Arthemy V. Kiselev

The geometry of variations in Batalin-Vilkovisky formalism

We explain why no sources of divergence are built into the Batalin–Vilkovisky (BV) Laplacian, whence there is no need to postulate any ad hoc conventions such as δ(0) = 0 and log δ(0) = 0 within BV-approach to quantisation of gauge systems. Remarkably, the geometry of iterated variations does not refer at all to the construction of Diracs δ-function as a limit of smooth kernels. We illustrate the reasoning by re-deriving –but not just formally postulating – the standard properties of BV-Laplacian and Schouten bracket and by verifying their basic inter-relations (e.g., cohomology preservation by gauge symmetries of the quantum master-equation).

Variational Lie algebroids and homological evolutionary vector fields

We define Lie algebroids over infinite jet spaces and obtain their equivalent representation in terms of homological evolutionary vector fields.

On the Geometry of Liouville Equation: Symmetries, Conservation Laws, and Bäcklund Transformations

Various geometrical structures related to the Liouville equation are considered. Properties of the symmetry algebra are discussed and local conserved currents are constructed. Bäcklund transformations for the Liouville equation are integrated and nontrivial generalizations of the latter are studied as well as the structures on them.

The calculus of multivectors on noncommutative jet spaces

The Leibniz rule for derivations is invariant under cyclic permutations of co-multiples within the arguments of derivations. We explore the implications of this principle: in effect, we construct a class of noncommutative bundles in which the sheaves of algebras of walks along a tessellated affine manifold form the base, whereas the fibres are free associative algebras or, at a later stage, such algebras quotients over the linear relation of equivalence under cyclic shifts. The calculus of variations is developed on the infinite jet spaces over such noncommutative bundles. In the frames of such field-theoretic extension of the Kontsevich formal noncommutative symplectic (super)geometry, we prove the main properties of the Batalin-Vilkovisky Laplacian and Schouten bracket. We show as by-product that the structures which arise in the classical variational Poisson geometry of infinite-dimensional integrable systems do actually not refer to the graded commutativity assumption

The Kontsevich tetrahedral flow revisited

We prove that the Kontsevich tetrahedral flow P=Qa:b(P), the right-hand side of which is a linear combination of two differential monomials of degree four in a bi-vector P on an affine real Poisson manifold Nn, does infinitesimally preserve the space of Poisson bi-vectors on Nn if and only if the two monomials in Qa:b(P) are balanced by the ratio a:b=1:6. The proof is explicit; it is written in the language of Kontsevich graphs.

Non-Abelian Lie algebroids over jet spaces

We associate Hamiltonian homological evolutionary vector fields – which are the non-Abelian variational Lie algebroids’ differentials – with Lie algebra-valued zero-curvature representations for partial differential equations.

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 ℤ2-graded commutative setup; an extension of the reasoning and assertion to the noncommutative geometry of cyclic words (see [1]) is immediate. The reasoning refers to the product bundle geometry of iterated variations (see [2]); no ad hoc regularizations occur anywhere in this theory.

On the variational noncommutative Poisson geometry

We outline the notions and concepts of the calculus of variational multivectors within the Poisson formalism over the spaces of infinite jets of mappings from commutative (non)graded smooth manifolds to the factors of noncommutative associative algebras over the equivalence under cyclic permutations of the letters in the associative words. We state the basic properties of the variational Schouten bracket and derive an interesting criterion for (non)commutative differential operators to be Hamiltonian (and thus determine the (non)commutative Poisson structures). We place the noncommutative jet-bundle construction at hand in the context of the quantum string theory.

On Conservation Laws for the Toda Equations

A description of the two-dimensional Toda equations Noether symmetries, assigned to conservation laws for the latter equations, is given. A continuum of recursion operators, both local and nonlocal, is obtained for the Toda equations symmetry algebras.

On the Kontsevich ★-product associativity mechanism

The deformation quantization by Kontsevich is a way to construct an associative noncommutative star-product (star = times + hbar {{ ,} _{rm P}} + overline o left( hbar right)) in the algebra of formal power series in h on a given finite-dimensional affine Poisson manifold: here × is the usual multiplication, {,} P ≠ 0 is the Poisson bracket, and h is the deformation parameter. The product ★ is assembled at all powers h k ≥ 0 via summation over a certain set of weighted graphs with k + 2 vertices; for each k > 0, every such graph connects the two co-multiples of ★ using k copies of {,} P . Cattaneo and Felder interpreted these topological portraits as genuine Feynman diagrams in the Ikeda–Izawa model for quantum gravity. By expanding the star-product up to (bar oleft( {{hbar ^3}} right)), i.e., with respect to graphs with at most five vertices but possibly containing loops, we illustrate the mechanism Assoc = ≦ (Poisson) that converts the Jacobi identity for the bracket {,} P into the associativity of ★.