Prop of ribbon hypergraphs and strongly homotopy involutive Lie bialgebras

in International Mathematics Research Notices (2022), rnac023

For any integer d we introduce a prop RHrad of d-oriented ribbon hypergraphs (in which "edges" can connect more than two vertices) and prove that there exists a canonical morphism Holieb⋄d⟶RHrad from the ... [more ▼]

For any integer d we introduce a prop RHrad of d-oriented ribbon hypergraphs (in which "edges" can connect more than two vertices) and prove that there exists a canonical morphism Holieb⋄d⟶RHrad from the minimal resolution Holieb⋄d of the (degree shifted) prop of involutive Lie bialgebras into the prop of ribbon hypergraphs which is non-trivial on each generator of Holieb⋄d. As an application we show that for any graded vector space W equipped with a family of cyclically (skew)symmetric higher products the associated vector space of cyclic words in elements of W has a combinatorial Holieb⋄d-structure. As an illustration we construct for each natural number N≥1 an explicit combinatorial strongly homotopy involutive Lie bialgebra structure on the vector space of cyclic words in N graded letters which extends the well-known Schedler's necklace Lie bialgebra structure from the formality theory of the Goldman-Turaev Lie bialgebra in genus zero. [less ▲]

From gravity to string topology

E-print/Working paper (2022)

The chain gravity properad introduced earlier by the author acts on the cyclic Hochschild of any cyclic A∞ algebra equipped with a scalar product of degree −d. In particular, it acts on the cyclic ... [more ▼]

The chain gravity properad introduced earlier by the author acts on the cyclic Hochschild of any cyclic A∞ 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 ST3−d of ribbon graphs which is in focus of this paper. We show that its cohomology properad H∙(ST3−d) is highly non-trivial and that it acts canonically on the reduced equivariant homology H¯S1∙(LM) of the loop space LM of any simply connected d-dimensional closed manifold M. By its very construction, the string topology properad H∙(ST3−d) comes equipped with a morphism from the gravity properad which is fully determined by the compactly supported cohomology of the moduli spaces Mg,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∙(ST3−d) is also a properad under the properad of involutive Lie bialgebras in degree 3−d whose induced action on H¯S1∙(LM) agrees precisely with the famous purely geometric construction of M. Chas and D. Sullivan, (ii) H∙(ST3−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∙(ST3−d) implying a purely algebraic counterpart of the geometric construction of C. Westerland establishing an action of the gravity operad on H¯S1∙(LM). [less ▲]

Gravity prop and moduli spaces Mg,n

E-print/Working paper (2021)

Let Mg,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(Mm+n) its compactly supported cohomology ... [more ▼]

Let Mg,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(Mm+n) its compactly supported cohomology group. We prove that the collection of S-bimodules {H∙−mc(Mg,m+n)} 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∙−1c(M0,1+n)}n≥2. Moreover, we prove that the generators of the 1-dimensional cohomology groups H∙−1c(M0,1+2), H∙−2c(M0,2+1) and H∙−3c(M0,3+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 ∏g,m,nH∙c(Mg,m+n)⊗Sopm×Sn(sgnm⊗Idn) 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). [less ▲]

On deformation quantization of quadratic Poisson structures

E-print/Working paper (2021)

We study the deformation complex of the dg wheeled properad of Z-graded quadratic Poisson structures and prove that it is quasi-isomorphic to the even M. Kontsevich graph complex. As a first application ... [more ▼]

We study the deformation complex of the dg wheeled properad of Z-graded quadratic Poisson structures and prove that it is quasi-isomorphic to the even M. Kontsevich graph complex. As a first application we show that the Grothendieck-Teichmüller group acts on the genus completion of that wheeled properad faithfully and essentially transitively. As a second application we classify all universal quantizations of Z-graded quadratic Poisson structures together with the underlying (so called) homogeneous formality maps. [less ▲]

Multi-oriented props and homotopy algebras with branes

in Letters in Mathematical Physics (2020), 110

We introduce a new category of differential graded {\em multi-oriented}\, props whose representations (called homotopy algebras with branes) in a graded vector space require a choice of a collection of $k ... [more ▼]

We introduce a new category of differential graded {\em multi-oriented}\, props whose representations (called homotopy algebras with branes) in a graded vector space require a choice of a collection of $k$ linear subspaces in that space, $k$ being the number of extra directions (if $k=0$ this structure recovers an ordinary prop); symplectic vector spaces equipped with $k$ Lagrangian subspaces play a distinguished role in this theory. Manin triples is a classical example of an algebraic structure (concretely, a Lie bialgebra structure) given in terms of a vector space and its subspace; in the context of this paper Manin triples are precisely symplectic Lagrangian representations of the {\em 2-oriented} generalization of the classical operad of Lie algebras. In a sense, the theory of multi-oriented props provides us with a far reaching strong homotopy generalization of Manin triples type constructions. [less ▲]

An explicit two step quantization of Poisson structures and Lie bialgebras

in Communications in Mathematical Physics (2018), 364(2), 505578

We develop a new approach to deformation quantizations of Lie bialgebras and Poisson structures which goes in two steps. In the first step one associates to any Poisson (resp. Lie bialgebra) structure a ... [more ▼]

We develop a new approach to deformation quantizations of Lie bialgebras and Poisson structures which goes in two steps. In the first step one associates to any Poisson (resp. Lie bialgebra) structure a so called quantizable Poisson (resp. Lie bialgebra) structure. We show explicit transcendental formulae for this correspondence. In the second step one deformation quantizes a quantizable Poisson (resp. Lie bialgebra) structure. We show again explicit transcendental formulae for this second step correspondence (as a byproduct we obtain configuration space models for biassociahedron and bipermutohedron). In the Poisson case the first step is the most non-trivial one and requires a choice of an associator while the second step quantization is essentially unique, it is independent of a choice of an associator and can be done by a trivial induction. We conjecture that similar statements hold true in the case of Lie bialgebras. The main new result is a surprisingly simple explicit universal formula (which uses only smooth differential forms) for universal quantizations of finite-dimensional Lie bialgebras. [less ▲]

The Frobenius operad is Koszul

in Duke Mathematical Journal (2016), 165(15), 2921-2989

We show Koszulness of the prop governing involutive Lie bialgebras and also of the props governing non-unital and unital-counital Frobenius algebras, solving a long-standing problem. This gives us minimal ... [more ▼]

We show Koszulness of the prop governing involutive Lie bialgebras and also of the props governing non-unital and unital-counital Frobenius algebras, solving a long-standing problem. This gives us minimal models for their deformation complexes, and for deformation complexes of their algebras which are discussed in detail. Using an operad of graph complexes we prove, with the help of an earlier result of one of the authors [W3], that there is a highly non-trivial action of the Grothendieck-Teichm¨uller group GRT on (completed versions of) the minimal models of the properads governing Lie bialgebras and involutive Lie bialgebras by automorphisms. As a corollary one obtains a large class of universal deformations of any (involutive) Lie bialgebra and any Frobenius algebra, parameterized by elements of the Grothendieck-Teichmueller Lie algebra. We also prove that, for any given homotopy involutive Lie bialgebra structure in a vector space, there is an associated homotopy Batalin-Vilkovisky algebra structure on the associated Chevalley-Eilenberg complex. [less ▲]

Props of ribbon graphs, involutive Lie bialgebras and moduli spaces of curves M_g,n

E-print/Working paper (2015)

We establish a new and surprisingly strong link between two previously unrelated theories: the theory of moduli spaces of curves $\cM_{g,n}$ (which, according to Penner, is controlled by the ribbon graph ... [more ▼]

We establish a new and surprisingly strong link between two previously unrelated theories: the theory of moduli spaces of curves $\cM_{g,n}$ (which, according to Penner, is controlled by the ribbon graph complex) and the homotopy theory of $E_d$ operads (controlled by ordinary graph complexes with no ribbon structure, introduced first by Kontsevich). The link between the two goes through a new intermediate {\em stable}\, ribbon graph complex which has roots in the deformation theory of quantum $A_\infty$ algebras and the theory of Kontsevich compactifications of moduli spaces of curves $\overline{\cM}_{g,n}^K$. Using a new prop of ribbon graphs and the fact that it contains the prop of involutive Lie bialgebras as a subprop we find new algebraic structures on the classical ribbon graph complex computing $H^\bu(\cM_{g,n})$. We use them to prove Comparison Theorems, and in particular to construct a non-trivial map from the ordinary to the ribbon graph cohomology. On the technical side, we construct a functor $\f$ from the category of prop(erad)s to the category of operads. If a properad $\cP$ is in addition equipped with a map from the properad governing Lie bialgebras (or graded versions thereof), then we define a notion of $\cP$-``graph'' complex, of stable $\cP$-graph complex and a certain operad, that is in good cases an $E_d$ operad. In the ribbon case, this latter operad acts on the deformation complexes of any quantum $A_\infty$-algebra. We also prove that there is a highly non-trivial, in general, action of the Grothendieck-Teichm\"uller group $GRT_1$ on the space of so-called {\em non-commutative Poisson structures}\, on any vector space $W$ equipped with a degree $-1$ symplectic form (which interpolate between cyclic $A_\infty$ structures in $W$ and ordinary polynomial Poisson structures on $W$ as an affine space). [less ▲]

Grothendieck-Teichmueller and Batalin-Vilkovisky

in Letters in Mathematical Physics (2014), 104(5), 625-634

It is proven that, for any affine supermanifold M equipped with a constant odd symplectic structure, there is a universal action (up to homotopy) of the Grothendieck-Teichmueller Lie algebra grt on the ... [more ▼]

It is proven that, for any affine supermanifold M equipped with a constant odd symplectic structure, there is a universal action (up to homotopy) of the Grothendieck-Teichmueller Lie algebra grt on the set of quantum BV structures (i. e.\ solutions of the quantum master equation) on M. [less ▲]

Operads, configuration spaces and quantization

in Bulletin of the Brazilian Mathematical Society (2011), 42(4), 683781

We review several well-known operads of compactified configuration spaces and construct several new such operads, $\overline{C}$, in the category of smooth manifolds with corners whose complexes of ... [more ▼]

We review several well-known operads of compactified configuration spaces and construct several new such operads, $\overline{C}$, in the category of smooth manifolds with corners whose complexes of fundamental chains give us (i) the 2-coloured operad of $A_\infty$-algebras and their homotopy morphisms, (ii) the 2-coloured operad of $L_\infty$-algebras and their homotopy morphisms, and (iii) the 4-coloured operad of open-closed homotopy algebras and their homotopy morphisms. Two gadgets --- a (coloured) operad of Feynman graphs and a de Rham field theory on $\overline{C}$ --- are introduced and used to construct quantized representations of the (fundamental) chain operad of $\overline{C}$ which are given by Feynman type sums over graphs and depend on choices of propagators. [less ▲]