Variational Lie algebroids and homological evolutionary vector fields

Abstract

We define Lie algebroids over infinite jet spaces and establish their equivalent representation through homological evolutionary vector fields. Keywords: Lie algebroid, BRST-differential, Poisson structure, integrable systems, string theory.Introduction. The construction of Lie algebroids over smooth manifolds is important in differential geometry (particularly, in Poisson geometry) and appears in various models of mathematical physics (e.g., in Poisson sigmamodels). We extend the classical definition of Lie algebroids over smooth manifolds [1] to the construction of variational Lie algebroids over infinite jet spaces. We define these structures in a standard way via vector bundles and also through homological vector fields Q 2 = 0 on infinite jet super-bundles, then proving the equivalence. Our generalization of the classical construction manifestly respects the geometry which appears under mappings between smooth manifolds. For this reason, the variational picture, which we develop here, more fully grasps the geometry of strings in space-time [2].First, we very briefly recall the definition of classical Lie algebroids; we refer to [1] for more details (see also [3] or the surveys [4] and references therein). The standard examples of Lie algebroids over usual manifolds are the tangent bundle or the Poisson algebroid structure of the cotangent bundle

The geometry of variations in Batalin–Vilkovisky formalism

“…Moreover, the no-derivatives reduction sometimes allows one to jump at conclusions which are correct; an integration by parts over the base manifold M n is restored -whenever possible-at the end of the day. Still this oversimplification is potentially dangerous because variational calculus of integral functionals conceptually exceeds any classical differential geometry on the fibres (see [33] for discussion and [28,34]). In the variational setup, the objects and their properties become geometrically different from their analogues on usual manifolds even if the terminology is kept unchanged.…”

The Jacobi identity for graded-commutative variational Schouten bracket revisited

“…

This short note contains an explicit proof of the Jacobi identity for variational Schouten bracket in Z 2 -graded commutative setup; an extension of the reasoning and assertion to the noncommutative geometry of cyclic words (see [1]) is immediate, still making the proof longer. We emphasize that for the reasoning to be rigorous, it must refer to the product bundle geometry of iterated variations (see [2]); on the other hand, no ad hoc regularizations occur anywhere in this theory.1 In fact, all these BV-, Poisson, or IST models are examples of variational Lie algebroids [4] and their encoding by Q 2 = 0. The construction of gauge automorphisms for the Q-cohomology determines the next generation of such structures, with new deformation quantization parameters beyond the Planck constant.2 Let all functionals that take field configurations to number be integral in this note; formal (sums of) products of functionals such as exp i S are dealt with by using the Leibniz rule, see [2, § 2.5].3 It is readily seen from the proof of theorem below and from example on p. i that composite-structure objects such as brackets of functionals retain a kind of memory of the way how they were produced; in effect, variational derivatives detect the traces of original objects' individual geometries, whence a variation within one of them does not mar any of the others.

…”

“…1 In fact, all these BV-, Poisson, or IST models are examples of variational Lie algebroids [4] and their encoding by Q 2 = 0. The construction of gauge automorphisms for the Q-cohomology determines the next generation of such structures, with new deformation quantization parameters beyond the Planck constant.…”

Introduction to graded geometry

“…The proof is exactly the same in the Z-graded case, since there are no local coordinates of even non-zero degree on G E [1] (see e.g. [8]). …”