray; Danny Stevenson; Raymond F. Vozzo

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In gauge theory, the Faddeev-Mickelsson-Shatashvili anomaly arises as a prolongation problem for the action of the gauge group on a bundle of projective Fock spaces. In this paper, we study this anomaly from the point of view of bundle gerbes and give several equivalent descriptions of the obstruction. These include lifting bundle gerbes with non-trivial structure group bundle and bundle gerbes related to the caloron correspondence.

Source: http://arxiv.org/abs/1112.1752v3

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Jun 28, 2018
06/18

by
Michael K. Murray; David Michael Roberts; Danny Stevenson; Raymond F. Vozzo

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We develop the theory of simplicial extensions for bundle gerbes and their characteristic classes with a view towards studying descent problems and equivariance for bundle gerbes. Equivariant bundle gerbes are important in the study of orbifold sigma models. We consider in detail two examples: the basic bundle gerbe on a unitary group and a string structure for a principal bundle. We show that the basic bundle gerbe is equivariant for the conjugation action and calculate its characteristic...

Topics: Differential Geometry, Mathematics, Category Theory, High Energy Physics - Theory

Source: http://arxiv.org/abs/1506.07931

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Sep 20, 2013
09/13

by
Alan L. Carey; Stuart Johnson; Michael K. Murray; Danny Stevenson; Bai-Ling Wang

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We develop the theory of Chern-Simons bundle 2-gerbes and multiplicative bundle gerbes associated to any principal $G$-bundle with connection and a class in $H^4(BG, \ZZ)$ for a compact semi-simple Lie group $G$. The Chern-Simons bundle 2-gerbe realises differential geometrically the Cheeger-Simons invariant. We apply these notions to refine the Dijkgraaf-Witten correspondence between three dimensional Chern-Simons functionals and Wess-Zumino-Witten models associated to the group $G$. We do...

Source: http://arxiv.org/abs/math/0410013v2