arXiv subjects
Publications and source records attributed to V. Schechtman.
We study the blow-ups of configuration spaces. These spaces have a structure of what we call an Orlik-Solomon manifold; it allows us to compute the intersection cohomology of certain flat connections with logarithmic singularities using some Aomoto type complexes of logarithmic forms. Using this construction we realize geometrically the sl_2 Bernstein - Gelfand - Gelfand resolution as an Aomoto complex.
We interpret the GL_n equivariant cohomology of a partial flag variety of flags of length N in \C^n as the Bethe algebra of a suitable gl_N[t] module associated with the tensor power (\C^N)^{\otimes n}.
We propose a formula expressing Perron - Frobenius eigenvectors of Cartan matrices in terms of products of values of the Gamma function.
We show that the conformal blocks constructed in the previous article by the first and the third author may be described as certain integrals in equivariant cohomology. When the bundles of conformal blocks have rank one, this construction may be compared with the old integral formulas of the second and the third author. The proportionality coefficients are some Selberg type integrals which are computed. Finally, a geometric construction of the tensor products of vector representations of the Lie algebra $\frak{gl}(m)$ is proposed.
Using a classical result of Avramov-Golod we strengthen a recent result of Gorodentsev, Khoroshkin and Rudakov on syzygies of highest weight orbit closure.
A new form of Bethe ansatz equations is introduced. A version of a separation of variables for the quantum $sl_3$ Gaudin model is presented.
We introduce an algebra given by quadratic relations in an algebra of polynomials in an infinite number of variables. Using this algebra, we prove some explicit formulas for the Sturm sequence of a polynomial.
We reproduce the quantum cohomology of toric varieties (and of some hypersurfaces in projective spaces) as the cohomology of certain vertex algebras with differential. The deformation technique allows us to compute the cohomology of the chiral de Rham complex over the projective space.
This article is a sequel to hep-th/9411050, q-alg/9412017, q-alg/9503013. Given a collection of $m$ finite factorizable sheaves $\{\CX_k\}$, we construct here some perverse sheaves over configuration spaces of points on a projective line $\BP^1$ with $m$ additional marked points. We announce here (with sketch proof) the computation of the cohomology spaces of these sheaves. They turn out to coincide with certain "semiinfinite" $\Tor$ spaces of the corresponding $\fu$-modules. As a corollary, we get a description of local systems of conformal blocks in WZW models in genus $0$ (cf. ~\cite{ms}) as natural subquotients of some semisimple local systems of geometric origin. In particular, these local systems are semisimple themselves.
This article is a sequel to hep-th/9411050, q-alg/9412017. In Chapter 1 we associate with every Cartan matrix of finite type and a non-zero complex number $ζ$ an abelian artinian category $\FS$. We call its objects {\em finite factorizable sheaves}. They are certain infinite collections of perverse sheaves on configuration spaces, subject to a compatibility ("factorization") and finiteness conditions. In Chapter 2 the tensor structure on $\FS$ is defined using functors of nearby cycles. It makes $\FS$ a braided tensor category. In Chapter 3 we define, using vanishing cycles functors, an exact tensor functor $$Φ:\FS\lra\CC$$ to the category $\CC$ connected with the corresponding quantum group. In Chapter 4 we show that $Φ$ is an equivalence. Some proofs are only sketched.
This paper is a sequel to "Localization of $\frak{u}$-modules. I", hep-th/9411050. We are starting here the geometric study of the tensor category $\cal{C}$ associated with a quantum group (corresponding to a Cartan matrix of finite type) at a root of unity. The main results establish isomorphisms between homogeneous components of irreducible objects in $\cal{C}$ and spaces of vanishing cycles at the origin of certain Goresky-MacPherson sheaves on configuration spaces; establish isomorphisms of the stalks at the origin of the above GM sheaves with certain Hochschild complexes (which compute the Hochschild homology of a certain "triangular" subalgebra of our quantum group with coefficients in the coresponding irreducible representation); establish the analogous results for tensor products of irreducibles. In geometry, the tensor product of representations corresponds to a "fusion" of sheaves on configuration spaces --- operation defined using the functor of nearby cycles.
This paper is the first in a series. The main goal of the series is to present a geometric construction of certain remarkable tensor categories arising from quantum groups coresponding to the value of deformation parameter $q$ equal to a root of unity. In the present paper we study perverse sheaves over a complex affine space which are smooth along the stratification determined by a finite arrangement of complex affine hyperplanes defined by real equations. In particular, we construct explicitely (in terms of combinatorial data) complexes computing cohomology of Goresky-MacPherson extensions of one-dimensional local systems over the complement of hyperplanes.
In this note we strenghten a theorem by Esnault-Schechtman-Viehweg which states that one can compute the cohomology of a complement of hyperplanes in a complex affine space with coefficients in a local system using only logarithmic global differential forms, provided certain "Aomoto non-resonance conditions" for monodromies are fulfilled at some "edges" (intersections of hyperplanes). We prove that it is enough to check these conditions on a smaller subset of edges. We show that for certain known one dimensional local systems over configuration spaces of points in a projective line defined by a root system and a finite set of affine weights (these local systems arise in the geometric study of Knizhnik-Zamolodchikov differential equations), the Aomoto resonance conditions at non-diagonal edges coincide with Kac-Kazhdan conditions of reducibility of Verma modules over affine Lie algebras.