Analytic Definition of Twisted K-Theory

The purpose of this work is to give a definition of a topological K-theory for dg-categories over$\mathbb{C}$and to prove that the Chern character map from algebraic K-theory to periodic cyclic homology descends naturally to this new invariant. This topological Chern map provides a natural candidate for the existence of a rational structure on the periodic cyclic homology of a smooth proper dg-algebra, within the theory of noncommutative Hodge structures. The definition of topological K-theory consists in two steps: taking the topological realization of algebraic K-theory and inverting the Bott element. The topological realization is the left Kan extension of the functor ‘space of complex points’ to all simplicial presheaves over complex algebraic varieties. Our first main result states that the topological K-theory of the unit dg-category is the spectrum$\mathbf{BU}$. For this we are led to prove a homotopical generalization of Deligne’s cohomological proper descent, using Lurie’s proper descent. The fact that the Chern character descends to topological K-theory is established by using Kassel’s Künneth formula for periodic cyclic homology and the proper descent. In the case of a dg-category of perfect complexes on a separated scheme of finite type, we show that we recover the usual topological K-theory of complex points. We show as well that the Chern map tensorized with$\mathbb{C}$is an equivalence in the case of a finite-dimensional associative algebra – providing a formula for the periodic homology groups in terms of the stack of finite-dimensional modules.

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Topological T-duality for torus bundles with monodromy

Reviews in Mathematical Physics ◽

10.1142/s0129055x15500087 ◽

2015 ◽

Vol 27 (03) ◽

pp. 1550008

Author(s):

David Baraglia

Keyword(s):

Sufficient Conditions ◽

General Setting ◽

Spectral Sequences ◽

Chern Classes ◽

Courant Algebroids ◽

Torus Bundles ◽

Twisted Cohomology ◽

K Theory ◽

Definition Of ◽

Necessary And Sufficient

We give a simplified definition of topological T-duality that applies to arbitrary torus bundles. The new definition does not involve Chern classes or spectral sequences, only gerbes and morphisms between them. All the familiar topological conditions for T-duals are shown to follow. We determine necessary and sufficient conditions for existence of a T-dual in the case of affine torus bundles. This is general enough to include all principal torus bundles as well as torus bundles with arbitrary monodromy representations. We show that isomorphisms in twisted cohomology, twisted K-theory and of Courant algebroids persist in this general setting. We also give an example where twisted K-theory groups can be computed by iterating T-duality.

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Algebraic K-Theory of ∞-Operads

Journal of K-theory K-theory and its Applications to Algebra Geometry and Topology ◽

10.1017/is014008019jkt277 ◽

2014 ◽

Vol 14 (3) ◽

pp. 614-641 ◽

Cited By ~ 1

Author(s):

Thomas Nikolaus

Keyword(s):

Homotopy Groups ◽

Combinatorial Model ◽

K Theory ◽

Definition Of ◽

Exact Sequences

AbstractThe theory of dendroidal sets has been developed to serve as a combinatorial model for homotopy coherent operads, see [MW07, CM13b]. An ∞-operad is a dendroidal setDsatisfying certain lifting conditions.In this paper we give a definition of K-groupsKn(D) for a dendroidal setD. These groups generalize the K-theory of symmetric monoidal (resp. permutative) categories and algebraic K-theory of rings. We establish some useful properties like invariance under the appropriate equivalences and long exact sequences which allow us to compute these groups in some examples. Using results from [Heu11b] and [BN12] we show that theK-theory groups ofDcan be realized as homotopy groups of a K-theory spectrum.

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Orientations on 2-vector Bundles and Determinant Gerbes

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-15482 ◽

2013 ◽

Vol 113 (1) ◽

pp. 63 ◽

Cited By ~ 1

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.

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On the definition and K-theory realization of a modular functor

Reviews in Mathematical Physics ◽

10.1142/s0129055x18500083 ◽

2018 ◽

Vol 30 (03) ◽

pp. 1850008

Author(s):

Igor Kriz ◽

Luhang Lai

Keyword(s):

Field Theory ◽

Point Of View ◽

Topological Field Theory ◽

Minimal Models ◽

Topological Field ◽

K Theory ◽

Modular Functor ◽

Definition Of

We present a definition of a (super)-modular functor which includes certain interesting cases that previous definitions do not allow. We also introduce a notion of topological twisting of a modular functor, and construct formally a realization by a 2-dimensional topological field theory valued in twisted [Formula: see text]-modules. We discuss, among other things, the [Formula: see text]-supersymmetric minimal models from the point of view of this formalism.

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Introductory paper

Symposium - International Astronomical Union ◽

10.1017/s0074180900053031 ◽

1966 ◽

Vol 24 ◽

pp. 3-5

Author(s):

W. W. Morgan

Keyword(s):

Line Intensity ◽

Classification Scheme ◽

Two Dimensional ◽

Spectral Classification ◽

Dimensional Array ◽

Rr Lyrae ◽

Metallic Line ◽

Introductory Paper ◽

Parameter Varying ◽

Definition Of

1. The definition of “normal” stars in spectral classification changes with time; at the time of the publication of theYerkes Spectral Atlasthe term “normal” was applied to stars whose spectra could be fitted smoothly into a two-dimensional array. Thus, at that time, weak-lined spectra (RR Lyrae and HD 140283) would have been considered peculiar. At the present time we would tend to classify such spectra as “normal”—in a more complicated classification scheme which would have a parameter varying with metallic-line intensity within a specific spectral subdivision.

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2.2. Working Group 2: Conceptual Definitions of Reference Coordinate Systems for Earth Dynamics

International Astronomical Union Colloquium ◽

10.1017/s0252921100050867 ◽

1975 ◽

Vol 26 ◽

pp. 21-26

Keyword(s):

Coordinate System ◽

Working Group ◽

General Character ◽

Coordinate Systems ◽

Reference Coordinate System ◽

Group 2 ◽

Definition Of ◽

Earth Dynamics

An ideal definition of a reference coordinate system should meet the following general requirements:1. It should be as conceptually simple as possible, so its philosophy is well understood by the users.2. It should imply as few physical assumptions as possible. Wherever they are necessary, such assumptions should be of a very general character and, in particular, they should not be dependent upon astronomical and geophysical detailed theories.3. It should suggest a materialization that is dynamically stable and is accessible to observations with the required accuracy.

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Symbiotic Stars at Optical, Infrared and Radio Wavelengths

International Astronomical Union Colloquium ◽

10.1017/s025292110007336x ◽

1979 ◽

Vol 46 ◽

pp. 125-149 ◽

Cited By ~ 3

Author(s):

David A. Allen

Keyword(s):

Temperature Regime ◽

Type Star ◽

Stellar Systems ◽

Symbiotic Stars ◽

Lower Excitation ◽

Definition Of ◽

Original Type

No paper of this nature should begin without a definition of symbiotic stars. It was Paul Merrill who, borrowing on his botanical background, coined the termsymbioticto describe apparently single stellar systems which combine the TiO absorption of M giants (temperature regime ≲ 3500 K) with He II emission (temperature regime ≳ 100,000 K). He and Milton Humason had in 1932 first drawn attention to three such stars: AX Per, CI Cyg and RW Hya. At the conclusion of the Mount Wilson Ha emission survey nearly a dozen had been identified, and Z And had become their type star. The numbers slowly grew, as much because the definition widened to include lower-excitation specimens as because new examples of the original type were found. In 1970 Wackerling listed 30; this was the last compendium of symbiotic stars published.

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Immunoferritin localization of intracellular antigens in positively stained frozen sections

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s042482010008170x ◽

1978 ◽

Vol 36 (2) ◽

pp. 168-169

Author(s):

K. T. Tokuyasu

Keyword(s):

Surface Tension ◽

Water Surface ◽

Thin Layers ◽

The Other ◽

Positive Staining ◽

Frozen Sections ◽

The Past ◽

Ultrathin Frozen Sections ◽

Definition Of

During the past investigations of immunoferritin localization of intracellular antigens in ultrathin frozen sections, we found that the degree of negative staining required to delineate u1trastructural details was often too dense for the recognition of ferritin particles. The quality of positive staining of ultrathin frozen sections, on the other hand, has generally been far inferior to that attainable in conventional plastic embedded sections, particularly in the definition of membranes. As we discussed before, a main cause of this difficulty seemed to be the vulnerability of frozen sections to the damaging effects of air-water surface tension at the time of drying of the sections.Indeed, we found that the quality of positive staining is greatly improved when positively stained frozen sections are protected against the effects of surface tension by embedding them in thin layers of mechanically stable materials at the time of drying (unpublished).

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