Related papers: $\hbar$-Riemann-Hilbert correspondence

A bi-variant theory $\mathbb B(X,Y)$ defined for a pair $(X,Y)$ is a theory satisfying properties similar to those of Fulton--MacPherson's bivariant theory $\mathbb B(X \xrightarrow f Y)$ defined for a morphism $f:X \to Y$. In this paper,…

If quantum mechanics were to be applicable to macroscopic objects, classical mechanics would have to be a limiting case of quantum mechanics. Then the category Set that packages classical mechanics has to be in some sense a 'limiting case'…

We derive two geometric approaches to categorification of quantum invariants of links associated to an arbitrary compact simple Lie group $^L{G}$. In part I, we describe the first approach, based on an equivariant derived category of…

Braverman and Finkelberg recently proposed the geometric Satake correspondence for the affine Kac-Moody group $G_\aff$ [Braverman A., Finkelberg M., arXiv:0711.2083]. They conjecture that intersection cohomology sheaves on the Uhlenbeck…

We address classical and quantum mechanics in a general setting of arbitrary time-dependent transformations. Classical non-relativistic mechanics is formulated as a particular field theory on smooth fibre bundles over a time axis.…

For a smooth, projective, complex algebraic variety $X$, the Riemann--Hilbert correspondence establishes a complex analytic isomorphism between the `Betti moduli space' of rank $n$ local systems on $X^\mathrm{an}$ and the `de Rham moduli…

We explicitly describe the Jacquet-Langlands correspondence at the level of modular forms. This gives a simpler and more flexible solution to Eichler's basis problem for general level than earlier work of Hijikata-Pizer-Shemanske for…

We provide several results on splice-quotient singularities: a combinatorial expression of the dimension of the first cohomology of all `natural' line bundles, an equivariant Campillo-Delgado-Gusein-Zade type formula about the dimension of…

A unified description of the relationship between the Hamiltonian structure of a large class of integrable hierarchies of equations and W-algebras is discussed. The main result is an explicit formula showing that the former can be…

Information geometry provides differential geometric concepts like a Riemannian metric, connections and covariant derivatives on spaces of probability distributions. We discuss here how these concepts apply to quantum field theories in the…

We establish an algebra isomorphism between the center of the category $\mathcal{O}$ for a hybrid quantum group at a root of unity $\zeta$ and the cohomology of $\zeta$-fixed locus on affine Grassmannian. A deformed version of this…

We define generalized bialgebras and Hopf algebras and on this basis we introduce quantum categories and quantum groupoids. The quantization of the category of linear (super)spaces is constructed. We establish a criterion for the classical…

All Lie bialgebra structures on the Heisenberg--Weyl algebra $[A_+,A_-]=M$ are classified and explicitly quantized. The complete list of quantum Heisenberg--Weyl algebras so obtained includes new multiparameter deformations, most of them…

The general construction of self-adjoint configuration space representations of the Heisenberg algebra over an arbitrary manifold is considered. All such inequivalent representations are parametrised in terms of the topology classes of flat…

We use twisted Fourier-Mukai transforms to study the relation between an abelian fibration on a holomorphic symplectic manifold and its dual fibration. Our reasoning leads to an equivalence between the derived category of coherent sheaves…

Let $(g,\delta_\hbar)$ be a Lie bialgebra. Let $(U_\hbar(g),\Delta_\hbar)$ a quantization of $(g,\delta_\hbar)$ through Etingof-Kazhdan functor. We prove the existence of a $L_\infty$-morphism between the Lie algebra $C(\g)=\Lambda(g)$ and…

We describe the derived category of coherent sheaves on the minimal resolution of the Kleinian singularity associated to a finite subgroup G of SL(2). Then, we give an application to the Euler-characteristic version of the Hall algebra of…

We show: the Floer homology over the Novikov ring of (nonexact!) rational Lagrangians in an (nonexact!) integral symplectic manifold can be computed in terms of exact Lagrangians in an exact filling of the prequantization bundle. As a…

Given a compact Hermitian complex space with isolated singular points, we construct a Dolbeault-type Hilbert complex whose cohomology is isomorphic to the cohomology of the structure sheaf. We show that the corresponding K-homology class…

Hilbert bimodules are morphisms between C*-algebraic models of quantum systems, while symplectic dual pairs are morphisms between Poisson geometric models of classical systems. Both of these morphisms preserve representation-theoretic…