Related papers: The Quantum $A_{\infty}$-Relations on the Elliptic…
We construct an algebra morphism from the elliptic quantum group $E_{\tau,\eta}(sl_2)$ to a certain elliptic version of the ``quantum groups in higher genus'' studied by V. Rubtsov and the first author. This provides an embedding of…
We show that four exceptional Fuchsian equations, each determined by the four parabolic singularities, known as the Chudnovsky equations, are transformed into each other by algebraic transformations. We describe equivalence of these…
The Hamiltonian of relativistic particles with electric and magnetic dipole moments that interact with an electromagnetic field is determined in the Foldy-Wouthuysen representation. Transition to the semiclassical approximation is carried…
Let $L\subset X$ be a not necessarily orientable relatively $Pin$ Lagrangian submanifold in a symplectic manifold $X$. Evaluation maps of moduli spaces of $J$-holomorphic disks with boundary in $L$ may not be relatively orientable. To deal…
We give an introductory account of functional determinants of elliptic operators on manifolds and Polyakov-type formulas for their infinitesimal and finite conformal variations. We relate this to extremal problems and to the Q-curvature on…
We show how classical and quantum dualities, as well as duality relations that appear only in a sector of certain theories ("emergent dualities"), can be unveiled, and systematically established. Our method relies on the use of morphisms of…
We define for an associative algebra an $A_{\infty}$ category whose objects are automorphisms of this algebra. This construction has some resemblance with Fukaya'a categories related to Floer cohomology.
We construct the algebra of operators acting on the Hilbert spaces of Quantum Mechanics for systems of $N$ identical particles from the field operators acting in the Fock space of Quantum Field Theory by providing the explicit relation…
The symplectic geometry of the phase space associated with a charged particle is determined by the addition of the Faraday 2-form to the standard structure on the Euclidean phase space. In this paper we describe the corresponding algebra of…
Using the formalism of discrete quantum group gauge theory, one can construct the quantum algebras of observables for the Hamiltonian Chern-Simons model. The resulting moduli algebras provide quantizations of the algebra of functions on the…
We investigate fundamental bounds on the curvature of quantum correlation functions in imaginary time. Focusing first on topological phases, we show that quantum geometry can qualitatively modify the imaginary-time decay of correlations,…
In this paper we introduce a notion of Feynman geometry on which quantum field theories could be properly defined. A strong Feynman geometry is a geometry when the vector space of $A_\infty$ structures is finite dimensional. A weak Feynman…
The previously introduced Quantum Arnold Transformation, a unitary operator mapping the solutions of the Schr\"odinger equation for time dependent quadratic Hamiltonians into the solutions for the free particle, is revised and some…
Consider a Hamiltonian action of a compact connected Lie group on a symplectic manifold $(M,\omega)$. Conjecturally, under suitable assumptions there exists a morphism of cohomological field theories from the equivariant Gromov-Witten…
We discuss the $q$ deformation of Weyl-Heisenberg algebra in connection with the von Neumann theorem in Quantum Mechanics. We show that the $q$-deformation parameter labels the Weyl systems in Quantum Mechanics and the unitarily…
Recent ideas on modular localization in local quantum physics are used to clarify the relation between on- and off-shell quantities in particle physics; in particular the relation between on-shell crossing symmetry and off-shell Einstein…
In the framework of Category Theory, we study the association between finite--dimensional representations of a compact quantum group and quantum vector bundles with linear connections for a given quantum principal bundle with a principal…
We discuss the hypothesis that the debate about the interpretation of the orthodox formalism of quantum mechanics (QM) might have been misguided right from the start by a biased metaphysical interpretation of the formalism and its inner…
The geometry of Quantum Mechanics in the context of uncertainty and complementarity, and probability is explored. We extend the discussion of geometry of uncertainty relations in wider perspective. Also, we discuss the geometry of…
Using technique of wheeled props we establish a correspondence between the homotopy theory of unimodular Lie 1-bialgebras and the famous Batalin-Vilkovisky formalism. Solutions of the so called quantum master equation satisfying certain…