Research ArticleNo Access

Higgs fields induced by Yang–Mills type Lagrangians on gauge-natural prolongations of principal bundles

Marcella Palese

http://orcid.org/0000-0001-6383-0325

Department of Mathematics, University of Torino, via C. Alberto 10, Torino I-10123, Italy

E-mail Address: marcella.palese@unito.it

Search for more papers by this author

and

Ekkehart Winterroth

Department of Mathematics, University of Torino, via C. Alberto 10, Torino I-10123, Italy

E-mail Address: ekkehart.winterroth@unito.it

Search for more papers by this author

https://doi.org/10.1142/S021988781950049XCited by:2 (Source: Crossref)

Abstract

We address some new issues concerning spontaneous symmetry breaking. We define classical Higgs fields for gauge-natural invariant Yang–Mills type Lagrangian field theories through the requirement of the existence of canonical covariant gauge-natural conserved quantities. As an illustrative example, we consider the ‘gluon Lagrangian’, i.e. a Yang–Mills Lagrangian on the $(1, 1)$ $(1, 1)$ -order gauge-natural bundle of $S U (3)$ -principal connections, and canonically define a ‘gluon’ classical Higgs field through the split reductive structure induced by the kernel of the associated gauge-natural Jacobi morphism.

Keywords:

AMSC: 55N30, 53Z05, 58A20, 55R10, 58A12, 58E30

References

1. L. Accornero and M. Palese, Higher variations and conservation laws; with applications to a Yang–Mills theory on a Minkowskian background, preprint (2017), arXiv:1710.09100. Google Scholar
2. D. V. Alekseevsky and P. W. Michor , Differential geometry of Cartan connections, Publ. Math. Debrecen 47(3–4) (1995) 349–375. Crossref, Web of Science, Google Scholar
3. S. Aloff and N. R. Wallach , An infinite family of distinct $7$ -manifolds admitting positively curved Riemannian structures, Bull. Amer. Math. Soc. 81(1) (1975) 93–97. Crossref, Web of Science, Google Scholar
4. M. F. Atiyah and R. Bott , The Yang–Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308(1505) (1983) 523–615. Crossref, Web of Science, Google Scholar
5. D. Bashkirov, G. Giachetta, L. Mangiarotti and G. Sardanashvily , Noether’s second theorem for BRST symmetries, J. Math. Phys. 46(5) (2005) 053517. Crossref, Web of Science, Google Scholar
6. D. Bashkirov, G. Giachetta, L. Mangiarotti and G. Sardanashvily , Noether’s second theorem in a general setting: Reducible gauge theories, J. Phys. A 38(23) (2005) 5329–5344. Crossref, Google Scholar
7. J. P. Bourguignon , Yang–Mills theory: The differential geometric side, in Differential Geometry, Lecture Notes Mathematics, Vol. 1263 (Springer, Berlin, 1987), pp. 13–54. Crossref, Google Scholar
8. D. J. Eck , Gauge-natural bundles and generalized gauge theories, Mem. Amer. Math. Soc. 247 (1981) 1–48. Google Scholar
9. L. Fatibene and M. Francaviglia , Natural and Gauge Natural Formalism for Classical Field Theories. A Geometric Perspective Including Spinors and Gauge Theories (Kluwer Academic Publishers, Dordrecht, 2003). Crossref, Google Scholar
10. L. Fatibene, M. Francaviglia and M. Palese , Conservation laws and variational sequences in gauge-natural theories, Math. Proc. Cambridge Philos. Soc. 130(3) (2001) 555–569. Crossref, Web of Science, Google Scholar
11. M. Ferraris, M. Francaviglia, M. Palese and E. Winterroth , Canonical connections in gauge-natural field theories, Int. J. Geom. Methods Mod. Phys. 5(6) (2008) 973–988. Link, Web of Science, Google Scholar
12. M. Ferraris, M. Francaviglia, M. Palese and E. Winterroth , Gauge-natural Noether currents and connection fields, Int. J. Geom. Methods Mod. Phys. 8(1) (2011) 177–185. Link, Web of Science, Google Scholar
13. M. Francaviglia, M. Palese and R. Vitolo , The Hessian and Jacobi morphisms for higher order calculus of variations, Differential Geom. Appl. 22 (2005) 105–120. Crossref, Web of Science, Google Scholar
14. M. Francaviglia, M. Palese and E. Winterroth , Generalized Bianchi identities in gauge-natural field theories and the curvature of variational principles, Rep. Math. Phys. 56(1) (2005) 11–22. Crossref, Web of Science, Google Scholar
15. H. Goldschmidt and S. Sternberg , The Hamilton–Cartan formalism in the calculus of variations, Ann. Inst. Fourier (Grenoble) 23 (1973) 203–267. Crossref, Google Scholar
16. J. Janyška and J. Vondra , Natural principal connections on the principal gauge prolongation of a principal bundle, Rep. Math. Phys. 64(3) (2009) 395–415. Crossref, Web of Science, Google Scholar
17. I. Kolář, P. W. Michor and J. Slovák , Natural Operations in Differential Geometry (Springer-Verlag, NY, 1993). Crossref, Google Scholar
18. Y. Kosmann, Dérivée de Lie de spineurs, C. R. Acad. Sci. Paris Sér. A-B 262 (1966) A289–A292; Dérivée de Lie de spineurs. Applications, C. R. Acad. Sci. Paris Sér. A-B 262 (1966) A394–A397. Google Scholar
19. D. Krupka , Variational sequences on finite order jet spaces, in Proc. Diff. Geom. and Its Appl., Brno, 1989, eds. J. Janyška and D. Krupka (World Scientific, Singapore, 1990), pp. 236–254. Google Scholar
20. E. Noether , Invariante variationsprobleme, Nachr. Ges. Wiss. Gött. Math. Phys. Kl. II (1918) 235–257. Google Scholar
21. M. Palese, O. Rossi, E. Winterroth and J. Musilová , Variational sequences, representation sequences and applications in physics, SIGMA 12 (2016) 45pp. Google Scholar
22. M. Palese and E. Winterroth , Covariant gauge-natural conservation laws, Rep. Math. Phys. 54(3) (2004) 349–364. Crossref, Web of Science, Google Scholar
23. M. Palese and E. Winterroth , Global generalized Bianchi identities for invariant variational problems on gauge-natural bundles, Arch. Math. (Brno) 41(3) (2005) 289–310. Google Scholar
24. M. Palese and E. Winterroth , The relation between the Jacobi morphism and the Hessian in gauge-natural field theories, Theoret. Math. Phys. 152(2) (2007) 1191–1200. Crossref, Web of Science, Google Scholar
25. M. Palese and E. Winterroth , Noether identities in Einstein–Dirac theory and the Lie derivative of spinor fields, in Proc. Differential Geometrics Appllication, Olomouc, 2007 (World Scientific Publ., Hackensack, NJ, 2008), pp. 643–653. Google Scholar
26. M. Palese and E. Winterroth , Lagrangian reductive structures on gauge-natural bundles, Rep. Math. Phys. 62(2) (2008) 229–239. Crossref, Web of Science, Google Scholar
27. M. Palese and E. Winterroth , Invariant variational problems and Cartan connections on gauge-natural bundles, AIP Conf. Proc. 1191 (2009) 160–165. Crossref, Google Scholar
28. M. Palese and E. Winterroth , A variational perspective on classical Higgs fields in gauge-natural theories, Theor. Math. Phys. 168(1) (2011) 1002–1008. Crossref, Web of Science, Google Scholar
29. M. Palese and E. Winterroth , Higgs fields on spinor gauge-natural bundles, J. Phys.: Conf. Ser. 411 (2013) 012025. Crossref, Google Scholar
30. M. Palese and E. Winterroth , Classical Higgs fields on gauge gluon bundles, EPJ Web of Conferences 129 (2016) 00016. Crossref, Google Scholar
31. M. Palese and E. Winterroth , Noether theorems and reality of motions, in Proc. of The Fourteenth Marcel Grossmann Meeting, Rome, 2015 (World Scientific, 2018), pp. 3331–3336. Google Scholar
32. M. Palese and E. Winterroth , Topological obstructions in Lagrangian field theories, with an application to 3D Chern-Simons gauge theory, J. Math. Phys. 58 (2017) 023502; https://doi.org/10.1063/1.4975336. Crossref, Web of Science, Google Scholar
33. M. Panák , Natural operators on the bundle of Cartan connections, in Differential Geometry and Applications (Masaryk University, Brno, 1999), pp. 285–292. Google Scholar
34. R. D. Peccei, Exact and broken symmetries in particle physics, preprint (2000), arXiv:hep-ph/0002225. Google Scholar
35. G. Sardanashvily , Geometry of classical Higgs fields, Int. J. Geom. Methods Mod. Phys. 3(1) (2006) 139–148. Link, Web of Science, Google Scholar
36. G. Sardanashvily , Supermetrics on supermanifolds, Int. J. Geom. Methods Mod. Phys. 5(2) (2008) 271–286. Link, Web of Science, Google Scholar
37. G. A. Sardanashvily , Classical Higgs fields, Theoret. Math. Phys. 181(3) (2014) 1599–1611. Crossref, Web of Science, Google Scholar
38. D. J. Saunders , The Geometry of Jet Bundles (Cambridge University Press, Cambridge, 1989). Crossref, Google Scholar
39. G. ’t Hooft , Topology of the gauge condition and new confinement phases in non-abelian gauge theories, Nucl. Phys. B 190 (1981) 455–478. Crossref, Web of Science, Google Scholar
40. R. W. Sharpe , Differential Geometry (Springer, New York, 1997). Google Scholar
41. J. Vondra, Natural prolongation of principal connections, PhD thesis, Masaryk University, Brno (2010). Google Scholar