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Nikita Markarian

Publications and source records attributed to Nikita Markarian.

17 recordsLinked to original sources
arXiv2026

Double shuffle relations via the Hodge defect

We interpret the convolution constructions of [arXiv:2604.26357] in Hodge-theoretic terms, extending the Hodge defect calculation of [arXiv:2609.19341]. For Hodge--Tate modules on $\mathbb C^*$ smooth outside $1$, the defect is the action of an element of the Deligne--Terasoma transport algebra expressed through the Drinfeld associator. For convolutions of iterated Kummer extensions, comparison of mixed Hodge structures produces a beta factor and yields the regularized double shuffle relations.

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arXiv2026

The Drinfeld associator and $A_\infty$ functors

We compare two mixed Hodge structures attached to a unipotent variation $\mathbb V$ of mixed Hodge--Tate structures on $X=\mathbb P^1\setminus{0,1,\infty}$: the cohomology of its extension by $*$ at $0$ and by $!$ at $\infty$, and the vanishing cycles at $1$. The semi-holonomy isomorphism constructed in [arXiv:2604.26357] identifies these two structures on their underlying rational structures, compatibly with their weight filtrations, but need not respect their Hodge filtrations. We compute the resulting Hodge defect. Its universal noncommutative series is given by a specific component of the inverse Drinfeld associator multiplied by an explicit local monodromy factor. We conclude with a formal $A_\infty$ interpretation of the coefficients of the Hodge defect in terms of weight-framed Tate extensions.

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arXiv2026

Harmonic sums and the Galois group of the Mellin-KZ difference equation

We identify the universal Galois group of the difference equation obtained by the Mellin transform of the Knizhnik-Zamolodchikov equation with the prounipotent group associated with the transport Hopf algebra of Deligne and Terasoma studied in \cite{Markarian}. At each finite weight, we realize the Picard-Vessiot ring by finite multiple harmonic sums. We also interpret the classical Gamma-corrected projected associator as the comparison between finite and regularized asymptotic fibers in this realization.

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arXiv2026

Multiplicative convolution and double shuffle relations

We develop a geometric approach to the regularized double shuffle relations for multiple zeta values, based on convolution of perverse sheaves on $\mathbb{C}^*$ and inspired by the approach of Deligne and Terasoma. We introduce semi-holonomy isomorphisms associated with pro-unipotent paths and show that their compatibility with multiplicative convolution is equivalent to a condition on the pro-unipotent fundamental group, the homological pentagon equation. We prove that this condition is equivalent to the regularized double shuffle relations, yielding a geometric proof that the pentagon equation implies these relations. The approach is purely topological and avoids Hodge-theoretic and Tannakian methods.

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arXiv2024

Multiplicative convolution and double shuffle relations: convolution

This is the first of two parts of a project devoted to a geometric interpretation of the Deligne-Terasoma approach to regularized double shuffle relations. The central fact of this approach is the isomorphism between vanishing cycles of multiplicative convolution of certain perverse sheaves and the tensor product of vanishing cycles, which may be written in two different ways. These isomorphisms depend on a choice of a functorial isomorphism $φ$ between vanishing cycles of a perverse sheaf on $\mathbb{C}^*$ and cohomology of its certain extension on $\mathbb{P}^1$. The isomorphism chosen in the present paper guarantees compatibilities with the isomorphisms. In the second part of the project, we will study other choices of $φ$. We will see that its compatibilities with convolution imply regularized double shuffle relations. In particular, associator relations imply them.

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arXiv2024

Compatible Poisson Brackets Associated with Elliptic Curves in $G(2,5)$

We prove that a pair of Feigin-Odesskii Poisson brackets on ${\mathbb P}^4$ associated with elliptic curves given as linear sections of the Grassmannian $G(2,5)$ are compatible if and only if this pair of elliptic curves is contained in a del Pezzo surface obtained as a linear section of $G(2,5)$.

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arXiv2024

On Conormal Lie Algebras of Feigin-Odesskii Poisson Structures

The main result of the paper is a description of conormal Lie algebras of Feigin-Odesskii Poisson structures. In order to obtain it we introduce a new variant of a definition of a Feigin-Odesskii Poisson structure: we define it using a differential on the second page of a certain spectral sequence. In the general case this spectral sequence computes morphisms and higher Ext's between filtered objects in an abelian category. Moreover, we use our definition to give another proof of the description of symplectic leaves of Feigin-Odesskii Poisson structures.

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arXiv2022

Compatible Feigin-Odesskii Poisson brackets

We prove that several Feigin-Odesskii Poisson brackets associated with normal elliptic curves in ${\mathbb P}^n$ are compatible if and only if they are contained in a scroll or in a Veronese surface in ${\mathbb P}^5$ (with an exception of one case when $n=3$). In the case $n=3$ we determine the quartic corresponding to the Schouten bracket of two (non-compatible) Poisson brackets associated with normal elliptic curves $E_1$ and $E_2$.

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arXiv2020

On algebra of big zeta values

The algebra of big zeta values we introduce in this paper is an intermediate object between multiple zeta values and periods of the multiple zeta motive. It consists of number series generalizing multiple zeta values, the simplest examples, which are not multiple zeta series, are Tornheim sums. We show that convergent big zeta values are periods of the moduli space of stable curves of genus zero on one hand and multiple zeta values on the other hand. It gives an alternative way to prove that any such period may be expressed as a rational linear combination of multiple zeta values and a simple algorithm for finding such an expression.

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arXiv2020

Weyl n-algebras and the Swiss cheese operad

We apply Weyl $n$-algebras to prove formality theorems for higher Hochschild cohomology. We present two approaches: via propagators and via the factorization complex. It is shown that the second approach is equivalent to the first one taken with a new family of propagators we introduce.

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arXiv2016

Weyl n-algebras

We introduce Weyl n-algebras and show how their factorization homology may be used to define invariants of manifolds. In the appendix we heuristically explain why these invariants must be perturbative Chern-Simons invariants.

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arXiv2016

Weyl n-algebras and the Kontsevich integral of the unknot

Given a Lie algebra with a scalar product, one may consider the latter as a symplectic structure on a $dg$-scheme, which is the spectrum of the Chevalley--Eilenberg algebra. In the first section we explicitly calculate the first order deformation of the differential on the Hochschild complex of the Chevalley--Eilenberg algebra. The answer contains the Duflo character. This calculation is used in the last section. There we sketch the definition of the Wilson loop invariant of knots, which is hopefully equal to the Kontsevich integral, and show that for unknot they coincide. As a byproduct we get a new proof of the Duflo isomorphism for a Lie algebra with a scalar product.

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arXiv2015

A note on the symplectic structure on the space of G-monopoles

Let $G$ be a semisimple complex Lie group with a Borel subgroup $B$. Let $X=G/B$ be the flag manifold of $G$. Let $C=P^1\ni\infty$ be the projective line. Let $α\in H_2(X,{\Bbb Z})$. The moduli space of $G$-monopoles of topological charge $α$ (see e.g. [Jarvis]) is naturally identified with the space $M_b(X,α)$ of based maps from $(C,\infty)$ to $(X,B)$ of degree $α$. The moduli space of $G$-monopoles carries a natural hyperkähler structure, and hence a holomorphic symplectic structure. We propose a simple explicit formula for the symplectic structure on $M_b(X,α)$. It generalizes the well known formula for $G=SL_2$ (see e.g. [Atiyah-Hitchin]). Let $P\supset B$ be a parabolic subgroup. The construction of the Poisson structure on $M_b(X,α)$ generalizes verbatim to the space of based maps $M=M_b(G/P,β)$. In most cases the corresponding map $T^*M\to TM$ is not an isomorphism, i.e. $M$ splits into nontrivial symplectic leaves. These leaves are explicilty described.

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arXiv2013

Manifoldic homology and Chern-Simons formalism

The aim of this note is to define for any $e_n$-algebra $A$ and a compact parallelizable n-manifold $M$ without borders a morphism from the homology of homotopy Lie algebra $A[n-1]$ to the topological chiral homology of $M$ with coefficients in $A$. This map plays a crucial role in the perturbative Chern-Simons theory.

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arXiv2012

Hypercommutative operad as a homotopy quotient of BV

We give an explicit formula for a quasi-isomorphism between the operads Hycomm (the homology of the moduli space of stable genus 0 curves) and BV/$Δ$ (the homotopy quotient of Batalin-Vilkovisky operad by the BV-operator). In other words we derive an equivalence of Hycomm-algebras and BV-algebras enhanced with a homotopy that trivializes the BV-operator. These formulas are given in terms of the Givental graphs, and are proved in two different ways. One proof uses the Givental group action, and the other proof goes through a chain of explicit formulas on resolutions of Hycomm and BV. The second approach gives, in particular, a homological explanation of the Givental group action on Hycomm-algebras.

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