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## Real moduli space of stable rational curves revisted

Cornell University
,
2019.
№ 1905.04499.

Khoroshkin A., Willwacher T.

We give a description of the operad formed by the real locus of the moduli space of stable genus zero curves with marked points $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$ in terms of a homotopy quotient of an operad of associative algebras. We use this model to find different Hopf models of the algebraic operad of Chains and homologies of $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$. In particular, we show that the operad $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$ is not formal. The manifolds $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$ are known to be Eilenberg-MacLane spaces for the so called pure Cacti groups.
As an application of the operadic constructions we prove that for each $n$ the cohomology ring $H^{\udot}(\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R}),{\mathbb{Q}})$ is a Koszul algebra and that the manifold $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$ is not formal but is a rational $K(\pi,1)$ space. We give the description of the Lie algebras associated with the lower central series filtration of the pure Cacti groups.

Khoroshkin A., Piontkovski D., / Cornell University. Series math "arxiv.org". 2014. No. arXiv:1202.5170.

Given an operad P with a finite Grobner basis of relations, we study the generating functions for the dimensions of its graded components P(n). Under moderate assumptions on the relations we prove that the exponential generating function for the sequence {dim P(n)} is differential algebraic, and in fact algebraic for P is a symmetrization of ...

Added: May 13, 2014

Markaryan N. S., / Cornell University. Series arXiv "math". 2020.

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. ...

Added: October 16, 2020

Soukhanov L., / Cornell University. Series math "arxiv.org". 2015.

We study the enumerative geometry of orbits of multidimensional toric action on projective algebraic varieties and develop a new cyclic differential-graded operad, conjecturally governing the real version of the enumerative geometry of these toric orbits. ...

Added: November 18, 2015

Khoroshkin, A., Markarian, N., Shadrin, S., Communications in Mathematical Physics 2013 Vol. 322 No. 3 P. 697-729

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 ...

Added: August 23, 2013

Khoroshkin A., International Mathematics Research Notices 2022 Vol. 2022 No. 4 P. 3106-3143

Given a symmetric operad $\mathcal{P}$ and a $\mathcal{P}$-algebra $V$, the associative universal enveloping algebra ${\mathsf{U}_{\mathcal{P}}}$ is an associative algebra whose category of modules is isomorphic to the abelian category of $V$-modules. We study the notion of PBW property for universal enveloping algebras over an operad.
In case $\mathcal{P}$ is Koszul a criterion ...

Added: February 13, 2021

Buryak A., Communications in Number Theory and Physics 2015 Vol. 9 No. 2 P. 239-271

In this paper we prove that the generating series of the Hodge integrals over the moduli space of stable curves is a solution of a certain deformation of the KdV hierarchy. This hierarchy is constructed in the framework of the Dubrovin-Zhang theory of the hierarchies of the topological type. It occurs that our deformation of ...

Added: September 29, 2020

Buryak A., Shadrin S., Advances in Mathematics 2011 Vol. 228 P. 22-42

We give a new proof of Faber's intersection number conjecture concerning the top intersections in the tautological ring of the moduli space of curves $\M_g$. The proof is based on a very straightforward geometric and combinatorial computation with double ramification cycles. ...

Added: October 1, 2020

Cherkasov A., Piontkovski D., , in : ISSAC '21: Proceedings of the 2021 on International Symposium on Symbolic and Algebraic Computation. : Association for Computing Machinery (ACM), 2021. P. 91-98.

Two operads are said to belong to the same Wilf class if they have the same generating series. We discuss possible Wilf classifications of non-symmetric operads with monomial relations. As a corollary, this would give the same classification for the operads with a finite Groebner basis.
Generally, there is no algorithm to decide whether two finitely ...

Added: September 27, 2021

Buryak A., Dubrovin B., Guere J. et al., Communications in Mathematical Physics 2018 Vol. 363 No. 1 P. 191-260

In this paper we continue the study of the double ramification hierarchy introduced by the first author. After showing that the DR hierarchy satisfies tau-symmetry we define its partition function as the (logarithm of the) tau-function of the string solution and show that it satisfies various properties (string, dilaton and divisor equations plus some important degree ...

Added: September 27, 2020

Buryak A., Mathematical Research Letters 2016 Vol. 23 No. 3 P. 675-683

In a previous paper we proved that after a simple transformation the generating series of the linear Hodge integrals on the moduli space of stable curves satisfies the hierarchy of the Intermediate Long Wave equation. In this paper we present a much shorter proof of this fact. Our new proof is based on an explicit ...

Added: September 28, 2020

Buryak A., Communications in Mathematical Physics 2015 Vol. 336 No. 3 P. 1085-1107

It this paper we present a new construction of a hamiltonian hierarchy associated to a cohomological field theory. We conjecture that in the semisimple case our hierarchy is related to the Dubrovin-Zhang hierarchy by a Miura transformation and check it in several examples. ...

Added: September 29, 2020

Dunin-Barkowski P., Popolitov A., Shabat G. et al., Differential Geometry and its Application 2015 Vol. 40 P. 86-102

We suggest a general method of computation of the homology of certain smooth covers $\widehat{\mathcal{M}}_{g,1}(\mathbb{C})$ of moduli spaces $\mathcal{M}_{g,1}\br{\mathbb{C}}$ of pointed curves of genus $g$. Namely, we consider moduli spaces of algebraic curves with level $m$ structures. The method is based on the lifting of the Strebel-Penner stratification of $\mathcal{M}_{g,1}\br{\mathbb{C}}$. We apply this method for ...

Added: March 3, 2015

Buryak A., Moscow Mathematical Journal 2017 Vol. 17 No. 1 P. 1-13

In this paper, using the formula for the integrals of the psi-classes over the double ramification cycles found by S. Shadrin, L. Spitz, D. Zvonkine and the author, we derive a new explicit formula for the n-point function of the intersection numbers on the moduli space of curves. ...

Added: September 27, 2020

Piontkovski D., , in : ISSAC '17 International Symposium on Symbolic and Algebraic Computation. Kaiserslautern, Germany — July 25 - 28, 2017. ACM, New York, 454 pp. : NY : ACM, 2017. P. 373-380.

We consider varieties of linear multioperator algebras, that is, classes of algebras with several multilinear operations satisfying certain identities. To each such a variety one can assign a numerical sequence called a sequence of codimensions. The n-th codimension is equal to the dimension of the vector space of all n-linear operations in the free algebra ...

Added: September 15, 2017

Khoroshkin A., Piontkovski D., Journal of Algebra 2015 Vol. 426 P. 377-429

Given an operad P with a finite Gröbner basis of relations, we study the generating functions for the dimensions of its graded components P(n). Under moderate assumptions on the relations we prove that the exponential generating function for the sequence {dimP(n)} is differential algebraic, and in fact algebraic if P is a symmetrization of a ...

Added: February 2, 2015

Kazaryan M., Zvonkine D., Lando S., International Mathematics Research Notices 2018 No. 22 P. 6817-6843

We consider families of curve-to-curve maps that have no singularities except those of genus 0 stable maps and that satisfy a versality condition at each singularity. We provide a universal expression for the cohomology class Poincaré dual to the locus of any given singularity. Our expressions hold for any family of curve-to-curve maps satisfying the ...

Added: July 10, 2017

Piontkovski D., Khoroshkin A., Journal of Algebra 2014 P. 1-53

For an operad P with a finite Gröbner basis of relations, we study the generating functions for the dimensions of its graded components P(n). Under moderate assumptions on the relations we prove that the exponential generating function for the sequence {dimP(n)} is differential algebraic, and in fact algebraic if P is a symmetrization of a ...

Added: March 5, 2015

Buryak A., Rossi P., Letters in Mathematical Physics 2016 Vol. 106 No. 3 P. 289-317

In this paper we define a quantization of the Double Ramification Hierarchies using intersection numbers of the double ramification cycle, the full Chern class of the Hodge bundle and psi-classes with a given cohomological field theory. We provide effective recursion formulae which determine the full quantum hierarchy starting from just one Hamiltonian, the one associated with ...

Added: September 28, 2020

Buryak A., Shadrin S., Spitz L. et al., American Journal of Mathematics 2015 Vol. 137 No. 3 P. 699-737

DR-cycles are certain cycles on the moduli space of curves. Intuitively, they parametrize curves that allow a map to the complex projective line with some specified ramification profile over two points. They are known to be tautological classes, but in general there is no known expression in terms of standard tautological classes. In this paper, ...

Added: September 30, 2020

Kochetkov Y., Функциональный анализ и его приложения 2010 Т. 44 № 2 С. 48-56

The cell structure of spaces M2,1 and M3,1 is considered. M2,1 and M3,1 are the spaces of complex curves of genus 2 and 3 with one marked point. For the space M2,1 nine cells of the highest dimension 8 are described and their adjacency is studied. For the space M3,1 the list of 1726 cells ...

Added: May 15, 2012

Markaryan N. S., Forum Mathematicum 2021 Vol. 33 No. 2 P. 531-545

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. ...

Added: March 15, 2021

Khoroshkin A., Piontkovski D., / Cornell University. Series math "arxiv.org". 2012. No. 1202.5170.

With any given operad $\mathcal{P}=\cup_{i=1}^{\infty}\mathcal{P}(n)$ we can associate a generating series of dimensions of the space of operations with the same arity. This article is an attempt to find a reasonable bounds and recursive relations for these generating series. Of course, for arbitrary operad the corresponding series may be transcendental, therefore we restrict our self ...

Added: September 29, 2013

Barannikov S., / hal.archives-ouvertes.fr (CNRS). Series HAL "math". 2018. No. 01804639.

The construction from [B06], see also [B10], of cohomology classes of compactified moduli spaces of Riemann surfaces, starting from a derivation of associative whose square is nonzero, is generalized to the case of A-infinity algebras. It is shown that the constructed cohomology classes define Cohomological Field Theory. ...

Added: October 25, 2018

Buryak A., Shadrin S., Zvonkine D., Journal of the European Mathematical Society 2016 Vol. 18 No. 12 P. 2925-2951

We describe the structure of the top tautological group in the cohomology of the moduli space of smooth genus g curves with n marked points. ...

Added: September 27, 2020