Arkady Vaintrob

Chiral de Rham Complex

Abstract: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.

PSL(n|n) sigma model as a conformal field theory

Abstract We discuss the sigma model on the PSL ( n | n ) supergroup manifold. We demonstrate that this theory is exactly conformal. The chiral algebra of this model is given by some extension of the Virasoro algebra, similar to the W algebra of Zamolodchikov. We also show that all group invariant correlation functions are coupling constant independent and can be computed in the free theory. The non-invariant correlation functions are highly non-trivial and coupling dependent. At the end we compare two- and three-point correlation functions of the PSL (1, 1|2) sigma model with the correlation functions in the boundary theory of AdS 3 × S 3 and find a qualitative agreement.

Moduli Spaces of Higher Spin Curves and Integrable Hierarchies

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 on this moduli space which allow one to construct a cohomological field theory of rank r−1 in all genera. In genus zero it produces a Frobenius manifold which is isomorphic to the Frobenius manifold structure on the base of the versal deformation of the singularity Ar−1. We prove analogs of the puncture, dilaton, and topological recursion relations by drawing an analogy with the construction of Gromov–Witten invariants and quantum cohomology.

Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations

We study the category of matrix factorizations for an isolated hypersurface singularity. We compute the canonical bilinear form on the Hochschild homology of this category. We find explicit expressions for the Chern character and the boundary-bulk maps and derive an analog of the Hirzebruch-Riemann-Roch formula for the Euler characteristic of the Hom-space between a pair of matrix factorizations. We also establish G-equivariant versions of these results.

A new matrix-tree theorem

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 interpreted topologically as an expression for the lowest order term of the Alexander-Conway polynomial of an algebraically split link. We also prove some algebraic properties of our Pfaffian-tree polynomial.

Milnor numbers, spanning trees, and the Alexander¿Conway polynomial

Abstract 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 m -component link L all of whose Milnor numbers μ i 1 … i p 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.

Melvin–Morton Conjecture and Primitive Feynman Diagrams

We give a very short proof of the Melvin–Morton conjecture relating the colored Jones polynomial and the Alexander polynomial of knots. The proof is based on explicit evaluation of the corresponding weight systems on primitive elements of the Hopf algebra of chord diagrams which, in turn, follows from simple identities between four-valent tensors on the Lie algebra sl2 and the Lie superalgebra gl(1|1). This shows that the miraculous connection between the Jones and Alexander invariants follows from the similarity (supersymmetry) between sl2 and gl(1|1).

The Universal Vassiliev Invariant for the Lie Superalgebra

Abstract: We compute the universal weight system for Vassiliev coming from the Lie superalgebra

Spin Gromov-Witten Invariants

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

Tensor products of Frobenius manifolds and moduli spaces of higher spin curves

, and we find a combinatorial expression for it in terms of the standard generators of Z. The resulting knot invariants generalize the Alexander-Conway polynomial.