Projective completions of Jordan pairs Part II. Manifold structures and symmetric spaces

We define symmetric spaces in arbitrary dimension and over arbitrary non-discrete topological fields \K, and we construct manifolds and symmetric spaces associated to topological continuous quasi-inverse Jordan pairs and -triple systems. This class of spaces, called smooth generalized projective geometries, generalizes the well-known (finite or infinite-dimensional) bounded symmetric domains as well as their ``compact-like'' duals. An interpretation of such geometries as models of Quantum Mechanics is proposed, and particular attention is paid to geometries that might be considered as "standard models" -- they are associated to associative continuous inverse algebras and to Jordan algebras of hermitian elements in such an algebra

Noncommutative del Pezzo surfaces and Calabi-Yau algebras

The hypersurface in a 3-dimensional vector space with an isolated quasi-homogeneous elliptic singularity of type E_r,r=6,7,8, has a natural Poisson structure. We show that the family of del Pezzo surfaces of the corresponding type E_r provides a semiuniversal Poisson deformation of that Poisson structure. We also construct a deformation-quantization of the coordinate ring of such a del Pezzo surface. To this end, we first deform the polynomial algebra C[x,y,z] to a noncommutative algebra with generators x,y,z and the following 3 relations (where [u,v]_t = uv- t.vu): [x,y]_t=F_1(z), [y,z]_t=F_2(x), [z,x]_t=F_3(y). This gives a family of Calabi-Yau algebras A(F) parametrized by a complex number t and a triple F=(F_1,F_2,F_3), of polynomials in one variable of specifically chosen degrees. Our quantization of the coordinate ring of a del Pezzo surface is provided by noncommutative algebras of the form A(F)/(g) where (g) stands for the ideal of A(F) generated by a central element g, which generates the center of the algebra A(F) if F is generic enough.Comment: The statement and proof of Theorem 2.4.1 corrected, Introduction expanded, several misprints fixe

Non-Commutative Chern Numbers for Generic Aperiodic Discrete Systems

The search for strong topological phases in generic aperiodic materials and meta-materials is now vigorously pursued by the condensed matter physics community. In this work, we first introduce the concept of patterned resonators as a unifying theoretical framework for topological electronic, photonic, phononic etc. (aperiodic) systems. We then discuss, in physical terms, the philosophy behind an operator theoretic analysis used to systematize such systems. A model calculation of the Hall conductance of a 2-dimensional amorphous lattice is given, where we present numerical evidence of its quantization in the mobility gap regime. Motivated by such facts, we then present the main result of our work, which is the extension of the Chern number formulas to Hamiltonians associated to lattices without a canonical labeling of the sites, together with index theorems that assure the quantization and stability of these Chern numbers in the mobility gap regime. Our results cover a broad range of applications, in particular, those involving quasi-crystalline, amorphous as well as synthetic (i.e. algorithmically generated) lattices.Comment: 44 pages, 4 figures. v2: typos corrected and references updated. v3: Minor changes, to appear in J. Phys. A (Mathematical and Theoretical