Papers by Arkady Vaintrob
The classical Matrix-Tree Theorem allows one to list the spanning trees of a graph by monomials i... more The classical Matrix-Tree Theorem allows one to list the spanning trees of a graph by monomials in the expansion of the determinant of a certain matrix. We prove that in the case of three-graphs (that is, hypergraphs whose edges have exactly three vertices) the spanning trees are generated by the Pfaffian of a suitably defined matrix. This result can be
We study complex-analytic properties of the augmented Teichmüller spaces Tg,n obtained by adding ... more We study complex-analytic properties of the augmented Teichmüller spaces Tg,n obtained by adding to the classical Teichmüller spaces Tg,n points corresponding to Riemann surfaces with nodal singularities. Unlike Tg,n, the space Tg,n is not a complex manifold (it is not even locally compact). We prove however that the quotient of the augmented Teichmüller space by any finite index subgroup of the Teichmüller modular group has a canonical structure of a complex orbifold. Using this structure we construct natural maps from T to stacks of admissible coverings of stable Riemann surfaces. This result is important for understanding the cup-product in stringy orbifold cohomology. We also establish some new technical results from the general theory of orbifolds which may be of independent interest.
We prove that the algebra A of chord diagrams, the dual to the associated graded algebra of Vassi... more We prove that the algebra A of chord diagrams, the dual to the associated graded algebra of Vassiliev knot invariants, is isomorphic to the universal enveloping algebra of a Casimir Lie algebra in a certain tensor category (the PROP for Casimir Lie algebras). This puts on a firm ground a known statement that the algebra A "looks and behaves like a universal enveloping algebra". An immediate corollary of our result is the conjecture of [BGRT] on the Kirillov-Duflo isomorphism for algebras of chord diagrams.
Communications in Mathematical Physics, 1999
We define natural sheaves of vertex algebras over smooth manifolds which may be regarded as semi-... more We define natural sheaves of vertex algebras over smooth manifolds which may be regarded as semi-infinite de Rham complexes of certain D-modules over the loop spaces. For Calabi-Yau manifolds they admit N = 2 supersymmetry. Connection with Wakimoto modules is discussed.
Compositio Mathematica, 2001
We prove the genus zero part of the generalized Witten conjecture, relating moduli spaces of high... more We prove the genus zero part of the generalized Witten conjecture, relating moduli spaces of higher spin curves to Gelfand–Dickey hierarchies. That is, we show that intersection numbers on the moduli space of stable r-spin curves assemble into a generating function which yields a solution of the semiclassical limit of the KdVr equations. We formulate axioms for a cohomology class
Advances in Mathematics, 2003
We study relations between the Alexander-Conway polynomial ∇ L and Milnor higher linking numbers ... more We study relations between the Alexander-Conway polynomial ∇ L and Milnor higher linking numbers of links from the point of view of finite-type (Vassiliev) invariants. We give a formula for the first non-vanishing coefficient of ∇ L of an mcomponent link L all of whose Milnor numbers µ i1...ip vanish for p ≤ n. We express this coefficient as a polynomial in Milnor numbers of L. Depending on whether the parity of n is odd or even, the terms in this polynomial correspond either to spanning trees in certain graphs or to decompositions of certain 3-graphs into pairs of spanning trees. Our results complement determinantal formulas of Traldi and Levine obtained by geometric methods.
Communications in Mathematical Physics, 1997
We compute the universal weight system for Vassiliev invariants coming from the Lie superalgebra ... more We compute the universal weight system for Vassiliev invariants coming from the Lie superalgebra gl(1|1) applying the construction of [13]. This weight system is a function from the space of chord diagrams to the center Z of the universal enveloping algebra of gl(1|1), and we find a combinatorial expression for it in terms of the standard generators of Z. The resulting knot invariants generalize the Alexander-Conway
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Papers by Arkady Vaintrob