G.Sardanashvily, Graded Lagrangian formalism, International Journal of Geometric Methods in Modern Physics 10 (2013) N5 1350016 #
Abstract. Graded Lagrangian formalism in terms of a Grassmann-graded variational bicomplex on graded manifolds is developed in a very general setting. This formalism provides the comprehensive description of reducible degenerate Lagrangian systems, characterized by hierarchies of non-trivial higher-order Noether identities and gauge symmetries. This is a general case of classical field theory and Lagrangian non-relativistic mechanics.
Conventional Lagrangian formalism on fibre bundles Y->X over a smooth manifold X is formulated in algebraic terms of a variational bicomplex of exterior forms on jet manifolds of sections of Y->X [2, 9, 16, 17, 19, 30, 36, 37]. The cohomology of this bicomplex provides the global first variational formula for Lagrangians and Euler – Lagrange operators, without appealing to the calculus of variations. For instance, this is the case of classical field theory if dimX>1 and non-autonomous mechanics if X=R [19, 20, 35].
However, this formalism is not sufficient in order to describe reducible degenerate Lagrangian systems whose degeneracy is characterized by a hierarchy of higher order Noether identities. They constitute the Kozul--Tate chain complex whose cycles are Grassmann-graded elements of certain graded manifolds [7, 8, 19]. Moreover, many field models also deal with Grassmann-graded fields, e.g., fermion fields, antifields and ghosts [19, 21, 35].
These facts motivate us to develop graded Lagrangian formalism of even and odd variables [8, 17, 19, 34].
Different geometric models of odd variables are described either on graded manifolds or supermanifolds. Both graded manifolds and supermanifolds are phrased in terms of sheaves of graded commutative algebras [5, 19]. However, graded manifolds are characterized by sheaves on smooth manifolds, while supermanifolds are constructed by gluing of sheaves on supervector spaces. Treating odd variables on a smooth manifold X, we follow the Serre – Swan theorem generalized to graded manifolds (Theorem 7). It states that, if a graded commutative C(X)-ring is generated by a projective C(X)-module of finite rank, it is isomorphic to a ring of graded functions on a graded manifold whose body is X. In accordance with this theorem, we describe odd variables in terms of graded manifolds [8, 17, 19, 34].
We consider a generic Lagrangian theory of even and odd variables on an n-dimensional smooth real manifold X. It is phrased in terms of the Grassmann-graded variational bicomplex (28) [4, 7, 8, 17, 19, 34]. Graded Lagrangians L and Euler – Lagrange operators are defined as elements of this bicomplex. Cohomology of the Grassmann-graded variational bicomplex (28) (Theorems 13 - 14) defines a class of variationally trivial graded Lagrangians (Theorem 15) and results in the global decomposition (33) of dL (Theorem 16), the first variational formula (37) and the first Noether Theorem 20.
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 obey first-stage Noether identities, which in turn are subject to the second-stage ones, and so on. Thus, there is a hierarchy of higher-stage Noether identities. In accordance with general analysis of Noether identities of differential operators , if certain conditions hold, one can associate to a graded Lagrangian system the exact antifield Koszul – Tate complex (62) possessing the boundary operator (60) whose nilpotentness is equivalent to all non-trivial Noether and higher-stage Noether identities [7, 8, 18].
It should be noted that the notion of higher-stage Noether identities has come from that of reducible constraints. The Koszul – Tate complex of Noether identities has been invented similarly to that of constraints under the condition that Noether identities are locally separated into independent and dependent ones [4, 13]. This condition is relevant for constraints, defined by a finite set of functions which the inverse mapping theorem is applied to. However, Noether identities unlike constraints are differential equations. They are given by an infinite set of functions on a Frechet manifold of infinite order jets where the inverse mapping theorem fails to be valid. Therefore, the regularity condition for the Koszul – Tate complex of constraints is replaced with homology regularity Condition
27 in order to construct the
Koszul – Tate complex (62) of Noether identities.
The second Noether theorems (Theorems 32 – 34) is formulated in homology terms, and it associates to this Koszul – Tate complex the cochain sequence of ghosts (71) with the ascent operator (72) whose components are non-trivial gauge and higher-stage gauge symmetries of Lagrangian theory.