相关论文: Introduction to homological geometry: part I
In the first part of this paper, we introduce the notion of "cyclic stratum" of a Frobenius manifold $M$. This is the set of points of the extended manifold $\mathbb C^*\times M$ at which the unit vector field is a cyclic vector for the…
The paper studies three classes of Frobenius manifolds: Quantum Cohomology (topological sigma-models), unfolding spaces of singularities (K. Saito's theory, Landau-Ginzburg models), and the recent Barannikov-Kontsevich construction starting…
Motivated by the work of Cappell, Deturck, Gluch and Miller, we extend the notion of cohomology of harmonic forms (of a compact manifold with boundary) to the abstract setting of Hilbert complexes. Then, we present some geometric…
The review is devoted to topological global aspects of quantal description. The treatment concentrates on quantizations of kinematical observables --- generalized positions and momenta. A broad class of quantum kinematics is rigorously…
The present thesis deals with some properties of classical and quantum scalar fields in an inhomogeneous and/or time-dependent background, focusing on models where the latter can be described as a curved space-time with an event horizon.…
This is an expanded version of the third author's lecture in String-Math 2015 at Sanya. It summarizes some of our works in quantum cohomology. After reviewing the quantum Lefschetz and quantum Leray--Hirsch, we discuss their applications to…
Quantum operators of coordinates and momentum components of a particle in Minkowski space-time belong to a noncommutative algebra and give rise to a quantum phase space. Under some constraints, in particular, the Lorentz invariance…
We explore and extend the application of homological algebra to describe quantum entanglement, initiated in arXiv:1901.02011, focusing on the Hodge-theoretic structure of entanglement cohomology in finite-dimensional quantum systems. We…
We introduce a structure of an infinite-dimensional Frobenius manifold on a subspace in the space of pairs of functions analytic inside/outside the unit circle with simple poles at 0/infinity respectively. The dispersionless 2D Toda…
A complete quantization of a homogeneous and isotropic spacetime with closed spatial sections coupled to a massive scalar field is provided, within the framework of Loop Quantum Cosmology. We identify solutions with their initial data on…
The big phase space, the geometric setting for the study of quantum cohomology with gravitational descendents, is a complex manifold and consists of an infinite number of copies of the small phase space. The aim of this paper is to define a…
Contents 1 Mappings and distortion 2 The mathematics of good behavior much of the time, and the BMO frame of mind 3 Finite polyhedra and combinatorial parameterization problems 4 Quantitative topology, and calculus on singular spaces 5…
We develop the noncommutative geometry (bundles, connections etc.) associated to algebras that factorise into two subalgebras. An example is the factorisation of matrices $M_2(\C)=\C\Z_2\cdot\C\Z_2$. We also further extend the coalgebra…
These notes are an introduction to the theory of quantum symmetries of finite and infinite sets, graphs, and locally compact spaces.
These are lecture notes from the IMPANGA 2010 Summer School. The lectures survey some of the main features of equivariant cohomology at an introductory level. The first part is an overview, including basic definitions and examples. In the…
This review gives an introduction to various attempts to understand the quantum nature of black holes. The first part focuses on thermodynamics of black holes, Hawking radiation, and the interpretation of entropy. The second part is devoted…
The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic treatment of Gromov-Witten classes, and a…
We consider Knapp-Vogan Hecke algebras in the quantum group setting. This allows us to produce a quantum analogue of the Bernstein functor as a first step towards the cohomological induction for quantum groups.
We define the notions of unital/counital/biunital infinitesimal anti-symmetric bialgebras and coFrobenius bialgebras and discuss their algebraic properties. We also define the notion of a graded 2D open-closed TQFT. These structures arise…
I construct lowest-energy representations of non-centrally extended algebras of Noether symmetries, including diffeomorphisms and reparametrizations of the observer's trajectory. This may be viewed as a new scheme for quantization. First…