We review the construction of homological evolutionary vector fields on infinite jet spaces and partial differential equations. We describe the applications of this concept in three tightly inter-related domains: the variational Poisson formalism (e.g., for equations of Korteweg-de Vries type), geometry of Liouville-type hyperbolic systems (including the 2D Toda chains), and Euler-Lagrange gauge theories (such as the Yang-Mills theories, gravity, or the Poisson sigma-models). Also, we formulate several open problems.
We present a systematic yet very basic review of the construction of homological evolutionary vector fields on the infinite jet spaces and on several natural classes of partial differential equations such as the gauge models. For a long time, this geometric structure developed in parallel in mathematics and physics. Since mid-70's, it has been being used intensively in theoretical physics: specifically, within the BRST-or BV-technique for the quantization of gauge-invariant systems [7,9,25,29,59]. In mathematics, the concept first stemmed over smooth manifolds in the problem of the homological vector field realizations of Lie algebroids [64], which encompass Lie algebras and vector bundles and are best known in the framework of symplectic and Poisson geometries. Although the notion of Lie algebroids in the geometry of manifolds had been known for a long time [49,54], the proper generalization of this omnipresent structure to the geometry of jet bundles (c.f. [46]) appeared in [37]. In principle, it could be contemporary to the discovery of the BRST-technique ( [9] vs [46]). In the meantime, a steady progress in the geometry of jet spaces [53,65] created the platform for important applications of the future concept. Let us name only a few of them:
• the variational Poisson dynamics for the KdV-type systems [15,23], also [10];
• modern revisions of the theory of Liouville-type hyperbolic equations [15,67], in particular, the 2D Toda chains associated with the Dynkin diagrams [48]; • the cohomological approach to gauge fields [3,4] and (quantum) Poisson theory [10].
In this paper, we recall the jet-bundle geometry that stands behind the structures Q 2 = 0. It allows us to address, from a unified standpoint [51,59], many relevant physical models ranging from the Korteweg-de Vries equation and its Poisson structures [50] to the Yang-Mills theories, gravity, and Poisson sigma-models [1,12].
In section 2 we first consider the evolutionary vector fields
dx σ (ϕ) • ∂/∂q σ whose generating sections ϕ x, [q] belong to the images of matrix linear operators A 1 , . . ., A N in total derivatives (and those derivatives are equal to d dx i = ∂ ∂x i + |σ|≥0 q σ+1 i • ∂/∂q σ ). The three above-mentioned classes of geometries are regular sources of collections of such operators. We impose the Berends-Burgers-van Dam hypothesis [11]: namely, we let the images of the operators be subject to the collective commutation closure. This involutive setup gives rise to the bi-differential structure constants [19] and bi-differential analogs of Christoffel symbols [38]. For any number N > 1 of the operators A 1 , . . ., A N , we reduce the setup at hand to the variational Lie algebroids [37] with one variational anchor. Having endowed the new bundle geometry with the Cartan connection, we reverse the parity of the fibres in it by brute force and then construct the odd, parity-reversing evolutionary vector field Q that encodes the entire initial geometry via the homological equality Q 2 = 0.
In section 3 we describe three natural examples of such geometries: the variational Poisson algebroids [27,37], 2D Toda chains viewed as variational Lie algebroids [37,39], and gauge algebroids [2,38].
First, in section 3.1 we recall the construction of necessary superbundles using the notion of the variational cotangent bundle [47] and derive the representation via Q 2 = 0 for the variational Poisson structures [23]. As usual, the renowned KdV equation [50] offers us the minimally possible nontrivial illustration. The odd vector fields Q are themselves Hamiltonian and we write the corresponding W -charges explicitly.
By exploiting the profound relation between the variational Poisson geometry and 2D Toda chains [15,35,39,67], we obtain the evolutionary fields Q for these models of Liouville type [62]. We emphasize that the differentials Q and their cohomology carry much more information than the ordinary Chevalley-Eilenberg differentials for the semi-simple Lie algebras at hand. Being obtained from the gauge-invariant Yang-Mills equations under a symmetry reduction [48], the 2D Toda chains represent the vast class of nonlinear Euler-Lagrange systems of Liouville type [58,62,67]. This class is very interesting by itself; moreover, the construction of the homological vector fields Q in section 3.2 remains valid uniformly for all such equations.
Finally, in section 3.3 we expose the true geometric nature of the BRST-differentials [9] or the ’longitudinal’ components of the BV-differentials [7] for gauge systems. In agreement with the Second Noether Theorem, the differential operators A 1 , . . ., A N which we study here emerge from the Noether relations between the equations of motion; we explain why the same approach of Q 2 = 0 can successfully grasp more subtle geometries without any further modifications. The gauge parameters (that is, the arguments of the operators A i ), being the parity-reversed neighbours of the ghosts, together with th
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