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Register a new userTopological Gauge Theory Of General Weitzenbock Manifolds Of Dislocations In Crystals
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Y. C. Huang
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General Weitzenbock material manifolds of dislocations in crystals Are proposed, the reference, idealized and deformation states of the bodies in general case are generally described by the general manifolds, the topological gauge field theory of dislocations is given in general case,true distributions and evolution of dislocations in crystals are given by the formulas describing dislocations in terms of the general manifolds,furthermore, their properties are discussed.
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We develop a non-singular theory of three-dimensional dislocation loops in a particular version of Mindlins anisotropic gradient elasticity with up to six length scale parameters. The theory is systematically developed as a generalization of the classical anisotropic theory in the framework of linearized incompatible elasticity. The non-singular version of all key equations of anisotropic dislocation theory are derived as line integrals, including the Burgers displacement equation with isolated solid angle, the Peach-Koehler stress equation, the Mura-Willis equation for the elastic distortion, and the Peach-Koehler force. The expression for the interaction energy between two dislocation loops as a double line integral is obtained directly, without the use of a stress function. It is shown that all the elastic fields are non-singular, and that they converge to their classical counterparts a few characteristic lengths away from the dislocation core. In practice, the non-singular fields can be obtained from the classical ones by replacing the classical (singular) anisotropic Greens tensor with the non-singular anisotropic Greens tensor derived by cite{Lazar:2015ja}. The elastic solution is valid for arbitrary anisotropic media. In addition to the classical anisotropic elastic constants, the non-singular Greens tensor depends on a second order symmetric tensor of length scale parameters modeling a weak non-locality, whose structure depends on the specific class of crystal symmetry. The anisotropic Helmholtz operator defined by such tensor admits a Greens function which is used as the spreading function for the Burgers vector density. As a consequence, the Burgers vector density spreads differently in different crystal structures.
This thesis is concerned with a realisation of topological theories in terms of statistical models known as Calabi-Yau crystals. The thesis starts with an introduction and review of topological field and string theories. Subsequently several new results are presented. The main focus of the thesis is on the topological string theory. In this case crystal models correspond to three-dimensional partitions and their relations with the topological vertex theory and knot invariants are studied. Two-dimensional crystal models corresponding to topological gauge theories on ALE spaces are also introduced and analysed. Essential mathematical tools are summarised in appendices.
We present a self-contained review of the theory of dislocation-mediated quantum melting at zero temperature in two spatial dimensions. The theory describes the liquid-crystalline phases with spatial symmetries in between a quantum crystalline solid and an isotropic superfluid: quantum nematics and smectics. It is based on an Abelian-Higgs-type duality mapping of phonons onto gauge bosons (stress photons), which encode for the capacity of the crystal to propagate stresses. Dislocations and disclinations, the topological defects of the crystal, are sources for the gauge fields and the melting of the crystal can be understood as the proliferation (condensation) of these defects, giving rise to the Anderson-Higgs mechanism on the dual side. For the liquid crystal phases, the shear sector of the gauge bosons becomes massive signaling that shear rigidity is lost. Resting on symmetry principles, we derive the phenomenological imaginary time actions of quantum nematics and smectics and analyze the full spectrum of collective modes. The quantum nematic is a superfluid having a true rotational Goldstone mode due to rotational symmetry breaking, and the origin of this deconfined mode is traced back to the crystalline phase. The two-dimensional quantum smectic turns out to be a dizzyingly anisotropic phase with the collective modes interpolating between the solid and nematic in a non-trivial way. We also consider electrically charged bosonic crystals and liquid crystals, and carefully analyze the electromagnetic response of the quantum liquid crystal phases. In particular, the quantum nematic is a real superconductor and shows the Meissner effect. Their special properties inherited from spatial symmetry breaking show up mostly at finite momentum, and should be accessible by momentum-sensitive spectroscopy.
The dislocation-mediated quantum melting of solids into quantum liquid crystals is extended from two to three spatial dimensions, using a generalization of boson-vortex or Abelian-Higgs duality. Dislocations are now Burgers-vector-valued strings that trace out worldsheets in spacetime while the phonons of the solid dualize into two-form (Kalb-Ramond) gauge fields. We propose an effective dual Higgs potential that allows for restoring translational symmetry in either one, two or three directions, leading to the quantum analogues of columnar, smectic or nematic liquid crystals. In these phases, transverse phonons turn into gapped, propagating modes while compressional stress remains massless. Rotational Goldstone modes emerge whenever translational symmetry is restored. We also consider electrically charged matter, and find amongst others that as a hard principle only two out of the possible three rotational Goldstone modes are observable using electromagnetic means.
We consider M-theory on compact spaces of G_2 holonomy constructed as orbifolds of the form (CY x S^1)/Z_2 with fixed point set Sigma on the CY. This describes N=1 SU(2) gauge theories with b_1(Sigma) chiral multiplets in the adjoint. For b_1=0, it generalizes to compact manifolds the study of the phase transition from the non-Abelian to the confining phase through geometrical S^3 flops. For b_1=1, the non-Abelian and Coulomb phases are realized, where the latter arises by desingularization of the fixed point set, while an S^2 x S^1 flop occurs. In addition, an extremal transition between G_2 spaces can take place at conifold points of the CY moduli space where unoriented membranes wrapped on CP^1 and RP^2 become massless.
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