On deformation quantization of quadratic Poisson structures

in Communications in Mathematical Physics (in press)

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 ▲]

From gravity to string topology

in Letters in Mathematical Physics (2023), 113

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 ▲]