Weyl's geometry and massive vector fields

Dirac's revitalization of Weyl's gauge invariant geometry is examined. Weyl's geometry is characterized by the breaking of the symmetries of charge conjugation (C) and time reversal (T), but not CT. When an attempt is made to introduce a very simple form of symmetry breaking material into the theory, it is found that the geometric role formerly played by the electromagnetic field is suddenly taken over by a very short range massive vector field.Research supported in part by the U.S. Atomic Energy Commission and the Research, Publications and Consulting Committee of West Virginia Institute of Technology.     

Noncommutative differential geometry, and the matrix representations of generalised algebras

The underly ing algebra I or a noncummutative geometry is taken to be a matrix algebra, and the set of derivatives the ad joint of a subset of traceless matrices. This is sufficient to calculate the dual 1-forms, and show that the space of 1-firms is at free module over the algebra of matrices. The concept of a generalised algebra is delined and it is shown that this is required in order for the space of 2-forms to exist, The exterior derivative is generalised for higher-order forms and these are also shown to he free modules over the matrix algebra. Examples of mappings that preserve the differential Structure are peen, Also giken are four examples of matrix generalised algebras, and the corresponding noncommutntive geometries, including the cases where the generalised algebra corresponds to a representation of a Lie algebra or a q-deformed algebra.     

General N=1 supersymmetric flux vacua of massive type IIA string theory

We derive conditions for the existence of four-dimensional N=1 supersymmetric flux vacua of massive type IIA string theory with general supergravity fluxes turned on. For an SU(3) singlet Killing spinor, we show that such flux vacua exist when the internal geometry is nearly K?hler. The geometry is not warped, all the allowed fluxes are proportional to the mass parameter, and the dilaton is fixed by a ratio of (quantized) fluxes. The four-dimensional cosmological constant, while negative, becomes small in the vacuum with the weak string coupling.     

Exceptional Calabi–Yau spaces: the geometry of backgrounds with flux

In this paper we define the analogue of Calabi–Yau geometry for generic , flux backgrounds in type II supergravity and M‐theory. We show that solutions of the Killing spinor equations are in one‐to‐one correspondence with integrable, globally defined structures in generalised geometry. Such “exceptional Calabi–Yau” geometries are determined by two generalised objects that parametrise hyper‐ and vector‐multiplet degrees of freedom and generalise conventional complex, symplectic and hyper‐Kähler geometries. The integrability conditions for both hyper‐ and vector‐multiplet structures are given by the vanishing of moment maps for the “generalised diffeomorphism group” of diffeomorphisms combined with gauge transformations. We give a number of explicit examples and discuss the structure of the moduli spaces of solutions. We then extend our construction to and flux backgrounds preserving eight supercharges, where similar structures appear, and finally discuss the analogous structures in generalised geometry.     

Topological Invariants for Lens Spaces and Exceptional Quantum Groups

The Reshetikhin–Turaev invariants arising from the quantum groups associated with the exceptional Lie algebras G2, F4 and E8 at odd roots of unity are constructed and explicitly computed for all the lens spaces.     

Classical and thermodynamic limits for generalised quantum spin systems

We prove that the rescaled upper and lower symbols for arbitrary generalised quantum spin systems converge in the classical limit. For a large class of models this enables us to derive the asyptotics of quantum free energies in the classical and in the thermodynamic limit.Research supported by a European Science Exchange Fellowship of the Royal Society, London     

The relation between the generalised Eshelby integral and the generalised BCS and DB models

The generalised BCS dislocation group model and the generalised DB atomic cohesive force zone model have obtained the same results on nonlinear fracture study of some one-,two-and three-dimensional quasicrystals.This work reveals some inherent connection between the two models,and finds that their common basis is the generalised Eshelby integral based on the generalised Eshelby energy-momentum tensor for quasicrystals.Further applications of the theory in solving nonlinear fracture problems of the materials are also discussed.     

Compatibility and generalised conditional symmetries

It is shown that the determining equations for generalised conditional symmetries (GCSs) of order n, of an evolution equation of arbitrary order, can be found as a consequence of compatibility with an nth-order invariant surface condition. The compatibility technique is demonstrated on a second-order nonlinear diffusion–convection equation with absorption and used to find new GCSs of a linear diffusion equation with nonlinear source.     

Miura maps and reductions

It is shown that the specialized forms of the generalized modified Korteweg-de Vries equations for the loop algebra A⁽¹⁾_n?1 reduce to the corresponding Korteweg-de Vries equations under certain Miura-type transformations.