Locally variational invariant field equations and global currents: Chern-Simons theories

Mauro Francaviglia; M. Palese; E. Winterroth

Communications in Mathematics (2012)

  • Volume: 20, Issue: 1, page 13-22
  • ISSN: 1804-1388

Abstract

top
We introduce the concept of conserved current variationally associated with locally variational invariant field equations. The invariance of the variation of the corresponding local presentation is a sufficient condition for the current beeing variationally equivalent to a global one. The case of a Chern-Simons theory is worked out and a global current is variationally associated with a Chern-Simons local Lagrangian.

How to cite

top

Francaviglia, Mauro, Palese, M., and Winterroth, E.. "Locally variational invariant field equations and global currents: Chern-Simons theories." Communications in Mathematics 20.1 (2012): 13-22. <http://eudml.org/doc/246897>.

@article{Francaviglia2012,
abstract = {We introduce the concept of conserved current variationally associated with locally variational invariant field equations. The invariance of the variation of the corresponding local presentation is a sufficient condition for the current beeing variationally equivalent to a global one. The case of a Chern-Simons theory is worked out and a global current is variationally associated with a Chern-Simons local Lagrangian.},
author = {Francaviglia, Mauro, Palese, M., Winterroth, E.},
journal = {Communications in Mathematics},
keywords = {local variational problem; global current; Chern-Simons theory; local variational problem; global current; Chern-Simons theory},
language = {eng},
number = {1},
pages = {13-22},
publisher = {University of Ostrava},
title = {Locally variational invariant field equations and global currents: Chern-Simons theories},
url = {http://eudml.org/doc/246897},
volume = {20},
year = {2012},
}

TY - JOUR
AU - Francaviglia, Mauro
AU - Palese, M.
AU - Winterroth, E.
TI - Locally variational invariant field equations and global currents: Chern-Simons theories
JO - Communications in Mathematics
PY - 2012
PB - University of Ostrava
VL - 20
IS - 1
SP - 13
EP - 22
AB - We introduce the concept of conserved current variationally associated with locally variational invariant field equations. The invariance of the variation of the corresponding local presentation is a sufficient condition for the current beeing variationally equivalent to a global one. The case of a Chern-Simons theory is worked out and a global current is variationally associated with a Chern-Simons local Lagrangian.
LA - eng
KW - local variational problem; global current; Chern-Simons theory; local variational problem; global current; Chern-Simons theory
UR - http://eudml.org/doc/246897
ER -

References

top
  1. Allemandi, G., Francaviglia, M., Raiteri, M., 10.1088/0264-9381/20/3/307, Classical Quant. Grav., 20, 3, 2003, 483-506 (2003) Zbl1023.83018MR1957170DOI10.1088/0264-9381/20/3/307
  2. Anderson, I.M., 10.2307/2006945, Ann. of Math. (2), 120, 2, 1984, 329-370 (1984) Zbl0565.58019MR0763910DOI10.2307/2006945
  3. Bañados, M., Teitelboim, C., Zanelli, J., 10.1103/PhysRevLett.69.1849, Phys. Rev. Lett., 69, 13, 1992, 1849-1851 (1992) Zbl0968.83514MR1181663DOI10.1103/PhysRevLett.69.1849
  4. Borowiec, A., Ferraris, M., Francaviglia, M., 10.1088/0305-4470/36/10/318, J. Phys. A., 36, 2003, 2589-2598 (2003) Zbl1027.83012MR1967520DOI10.1088/0305-4470/36/10/318
  5. Borowiec, A., Ferraris, M., Francaviglia, M., Palese, M., Conservation laws for non-global Lagrangians, Univ. Iagel. Acta Math., 41, 2003, 319-331 (2003) Zbl1060.70034MR2084774
  6. Brajerčík, J., Krupka, D., 10.1063/1.1901323, J. Math. Phys., 46, 5, 2005, 052903, 15 pp. (2005) Zbl1110.58011MR2143003DOI10.1063/1.1901323
  7. Chamseddine, A.H., 10.1016/0550-3213(90)90245-9, Nucl. Phys. B, 346, 1, 1990, 213-234 (1990) MR1076761DOI10.1016/0550-3213(90)90245-9
  8. Chern, S.S., Simons, J., 10.1073/pnas.68.4.791, Proc. Nat. Acad. Sci. U.S.A., 68, 4, 1971, 791-794 (1971) Zbl0209.25401MR0279732DOI10.1073/pnas.68.4.791
  9. Chern, S.S., Simons, J., 10.2307/1971013, Ann. Math., 99, 1974, 48-69 (1974) Zbl0283.53036MR0353327DOI10.2307/1971013
  10. Deser, S., Jackiw, R., Templeton, S., 10.1016/0003-4916(82)90164-6, Ann. Physics, 140, 2, 1982, 372-411 (1982) MR0665601DOI10.1016/0003-4916(82)90164-6
  11. Eck, D.J., Gauge-natural bundles and generalized gauge theories, Mem. Amer. Math. Soc., 247, 1981, 1-48 (1981) Zbl0493.53052MR0632164
  12. Fatibene, L., Ferraris, M., Francaviglia, M., 10.1142/S0219887805000557, Int. J. Geom. Methods Mod. Phys., 2, 3, 2005, 373-392 (2005) Zbl1133.70335MR2152166DOI10.1142/S0219887805000557
  13. Ferraris, M., Francaviglia, M., Palese, M., Winterroth, E., 10.1142/S0219887808003144, Int. J. Geom. Methods Mod. Phys., 5, 6, 2008, 973-988 (2008) Zbl1175.58006MR2453935DOI10.1142/S0219887808003144
  14. Ferraris, M., Francaviglia, M., Palese, M., Winterroth, E., 10.1142/S0219887811005075, Int. J. Geom. Methods Mod. Phys., 8, 1, 2011, 177-185 (2011) Zbl1215.58005MR2782884DOI10.1142/S0219887811005075
  15. Ferraris, M., Francaviglia, M., Raiteri, M., 10.1088/0264-9381/20/18/312, Classical Quant. Grav., 20, 2003, 4043-4066 (2003) MR2017333DOI10.1088/0264-9381/20/18/312
  16. Ferraris, M., Palese, M., Winterroth, E., Local variational problems and conservation laws, Diff. Geom. Appl, 29, 2011, S80-S85 (2011) Zbl1233.58002MR2832003
  17. Francaviglia, M., Palese, M., Winterroth, E., Second variational derivative of gauge-natural invariant Lagrangians and conservation laws, 2005, Differential geometry and its applications, Matfyzpress, Prague, 591-604 (2005) Zbl1109.58005MR2268969
  18. Francaviglia, M., Palese, M., Winterroth, E., Variationally equivalent problems and variations of Noether currents, Int. J. Geom. Methods Mod. Phys., 10, 1, 2013, 1220024 (10 pages). (2013) Zbl1271.58008MR2998326
  19. Francaviglia, M., Palese, M., Vitolo, R., 10.1023/A:1021735824163, Czech. Math. J., 52, 1, 2002, 197-213 (2002) Zbl1006.58014MR1885465DOI10.1023/A:1021735824163
  20. Krupka, D., Variational Sequences on Finite Order Jet Spaces, Proc. Diff. Geom. Appl., 1990, 236-254, J. Janyška, D. Krupka eds., World Sci., Singapore (1990) Zbl0813.58014MR1062026
  21. Krupka, D., Sedenkova, J., Variational sequences and Lepage forms, Differential geometry and its applications, Matfyzpress, Prague, 2005, 617-627 (2005) Zbl1115.35349MR2271823
  22. Krupkova, O., Lepage forms in the calculus of variations, Variations, geometry and physics, Nova Sci. Publ., New York, 2009, 27-55 (2009) Zbl1208.58019MR2523431
  23. Lepage, Th.H.J., Sur les champ geodesiques du Calcul de Variations, I, II, Bull. Acad. Roy. Belg., Cl. Sci., 22, 1936, 716-729, 1036--1046 (1936) 
  24. Musilová, J., Lenc, M., Lepage forms in variational theories from Lepage's idea to the variational sequence, Variations, geometry and physics, Nova Sci. Publ., New York, 2009, 3-26 (2009) Zbl1208.58001MR2523430
  25. Noether, E., Invariante Variationsprobleme, Nachr. Ges. Wiss. Gött., Math. Phys. Kl., II, 1918, 235-257 (1918) 
  26. Palese, M., Winterroth, E., Global Generalized Bianchi Identities for Invariant Variational Problems on Gauge-natural Bundles, Arch. Math. (Brno), 41, 3, 2005, 289-310 (2005) Zbl1112.58005MR2188385
  27. Palese, M., Winterroth, E., 10.1016/S0034-4877(04)80024-7, Rep. Math. Phys., 54, 3, 2004, 349-364 (2004) Zbl1066.58009MR2115744DOI10.1016/S0034-4877(04)80024-7
  28. Palese, M., Winterroth, E., On the relation between the Jacobi morphism and the Hessian in gauge-natural field theories, Teoret. Mat. Fiz., 152, 2, 2007, 377-389, transl. Theoret. and Math. Phys. 152 (2007) 1191--1200 (2007) MR2429287
  29. Palese, M., Winterroth, E., Variational Lie derivative and cohomology classes, AIP Conf. Proc., 1360, 2011, 106-112 (2011) Zbl1276.70012
  30. Palese, M., Winterroth, E., Garrone, E., Second variational derivative of local variational problems and conservation laws, Arch. Math. (Brno), 47, 5, 2011, 395-403 (2011) Zbl1265.58008MR2876943
  31. Sardanashvily, G., Noether conservation laws issue from the gauge invariance of an Euler-Lagrange operator, but not a Lagrangian, arXiv:math-ph/0302012 
  32. Witten, E., 10.1016/0550-3213(88)90143-5, Nucl. Phys., B 311, 1, 1988, 46-78, E. Witten: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121 (1989) 351-399 (1988) MR0974271DOI10.1016/0550-3213(88)90143-5

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.