arXiv subjects

Aliaksandr Hancharuk

Publications and source records attributed to Aliaksandr Hancharuk.

4 recordsLinked to original sources

arXiv2026

On construction of differential $\mathbb Z$-graded varieties

Given a commutative unital algebra $\mathcal O$, a proper ideal $\mathcal I$ in $\mathcal O$, and a positively graded differential variety over $\mathcal O/\mathcal I$, we provide a $\mathbb Z$-graded extension, whose negative part is an arborescent Koszul-Tate resolution of $\mathcal O/ \mathcal I$. This extension is obtained through an algorithm exploiting the explicit homotopy retract data of the arborescent Koszul-Tate resolution, so that the number of homological computations in the construction is significantly reduced. For a positively graded differential variety over $\mathcal O$ that preserves the ideal $\mathcal I$, the extension admits a manifest description in terms of decorated trees and computed data. As a by-product, to every Lie-Rinehart algebra over the coordinate ring of an affine variety $ W \subseteq M = \mathbb{C}^d$, one associates an explicit differential $\mathbb{Z}$-graded variety over $M$ whose negative component is the arborescent Koszul-Tate resolution of the coordinate ring $\mathbb C[x_1, \ldots, x_d]/\mathcal I_W$ of $W$, and whose positive component is the universal dg-variety of the given Lie-Rinehart algebra. Concrete examples are given.

math-ph↗

arXiv2024

Koszul-Tate resolutions and decorated trees

Given a commutative algebra $\mathcal O$, a proper ideal $\mathcal I$, and a resolution of $\mathcal O/ \mathcal I$ by projective $\mathcal O $-modules, we construct an explicit Koszul-Tate resolution. We call it the arborescent Koszul-Tate resolution since it is indexed by decorated trees. When the $ \mathcal O$-module resolution has finite length, only finitely many operations are needed in our constructions -- this is to be compared with the classical Tate algorithm, which requires infinitely many such computations if $ \mathcal I$ is not a complete intersection. As a by-product of our construction, the initial projective $\mathcal O $-module resolution becomes equipped with an explicit $A_\infty$-algebra.

math.AC↗

arXiv2022

BFV extensions for mechanical systems with Lie-2 symmetry

We consider mechanical systems on $T^*M$ with possibly irregular and reducible first class contraints linear in the momenta, which thus correspond to singular foliations on $M$. According to a recent result, the latter ones have a Lie-infinity algebroid $(\cal M,Q)$ covering them, where we restrict to the case of Lie-2 algebroids. We propose to consider $T^*\cal M$ as a potential BFV extended phase space of the constrained system, such that the canonical lift of the nilpotent vector field $Q$ yields automatically a solution to the BFV master equation. We show that in this case, the BFV extension of the Hamiltonian, providing a second corner stone of the BFV formalism, may be obstructed. We identify the corresponding complex governing this second extension problem explicitly (the first extension problem was circumvented by means of the lift of the Lie-2 algebroid structure). We repeatedly come back to the example of angular momenta on $T^*\mathbb R^3$: in this procedure, the standard free Hamiltonian does not have a BFV extension -- while it does so on $T^*(\mathbb R^3 \backslash \{0 \})$, with a relatively involved ghost contribution singular at the origin.

hep-th↗

arXiv2018

On the structure of the conformal higher-spin wave operators

We study conformal higher spin (CHS) fields on constant curvature backgrounds. By employing parent formulation technique in combination with tractor description of GJMS operators we find a manifestly factorized form of the CHS wave operators for symmetric fields of arbitrary integer spin $s$ and gauge invariance of arbitrary order $t\leq s$. In the case of the usual Fradkin-Tseytlin fields $t=1$ this gives a systematic derivation of the factorization formulas known in the literature while for $t>1$ the explicit formulas were not known. We also relate the gauge invariance of the CHS fields to the partially-fixed gauge invariance of the factors and show that the factors can be identified with (partially gauge-fixed) wave operators for (partially)-massless or special massive fields. As a byproduct, we establish a detailed relationship with the tractor approach and, in particular, derive the tractor form of the CHS equations and gauge symmetries.

hep-th↗