Biframe bundle geometry: An extension of Rainich-Misner-Wheeler theory to include sources

Recently the original theory of Rainich, Misner, and Wheeler (RMW) has been shown to have a natural reformulation in terms of a new principal fiber bundle, namely the bundle of biframesL ² M over spacetime. We extend this new formalism further and show that the original RMW program can be generalized to include Einstein-Maxwell spacetimes with geometrical sources. The assumptions of a Riemannian connection one-form on the linear frame bundleLM and a general connection one-form onL ² M necessarily imply the existence of a difference formK. A generalization of the standard RMW theorem is developed which provides the necessary and sufficient conditions on an arbitrary triple (M, g, K) in order for this triple to be an Einstein-Maxwell spacetime with geometrical sources. All sources for the field equations associated with such spacetimes are geometrical, as they are constructible from the metricg, the difference formK, and their derivatives. The extension of the RMW program presented here introduces a second complexion vector, in addition to the standard RMW complexion vector, and the formalism reduces, in the special case of no sources, to the standard RMW program.     

Biframe bundle geometry and an extension of RMW theory: Application to a charged perfect fluid

An extension of the original Rainich-Misner-Wheeler (RMW) theorem to include Einstein-Maxwell spacetimes with geometrical sources has recently been accomplished by generalizing the geometrical arena from the linear frame bundleLM to the bundle of biframesL ² M. The assumptions of a Riemannian connection one-form onLM and a general connection one-form onL ² M necessarily implies the existence of a difference formK. We provide new algebraic and differential conditions on an arbitrary triple (M, g, K), in addition to those already imposed by the generalization of the RMW theorem, which guarantee the form of the coupled Einstein-Maxwell field equations associated with a charged perfect fluid spacetime. All physical quantities associated with these field equations, namely the Maxwell field strength, the mass-energy density, the pressure, the electric and magnetic charge to mass ratios, and the unit four velocity of the fluid, can be recovered from the geometry as they are constructible entirely from the metricg, the difference formK, and their derivatives.     

Dirac-Hestenes spinor fields on Riemann-Cartan manifolds

In this paper we study Dirac-Hestenes spinor fields (DHSF) on a four-dimensional Riemann-Cartan spacetime (RCST). We prove that these fields must be defined as certain equivalence classes of even sections of the Clifford bundle (over the RCST), thereby being certain particular sections of a new bundle named the spin-Clifford bundle (SCB). The conditions for the existence of the SCB are studied and are shown to be equivalent to Geroch's theorem concerning the existence of spinor structures in a Lorentzian spacetime. We introduce also the covariant and algebraic Dirac spinor fields and compare these with DHSF, showing that all three kinds of spinor fields contain the same mathematical and physical information. We clarify also the notion of (Crumeyrolle's) amorphous spinors (Dirac-Kähler spinor fields are of this type), showing that they cannot be used to describe fermionic fields. We develop a rigorous theory for the covariant derivatives of Clifford fields (sections of the Clifford bundle, CB) and of Dirac-Hestenes spinor fields. We show how to generalize the original Dirac-Hestenes equation in Minkowski spacetime for the case of RCST. Our results are obtained from a variational principle formulated through the multiform derivative approach to Lagrangian field theory in the Clifford bundle.     

The algebraic structure of cohomological field theory

The algebraic foundation of cohomological field theory is presented. It is shown that these theories are based upon realizations of an algebra which contains operators for both BRST and vector supersymmetry. Through a localization of this algebra, we construct a theory of cohomological gravity in arbitrary dimensions. The observables in the theory are polynomial, but generally non-local operators, and have a natural interpretation in terms of a universal bundle for gravity. As such, their correlation functions correspond to cohomology classes on moduli spaces of Riemannian connections. In this uniformization approach, different moduli spaces are obtained by introducing curvature singularities on codimension two submanifolds via a puncture operator. This puncture operator is constructed from a naturally occuring differential form of co-degree two in the theory, and we are led to speculate on connections between this continuum quantum field theory, and the discrete Regge calculus.     

On the algebraic structure of quantum mechanics

We present a reformulation of the axiomatic basis of quantum mechanics with particular reference to the manner in which the usual algebraic structures arise from certain natural physical requirements. Care is taken to distinguish between features of physical significance and those introduced for mathematical convenience. Our conclusion is that the usual algebraic structures cannot be significantly generalised without conflicting with our current experimental picture of processes occurring at the quantum level.     

Vanishing of the cosmological constant in non-Abelian Kaluza-Klein theories

We present a new approach to the unification of gravity and non-Abelian gauge fields in the framework of Kaluza-Klein theory. It consists in introducing a new connection on the (n + 4)-dimensional manifoldP (metrized principal fiber bundle). This connection is metrical, but with nonvanishing torsion. An enormous cosmological term in the Einstein equations vanishes due to this connection. The new connection simultaneously cancels Planck's mass term in the Dirac equation for the five-dimensional case. The usual interpretation of geodesic equations is still valid.     

Gravitational Descendants in Symplectic Field Theory

It was pointed out by Y. Eliashberg in his ICM 2006 plenary talk that the rich algebraic formalism of symplectic field theory leads to a natural appearance of quantum and classical integrable systems, at least in the case when the contact manifold is the prequantization space of a symplectic manifold. In this paper we generalize the definition of gravitational descendants in SFT from circle bundles in the Morse-Bott case to general contact manifolds. After we have shown using the ideas in Okounkov and Pandharipande (Ann Math 163(2):517–560, 2006) that for the basic examples of holomorphic curves in SFT, that is, branched covers of cylinders over closed Reeb orbits, the gravitational descendants have a geometric interpretation in terms of branching conditions, we follow the ideas in Cieliebak and Latschev ( [math.s6], 2007) to compute the corresponding sequence of Poisson-commuting functions when the contact manifold is the unit cotangent bundle of a Riemannian manifold.     

String theory and algebraic geometry of moduli spaces

It is shown how the algebraic geometry of the moduli space of Riemann surfaces entirely determines the partition function of Polyakov's string theory. This is done by using elements of Arakelov's intersection theory applied to determinants of families of differential operators parametrized by moduli space. As a result we write the partition function in terms of exponentials of Arakelov's Green functions and Faltings' invariant on Riemann surfaces. Generalizing to arithmetic surfaces, i.e. surfaces which are associated to an algebraic number fieldK, we establish a connection between string theory and the infinite primes ofK. As a result we conjecture that the usual partition function is a special case of a new partition function on the moduli space defined overK.     

On generally covariant quantum field theory and generalized causal and dynamical structures

We give an example of a generally covariant quasilocal algebra associated with the massive free field. Maximal, two-sided ideals of this algebra are algebraic representatives of external metric fields. In some sense, this algebra may be regarded as a concrete realization of Ekstein's ideas of presymmetry in quantum field theory. Using ideas from our example and from usual algebraic quantum field theory, we discuss a generalized scheme, in which maximal ideals are viewed as algebraic representatives of dynamical equations or Lagrangians. The considered frame is no quantum gravity, but may lead to further insight into the relation between quantum theory and space-time geometry.     

Remarks on the Renormalization Properties of Lorentz- and CPT-Violating Quantum Electrodynamics

In this work, we employ algebraic renormalization technique to show the renormalizability to all orders in perturbation theory of the Lorentz- and CPT-violating QED. Essentially, we control the breaking terms by using a suitable set of external sources. Thus, with the symmetries restored, a perturbative treatment can be consistently employed. After showing the renormalizability, the external sources attain certain physical values, which allow the recovering of the starting physical action. The main result is that the original QED action presents the three usual independent renormalization parameters. The Lorentz-violating sector can be renormalized by 19 independent parameters. Moreover, vacuum divergences appear with extra independent renormalization. Remarkably, the bosonic odd sector (Chern-Simons-like term) does not renormalize and is not radiatively generated. One-loop computations are also presented and compared with the existing literature.     

Extended Connection in Yang-Mills Theory

The three fundamental geometric components of Yang-Mills theory –gauge field, gauge fixing and ghost field– are unified in a new object: an extended connection in a properly chosen principal fiber bundle. To do this, it is necessary to generalize the notion of gauge fixing by using a gauge fixing connection instead of a section. From the equations for the extended connection’s curvature, we derive the relevant BRST transformations without imposing the usual horizontality conditions. We show that the gauge field’s standard BRST transformation is only valid in a local trivialization and we obtain the corresponding global generalization. By using the Faddeev-Popov method, we apply the generalized gauge fixing to the path integral quantization of Yang-Mills theory. We show that the proposed gauge fixing can be used even in the presence of a Gribov’s obstruction.     

On the natural line bundle on the moduli space of stable parabolic bundles

We construct the natural holomorphic line bundle on the moduli space of stable parabolic bundles on a compact marked Riemann surface, which is the prequantum line bundle for the Chern-Simons gauge theory. The fusion rule in the Chern-Simons gauge theory can be viewed as the existence condition of this line bundle.     

Off-shell states in dual resonance theory

We discuss ooff-shell states guided by an analogue model approach. This leads us to a more complete understanding of a model proposed recently by Schwarz with critical dimension 16. We are led, by algebraic considerations, to off-shell states in the Neveu- Schwarz-Ramond model, which obey the gauge conditions in the same critical dimension as the on-shell theory, the amplitudes factorizing on the usual positive definite states in 10 dimensions. Brief calculations reveal that some of the divergences present in the orbital model disappear in the fermion theory.     

Lifting geometric objects to a cotangent bundle, and the geometry of the cotangent bundle of a tangent bundle

We show that the cotangent bundle T^*T of the tangent bundle of any differentiable manifold carries an integrable almost tangent structure which is generated by a natural lifting procedure from the canonical almost tangent structure (vertical endomorphism) of T . Using this almost tangent structure we show that T^*T is diffeomorphic to a tangent bundle, namely TT^* . This provides a new and geometrically instructive proof of a result of Tulczyjew, which has applications in Lagrangian and Hamiltonian dynamics and in field theory The requisite general definitions and results concerning liftings of geometric objects from a manifold to its cotangent bundle are given. As an application, we shed new light on the meaning of so-called adjoint symmetries of second-order differential equations.     

Unoriented WZW Models and Holonomy of Bundle Gerbes

The Wess-Zumino term in two-dimensional conformal field theory is best understood as a surface holonomy of a bundle gerbe. We define additional structure for a bundle gerbe that allows to extend the notion of surface holonomy to unoriented surfaces. This provides a candidate for the Wess-Zumino term for WZW models on unoriented surfaces. Our ansatz reproduces some results known from the algebraic approach to WZW models. manche meinen lechts und rinks kann man nicht velwechsern werch ein illtum Ernst Jandl [Jan95] K.W. is supported with scholarships by the German Israeli Foundation (GIF) and by the Rudolf und Erika Koch–Stiftung.     

Algebraic Approach to Gauge Theory and Its Non-Commutative Extension

We propose an alternative algebraic construction of the connection theory on the fiber bundle, and find its non-commutative extension, i.e., the connection theory based on guantum groups. In thb construction, the Yang-Baxter equation arises ae the exchange relations of connections and a deformed trace of quantum groups is introduced to build up the characteristic classes.     

Morse homotopy and Chern-Simons perturbation theory

We define and invariant of a three manifold equipped with a flat bundle with vanishing homology. The construction is based on Morse theory using several Morse functions simultaneously and is regarded as a higher loop analogue of various product operations in algebraic topology. There is a heuristic argument that this invariant is related to perturbative Chern-Simons Gauge theory by Axelrod-Singer, etc. There is also a theorem which gives a relation of the construction to open string theory on the cotangent bundle.Partially supported by Grants-in-Aid for Scientific Research on Priority Areas 231 Infinite Analysis.     

Scaling behaviour in the fracture of fibrous materials

We study the existence of distinct failure regimes in a model for fracture in fibrous materials. We simulate a bundle of parallel fibers under uniaxial static load and observe two different failure regimes: a catastrophic and a slowly shredding. In the catastrophic regime the initial deformation produces a crack which percolates through the bundle. In the slowly shredding regime the initial deformations will produce small cracks which gradually weaken the bundle. The boundary between the catastrophic and the shredding regimes is studied by means of percolation theory and of finite-size scaling theory. In this boundary, the percolation density scales with the system size L, which implies the existence of a second-order phase transition with the same critical exponents as those of usual percolation. Received 24 June 1999     

Even dimensional generalization of Chern-Simons action and new gauge symmetry

We propose a new even-dimensional action which shares close algebraic similarities with the Chern-Simons action and thus possesses a topological nature. This action has a new type of gauge symmetry in the sense that adjoint representation is not enough to close the gauge transformation and gauge fermions are incorporated. Quaternionic structure emerges as a natural algebra to control the different natures of even forms, odd forms, bosons and fermions. We claim that the bundle structure in consideration is mathematically a new object.