**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.

**Introduction**

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 dim*X>1*and non-autonomous mechanics if*X=*[19, 20, 35].**R**
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 [33], 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.

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