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 ▲]

Modular forms of degree 2 and curves of genus 2 in characteristic 2

in International Mathematics Research Notices (2022)

Multi-directed Graph Complexes and Quasi-isomorphisms Between Them II: Sourced Graphs

in International Mathematics Research Notices (2019), 00(0), 1-57

We prove that the projection from graph complex with at least one source to oriented graph complex is a quasi-isomorphism, showing that homology of the “sourced” graph complex is also equal to the ... [more ▼]

We prove that the projection from graph complex with at least one source to oriented graph complex is a quasi-isomorphism, showing that homology of the “sourced” graph complex is also equal to the homology of standard Kontsevich’s graph complex. This result may have applications in theory of multi-vector fields T≥1poly of degree at least one, and to the hairy graph complex that computes the rational homotopy of the space of long knots. The result is generalized to multi-directed graph complexes, showing that all such graph complexes are quasi-isomorphic. These complexes play a key role in the deformation theory of multi-oriented props recently invented by Sergei Merkulov. We also develop a theory of graph complexes with arbitrary edge types. [less ▲]

Prescribing metrics on the boundary of AdS 3-manifolds

in International Mathematics Research Notices (2016)

We prove that given two metrics g+ and g− with curvature κ<−1 on a closed, oriented surface S of genus τ≥2, there exists an AdS manifold N with smooth, space-like, strictly convex boundary such that the ... [more ▼]

We prove that given two metrics g+ and g− with curvature κ<−1 on a closed, oriented surface S of genus τ≥2, there exists an AdS manifold N with smooth, space-like, strictly convex boundary such that the induced metrics on the two connected components of ∂N are equal to g+ and g−. Using the duality between convex space-like surfaces in AdS3, we obtain an equivalent result about the prescription of the third fundamental form. [less ▲]

Wilson Lines from Representations of NQ-Manifolds

in International Mathematics Research Notices (2013), 2013(24),

An NQ-manifold is a nonnegatively graded supermanifold with a degree 1 homological vector field. The focus of this paper is to define the Wilson loops/lines in the context of NQ-manifolds and to study ... [more ▼]

An NQ-manifold is a nonnegatively graded supermanifold with a degree 1 homological vector field. The focus of this paper is to define the Wilson loops/lines in the context of NQ-manifolds and to study their properties. The Wilson loops/lines, which give the holonomy or parallel transport, are familiar objects in usual differential geometry, we analyze the subtleties in the generalization to the NQ-setting and we also sketch some possible applications of our construction. [less ▲]

The A_infty de Rham theorem and integration of representations up to homotopy

in International Mathematics Research Notices (2013), 2013(16), 3790-3855

We use Chen's iterated integrals to integrate representations up to homotopy. That is, we construct an A-infinity functor from the representations up to homotopy of a Lie algebroid A to those of its ... [more ▼]

We use Chen's iterated integrals to integrate representations up to homotopy. That is, we construct an A-infinity functor from the representations up to homotopy of a Lie algebroid A to those of its infinity groupoid. This construction extends the usual integration of representations in Lie theory. We discuss several examples including Lie algebras and Poisson manifolds. The construction is based on an A-infinity version of de Rham's theorem due to Gugenheim. The integration procedure we explain here amounts to extending the construction of parallel transport for superconnections, introduced by Igusa and Block-Smith, to the case of certain differential graded manifolds. [less ▲]

Tame Galois realizations of $ GSp_4(\Bbb F_łl)$ over $\Bbb Q$

in International Mathematics Research Notices (2011), (9), 2028--2046

In this paper, we obtain realizations of the 4-dimensional general symplectic group over a prime field of characteristic l> 3 as the Galois group of a tamely ramified Galois extension of Q. The strategy ... [more ▼]

In this paper, we obtain realizations of the 4-dimensional general symplectic group over a prime field of characteristic l> 3 as the Galois group of a tamely ramified Galois extension of Q. The strategy is to consider the Galois representation ρ_l attached to the Tate module at l of a suitable abelian surface. We need to choose the abelian surfaces carefully in order to ensure that the image of ρ_l is large and simultaneously maintain a control on the ramification of the corresponding Galois extension. We obtain an explicit family of curves of genus 2 such that the Galois representation attached to the l-torsion points of their Jacobian varieties provides tame Galois realizations of the desired symplectic groups. [less ▲]

Higher Schläfli formulas and applications. II. Vector-valued differential relations

in International Mathematics Research Notices (2008)