| Title:
|
Locally variational invariant field equations and global currents: Chern-Simons theories (English) |
| Author:
|
Francaviglia, M. |
| Author:
|
Palese, M. |
| Author:
|
Winterroth, E. |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
|
1804-1388 |
| Volume:
|
20 |
| Issue:
|
1 |
| Year:
|
2012 |
| Pages:
|
13-22 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
local variational problem |
| Keyword:
|
global current |
| Keyword:
|
Chern-Simons theory |
| MSC:
|
55N30 |
| MSC:
|
55R10 |
| MSC:
|
58A12 |
| MSC:
|
58A20 |
| MSC:
|
58E30 |
| MSC:
|
70S10 |
| idZBL:
|
Zbl 06202715 |
| idMR:
|
MR3001628 |
| . |
| Date available:
|
2012-11-27T16:27:16Z |
| Last updated:
|
2014-01-27 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143077 |
| . |
| Related article:
|
http://dml.cz/handle/10338.dmlcz/143590 |
| . |
| Reference:
|
[1] Allemandi, G., Francaviglia, M., Raiteri, M.: Covariant charges in Chern-Simons $AdS_3$ gravity.Classical Quant. Grav., 20, 3, 2003, 483-506 Zbl 1023.83018, MR 1957170, 10.1088/0264-9381/20/3/307 |
| Reference:
|
[2] Anderson, I.M.: Natural variational principles on Riemannian manifolds.Ann. of Math. (2), 120, 2, 1984, 329-370 Zbl 0565.58019, MR 0763910, 10.2307/2006945 |
| Reference:
|
[3] Bañados, M., Teitelboim, C., Zanelli, J.: Black hole in three-dimensional spacetime.Phys. Rev. Lett., 69, 13, 1992, 1849-1851 Zbl 0968.83514, MR 1181663, 10.1103/PhysRevLett.69.1849 |
| Reference:
|
[4] Borowiec, A., Ferraris, M., Francaviglia, M.: A covariant Formalism for Chern-Simons Gravity.J. Phys. A., 36, 2003, 2589-2598 Zbl 1027.83012, MR 1967520, 10.1088/0305-4470/36/10/318 |
| Reference:
|
[5] Borowiec, A., Ferraris, M., Francaviglia, M., Palese, M.: Conservation laws for non-global Lagrangians.Univ. Iagel. Acta Math., 41, 2003, 319-331 Zbl 1060.70034, MR 2084774 |
| Reference:
|
[6] Brajerčík, J., Krupka, D.: Variational principles for locally variational forms.J. Math. Phys., 46, 5, 2005, 052903, 15 pp. Zbl 1110.58011, MR 2143003, 10.1063/1.1901323 |
| Reference:
|
[7] Chamseddine, A.H.: Topological gravity and supergravity in various dimensions.Nucl. Phys. B, 346, 1, 1990, 213-234 MR 1076761, 10.1016/0550-3213(90)90245-9 |
| Reference:
|
[8] Chern, S.S., Simons, J.: Some cohomology classes in principal fiber bundles and their application to riemannian geometry.Proc. Nat. Acad. Sci. U.S.A., 68, 4, 1971, 791-794 Zbl 0209.25401, MR 0279732, 10.1073/pnas.68.4.791 |
| Reference:
|
[9] Chern, S.S., Simons, J.: Characteristic forms and geometric invariants.Ann. Math., 99, 1974, 48-69 Zbl 0283.53036, MR 0353327, 10.2307/1971013 |
| Reference:
|
[10] Deser, S., Jackiw, R., Templeton, S.: Topologically massive gauge theories.Ann. Physics, 140, 2, 1982, 372-411 MR 0665601, 10.1016/0003-4916(82)90164-6 |
| Reference:
|
[11] Eck, D.J.: Gauge-natural bundles and generalized gauge theories.Mem. Amer. Math. Soc., 247, 1981, 1-48 Zbl 0493.53052, MR 0632164 |
| Reference:
|
[12] Fatibene, L., Ferraris, M., Francaviglia, M.: Augmented variational principles and relative conservation laws in classical field theory.Int. J. Geom. Methods Mod. Phys., 2, 3, 2005, 373-392 Zbl 1133.70335, MR 2152166, 10.1142/S0219887805000557 |
| Reference:
|
[13] Ferraris, M., Francaviglia, M., Palese, M., Winterroth, E.: Canonical connections in gauge-natural field theories.Int. J. Geom. Methods Mod. Phys., 5, 6, 2008, 973-988 Zbl 1175.58006, MR 2453935, 10.1142/S0219887808003144 |
| Reference:
|
[14] Ferraris, M., Francaviglia, M., Palese, M., Winterroth, E.: Gauge-natural Noether currents and connection fields.Int. J. Geom. Methods Mod. Phys., 8, 1, 2011, 177-185 Zbl 1215.58005, MR 2782884, 10.1142/S0219887811005075 |
| Reference:
|
[15] Ferraris, M., Francaviglia, M., Raiteri, M.: Conserved Quantities from the Equations of Motion (with applications to natural and gauge natural theories of gravitation).Classical Quant. Grav., 20, 2003, 4043-4066 MR 2017333, 10.1088/0264-9381/20/18/312 |
| Reference:
|
[16] Ferraris, M., Palese, M., Winterroth, E.: Local variational problems and conservation laws.Diff. Geom. Appl, 29, 2011, S80-S85 Zbl 1233.58002, MR 2832003 |
| Reference:
|
[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 Zbl 1109.58005, MR 2268969 |
| Reference:
|
[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). Zbl 1271.58008, MR 2998326 |
| Reference:
|
[19] Francaviglia, M., Palese, M., Vitolo, R.: Symmetries in finite order variational sequences.Czech. Math. J., 52, 1, 2002, 197-213 Zbl 1006.58014, MR 1885465, 10.1023/A:1021735824163 |
| Reference:
|
[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 Zbl 0813.58014, MR 1062026 |
| Reference:
|
[21] Krupka, D., Sedenkova, J.: Variational sequences and Lepage forms.Differential geometry and its applications, Matfyzpress, Prague, 2005, 617-627 Zbl 1115.35349, MR 2271823 |
| Reference:
|
[22] Krupkova, O.: Lepage forms in the calculus of variations.Variations, geometry and physics, Nova Sci. Publ., New York, 2009, 27-55 Zbl 1208.58019, MR 2523431 |
| Reference:
|
[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 |
| Reference:
|
[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 Zbl 1208.58001, MR 2523430 |
| Reference:
|
[25] Noether, E.: Invariante Variationsprobleme.Nachr. Ges. Wiss. Gött., Math. Phys. Kl., II, 1918, 235-257 |
| Reference:
|
[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 Zbl 1112.58005, MR 2188385 |
| Reference:
|
[27] Palese, M., Winterroth, E.: Covariant gauge-natural conservation laws.Rep. Math. Phys., 54, 3, 2004, 349-364 Zbl 1066.58009, MR 2115744, 10.1016/S0034-4877(04)80024-7 |
| Reference:
|
[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 MR 2429287 |
| Reference:
|
[29] Palese, M., Winterroth, E.: Variational Lie derivative and cohomology classes.AIP Conf. Proc., 1360, 2011, 106-112 Zbl 1276.70012 |
| Reference:
|
[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 Zbl 1265.58008, MR 2876943 |
| Reference:
|
[31] Sardanashvily, G.: Noether conservation laws issue from the gauge invariance of an Euler-Lagrange operator, but not a Lagrangian.arXiv:math-ph/0302012 |
| Reference:
|
[32] Witten, E.: $2+1$-dimensional gravity as an exactly soluble system.Nucl. Phys., B 311, 1, 1988, 46-78, E. Witten: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121 (1989) 351-399 MR 0974271, 10.1016/0550-3213(88)90143-5 |
| . |
