Twisted differential cohomology

Bunke, Ulrich; Nikolaus, Thomas

doi:10.2140/agt.2019.19.1631

Cited by 19 publications

(54 citation statements)

References 37 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%

“…Remark 5.33. We refer to [46][47][48][49] for precise statements, constructions, theorems, and proofs as well as additional references. Presumably more work is needed to develop the full "categorified integration theory" we use here.…”

Section: Integrationmentioning

confidence: 99%

“…The work of Hopkins-Singer [46] extends the differential theory to generalized cohomology theories and also develops a version with geometric representatives of differential cohomology classes. See [47][48][49] and the references therein for modern developments. Our discursive treatment of low degree classes is meant as background for the reader, who should pursue the subject in these and other references.…”

Section: Introduction To Differential Cohomologymentioning

confidence: 99%

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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%

Section: Integrationmentioning

confidence: 99%

Section: Introduction To Differential Cohomologymentioning

confidence: 99%

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

et al. 2020

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“…It is interesting to see where the gerbe data appears in defining the twisted theory. In fact, as observed in [BN14], a crucial ingredient in defining twisted differential theories is the analogue of the de Rham isomorphism theorem for twisted cohomology. In the untwisted case, recall that the locally constant sheaf R admits an acyclic resolution via the de Rham complex…”

Section: Introductionmentioning

confidence: 96%

“…There has been a lot of recent activity on modifying generalized cohomology theories to include twists and geometric refinements, in order to account for automorphisms and include geometric data. Twisted differential generalized cohomology theories are established at the general axiomatic level [BN14]. However, working out these theories explicitly is in practice not a straightforward task.…”

Section: Introductionmentioning

confidence: 99%

Higher-twisted periodic smooth Deligne cohomology

Grady

Sati

2019

Homology, Homotopy and Applications

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Degree one twisting of Deligne cohomology, as a differential refinement of integral cohomology, was established in previous work. Here we consider higher degree twists. The Rham complex, hence de Rham cohomology, admits twists of any odd degree. However, in order to consider twists of integral cohomology we need a periodic version. Combining the periodic versions of both ingredients leads us to introduce a periodic form of Deligne cohomology. We demonstrate that this theory indeed admits a twist by a gerbe of any odd degree. We present the main properties of the new theory and illustrate its use with examples and computations, mainly via a corresponding twisted differential Atiyah-Hirzebruch spectral sequence.1 This is sometimes also denoted Z 8 D pnq or Zpnq 8 D . We are in the smooth setting throughout, so we will not need extra decorations. 2 This would be H n pM ; Dpnqq if we use the opposite convention. However, the one we use is positively graded, hence better adapted for stacks.where L is the stackification functor. 11 Let φ : R n Ñ M be a local chart. For a convex open subset U Ă R n , this stack can be evaluated on the corresponding open subset V " φpU q via MappV, B 2 U p1q ∇ q » DK`C 8 pV, U p1qq d log Ý ÝÝ Ñ Ω 1 pV q d ÝÑ Ω 2 pV q˘. More generally, descent for the stack B 2 U p1q ∇ implies that, for any choice of good open cover tU α u of M , the space of maps MappM, B 2 U p1q ∇ q can be identified by replacing M with theČech nerveČptU α uq of tU α u and considering instead the space of maps Map´ČptU α uq, DK`U p1q d logBy the basic properties of the Dold-Kan correspondence we have an isomorphism 12By [Br93, Theorem 5.3.11], the elements on the right parametrize the homotopy classes of the gerbes with connective structure and curving considered in Example 6.11 This is a functor which turns a prestack into a stack, analogously to the way a sheafification functor turns a presheaf into a sheaf. See [Lu09, Sec. 6.5.3] for details. 12 The shift in degree occurs because on the left we consider the complex U p1q d log Ý ÝÝ Ñ Ω 1 d Ý Ý Ñ Ω 2 as being shifted up two degrees relative to the complex appearing on the right. 15

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Discrete and higher-form symmetries in SCFTs from wrapped M5-branes

Bah

Bonetti

Minasian

2021

J. High Energ. Phys.

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We analyze topological mass terms of BF type arising in supersymmetric M-theory compactifications to AdS5. These describe spontaneously broken higher-form gauge symmetries in the bulk. Different choices of boundary conditions for the BF terms yield dual field theories with distinct global discrete symmetries. We discuss in detail these symmetries and their ’t Hooft anomalies for 4d $$ \mathcal{N} $$ N = 1 SCFTs arising from M5-branes wrapped on a Riemann surface without punctures, including theories from M5-branes at a ℤ2 orbifold singularity. The anomaly polynomial is computed via inflow and contains background fields for discrete global 0-, 1-, and 2-form symmetries and continuous 0-form symmetries, as well as axionic background fields. The latter are properly interpreted in the context of anomalies in the space of coupling constants.

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Twisted differential cohomology

Cited by 19 publications

References 37 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

Higher-twisted periodic smooth Deligne cohomology

Discrete and higher-form symmetries in SCFTs from wrapped M5-branes

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