I. Overview of the computation Reviewing the setup from our lecture notes of last year [S1], we let R denote a crystallographic root system with V the ambient real Euclidean space, along with the usual choices (simple roots, positive roots,...), and W the Weyl group. Given a reduced expression for the longest element of the Weyl group, say w0 = si1 · · · sil , there is an induced ordering β ∨ 1...

We report here on the structure of reversible quantum cellular automata with the additional restriction that these are also Clifford operations. This means that tensor products of Weyl operators (projective representation of a finite abelian symplectic group) are mapped to multiples of tensor products of Weyl operators. Therefore Clifford quantum cellular automata are induced by symplectic cell...

The quantum state of a particle can be completely specified by a position at one instant of time. This implies a lack of information, hence a symmetry, as to where the particle will move. We here study the consequences for free particles of spin 0 and spin 1/2. On a cubic spatial lattice a hopping equation is derived, and the continuum limit taken. Spin 0 leads to the Schrödinger equation, and ...

We relate the Bohr-Sommerfeld conditions established in λ-microlocal analysis to formal deformation quantization of symplectic manifolds by classifying star products adapted to some Lagrangian sub-manifold L, i.e. products preserving the classical vanishing ideal IL of L up to IL-preserving equivalences.

Abelian deformations of ordinary algebras of functions are studied. The role of Harrison cohomology in classifying such deformations is illustrated in the context of simple examples chosen for their relevance to physics. It is well known that Harrison cohomology is trivial on smooth manifolds and that, consequently, abelian ∗-products on such manifolds are trivial to first order in the deformat...

In this paper we consider algebras with involution over a ring C which is given by the quadratic extension by i of an ordered ring R. We discuss the -representation theory of such -algebras on pre-Hilbert spaces over C and develop the notions of Rieffel induction and formal Morita equivalence for this category analogously to the situation for C-algebras. Throughout this paper the notion of posi...

Let A be a star product on a symplectic manifold (M,ω0), 1 t [ω] its Fedosov class, where ω is a deformation of ω0. We prove that for a complex polarization of ω there exists a commutative subalgebra, O, in A that is isomorphic to the algebra of functions constant along the polarization. Let F (A) consists of elements of A whose commutator with O belongs to O. Then, F (A) is a Lie algebra which...

We study complex 4-manifolds with holomorphic self-dual conformal structures, and we obtain an interpretation of the Weyl tensor of such a manifold as the projective curvature of a field of cones on the ambitwistor space. In particular, its vanishing is implied by the existence of some compact, simply-connected, null-geodesics. We also relate the Cotton-York tensor of an umbilic hypersurface to...

We generalize to Hilbert modular varieties of arbitrary dimension the work of W. Duke [14] on the equidistribution of Heegner points and of primitive positively oriented closed geodesics in the Poincaré upper half plane, subject to certain subconvexity results. We also prove vanishing results for limits of cuspidal Weyl sums associated with analogous problems for the Siegel upper half space of ...

We study the topological properties of magnon excitations in a wide class of three dimensional (3D) honeycomb lattices with ferromagnetic ground states. It is found that they host nodal ring magnon excitations. These rings locate on the same plane in the momentum space. The rings can be gapped by Dzyaloshinskii-Moriya (DM) interactions to form two Weyl points with opposite charges. We explicitl...