Related papers: Derived KZ equations
Pulling back the weight systems associated with the exceptional Lie algebras and their standard representations by a modification of the universal Vassiliev-Kontsevich invariant yields link invariants; extending them to coloured 3-nets, we…
We review results on the Knizhnik-Zamolodchikov (KZ) and dynamical equations, both differential and difference, in the context of the $(gl_k,gl_n)$ duality, and their implications for hypergeometric integrals. The KZ and dynamical equations…
In the context of infinity categories, we rethink the notion of derived functor in terms of correspondences. This is especially convenient for the description of a passage from an adjoint pair (F,G) of functors to a derived adjoint pair…
We prove that derived equivalent algebras have isomorphic differential calculi in the sense of Tamarkin--Tsygan.
We discuss the correspondence between the Knizhnik-Zamolodchikov equations associated with $GL(N)$ and the $n$-particle quantum Calogero model in the case when $n$ is not necessarily equal to $N$. This can be viewed as a natural…
Using a theorem of partial differential equations, we present a general way of deriving the conserved quantities associated with a given classical point mechanical system, denoted by its Hamiltonian. Some simple examples are given to…
We show that Derksen-Weyman-Zelevinsky's mutations of quivers with potential yield equivalences of suitable 3-Calabi-Yau triangulated categories. Our approach is related to that of Iyama-Reiten and Koszul dual to that of…
We derive a generalized Knizhnik-Zamolodchikov equation for the correlation function of the intertwiners of the vector and the MacMahon representations of Ding-Iohara-Miki algebra. These intertwiners are cousins of the refined topological…
We explicitly write dowm integral formulas for solutions to Knizhnik-Zamolodchikov equations with coefficients in non-bounded -- neither highest nor lowest weight -- $\gtsl_{n+1}$-modules. The formulas are closely related to WZNW model at a…
The aim of this article is to provide a complementary understanding to some results of the second author using the machinery of Koszul complexes, and to explain how this approach can provide a new description of projective derived…
An elliptic analogue of the $q$ deformed Knizhnik-Zamolodchikov equations is introduced. A solution is given in the form of a Jackson-type integral of Bethe vectors of the XYZ-type spin chains.
Drinfeld defined the Knizhinik--Zamolodchikov (KZ) associator $\Phi_{\rm KZ}$ by considering the regularized holonomy of the KZ connection along the {\em droit chemin} $[0,1]$. The KZ associator is a group-like element of the free…
In this note a generalized Gauss-Manin connection is constructed for cohomology of Lie-Rinehart algebras, generalizing the classical Gauss-Manin connection. As an application a Gysin-map between K-groups of flat connections is constructed.…
The Kashiwara-Vergne (KV) conjecture states the existence of solutions of a pair of equations related with the Campbell-Baker-Hausdorff series. It was solved by Meinrenken and the first author over the real numbers, and in a formal version,…
A simple relation between inhomogeneous transfer matrices and boundary quantum KZ equations is exhibited for quantum integrable systems with reflecting boundary conditions, analogous to an observation by Gaudin for periodic systems. Thus…
Deformed and undeformed KZ equations are considered for $k=0$. It is shown that they allow the same number of solutions, one being the asymptotics of others. Essential difference in analitical properties of the solutions is explained.
We define Getzler's Gauss-Manin connection in cyclic homology at the level of chains and outline some relations of this construction to noncommutative calculus.
We extend the notion of a Thomas projective connection (a projective equivalence class of linear connections) for supermanifolds. As a by-product, we arrive at a generalisation of the multidimensional Schwarzian derivative for the super…
In this note, we generalize results of Donagi and Pantev on twisted derived equivalences between elliptically fibered surfaces to higher dimensions. First, we establish a twisted derived equivalence between torsors under abelian schemes…
The Gaudin models based on the face-type elliptic quantum groups and the $XYZ$ Gaudin models are studied. The Gaudin model Hamiltonians are constructed and are diagonalized by using the algebraic Bethe ansatz method. The corresponding…