Abstract: Noether theorems in a general setting

This is an Abstract of my invited lecture: Noether theorems in a general setting. Reducible graded Lagrangians, in the Conference: Geometry of Jets and Fields (10-16 May 2015, Bedlewo, Poland).

Noether theorems are formulated in a general case of reducible degenerate Grassmann-graded Lagrangian theory of even and odd variables on graded bundles. A problem is that any Euler-Lagrange operator satisfies Noether identities, which therefore must be separated into the trivial and non-trivial ones. These Noether identities can obey first-stage Noether identities, which in turn are subject to the second-stage ones, and so on. Thus, there is a hierarchy of non-trivial Noether and higher-stage Noether identities. This hierarchy is described in homology terms. If a certain homology regularity conditions holds, one can associate to a reducible degenerate Lagrangian the exact Koszul-Tate chain complex possessing the boundary operator whose nilpotentness is equivalent to all complete non-trivial Noether and higher-stage Noether identities. Since this complex is necessarily Grassmann-graded, Lagrangian theory on graded bundles is considered from the beginning, and is formulated in terms of the Grassmann-graded variational bicomplex. Its cohomology defines a first variational formula whose straightforward corollary is the first Noether theorem. Second Noether theorems associate to the above mentioned  Koszul-Tate complex a certain cochain sequence whose ascent operator consists of the gauge and higher-order gauge symmetries of a Lagrangian system. If gauge symmetries are algebraically closed, this ascent operator is prolonged to the ilpotent BRST operator which brings the gauge cochain sequence into a BRST complex, and thus provides a BRST extension of an original Lagrangian system. [G.Sardanashvily,  arXiv: 1411.2910]

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