Geometry and Physics Today

“Geometry,” in the sense of the classical differential geometry of smooth manifolds (CDG), is put under scrutiny from the point of view of Abstract Differential Geometry (ADG). We explore potential physical implications of viewing things under the light of ADG, especially matters concerning the “gauge theories” of modern physics, when the latter are viewed (as they are actually regarded currently) as “physical theories of a geometrical character.” Thence, “physical geometry,” in connection with physical laws and the associated with them, within the background spacetime manifoldless context of ADG, “differential” equations, are also being discussed.     

Spectral action and big desert

The values of the Higgs mass are obtained for two possibilities of extending the standard model in a way compatible with the existence of a noncommutative structure at high energies. We assume the existence of a big desert between the low energy electroweak scale and the high energy scale Λ=1.1×10¹⁷ GeV, where noncommutative features become relevant. We conclude that it is extremely difficult to depart from the Higgs mass value obtained from noncommutative geometry for the standard model with three generations only.     

Characteristic classes of bad orbifold vector bundles

We show that every bad orbifold vector bundle can be realized as the restriction of a good orbifold vector bundle to a suborbifold of the base space. We give an explicit construction of this result in which the Chen–Ruan orbifold cohomologies of the two base spaces are isomorphic (as additive groups). This construction is used to indicate an extension of the Chern–Weil construction of characteristic classes to bad orbifold vector bundles. In particular, we apply this construction to the orbifold Euler class and demonstrate that it acts as an obstruction to the existence of nonvanishing sections.     

Evolution of Grassmannian invariant-angle coherent states and nonadiabatic Hannay's angle

We show how the exact evolution and nonadiabatic Hannay's angle of Grassmannian classical mechanics of spin one half in a varying external magnetic field is associated with the evolution of Grassmannian invariant-angle coherent states. Received: 18 February 1998     

Two dimensional gauge theories revisited

Two dimensional quantum Yang-Mills theory is reexamined using a non-abelian version of the Duistermaat-Heckman integration formula to carry out the functional integral. This makes it possible to explain properties of the theory that are inaccessible to standard methods and to obtain general expressions for intersection pairings on moduli spaces of flat connections on a two dimensional surface. The latter expressions agree, for gauge group SU (2), with formulas obtained recently by several methods.     

Self-Dual Solutions for SU(2) and SU(3) Gauge Fields on Euclidean Space

Self-dual solutions for SU(2) gauge fields on Euclidean space that satisfy Yang's ansatz are generalized by considering as a function of for a special case when is a complex analytic function and for SU(3) when _i, i = 1, 2, 3, are complex analytic functions.     

Quantization of constrained systems: A comparative look at three methods

Constrained Hamiltonian systems are investigated using three different methods: Hamilton-Jacobi, Batalin-Fradkin-Tyutin and gauge unfixing methods. The abelian Proca model is analyzed and the involutive Hamiltonian is obtained by the three methods.     

On eleven-dimensional supergravity and Chern-Simons theory

We probe in some depth into the structure of eleven-dimensional, osp(32|1)-based Chern-Simons supergravity, as put forward by Troncoso and Zanelli (TZ) in 1997. We find that the TZ Lagrangian may be cast as a polynomial in 1/l, where l is a length, and compute explicitly the first three dominant terms. The term proportional to 1/l⁹ turns out to be essentially the Lagrangian of the standard 1978 supergravity theory of Cremmer, Julia and Scherk, thus establishing a previously unknown relation between the two theories. The computation is nontrivial because, when written in a sufficiently explicit way, the TZ Lagrangian has roughly one thousand non-explicitly Lorentz-covariant terms. Specially designed algebraic techniques are used to accomplish the results.     

Geometric Formulation of Gauge Theory of Gravity

Differential geometric formulation of quantum gauge theory of gravity is studied in this paper. The quantumgauge theory of gravity is formulated completely in the framework of traditional quantum field theory. In order to studythe relationship between quantum gauge theory of gravity and traditional quantum gravity which is formulated in curvedspace, it is important to set up the geometry picture of quantum gauge theory of gravity. The correspondence betweenquantum gauge theory of gravity and differential geometry is discussed and the geometry picture of quantum gaugetheory of gravity is studied.     

Formal and Analytic Deformations of the Witt Algebra

When studying gauge theories (e.g. with finite energy conditions), attention is traditionally restricted to the subset of irreducible connections, which is open and dense in the full space of connections. We point out that generally the residual set of reducible connections contains critical points of the gauge functionals which, moreover, are the only ones common to all theories with a given symmetry, i.e. those determined by the symmetry and geometry of the problem alone, and not by the specific choice of functional.     

Inequivalent quantizations in multiply connected spaces

Summary The novel features which show up in the process of quantization of a dynamical system on a multiply (nonsimply)-connected configuration space are analysed in the present paper. After rediscussing the path integral approach to the problem, we show how one can give a fiber bundle classification of the inequivalent quantizations associated with nonsimply connected spaces. We discuss various examples, and the generalization to the case in which, due to internal symmetries, a system can admit of nonscalar quantizations. To speed up publication, the authors have agreed not to receive proofs which have been supervised by the Scientific Committee.     

Monopole and Coulomb Field as Duals within the Unifying Reissner-NordstrSm Geometry

We are going to prove that the Monopole and the Coulomb fields are duals within the unifying structure provided by the Reissner-Nordstrom spacetime. This is accomplished when noticing that in order to produce the tetrad that locally and covariantly diagonalizes the stress-energy tensor, both the Monopole and the Coulomb fields are necessary in the construction. Without any of them it would be impossible to express the tetrad vectors that locally and covariantly diagonalize the stress-energy tensor. Then, both electromagnetic fields are an integral part of the same structure, the Reissner-Nordstrom geometry.