Differential cohomology theories as sheaves of spectra

Bunke, Ulrich; Nikolaus, Thomas; Völkl, Michael

doi:10.48550/arxiv.1311.3188

Cited by 9 publications

(15 citation statements)

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“…Other prominent models are, for instance, based on sheaves of groupoids [Bry93] and predate the discovery of bundle gerbes. Models that generalise to other differential cohomology theories have been developed in [BNV16,HS05]. For us, however, bundle gerbes are the most convenient model, for they allow for a straightforward interpretation as a categorification of line bundles and their sections.…”

Section: Bundle Gerbes On R Dmentioning

confidence: 99%

Geometry and 2-Hilbert space for nonassociative magnetic translations

Bunk

Müller²,

Szabo³

2019

Lett Math Phys

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We suggest a geometric approach to quantisation of the twisted Poisson structure underlying the dynamics of charged particles in fields of generic smooth distributions of magnetic charge, and dually of closed strings in locally non-geometric flux backgrounds, which naturally allows for representations of nonassociative magnetic translation operators. We show how one can use the 2-Hilbert space of sections of a bundle gerbe in a putative framework for canonical quantisation. We define a parallel transport on bundle gerbes on R d and show that it naturally furnishes weak projective 2-representations of the translation group on this 2-Hilbert space. We obtain a notion of covariant derivative on a bundle gerbe and a novel perspective on the fake curvature condition.

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Section: Bundle Gerbes On R Dmentioning

confidence: 99%

Geometry and 2-Hilbert space for nonassociative magnetic translations

Bunk

Müller²,

Szabo³

2019

Lett Math Phys

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“…(iii) (Mayer-Vietoris) Again applying the general construction of [BNV16] to our case, if M a smooth manifold and tU, V u an open cover, we also have a Mayer-Vietoris sequence…”

Section: Rr Fields As Twisted Differential K-theory Classes 21 Differ...mentioning

confidence: 96%

“…The construction of the Hopkins-Singer refinement of the K-theory spectrum proceeds by applying the cohesive 8-adjoints pδ : $ Γ $ δ $ Πq as introduced in [Sc13] to the sheaf of spectra kU ∇ . The topological realization 3 δΠ induces a morphism of sheaves of spectra (see [BNV16]) cyc :" δΠ : kU ∇ » kU ÝÑ KU.…”

Section: Rr Fields As Twisted Differential K-theory Classes 21 Differ...mentioning

confidence: 99%

Ramond-Ramond fields and twisted differential K-theory

Grady¹,

Sati²

2019

Preprint

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We provide a systematic approach to describing the Ramond-Ramond (RR) fields as elements in twisted differential K-theory. This builds on a series of constructions by the authors on geometric and computational aspects of twisted differential K-theory, which to a large extent were originally motivated by this problem. In addition to providing a new conceptual framework and a mathematically solid setting, this allows us to uncover interesting and novel effects. Explicitly, we use our recently constructed Atiyah-Hirzebruch spectral sequence (AHSS) for twisted differential K-theory to characterize the RR fields and their quantization, which involves interesting interplay between geometric and topological data. We illustrate this with the examples of spheres, tori, and Calabi-Yau threefolds.

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“…There are various constructions of differential cohomologies ( [4,5,13]). A particular simple construction to us was given by Harvey and Lawson through their theory of spark complexes ( [10,11,12,13,15,20]) which unifies many known results.…”

Section: Introductionmentioning

confidence: 99%