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.
from the usual Leibniz rule for the derivations Q ξ acting on the bracket [[η, ω]]. Later on, we focus on the noncommutative variational Poisson bivectors π such that [[π, π]] = 0 or, equivalently, 1 2 Q π 2 = 0. We derive an interesting criterion for a (non)commutative linear operator A to be Hamiltonian (resp., for the bivector π = 1 2 b, A(b) to satisfy [[π, π]] = 0).
- Noncommutative jets. Let M n be a smooth oriented R-manifold and x ∈ M n be a point. Let A be a noncommutative associative algebra of dimension m; denote by a a basis in A. Consider the maps M n → A and construct the infinite jet space J ∞ (M n → A) =: J ∞ (π nC ), see [5]. Denote by (a•) and (•a) the operators of left-and right-multiplication by a word a ∈ A that is always read from left to right. The total derivative w.r.t. ϕ , the value p, ϕ is well-defined irrespective of the normalization in p.)
Let A : p → ϕ be a Noether noncommutative linear matrix operator in total derivatives. The adjoint operator A † is defined from the equality p 1 , A(p 2 ) = p 2 , A † (p 1 ) , in which we first integrate by parts and then transport the even covector p 2 around the circle. 2. Noncommutative multivectors. The covectors p x, [a] were even. We reverse their parity, Π : p → b x, [a] , preserving the topology but endowing the space of differential functions f that depend on b with a new ring structure: now, each f is polynomial in finitely many derivatives of b. Next, we consider the noncommutative variational cotangent superspace [3,4] and [5]. In effect, we declare that
). The value of the k-vector ξ on k arbitrary covectors p i is ξ(p 1 , . . . , p k ) = s∈S k (-) |s| p s(1) , A(p s(2) , . . . , p s(k) ) /k!; we emphasize that we shuffle the arguments but never swap their slots, which are built into the cyclic word ξ.
- Noncommutative Schouten bracket. The concatenation × of densities of two multivectors provides an ill-defined product in Hn Π π nC π , where the genuine multiplication is the noncommutative antibracket. We fix the Dirac ordering δa ∧ δb over each x in J ∞ Π π nC π → M n ; note that δa is a covector and δb is an odd vector so that their coupling equals +1•dx. The noncommutative variational Schouten bracket of two multivectors ξ and η is
) all the derivatives are thrown off the variations δa and δb via the integration by parts, then (2) the letters a σ , b τ , δa, and δb, which are thread on the two circles δξ and δη, spin along these rosaries so that the variations δa and δb match in all possible combinations, and finally, (3) the variations δa and δb detach from the circles and couple, while the loose ends of the two remaining open strings join and form the new circle.
The Schouten bracket is shifted-graded skew-symmetric: if ξ is a k-vector and η an
] a bi-derivation? In the notation of §3, define the evolutionary vector field
, see [4,5].
Freeze the coordinates, fix the volume form on M n , and choose any representatives ξ and η of the cohomology classes in Hn Π π nC π . The derivation Q ξ acts on the word η by the graded Leibniz rule, inserting d |σ| dx σ Q ξ (q) instead of each letter q σ (here q is a or b). Next, promote the letter q to the zero-or one-vector q • dx ∈ Λn Π π nC π and use the Leibniz rule again to expand the entries [[ξ, q]] = (ξ)
← -
becomes a derivation in each argument. However, the calculation of [[ξ, η]] is ill-defined if one permits the addition of d-exact terms (e.g., stemming from the integration by parts) to the entries [[ξ, q]]. Besides, the normalization of the final result is a must in order to let us compare any given multivectors; in fact, the usual commutator of one-vectors is always transformed to b, -∂ (a)
, with the graded commutator in its l.-h.s. and the noncommutative variational Schouten bracket in the r.-h.s., proves that the Leibniz rule
Each skew-adjoint noncommutative linear total differential operator A : p → ȧ = ϕ yields the bivector π = 1 2 b, A(b) . Let H 1 , H 2 , H 3 be zero-vectors, i.e., H i = h i (x, [a]) dx . By definition, put {H i , H j } A := π δH i /δa, δH j /δa , which equals δH i /δa, A δH j /δa = ∂ (a)
A( δH j /δa) (H i ) (mod im d). The bracket { , } A is bilinear and skew-symmetric (but it does not restrict as a bi-derivation to the cohomology w.r.t. d); it becomes Poisson if it satisfies the Jacobi identity Criterion (see [5,7] on 1 2 Q π 2 = 0). A skew -adjoint (non)commutative linear matrix operator A : p → ȧ = ϕ in total derivatives is Hamiltonian -i.e., the bivector π is Poisson-if and only if its image is involutive: [im A, im A] ⊆ im A.
Remark. The construction of [[ , ]] in §3 is the standard string theory’s pair of pants Σ |i • Σ |j → Σ |i•j flying over the Minkowski space-time M 3,1 ; the dimension reduction is not required. Neither the diameters of the circles Σ ≃ S 1 that carry the words nor their stretching or oscillations on them, but it is the information about the cyclic order that matters.
The linking of the words into circles i
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