scholarly journals A version of smooth K-theory adapted to the total Chern class

2010 ◽  
Vol 6 (2) ◽  
pp. 197-230 ◽  
Author(s):  
Alain Berthomieu
Keyword(s):  
Chern Class ◽  
Chern Character ◽  
Chern Classes ◽  
New Model ◽  
K Theory ◽  
Chern Simons ◽  
Natural Way

AbstractA new model of smooth K0-theory ([5] [1]) is constructed, with the help of the total Chern class (contrary to the theories considered in ]1], [5], [12] and [13] which use the Chern character). The correspondence with the earlier model [1] is obtained by adapting, at the level of transgression forms, the usual formulae which express the Chern character in terms of the Chern classes and vice versa.The advantage of this new model is that it allows constructing Chern classes with values in integral Chern-Simons characters in a natural way: this construction answers a question asked by U. Bunke [4].

scholarly journals Wilson loop algebras and quantum K-theory for Grassmannians

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Hans Jockers ◽  
Peter Mayr ◽  
Urmi Ninad ◽  
Alexander Tabler
Keyword(s):  
Wilson Loop ◽  
Gauge Theories ◽  
Wilson Line ◽  
Quantum Cohomology ◽  
Three Dimensional ◽  
Loop Algebra ◽  
Loop Algebras ◽  
Verlinde Algebra ◽  
K Theory ◽  
Chern Simons

Abstract We study the algebra of Wilson line operators in three-dimensional $$ \mathcal{N} $$ N = 2 supersymmetric U(M ) gauge theories with a Higgs phase related to a complex Grassmannian Gr(M, N ), and its connection to K-theoretic Gromov-Witten invariants for Gr(M, N ). For different Chern-Simons levels, the Wilson loop algebra realizes either the quantum cohomology of Gr(M, N ), isomorphic to the Verlinde algebra for U(M ), or the quantum K-theoretic ring of Schubert structure sheaves studied by mathematicians, or closely related algebras.

scholarly journals Higher spectral flow and an entire bivariant JLO cocycle

2012 ◽  
Vol 11 (1) ◽  
pp. 183-232 ◽  
Author(s):  
Moulay-Tahar Benameur ◽  
Alan L. Carey
Keyword(s):  
Dirac Operator ◽  
Chern Character ◽  
Dirac Operators ◽  
Cyclic Cohomology ◽  
Closed Manifold ◽  
Spectral Flow ◽  
Smooth Functions ◽  
Base Manifold ◽  
K Theory ◽  
Higher Spectral Flow

AbstractFor a single Dirac operator on a closed manifold the cocycle introduced by Jaffe-Lesniewski-Osterwalder [19] (abbreviated here to JLO), is a representative of Connes' Chern character map from the K-theory of the algebra of smooth functions on the manifold to its entire cyclic cohomology. Given a smooth fibration of closed manifolds and a family of generalized Dirac operators along the fibers, we define in this paper an associated bivariant JLO cocycle. We then prove that, for any l ≥ 0, our bivariant JLO cocycle is entire when we endow smoooth functions on the total manifold with the Cl+1 topology and functions on the base manifold with the Cl topology. As a by-product of our theorem, we deduce that the bivariant JLO cocycle is entire for the Fréchet smooth topologies. We then prove that our JLO bivariant cocycle computes the Chern character of the Dai-Zhang higher spectral flow.

scholarly journals Chern character for twisted K-theory of orbifolds

2006 ◽  
Vol 207 (2) ◽  
pp. 455-483 ◽  
Author(s):  
Jean-Louis Tu ◽  
Ping Xu
Keyword(s):  
Chern Character ◽  
K Theory

scholarly journals Chern Character in Twisted K -Theory: Equivariant and Holomorphic Cases

2003 ◽  
Vol 236 (1) ◽  
pp. 161-186 ◽  
Author(s):  
Varghese Mathai ◽  
Danny Stevenson
Keyword(s):  
Chern Character ◽  
K Theory

On the β-Construction in K-Theory

1972 ◽  
Vol 24 (5) ◽  
pp. 819-824
Author(s):  
C. M. Naylor
Keyword(s):  
Lie Group ◽  
Chern Character ◽  
Complex Representation ◽  
Group Homomorphism ◽  
Unique Element ◽  
Compact Lie Group ◽  
Fundamental Representation ◽  
K Theory ◽  
Simply Connected

The β-construction assigns to each complex representation φ of the compact Lie group G a unique element β(φ) in (G). For the details of this construction the reader is referred to [1] or [5]. The purpose of the present paper is to determine some of the properties of the element β(φ) in terms of the invariants of the representation φ. More precisely, we consider the following question. Let G be a simple, simply-connected compact Lie group and let f : S3 →G be a Lie group homomorphism. Then (S3) ⋍ Z with generator x = β(φ1), φ1 the fundamental representation of S3 , so that if φ is a representation of G,f*(φ) = n(φ)x, where n(φ) is an integer depending on φ and f . The problem is to determine n(φ).Since G is simple and simply-connected we may assume that ch2, the component of the Chern character in dimension 4 takes its values in H4(SG,Z)≅Z. Let u be a generator of H4(SG,Z) so that ch2(β (φ)) = m(φ)u, m(φ) an integer depending on φ.

Intersection K-theory for isolated conical singularities

2011 ◽  
Vol 9 (1) ◽  
pp. 177-200
Author(s):  
André Legrand ◽  
David Poutriquet
Keyword(s):  
Chern Character ◽  
Intersection Cohomology ◽  
Conical Singularities ◽  
K Theory

AbstractStarting from the Karoubi multiplicative K-theory, we construct a Chern-Weil theory adapted to isolated conical singularities. The Chern character takes its values in the intersection cohomology of Goresky-MacPherson. We also propose an integer intersection K-theory for such singularities.

On the relation of connective K-theory to homology

1970 ◽  
Vol 68 (3) ◽  
pp. 637-639 ◽  
Author(s):  
Larry Smith
Keyword(s):  
Exact Sequence ◽  
Natural Transformation ◽  
Homology Theory ◽  
Finite Complex ◽  
Ring Spectrum ◽  
Bott Periodicity ◽  
Ring Spectra ◽  
K Theory ◽  
Image Position ◽  
Natural Way

Let us denote by k*( ) the homology theory determined by the connective BU spectrum, bu, that is, in the notations of (1) and (9), bu2n = BU(2n,…,∞), bu2n+1 = U(2n + 1,…, ∞) with the spectral maps induced via Bott periodicity. The resulting spectrum, bu, is a ring spectrum. Recall that k*(point) ≅ Z[t], degree t = 2. There is a natural transformation of ring spectrainducing a morphismof homology functors. It is the objective of this note to establish: Theorem. Let X be a finite complex. Then there is a natural exact sequencewhere Z is viewed as a Z[t] module via the augmentationand, is induced by η*in the natural way.

scholarly journals Transferred Chern classes in Morava $K$-theory

2003 ◽  
Vol 132 (6) ◽  
pp. 1855-1860 ◽  
Author(s):  
Malkhaz Bakuradze ◽  
Stewart Priddy
Keyword(s):  
Chern Classes ◽  
K Theory
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