Spin geometry and grand unification

Gravitation becomes unified with quantum mechanics when we recognize that the spacetime tetrads and the matter fields of Fermions are the integral and half-integral spin representations of theEinstein group, E, the global extension of the Poincaré group to a curved spacetimeM. There are8 fundamental spinor representations of theE group, interchanged byP, T, andC: the degree-one maps of spin space overM. Tensor products of2 spinor fields buildClifford vectors or 1 forms, e.g. the spacetime tetrads. It takes tensor products of all8 spinor fields to build a natural 4 form; in particular, ourE-invariant Lagrangian density . We propose a simple form for : the8-spinor factorization of theMaurer-Cartan 4-form, Ω⁴. Thespin connections Ω_α step off the conjoined left and right internalgl (2, ?) phase increments over aspacetime incremente _α. Our actionS _g measures the covering number of the spinor phases over spacetimeM∪D _J; theD _J aresingular domains or caustics, whereJ=1, 2, and 3 chiral pairs of spin waves cross. Here, the massive Dirac equations emerge to govern the mass scattering that keep the “null zig-zags” of a bispinor particle confined to a timelike worldtube. We identify the coupled envelopes of 1, 2, and 3 chiral bispinor pairs as the leptons, mesons, and hadrons, respectively. These source topologically —nontrivialgl (2,C) phase distributions in the far-field region, which appear aseffective vector potentials. Their vorticities are thespin curvatures, whose Hermitian parts —thegravitational curvatures —specify how our spacetime manifoldM must expand and curve to accommodate such anholonomic differentials. The anti-Hermitian parts reproduce the standard electroweak and strong fields, together with their actions. also contains some new cross terms between electroweak potentials and gravitational curvatures. Do these signal a failure of unification, or predict new phenomena?     

About positive invariance and asymptotic stability

In this note we consider the differential equation , t0 with initial conditionx(0)G, whereG is a closed pointed convex cone. Our results pertain to positive invariance and asymptotic stability. A result by Stern, [3], will be improved. Subtangentiality will also be discussed as this notion is important in deriving results about positive invariance.     

The Wu and Yang monopole solution in curved spacetime

Abstract

A simplified treatment of the derivation of the Wu and Yang monopole solution on a curved spacetime is given and the similarity to Maxwell theory with minimal coupling is pointed out.¹     

Topological conformal field theories and Calabi-Yau categories

This is the first of two papers which construct a purely algebraic counterpart to the theory of Gromov-Witten invariants (at all genera). These Gromov-Witten type invariants depend on a Calabi-Yau A_∞ category, which plays the role of the target in ordinary Gromov-Witten theory. When we use an appropriate A_∞ version of the derived category of coherent sheaves on a Calabi-Yau variety, this constructs the B model at all genera. When the Fukaya category of a compact symplectic manifold X is used, it is shown, under certain assumptions, that the usual Gromov-Witten invariants are recovered. The assumptions are that open-closed Gromov-Witten theory can be constructed for X, and that the natural map from the Hochschild homology of the Fukaya category of X to the ordinary homology of X is an isomorphism.     

Non-autonomous systems: asymptotic behaviour and weak invariance principles

Results pertaining to asymptotic behaviour of solutions of non-autonomous ordinary differential equations with locally integrably bounded right-hand sides are presented. Ramifications for weakly asymptotically autonomous systems and adaptively controlled systems are highlighted.     

Factorizable ribbon quantum groups in logarithmic conformal field theories

We review the properties of quantum groups occurring as the Kazhdan-Lusztig dual to logarithmic conformal field theory models. These quantum groups at even roots of unity are not quasitriangular but are factorizable and have a ribbon structure; the modular group representation on their center coincides with the representation on generalized characters of the chiral algebra in logarithmic conformal field models. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 3, pp. 510–535, March, 2008.     

q-Deformation of the Virasoro algebra and lattice conformal theories

A natural definition of the q-deformation of the Virasoro and superconformal algebras is suggesteded. New Liealgebraic symmetries describe a lattice version of the continual theory. A close link between the deformation constructed in this paper and the lattice version of the Faddeev-Takhtadjian-Volkov Virasaro algebra is shown. 18 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 235, 1900, pp. 217–227.     

Universality in the 2D Ising model and conformal invariance of fermionic observables

It is widely believed that the celebrated 2D Ising model at criticality has a universal and conformally invariant scaling limit, which is used in deriving many of its properties. However, no mathematical proof has ever been given, and even physics arguments support (a priori weaker) M?bius invariance. We introduce discrete holomorphic fermions for the 2D Ising model at criticality on a large family of planar graphs. We show that on bounded domains with appropriate boundary conditions, those have universal and conformally invariant scaling limits, thus proving the universality and conformal invariance conjectures.     

Gauge invariance and asymptotic behavior for the Ginzburg-Landau equations of superconductivity

In this paper we study the gauge invariance of the time-dependent Ginzburg-Landau equations through the introduction of a model which uses observable variables. We observe that the various choices of gauge lead to a different representation of such variables and therefore to a different definition of the weak solution of the problem. With a suitable decomposition of the unknown fields, related to the choice of London gauge, we examine the Ginzburg-Landau equations and deduce some energy estimates which prove the existence of a maximal attractor for the system.