A Survey on the Differential and Symplectic Geometry of Linking Numbers

Spera, Mauro

doi:10.1007/s00032-006-0061-5

Cited by 15 publications

(23 citation statements)

References 89 publications

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“…We therefore hope that by refining our arguments one can establish results on knot invariants and Jones polynomials, as obtained by Witten in topological quantum field theory [13], also in case of relativistic quantum field theories with non-abelian gauge groups. Since the notion of homology, used in Lemma 3.2, is insensitive to knots, such an analysis has to be based on the finer concept of isotopy [1,10], however.…”

Section: Discussionmentioning

confidence: 99%

Linking numbers in local quantum field theory

Buchholz¹,

Ciolli²,

Ruzzi³

et al. 2018

Lett Math Phys

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Linking numbers appear in local quantum field theory in the presence of tensor fields, which are closed two-forms on Minkowski space. Given any pair of such fields, it is shown that the commutator of the corresponding intrinsic (gauge invariant) vector potentials, integrated about spacelike separated, spatial loops, are elements of the center of the algebra of all local fields. Moreover, these commutators are proportional to the linking numbers of the underlying loops. If the commutators are different from zero, the underlying two-forms are not exact (there do not exist local vector potentials for them). The theory then necessarily contains massless particles. A prominent example of this kind, due to J.E. Roberts, is given by the free electromagnetic field and its Hodge dual. Further examples with more complex mass spectrum are presented in this article.Mathematics Subject Classification. 81T05, 83C47, 57T15

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Section: Discussionmentioning

confidence: 99%

Linking numbers in local quantum field theory

Buchholz¹,

Ciolli²,

Ruzzi³

et al. 2018

Lett Math Phys

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“…ℓ(i, j), with ℓ(i, j) = ℓ(j, i) being the Gauss linking number of components L i and L j if i j and ℓ(j, j) the framing of L j , equal to ℓ(L j , L ′ j ) with L ′ j being a section of the normal bundle of L j ; see, for example, [35,41,43,48] and below. A regular projection of a link onto a plane produces a natural framing called the blackboard framing.…”

Section: A Hamiltonian 1-form For Linksmentioning

confidence: 99%

“…Here we wish to apply some recently emerged concepts in multisymplectic geometry (mostly building on [7,45,46]) and construct an explicit homotopy co-momentum map [7] in a hydrodynamical setting, leading to a multisymplectic interpretation of the so-called higher-order linking numbers, viewed à la Massey [23,39,48]. The construction is generalized to cover connected compact oriented Riemannian 2 A. M. Miti and M. Spera [2] manifolds having vanishing intermediate de Rham groups.…”

Section: Introductionmentioning

confidence: 99%

A Hydrodynamical Homotopy Co-Momentum Map and a Multisymplectic Interpretation of Higher-Order Linking Numbers

Miti

Spera

2021

J. Aust. Math. Soc.

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In this paper a homotopy co-momentum map (à la Callies, Frégier, Rogers and Zambon) transgressing to the standard hydrodynamical co-momentum map of Arnol’d, Marsden, Weinstein and others is constructed and then generalized to a special class of Riemannian manifolds. Also, a covariant phase space interpretation of the coadjoint orbits associated to the Euler evolution for perfect fluids, and in particular of Brylinski’s manifold of smooth oriented knots, is discussed. As an application of the above homotopy co-momentum map, a reinterpretation of the (Massey) higher-order linking numbers in terms of conserved quantities within the multisymplectic framework is provided and knot-theoretic analogues of first integrals in involution are determined.

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“…We point out [104], [103] for a recent application to the issue of quantum monodromy. The full range of applications is however enormous, so we confine ourselves to mentioning in addition to the references already given, [47,[83][84][85]107] for applications to Kepler-type systems, [21,[91][92][93][94][95]114] for applications to fluid mechanics and [99,112,[117][118][119][120]128] for the geometry of infinite dimensional Grassmannians and related issues. These topics were indeed touched upon in the lectures.…”

Section: A Glimpse At Geometric Quantizationmentioning

confidence: 99%