Abstract:
Some physically interesting weak-gravitational effects and phenomena are reviewed and briefly discussed: particle geometric phases due to the time-dependent spin-rotation couplings, non-inertial gravitational wave in rotating reference of frame, hyperbolical geometric quantum phases and topological dual mass as well.

Abstract:
On the basis of the principle that topological quantum phases arise from the scattering around space-time defects in higher dimensional unification, a geometric model is presented that associates with each quantum phase an element of a transformation group.

Abstract:
We study smooth projective varieties with small dual variety using methods from symplectic topology. We prove the affine parts of such varieties are subcritical, and that the hyperplane class is invertible in their quantum cohomology. We derive several topological and algebraic geometric consequences from that. The main tool in our work is the Seidel representation associated to Hamiltonian fibrations.

Abstract:
We report on the experimental realization of an optical analogue of a quantum geometric potential for light wave packets constrained on thin dielectric guiding layers fabricated in silica by the femtosecond laser writing technology. We further demonstrate the optical version of a topological crystal, with the observation of Bloch oscillations and Zener tunneling of purely geometric nature.

Abstract:
Geometric relativistc interactions in a new geometric unified theory are classified using the dynamic holonomy groups of the connection. Physical meaning may be given to these interactions if the frame excitations represent particles. These excitations have algebraic and topological quantum numbers. The proton, electron and neutrino may be associated to the frame excitations of the three dynamical holonomy subgroups. In particular, the proton excitation has a dual mathematical structure of a triplet of subexcitations. Hadronic, leptonic and gravitational interactions correspond to the same subgroups. The background geometry determines non trivial fiber bundles where excitations live, introducing topological quantum numbers that classify families of excitations. From these three particles, the only stable ones, it may be possible, as suggested by Barut, to build the rest of the particles. The combinations of the three fundamental excitations display SU(3)xSU(2)xU(1) symmetries.

Abstract:
In a topological quantum computer, braids of non-Abelian anyons in a (2+1)-dimensional space-time form quantum gates, whose fault tolerance relies on the topological, rather than geometric, properties of the braids. Here we propose to create and exploit redundant geometric degrees of freedom to improve the theoretical accuracy of topological single- and two-qubit quantum gates. We demonstrate the power of the idea using explicit constructions in the Fibonacci model. We compare its efficiency with that of the Solovay-Kitaev algorithm and explain its connection to the leakage errors reduction in an earlier construction [Phys. Rev. A 78, 042325 (2008)].

Abstract:
In the last decade, the development of new ideas in quantum theory, including geometric and deformation quantization, the non-Abelian Berry factor, super- and BRST symmetries, non-commutativity, has called into play the geometric techniques based on the deep interplay between algebra, differential geometry and topology. The present book aims at being a guide to advanced differential geometric and topological methods in quantum mechanics. Their main peculiarity lies in the fact that geometry in quantum theory speaks mainly the algebraic language of rings, modules, sheaves and categories. Geometry is by no means the primary scope of the book, but it underlies many ideas in modern quantum physics and provides the most advanced schemes of quantization.

Abstract:
We introduce a toric code model on the dice lattice which is exactly solvable and displays topological order at zero temperature. In the presence of a magnetic field, the flux dynamics is mapped to the highly frustrated transverse field Ising model on the kagome lattice. This correspondence suggests an intriguing disorder by disorder phenomenon in a topologically ordered system implying that the topological order is extremely robust due to the geometric frustration. Furthermore, a connection between fully frustrated transverse field Ising models and topologically ordered systems is demonstrated which opens an exciting physical playground due to the interplay of topological quantum order and geometric frustration.

Abstract:
We discuss a non-fermi liquid gapless metallic surface state of the topological band insulator. It has an odd number of gapless Dirac fermions coupled to a non-compact U(1) gauge field. This can be viewed as a vortex dual to the conventional Dirac fermion surface state. This surface duality is a reflection of a bulk dual description discussed recently for the gauged topological insulator. All the other known surface states can be conveniently accessed from the dual Dirac liquid, including the surface quantum hall state, the Fu-Kane superconductor, the gapped symmetric topological order and the "composite Dirac liquid". We also discuss the physical properties of the dual Dirac liquid, and its connection to the half-filled Landau level.

Abstract:
Chiral topological insulator (AIII-class) with Landau levels is constructed based on the Nambu 3-algebraic geometry. We clarify the geometric origin of the chiral symmetry of the AIII-class topological insulator in the context of non-commutative geometry of 4D quantum Hall effect. The many-body groundstate wavefunction is explicitly derived as a $(l,l,l-1)$ Laughlin-Halperin type wavefunction with unique $K$-matrix structure. Fundamental excitation is identified with anyonic string-like object with fractional charge ${1}/({1+2(l-1)^2})$. The Hall effect of the chiral topological insulators turns out be a color version of Hall effect, which exhibits a dual property of the Hall and spin-Hall effects.