Differential cohomology theories as sheaves of spectra

Bunke, Ulrich; Nikolaus, Thomas; Völkl, Michael

doi:10.1007/s40062-014-0092-5

Cited by 56 publications

(77 citation statements)

References 20 publications

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“…Our introduction here is offered as a gateway into the literature. We remark that some of the earliest appearances of differential cohomology in physics, implicitly and explicitly, are [44,45]; the mathematical theory is now highly developed; see [46][47][48][49] and the references therein.…”

Section: Examples and Summarymentioning

confidence: 99%

Anomalies in the space of coupling constants and their dynamical applications I

et al. 2020

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It is customary to couple a quantum system to external classical fields. One application is to couple the global symmetries of the system (including the Poincaré symmetry) to background gauge fields (and a metric for the Poincaré symmetry). Failure of gauge invariance of the partition function under gauge transformations of these fields reflects 't Hooft anomalies. It is also common to view the ordinary (scalar) coupling constants as background fields, i.e. to study the theory when they are spacetime dependent. We will show that the notion of 't Hooft anomalies can be extended naturally to include these scalar background fields. Just as ordinary 't Hooft anomalies allow us to deduce dynamical consequences about the phases of the theory and its defects, the same is true for these generalized 't Hooft anomalies. Specifically, since the coupling constants vary, we can learn that certain phase transitions must be present. We will demonstrate these anomalies and their applications in simple pedagogical examples in one dimension (quantum mechanics) and in some two, three, and four-dimensional quantum field theories. An anomaly is an example of an invertible field theory, which can be described as an object in (generalized) differential cohomology. We give an introduction to this perspective. Also, we use Quillen's superconnections to derive the anomaly for a free spinor field with variable mass. In a companion paper we will study four-dimensional gauge theories showing how our view unifies and extends many recently obtained results.2 In condensed matter physics, symmetry protected topological orders (SPTs) are also characterized at low energies by such actions. Depending on the precise definitions and context, "SPT" may be synonymous with "invertible field theory", or may instead refer to the deformation class of an invertible field theory, i.e. the equivalence class of invertible theories obtained by continuously varying parameters.3 In certain cases, there is no Y such that BY " X and A on X is extended into Y . Then, one can construct an anomaly free partition function by assuming that X is a component of the boundary of Y and Y has additional boundary components.

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Section: Examples and Summarymentioning

confidence: 99%

Anomalies in the space of coupling constants and their dynamical applications I

et al. 2020

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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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“…Finally, a more radical generalisation such as working with the sheaves of spectra of [90] might be necessary. In the latter framework, the discussion of twisted differential K-theory seems to be less restrictive.…”

Section: Outlook: the Non-torsion Casementioning

confidence: 99%

The 2-Hilbert space of a prequantum bundle gerbe

2018

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We construct a prequantum 2-Hilbert space for any line bundle gerbe whose Dixmier-Douady class is torsion. Analogously to usual prequantisation, this 2-Hilbert space has the category of sections of the line bundle gerbe as its underlying 2-vector space. These sections are obtained as certain morphism categories in Waldorf's version of the 2-category of line bundle gerbes. We show that these morphism categories carry a monoidal structure under which they are semisimple and abelian. We introduce a dual functor on the sections, which yields a closed structure on the morphisms between bundle gerbes and turns the category of sections into a 2-Hilbert space. We discuss how these 2-Hilbert spaces fit various expectations from higher prequantisation. We then extend the transgression functor to the full 2-category of bundle gerbes and demonstrate its compatibility with the additional structures introduced. We discuss various aspects of Kostant-Souriau prequantisation in this setting, including its dimensional reduction to ordinary prequantisation.

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Differential cohomology theories as sheaves of spectra

Cited by 56 publications

References 20 publications

Anomalies in the space of coupling constants and their dynamical applications I

Anomalies in the space of coupling constants and their dynamical applications I

Geometry and 2-Hilbert space for nonassociative magnetic translations

The 2-Hilbert space of a prequantum bundle gerbe

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