scholarly journals Orientations on 2-vector Bundles and Determinant Gerbes

2013 ◽  
Vol 113 (1) ◽  
pp. 63 ◽  
Author(s):  
Thomas Kragh
Keyword(s):  
Vector Bundles ◽  
Magnetic Monopole ◽  
Monoidal Categories ◽  
Continuous Map ◽  
K Theory ◽  
Definition Of

In a paper from 2009, a half magnetic monopole was discovered by Ausoni, Dundas, and Rognes. This describes an obstruction to the existence of a continuous map $K(ku) \to B(ku^*)$ with determinant like properties. This magnetic monopole is in fact an obstruction to the existence of a map from $K(ku)$ to $K(\mathsf{Z},3)$, which is a retract of the natural map $K(\mathsf{Z},3) \to K(ku)$; and any sensible definition of determinant like should produce such a retract. In this paper we describe this obstruction precisely using monoidal categories. By a result from 2011 by Baas, Dundas, Richter and Rognes $K(ku)$ classifies 2-vector bundles. We thus define the notion of oriented 2-vector bundles, which removes the obstruction by the magnetic monopole. We use this to define an oriented K-theory of 2-vector bundles with a lift of the natural map from $K(\mathsf{Z},3)$. It is then possible to define a retraction of this map and since $K(\mathsf{Z},3)$ classifies complex gerbes we call this a determinant gerbe map.

scholarly journals On q-th Derivative of Vector Bundles

1967 ◽  
Vol 29 ◽  
pp. 121-126
Author(s):  
Akikuni Kato
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Present Note ◽  
Higher Order ◽  
Enumerative Geometry ◽  
Sheaf Theory ◽  
Definition Of

In the present note we shall be concerned with the improvement of fundamental definitions in higher order enumerative geometry which has been recently given by W. F. Pohl. Pohl’s definition of q-th derivative of vector bundle is very complicated. We shall give a simpler and more reasonable definition of the q-th derivative of vector bundle in terms of sheaf theory and simplify the proofs in [P]. We shall also give a definition of higher order singularity of map.

K-Theory and the Jones Polynomial

2018 ◽  
Vol 30 (06) ◽  
pp. 1840002
Author(s):  
Michael F. Atiyah ◽  
Carlos Zapata-Carratala
Keyword(s):  
Jones Polynomial ◽  
New Approach ◽  
K Theory ◽  
Definition Of

In this paper, we present a new approach to the definition of the Jones polynomial using equivariant K-theory. Dedicated to Ludwig Faddeev

scholarly journals The group structure of the homotopy set whose target is the automorphism group of the Cuntz algebra

2019 ◽  
Vol 30 (11) ◽  
pp. 1950057 ◽  
Author(s):  
M. Izumi ◽  
T. Sogabe
Keyword(s):  
Automorphism Group ◽  
Vector Bundles ◽  
Group Structure ◽  
Cuntz Algebra ◽  
Classifying Space ◽  
Cuntz Algebras ◽  
K Theory ◽  
Continuous Fields

We determine the group structure of the homotopy set whose target is the automorphism group of the Cuntz algebra [Formula: see text] for finite [Formula: see text] in terms of K-theory. We show that there is an example of a space for which the homotopy set is a noncommutative group, and hence, the classifying space of the automorphism group of the Cuntz algebra for finite [Formula: see text] is not an H-space. We also make an improvement of Dadarlat’s classification of continuous fields of the Cuntz algebras in terms of vector bundles.

K-Theory of Vector Bundles, of Modules, and of Idempotents

2007 ◽  
pp. 45-54
Author(s):  
D. Husemöller ◽  
M. Joachim ◽  
B. Jurčo ◽  
M. Schottenloher
Keyword(s):  
Vector Bundles ◽  
K Theory

scholarly journals The Weil Algebra of a Double Lie Algebroid

2020 ◽  
Author(s):  
Eckhard Meinrenken ◽  
Jeffrey Pike
Keyword(s):  
Vector Fields ◽  
Vector Bundles ◽  
Lie Algebroid ◽  
Weil Algebra ◽  
Algebra Of Functions ◽  
Double Vector Bundles ◽  
Definition Of ◽  
Deformation Complex ◽  
Double Vector Bundle ◽  
Gerstenhaber Bracket

Abstract Given a double vector bundle $D\to M$, we define a bigraded bundle of algebras $W(D)\to M$ called the “Weil algebra bundle”. The space ${\mathcal{W}}(D)$ of sections of this algebra bundle ”realizes” the algebra of functions on the supermanifold $D[1,1]$. We describe in detail the relations between the Weil algebra bundles of $D$ and those of the double vector bundles $D^{\prime},\ D^{\prime\prime}$ obtained from $D$ by duality operations. We show that ${\mathcal{V}\mathcal{B}}$-algebroid structures on $D$ are equivalent to horizontal or vertical differentials on two of the Weil algebras and a Gerstenhaber bracket on the 3rd. Furthermore, Mackenzie’s definition of a double Lie algebroid is equivalent to compatibilities between two such structures on any one of the three Weil algebras. In particular, we obtain a ”classical” version of Voronov’s result characterizing double Lie algebroid structures. In the case that $D=TA$ is the tangent prolongation of a Lie algebroid, we find that ${\mathcal{W}}(D)$ is the Weil algebra of the Lie algebroid, as defined by Mehta and Abad–Crainic. We show that the deformation complex of Lie algebroids, the theory of IM forms and IM multi-vector fields, and 2-term representations up to homotopy all have natural interpretations in terms of our Weil algebras.

scholarly journals Equivariant KK-theory for generalised actions and Thom isomorphism in groupoid twisted K-theory

2013 ◽  
Vol 13 (1) ◽  
pp. 83-113 ◽  
Author(s):  
El-Kaïoum M. Moutuou
Keyword(s):  
Vector Bundles ◽  
Isomorphism Theorem ◽  
Locally Compact ◽  
Bott Periodicity ◽  
C Algebras ◽  
Thom Isomorphism ◽  
K Theory ◽  
Thom Isomorphism Theorem

AbstractWe develop equivariant KK–theory for locally compact groupoid actions by Morita equivalences on real and complex graded C*-algebras. Functoriality with respect to generalised morphisms and Bott periodicity are discussed. We introduce Stiefel-Whitney classes for real or complex equivariant vector bundles over locally compact groupoids to establish the Thom isomorphism theorem in twisted groupoid K–theory.

scholarly journals Logarithmic Connections, WZNW Action, and Moduli of Parabolic Bundles on the Sphere

2021 ◽  
Author(s):  
Claudio Meneses ◽  
Leon A. Takhtajan
Keyword(s):  
Moduli Spaces ◽  
Vector Bundles ◽  
Riemann Sphere ◽  
Parabolic Bundle ◽  
Parabolic Bundles ◽  
Logarithmic Connection ◽  
Definition Of ◽  
Suitable Notion ◽  
Marked Points ◽  
Explicit Relation

AbstractModuli spaces of stable parabolic bundles of parabolic degree 0 over the Riemann sphere are stratified according to the Harder–Narasimhan filtration of underlying vector bundles. Over a Zariski open subset $$\mathscr {N}_{0}$$ N 0 of the open stratum depending explicitly on a choice of parabolic weights, a real-valued function $$\mathscr {S}$$ S is defined as the regularized critical value of the non-compact Wess–Zumino–Novikov–Witten action functional. The definition of $$\mathscr {S}$$ S depends on a suitable notion of parabolic bundle ‘uniformization map’ following from the Mehta–Seshadri and Birkhoff–Grothendieck theorems. It is shown that $$-\mathscr {S}$$ - S is a primitive for a (1,0)-form $$\vartheta $$ ϑ on $$\mathscr {N}_{0}$$ N 0 associated with the uniformization data of each intrinsic irreducible unitary logarithmic connection. Moreover, it is proved that $$-\mathscr {S}$$ - S is a Kähler potential for $$(\Omega -\Omega _{\mathrm {T}})|_{\mathscr {N}_{0}}$$ ( Ω - Ω T ) | N 0 , where $$\Omega $$ Ω is the Narasimhan–Atiyah–Bott Kähler form in $$\mathscr {N}$$ N and $$\Omega _{\mathrm {T}}$$ Ω T is a certain linear combination of tautological (1, 1)-forms associated with the marked points. These results provide an explicit relation between the cohomology class $$[\Omega ]$$ [ Ω ] and tautological classes, which holds globally over certain open chambers of parabolic weights where $$\mathscr {N}_{0} = \mathscr {N}$$ N 0 = N .

Analytic Definition of Twisted K-Theory

2007 ◽  
pp. 251-253
Author(s):  
D. Husemöller ◽  
M. Joachim ◽  
B. Jurčo ◽  
M. Schottenloher
Keyword(s):  
K Theory ◽  
Definition Of
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