Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T14:43:43.436Z Has data issue: false hasContentIssue false
  • English
  • Français

A HYDRODYNAMICAL HOMOTOPY CO-MOMENTUM MAP AND A MULTISYMPLECTIC INTERPRETATION OF HIGHER-ORDER LINKING NUMBERS

Part of: Symplectic geometry, contact geometry Operations and obstructions

Published online by Cambridge University Press:  11 February 2021

ANTONIO MICHELE MITI Open the ORCID record for ANTONIO MICHELE MITI [Opens in a new window]  and
MAURO SPERA Open the ORCID record for MAURO SPERA [Opens in a new window]
ANTONIO MICHELE MITI*
Affiliation:
Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via dei Musei 41, 25121Brescia, Italy and Department of Mathematics, KU Leuven, Celestijnenlaan 200B Box 2400, 3001Leuven, Belgium
MAURO SPERA
Affiliation:
Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via dei Musei 41, 25121Brescia, Italy e-mail: mauro.spera@unicatt.it
*
e-mail: antoniomichele.miti@unicatt.it
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

higher-order linking numbershomotopy co-momentum mapshydrodynamicssymplectic and multisymplectic geometry

MSC classification

Primary: 53D20: Momentum maps; symplectic reduction
Secondary: 53D99: None of the above, but in this section 55S30: Massey products
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 112 , Issue 3 , June 2022 , pp. 335 - 354
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© 2021 Australian Mathematical Publishing Association Inc.

Footnotes

Communicated by Robert Yuncken

References

Abraham, R. and Marsden, J., Foundations of Mechanics, Benjamin/Cummings, Reading, MA, 1978.Google Scholar
Arnol’d, V. I., ‘Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits’, Ann. Inst. Fourier (Grenoble) 16 (1966) fasc. 1, 319–361.CrossRefGoogle Scholar
Arnol’d, V. I. and Khesin, B., Topological Methods in Hydrodynamics (Springer, Berlin, 1998).CrossRefGoogle Scholar
Besana, A. and Spera, M., ‘On some symplectic aspects of knots framings’, J. Knot Theory Ramifications 15 (2006), 883–912.CrossRefGoogle Scholar
Bott, R. and Tu, L., Differential Forms in Algebraic Topology (Springer, Berlin, 1982).CrossRefGoogle Scholar
Brylinski, J.-L., Loop Spaces, Characteristic Classes and Geometric Quantization (Birkhäuser, Basel, 1993).CrossRefGoogle Scholar
Callies, M., Frégier, Y., Rogers, C. L. and Zambon, M., ‘Homotopy moment maps’, Adv. Math. 303 (2016), 954–1043.CrossRefGoogle Scholar
Cantrijn, F., Ibort, L. A. and De León, M., ‘On the geometry of multisymplectic manifolds’, J. Aust. Math. Soc. A 66(3) (1999), 303–330.CrossRefGoogle Scholar
Chen, K.-T., ‘Iterated path integrals’, Bull. Amer. Math. Soc. 83 (1977), 831–879.CrossRefGoogle Scholar
Chen, K.-T., Collected Papers of K.-T. Chen (eds. Tondeur, P. and Hain, R.) (Birkäuser, Boston, MA, 2001).Google Scholar
Crnković, Č., ‘Symplectic geometry of the covariant phase space’, Classical and Quantum Gravity 5 (1988), 1557–1575.CrossRefGoogle Scholar
de Rham, G., Variétés Différentiables (Hermann, Paris, 1954).Google Scholar
Ebin, D. and Marsden, J., ‘Groups of diffeomorphisms and the motion of incompressible fluids’, Ann. of Math. 92 (1970), 102–163.CrossRefGoogle Scholar
Fenn, R. A., Techniques of Geometric Topology, London Mathematical Society Lecture Notes Series, 57 (Cambridge University Press, Cambridge, 1983).Google Scholar
Forger, M. and Romero, S. V., ‘Covariant Poisson brackets in geometric field theory’, Comm. Math. Phys. 256 (2005), 375–410.CrossRefGoogle Scholar
Frégier, Y., Laurent-Gengoux, C. and Zambon, M., ‘A cohomological framework for homotopy moment maps’, J. Geom. Phys. 97 (2015), 119–132.CrossRefGoogle Scholar
Goldin, G., ‘Non-relativistic current algebras as unitary representations of groups’, J. Math. Phys. 12 (1971), 462–488.CrossRefGoogle Scholar
Goldin, G., ‘Parastatistics, θ-statistics, and topological quantum mechanics from unitary representations of diffeomorphism groups’, in: Proceedings of the XV International Conference on Differential Geometric Methods in Physics (eds. Doebner, H. D. and Henning, J. D.) (World Scientific, Singapore, 1987), 197–207.Google Scholar
Goldin, G., ‘Diffeomorphism groups and nonlinear quantum mechanics’, J. Phys. Conf. Ser. 343 (2012), 012006.CrossRefGoogle Scholar
Gotay, M. J., Isenberg, J., Marsden, J. E. and Montgomery, R., ‘Momentum maps and classical fields. Part I: Covariant field theory’, Preprint, 1998, arXiv:physics/9801019v2 [math-ph], ‘Momentum maps and classical fields. Part II: Canonical analysis of field theories’, Preprint, 2004, arXiv:math-phys/0411036 [math-ph].Google Scholar
Guillemin, V. and Sternberg, S., Symplectic Techniques in Physics (Cambridge University Press, Cambridge, 1984).Google Scholar
Hain, R., ‘The geometry of the mixed Hodge structure on the fundamental group’, Proc. Sympos. Pure Math. 46 (1987), 247–282.CrossRefGoogle Scholar
Hebda, J. J. and Tsau, C. M., ‘An approach to higher order linking invariants through holonomy and curvature’, Trans. Amer. Math. Soc. 364 (2012), 4283–4301.CrossRefGoogle Scholar
Herman, J., ‘Existence and uniqueness of weak homotopy moment maps’, J. Geom. Phys. 131 (2018), 52–65.CrossRefGoogle Scholar
Khesin, B., ‘Topological fluid dynamics: theory and applications. The vortex filament equation in any dimension’, Procedia IUTAM 7 (2013), 135–140.CrossRefGoogle Scholar
Kijowski, J. and Szczyrba, V., ‘A canonical structure for classical field theories’, Comm. Math. Phys. 46 (1976), 183–206.CrossRefGoogle Scholar
Kirillov, A., ‘Geometric quantization’, in: Dynamical Systems IV, Encyclopaedia of Mathematical Sciences, 4 (Springer, Berlin, 2001), 139–176.CrossRefGoogle Scholar
Kostant, B., ‘Quantization and unitary representations’, in: Lectures in Modern Analysis and Applications, Lecture Notes in Mathematics, 170 (Springer, Berlin, 1970), pp. 87–208.Google Scholar
Kriegl, A. and Michor, P. W., The Convenient Setting of Global Analysis, Mathematical Surveys and Monographs, 53 (American Mathematical Society, Providence, RI, 1997).CrossRefGoogle Scholar
Kuznetsov, E. A. and Mikhailov, A. V., ‘On the topological meaning of canonical Clebsch variables’, Phys. Lett. A 77 (1980), 37–38.CrossRefGoogle Scholar
Mammadova, L. and Ryvkin, L., ‘On the extension problem for weak moment maps’, Preprint, 2020, arXiv:2001.00264.Google Scholar
Marsden, J. and Weinstein, A., ‘Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids’, Physica 7 D (1983), 305–323.Google Scholar
Marsden, J. E., Pekarsky, S., Shkoller, S. and West, M., ‘Variational methods, multisymplectic geometry and continuum mechanics’, J. Geom. Phys. 38 (2001), 253–284.CrossRefGoogle Scholar
Miti, A. M. and Spera, M., On some (multi)symplectic aspects of link invariants, Preprint, arXiv:1805.01696 [math.DG] v2.Google Scholar
Moffatt, H. K. and Ricca, R. L., ‘Helicity and the Călugăreanu invariant’, Proc. R. Soc. Lond. A 439 (1992), 411–429.Google Scholar
Penna, V. and Spera, M., ‘A geometric approach to quantum vortices’, J. Math. Phys. 30 (1989), 2778–2784.CrossRefGoogle Scholar
Penna, V. and Spera, M., ‘On coadjoint orbits of rotational perfect fluids’, J. Math. Phys. 33 (1992), 901–909.CrossRefGoogle Scholar
Penna, V. and Spera, M., ‘String limit of vortex current algebra’, Phys. Rev. B 62 (2000), 14547–14553.CrossRefGoogle Scholar
Penna, V. and Spera, M., ‘Higher order linking numbers, curvature and holonomy’, J. Knot Theory Ramifications 11 (2002), 701–723.CrossRefGoogle Scholar
Rasetti, M. and Regge, T., ‘Vortices in He-II, current algebras and quantum knots’, Physica A 80 (1975), 217–233.CrossRefGoogle Scholar
Ricca, R. L. and Nipoti, B., ‘ “Gauss” linking number revisited’, J. Knot Theory Ramifications 20 (2011), 1325–1343.CrossRefGoogle Scholar
Rogers, C. L., ‘ ${L}_{\infty }$ -algebras from multisymplectic geometry’, Lett. Math. Phys. 100 (2012), 29–50.CrossRefGoogle Scholar
Rolfsen, D., Knots and Links (Publish or Perish, Berkeley, CA, 1976).Google Scholar
Ryvkin, L. and Wurzbacher, T., ‘Existence and unicity of co-moments in multisymplectic geometry’, Differential Geom. Appl. 41 (2015), 1–11.CrossRefGoogle Scholar
Ryvkin, L. and Wurzbacher, T., ‘An invitation to multisymplectic geometry’, J. Geom. Phys. 142 (2019), 9–36.CrossRefGoogle Scholar
Ryvkin, L., Wurzbacher, T. and Zambon, M., ‘Conserved quantities on multisymplectic manifolds’, J. Aust. Math. Soc. 108 (2020), 120–144.CrossRefGoogle Scholar
Souriau, J.-M., Structure des systèmes dynamiques (Dunod, Paris, 1970).Google Scholar
Spera, M., ‘A survey on the differential and symplectic geometry of linking numbers’, Milan J. Math. 74 (2006), 139–197.CrossRefGoogle Scholar
Spera, M., ‘Moment map and gauge geometric aspects of the Schrödinger and Pauli equations’, Int. J. Geom. Methods Mod. Phys. 13(4) (2016), 1630004.CrossRefGoogle Scholar
Tavares, J. N., ‘Chen integrals, generalized loops and loop calculus’, Int. J. Mod. Phys. A 9 (1994), 4511–4548.CrossRefGoogle Scholar
Warner, F., Foundations of Differentiable Manifolds and Lie Groups, Graduate Texts in Mathematics, 94 (Springer, Berlin, 1983).CrossRefGoogle Scholar
Zuckerman, G. J., ‘Action principles and global geometry’, Proceedings of the Conference on Mathematical Aspects of String Theory, San Diego, CA, 21 July–2 August 1986 (ed. Yau, S. T.) (World Scientific, Singapore, 1987), 259–284.CrossRefGoogle Scholar