Superconnections and index theory

Kahle, Alexander

doi:10.1016/j.geomphys.2011.04.004

Cited by 7 publications

(13 citation statements)

References 24 publications

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“…Remark 7.28. We explain briefly why odd-dimensional massive spinor fields are not covered by the theorems in [51] in a similar way. For definiteness consider the 3-dimensional case, as in section 3.2.…”

Section: Massive Spinor Fields Partmentioning

confidence: 99%

“…The new twist is to use Quillen's superconnections to encode the mass. Some cases (section 7.3) are covered by theorems in the existing literature [51]; for the general case (section 7.4) we make a conjecture.…”

Section: Examples and Summarymentioning

confidence: 99%

“…Here, as usual with spinor fields, geometric index theory determines the precise form of the anomaly. We treat the 4-dimensional theory in section 7.3 by directly applying a theorem of Kahle [51]. In section 7.4 we conjecture a general formula for the isomorphism class of the anomaly of a free spinor field in any dimension.…”

Section: Anomalies and Differential Cohomologymentioning

confidence: 99%

“…Here we indicate how to apply Quillen's superconnections [88] to compute the precise anomaly in the massive case. In this section we treat the 4-dimensional theory (section 3.3), since it fits cleanly into existing work in geometric index theory [51]. We make some remarks about the 3-dimensional case as well, but further work is needed to complete the story in odd dimensions in these terms.…”

Section: Massive Spinor Fields Partmentioning

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: Massive Spinor Fields Partmentioning

confidence: 99%

Section: Examples and Summarymentioning

confidence: 99%

Section: Anomalies and Differential Cohomologymentioning

confidence: 99%

Section: Massive Spinor Fields Partmentioning

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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“…The complex analogue of (3.62) generalizes the twisted Dirac operators defined in [Kah11]. In that paper, the author considers the case where we have a Z2-graded hermitian vector bundle E equipped with a Quillen's superconnection ∇ ∇ E , and Spin c -Dirac operators twisted by ∇ ∇ E .…”

Section: Moreover the Homomorphismmentioning

confidence: 99%

Differential $KO$-theory via gradations and mass terms

Gomi

Yamashita

2021

Preprint

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We construct models of the differential KO-theory and the twisted differential KO-theory, by refining Karoubi's in terms of gradations on Clifford modules. In order for this, we set up the generalized Clifford superconnection formalism which generalizes the Quillen's superconnection formalism [Qui85]. One of our models can be regarded as classifying "fermionic mass terms" in physics.A + The twisted groups KOA + 5. The model in terms of skew-adjoint operators 5.1. Algebraic preparations 5.2. The topological model KO − 5.3. The Pontryagin character forms for mass terms 5.4. The differential model 6. The complex case : K + and K − 6.1. The complex superconnections 6.2. The definitions of differential K-theories

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Higher Berry phase of fermions and index theorem

Choi

Ohmori

2022

J. High Energ. Phys.

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When a quantum field theory is trivially gapped, its infrared fixed point is an invertible field theory. The partition function of the invertible field theory records the response to various background fields in the long-distance limit. The set of background fields can include spacetime-dependent coupling constants, in which case we call the corresponding invertible theory a parameterized invertible field theory. We study such parameterized invertible field theories arising from free Dirac fermions with spacetime-dependent mass parameters using the Atiyah-Patodi-Singer index theorem for superconnections. In particular, the response to an infinitesimal modulation of the mass is encoded into a higher analog of the Berry curvature, for which we provide a general formula. When the Berry curvature vanishes, the invertible theory can still be nontrivial if there is a remaining torsional Berry phase, for which we list some computable examples.

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Superconnections and index theory

Cited by 7 publications

References 24 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

Differential $KO$-theory via gradations and mass terms

Higher Berry phase of fermions and index theorem

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