We discuss the geometry behind some integrals related to structure constants of the Liouville conformal field theory.
The toric surfaces for octonions and related objects are discussed.
In this (mostly historical) note we show how a unified Kummer-Artin-Schreier sequence from [W], [SOS] may be recovered from the relativistic velocity addition law.
In this paper we strengthen the results of [SV] by presenting their derived version. Namely, we define a derived Knizhnik - Zamolodchikov connection and identify it with a derived Gauss - Manin connection.
The rings of $p$-typical Witt vectors are interpreted as spaces of vanishing cycles for some perverse sheaves over a disc. This allows to localize an isomorphism emerging in Drinfelds theory of prismatization [Dr], Prop. 3.5.1, namely to express it a
s an integral of a standard exact triangle on the disc.
In the work of Mukhin and Varchenko from 2002 there was introduced a Wronskian map from the variety of full flags in a finite dimensional vector space into a product of projective spaces. We establish a precise relationship between this map and the P
lucker map. This allows us to recover the result of Varchenko and Wright saying that the polynomials appearing in the image of the Wronsky map are the initial values of the tau-functions for the Kadomtsev-Petviashvili hierarchy.
A positive structure on the varieties of critical points of master functions for KZ equations is introduced. It comes as a combination of the ideas from classical works by G.Lusztig and a previous work by E.Mukhin and the second named author.
Motivated by applications to perverse sheaves, we study combinatorics of two cell decompositions of the symmetric product of the complex line, refining the complex stratification by multiplicities. Contingency matrices, appearing in classical statist
ics, parametrize the cells of one such decomposition, which has the property of being quasi-regular. The other, more economical, decomposition, goes back to the work of Fox-Neuwirth and Fuchs on the cohomology of braid groups. We give a criterion for a sheaf constructible with respect to the contingency decomposition to be constructible with respect to the complex stratification. We also study a polyhedral ball which we call the stochastihedron and whose boundary is dual to the two-sided Coxeter complex (for the root system $A_n$) introduced by T.K. Petersen. The Appendix by P. Etingof studies enumerative aspects of contingency matrices. In particular, it is proved that the meta-matrix formed by the numbers of contingency matrices of various sizes, is totally positive.
Let $mathfrak{g}$ be a simple Lie algebra of rank $r$ over $mathbb{C}$, $mathfrak{h} subset mathfrak{g}$ a Cartan subalgebra. We construct a family of $r$ commuting Hermitian operators acting on $mathfrak{h}$ whose eigenvalues are equal to the coordi
nates of the eigenvectors of the Cartan matrix of $mathfrak{g}$.
Using the vanishing cycles of simple singularities, we study the eigenvectors of Cartan matrices of finite root systems, and of q-deformations of these matrices.
