Computations and Applications of η Invariants

Goette, Sebastian

doi:10.1007/978-3-642-22842-1_13

Cited by 11 publications

(11 citation statements)

References 78 publications

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“…These metrics, however, do not admit any isometries, in general, and the same thing should be expected to happen here. Application of the index theorem in these backgrounds will then require an equivariant generalization of the η-invariant based on the spectrum of the Dirac operator on lens spaces [40] (and generalizations thereof), as for the Eguchi-Hanson metric in Einstein gravity where Σ 3 ≃ S 3 /Z 2 (see, for instance, [87] for an up to date exposition of this subject with several references to the mathematics literature).…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Axial anomalies of Lifshitz fermions

Bakas

Lüst

2011

Fortschritte der Physik

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We compute the axial anomaly of a Lifshitz fermion theory with anisotropic scaling z=3 which is minimally coupled to geometry in 3+1 space-time dimensions. We find that the result is identical to the relativistic case using path integral methods. An independent verification is provided by showing with spectral methods that the eta-invariant of the Dirac and Lifshitz fermion operators in three dimensions are equal. Thus, by the integrated form of the anomaly, the index of the Dirac operator still accounts for the possible breakdown of chiral symmetry in non-relativistic theories of gravity. We apply this framework to the recently constructed gravitational instanton backgrounds of Horava-Lifshitz theory and find that the index is non-zero provided that the space-time foliation admits leaves with harmonic spinors. Using Hitchin's construction of harmonic spinors on Berger spheres, we obtain explicit results for the index of the fermion operator on all such gravitational instanton backgrounds with SU(2)xU(1) isometry. In contrast to the instantons of Einstein gravity, chiral symmetry breaking becomes possible in the unimodular phase of Horava-Lifshitz theory arising at lambda = 1/3 provided that the volume of space is bounded from below by the ratio of the Ricci to Cotton tensor couplings raised to the third power. Some other aspects of the anomalies in non-relativistic quantum field theories are also discussed.Comment: 114 pages, 6 figures; minor improvements made in v2 (version to appear in Fortschritte der Physik

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Section: Conclusion and Discussionmentioning

confidence: 99%

Axial anomalies of Lifshitz fermions

Bakas

Lüst

2011

Fortschritte der Physik

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“…It would be interesting to find an intrinsic definition of the generalised Eells–Kuiper invariant when

p_{M} \neq 0 \in H^{4} false(M; double-struckQ false)

. For further information about the role of eta invariants in the classification of 7‐manifolds, we refer the reader to [, §4].…”

Section: Invariantsmentioning

confidence: 99%

“…Let g W : S dπ → Q/d π 4 Z be the Gauss refinement of (H 4 (M ), q • M , p M ) defined by the characteristic form (F H 4 (W, ∂W ), λ W , p W ). Applying (18), this means that for n ∈ H 4…”

Section: The Generalised Eells-kuiper Invariantmentioning

confidence: 99%

The classification of 2‐connected 7‐manifolds

Crowley

Nordström

2018

Proc. London Math. Soc.

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We present a comprehensive classification of closed smooth 2‐connected manifolds of dimension 7. This builds on the almost‐smooth classification from the first author's thesis. The main new ingredient is a generalisation of the Eells–Kuiper invariant that is defined for any closed spin 7‐manifold M, regardless of whether the spin characteristic class pM∈H4false(Mfalse) is torsion. We also determine the inertia group of 2‐connected M — equivalently the number of oriented smooth structures on the underlying topological manifold — in terms of pM and the torsion linking form.

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“…Certainly there are many other implications of Theorem 2.1 and the η-invariant which are not touched in this very brief account. We refer to the survey papers [82] and [64] for some of the topics we did not discuss above.…”

Section: η-Invariant and The Index Theorem On Manifolds With Boundarymentioning

confidence: 99%