arXiv subjects

Sergei A. Merkulov

Publications and source records attributed to Sergei A. Merkulov.

5 recordsLinked to original sources

arXiv2026

Homotopy Poisson and BV$_\infty$ structures on derived moduli spaces of flat connections

We show that quasi-Poisson and quasi-BV structures on (super)manifolds $M$ admitting an action of a Lie (super)group $G$ can be identified with a special class of homotopy Poisson and respectively $BV_\infty$ structures on the associated differential (super)manifolds of the form ${\mathcal M}:=M\times Πg$, $g$ being the Lie (super)algebra of $G$. The construction is based on the proven existence of an essentially unique family of higher Jacobiators for any Lie (super)algebra $g$ with an invariant (odd or even) scalar product. We apply these arguments to the moduli spaces $M$ of flat $G$-connections on a surface with boundaries using Fock-Rosly construction.

math.DG↗

arXiv2026

On dg properads of pre-CY algebras

We present new small models for the dg properads that govern pre-CY algebras: one for general pre-CY algebras and one for pre-CY algebras without curvature terms. These small models have only four and, respectively, three generators with valencies $\leq 4$ (in contrast to the original properads which have infinitely many generators with arbitrarily large valencies). We prove that the valency upper bound $4$ is sharp. We study the cohomology group of the deformation complex of the dg properad governing general pre-CY algebras and prove that it contains the totality $\prod_{g\geq 1}H^\bu(\M_{g,1})$ of cohomology groups of moduli spaces $M_{g,1}$ of genus $g$ algebraic curves with one marked point. As an application of this result we prove that the Tradler-Zeinalian properad of $V^{(d)}$-algebras is not Koszul.

math.QA↗

arXiv2026

A low-valence ribbon graph complex computing the cohomology of $M_{g,m}$

It is proven that every cohomology class of the moduli space $M_{g,m}$ for any $2g+m\geq 3$, $m\geq 1$ can be represented combinatorially by a ribbon quiver with at most four-valent vertices. The "at most four"-valency condition is sharp.

math.AG↗

arXiv2025

Gravity prop and moduli spaces $\mathcal{M}_{g,n}$

Let $\mathcal{M}_{g,n}$ be the moduli space of algebraic curves of genus $g$ with $m+n$ marked points decomposed into the disjoint union of two sets of cardinalities $m$ and $n$, and $H_c^{\bullet}(\mathcal{M}_{m+n})$ its compactly supported cohomology group. We prove that the collection of $S$-bimodules $\left\{H_c^{\bullet-m}(\mathcal{M}_{g,m+n})\right\}$ has the structure of a properad (called the gravity properad) such that it contains the (degree shifted) E. Getzler's gravity operad as the sub-collection $\{H_c^{\bullet-1}(\mathcal{M}_{0,1+n})\}_{n\geq 2}$. Moreover, we prove that the generators of the 1-dimensional cohomology groups $H_c^{\bullet-1}(\mathcal{M}_{0,1+2})$, $H_c^{\bullet-2}(\mathcal{M}_{0,2+1})$ and $H_c^{\bullet-3}(\mathcal{M}_{0,3+0})$ satisfy with respect to this properadic structure the relations of the (degree shifted) quasi-Lie bialgebra, a fact making the totality of cohomology groups $ \prod_{g,m,n} H_c^{\bullet}(\mathcal{M}_{g,m+n})\otimes_{S_m^{op}\times S_n} (sgn_m\otimes Id_n)$ into a complex with the differential fully determined by the just mentioned three cohomology classes . It is proven that this complex contains infinitely many cohomology classes, all coming from M. Kontsevich's odd graph complex. The gravity prop structure is established with the help of T. Willwacher's twisting endofunctor (in the category of properads under the operad of Lie algebras) and K. Costello's theory of moduli spaces of nodal disks with marked boundaries and internal marked points (such that each disk contains at most one internal marked point).

math.AG↗

arXiv2022

From gravity to string topology

The chain gravity properad introduced earlier by the author acts on the cyclic Hochschild of any cyclic $A_\infty$ algebra equipped with a scalar product of degree $-d$. In particular, it acts on the cyclic Hochschild complex of any Poincare duality algebra of degree $d$, and that action factors through a quotient dg properad $ST_{3-d}$ of ribbon graphs which is in focus of this paper. We show that its cohomology properad $H^\bullet(ST_{3-d})$ is highly non-trivial and that it acts canonically on the reduced equivariant homology $\bar{H}_\bullet^{S^1}(LM)$ of the loop space $LM$ of any simply connected $d$-dimensional closed manifold $M$. By its very construction, the string topology properad $H^\bullet(ST_{3-d})$ comes equipped with a morphism from the gravity properad which is fully determined by the compactly supported cohomology of the moduli spaces $M_{g,n}$ of stable algebraic curves of genus $g$ with marked points. This result gives rise to new universal operations in string topology as well as reproduces in a unified way several known constructions: we show that (i) $H^\bullet(ST_{3-d})$ is also a properad under the properad of involutive Lie bialgebras in degree $3-d$ whose induced action on $\bar{H}_\bullet^{S^1}(LM)$ agrees precisely with the famous purely geometric construction of M. Chas and D. Sullivan, (ii) $H^\bullet(ST_{3-d})$ is a properad under the properad of homotopy involutive Lie bialgebras in degree $2-d$; (iii) E. Getzler's gravity operad injects into $H^\bullet(ST_{3-d})$ implying a purely algebraic counterpart of the geometric construction of C. Westerland establishing an action of the gravity operad on $\bar{H}_\bullet^{S^1}(LM)$.

math.AT↗