### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 098, 24 pages      arXiv:1607.01626      https://doi.org/10.3842/SIGMA.2016.098

### Variational Tricomplex, Global Symmetries and Conservation Laws of Gauge Systems

Alexey A. Sharapov
Physics Faculty, Tomsk State University, Lenin ave. 36, Tomsk 634050, Russia

Received July 12, 2016, in final form September 30, 2016; Published online October 03, 2016

Abstract
Using the concept of variational tricomplex endowed with a presymplectic structure, we formulate the general notion of symmetry. We show that each generalized symmetry of a gauge system gives rise to a sequence of conservation laws that are represented by on-shell closed forms of various degrees. This extends the usual Noether's correspondence between global symmetries and conservation laws to the case of lower-degree conservation laws and not necessarily variational equations of motion. Finally, we equip the space of conservation laws of a given degree with a Lie bracket and establish a homomorphism of the resulting Lie algebra to the Lie algebra of global symmetries.

Key words: variational bicomplex; BRST differential; presymplectic structure; lower-degree conservation laws.

pdf (465 kb)   tex (31 kb)

References

1. Alkalaev K.B., Grigoriev M., Frame-like Lagrangians and presymplectic AKSZ-type sigma models, Internat. J. Modern Phys. A 29 (2014), 1450103, 33 pages, arXiv:1312.5296.
2. Anderson I.M., Introduction to the variational bicomplex, in Mathematical Aspects of Classical Field Theory (Seattle, WA, 1991), Contemp. Math., Vol. 132, Amer. Math. Soc., Providence, RI, 1992, 51-73.
3. Anderson I.M., Torre C.G., Asymptotic conservation laws in classical field theory, Phys. Rev. Lett. 77 (1996), 4109-4113, hep-th/9608008.
4. Barnich G., Brandt F., Covariant theory of asymptotic symmetries, conservation laws and central charges, Nuclear Phys. B 633 (2002), 3-82, hep-th/0111246.
5. Barnich G., Brandt F., Henneaux M., Local BRST cohomology in gauge theories, Phys. Rep. 338 (2000), 439-569, hep-th/0002245.
6. Barnich G., Henneaux M., Isomorphisms between the Batalin-Vilkovisky antibracket and the Poisson bracket, J. Math. Phys. 37 (1996), 5273-5296, hep-th/9601124.
7. Bridges T.J., Hydon P.E., Lawson J.K., Multisymplectic structures and the variational bicomplex, Math. Proc. Cambridge Philos. Soc. 148 (2010), 159-178.
8. Bryant R.L., Griffiths P.A., Characteristic cohomology of differential systems. I. General theory, J. Amer. Math. Soc. 8 (1995), 507-596.
9. Cattaneo A.S., Schätz F., Introduction to supergeometry, Rev. Math. Phys. 23 (2011), 669-690, arXiv:1011.3401.
10. Crnković Č., Witten E., Covariant description of canonical formalism in geometrical theories, in Three Hundred Years of Gravitation, Cambridge University Press, Cambridge, 1987, 676-684.
11. Dickey L.A., Soliton equations and Hamiltonian systems, Advanced Series in Mathematical Physics, Vol. 12, World Scientific Publishing Co., Inc., River Edge, NJ, 1991.
12. Grigoriev M., Presymplectic structures and intrinsic Lagrangians, arXiv:1606.07532.
13. Grigoriev M.A., Semikhatov A.M., Tipunin I.Yu., Becchi-Rouet-Stora-Tyutin formalism and zero locus reduction, J. Math. Phys. 42 (2001), 3315-3333, hep-th/0001081.
14. Henneaux M., Knaepen B., Schomblond C., Characteristic cohomology of $p$-form gauge theories, Comm. Math. Phys. 186 (1997), 137-165, hep-th/9606181.
15. Henneaux M., Teitelboim C., Quantization of gauge systems, Princeton University Press, Princeton, NJ, 1992.
16. Kaparulin D.S., Lyakhovich S.L., Sharapov A.A., Rigid symmetries and conservation laws in non-Lagrangian field theory, J. Math. Phys. 51 (2010), 082902, 22 pages, arXiv:1001.0091.
17. Kaparulin D.S., Lyakhovich S.L., Sharapov A.A., Local BRST cohomology in (non-)Lagrangian field theory, J. High Energy Phys. 2011 (2011), no. 9, 006, 34 pages, arXiv:1106.4252.
18. Kazinski P.O., Lyakhovich S.L., Sharapov A.A., Lagrange structure and quantization, J. High Energy Phys. 2005 (2005), no. 7, 076, 42 pages, hep-th/0506093.
19. Khavkine I., Presymplectic current and the inverse problem of the calculus of variations, J. Math. Phys. 54 (2013), 111502, 11 pages, arXiv:1210.0802.
20. Khavkine I., Covariant phase space, constraints, gauge and the Peierls formula, Internat. J. Modern Phys. A 29 (2014), 1430009, 74 pages, arXiv:1402.1282.
21. Kosmann-Schwarzbach Y., The Noether theorems. Invariance and conservation laws in the twentieth century, Sources and Studies in the History of Mathematics and Physical Sciences, Springer, New York, 2011.
22. Lyakhovich S.L., Sharapov A.A., BRST theory without Hamiltonian and Lagrangian, J. High Energy Phys. 2005 (2005), no. 3, 011, 22 pages, hep-th/0411247.
23. Mehta R.A., Supergroupoids, double structures, and equivariant cohomology, Ph.D. Thesis, University of California, Berkeley, 2006, math.DG/0605356.
24. Olver P.J., Applications of Lie groups to differential equations, Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York, 1986.
25. Roytenberg D., On the structure of graded symplectic supermanifolds and Courant algebroids, in Quantization, Poisson Brackets and Beyond (Manchester, 2001), Contemp. Math., Vol. 315, Amer. Math. Soc., Providence, RI, 2002, 169-185, math.SG/0203110.
26. Saunders D.J., The geometry of jet bundles, London Mathematical Society Lecture Note Series, Vol. 142, Cambridge University Press, Cambridge, 1989.
27. Sharapov A.A., Variational tricomplex of a local gauge system, Lagrange structure and weak Poisson bracket, Internat. J. Modern Phys. A 30 (2015), 1550152, 32 pages, arXiv:1506.04652.
28. Sharapov A.A., On presymplectic structures for massless higher-spin fields, Eur. Phys. J. C Part. Fields 76 (2016), 305, 16 pages, arXiv:1602.06393.
29. Torre C.G., Local cohomology in field theory (with applications to the Einstein equations), Lectures given at 2nd Mexican School on Gravitation and Mathematical Physics (December 1-7, 1996, Tlaxcala, Mexico), hep-th/9706092.
30. Tsujishita T., Homological method of computing invariants of systems of differential equations, Differential Geom. Appl. 1 (1991), 3-34.
31. Verbovetsky A., Notes on the horizontal cohomology, in Secondary Calculus and Cohomological Physics (Moscow, 1997), Contemp. Math., Vol. 219, Amer. Math. Soc., Providence, RI, 1998, 211-231, math.DG/9803115.
32. Vinogradov A.M., The ${\mathcal C}$-spectral sequence, Lagrangian formalism, and conservation laws. I. The linear theory, J. Math. Anal. Appl. 100 (1984), 1-40.
33. Vinogradov A.M., The ${\mathcal C}$-spectral sequence, Lagrangian formalism, and conservation laws. II. The nonlinear theory, J. Math. Anal. Appl. 100 (1984), 41-129.
34. Voronov T., Graded manifolds and Drinfeld doubles for Lie bialgebroids, in Quantization, Poisson Brackets and Beyond (Manchester, 2001), Contemp. Math., Vol. 315, Amer. Math. Soc., Providence, RI, 2002, 131-168, math.DG/0105237.
35. Zuckerman G.J., Action principles and global geometry, in Mathematical Aspects of String Theory (San Diego, Calif., 1986), Adv. Ser. Math. Phys., Vol. 1, World Sci. Publishing, Singapore, 1987, 259-284.