Non-trivial Curvature Research Articles

Primordial bispectrum from multifield inflation with nonminimal couplings

Realistic models of high-energy physics include multiple scalar fields. Renormalization requires that the fields have nonminimal couplings to the spacetime Ricci curvature scalar, and the couplings can be large at the energy scales of early-universe inflation. The nonminimal couplings induce a nontrivial field-space manifold in the Einstein frame, and they also yield an effective potential in the Einstein frame with nontrivial curvature. The ridges or bumps in the Einstein-frame potential can lead to primordial non-Gaussianities of observable magnitude. We develop a covariant formalism to study perturbations in such models and calculate the primordial bispectrum. As in previous studies of non-Gaussianities in multifield models, our results for the bispectrum depend sensitively on the fields' initial conditions.

Rigid flat webs on the projective plane

This paper studies global webs on the projective plane with vanishing curvature. The study is based on an interplay of local and global arguments. The main local ingredient is a criterium for the regularity of the curvature at the neighborhood of a generic point of the discriminant. The main global ingredient, the Legendre transform, is an avatar of classical projective duality in the realm of differential equations. We show that the Legendre transform of what we call reduced convex foliations are webs with zero curvature, and we exhibit a countable infinity family of convex foliations which give rise to a family of webs with zero curvature not admitting non-trivial deformations with zero curvature.

On Breakdown Criteria for Nonvacuum Einstein Equations

The recent “breakdown criterion” result of Klainerman and Rodnianski (On the breakdown criterion in general relativity, Preprint, 2009) stated roughly that an Einstein-vacuum spacetime, given as a CMC foliation, can be further extended in time if the second fundamental form and the derivative of the lapse of the foliation are uniformly bounded. This theorem and its proof were extended to Einstein-scalar and Einstein–Maxwell spacetimes in the thesis (Shao in Breakdown criteria for nonvacuum Einstein equations, Ph.D. thesis, Princeton University, 2010). In this paper, we state the main results of Shao (2010), and we summarize and discuss their proofs. In particular, we will discuss the various issues resulting from nontrivial Ricci curvature and the coupling between the Einstein and the field equations.

Quantum backreaction in evolving FLRW spacetimes

Quantum fluctuations of a nonminimally coupled scalar field in $D$-dimensional homogeneous and isotropic background are calculated within the operator formalism in curved models with time evolutions of the scale factor that allow smooth transitions between contracting and expanding and between decelerating and accelerating regimes. The coincident propagator is derived and used to compute the one-loop backreaction from the scalar field. The inflationary infrared divergences are absent in Bunch-Davies vacuum when taking into account a preceding cosmological era or spatial curvature which can be either positive or negative. It is found that asymptotically, the backreaction energy density in the minimally coupled case grows logarithmically with the scale factor in quasi--de Sitter space and in a class of models decays in slow-roll inflation and grows as a power law during superinflation. The backreaction increases generically in a contracting phase or in the presence of a negative nonminimal coupling. The effects of the coupling and renormalization scale upon the quantum fluctuations together with the novel features due to nontrivial time evolution and spatial curvature are clarified with exact solutions and numerical examples.

Formula omitted]-duality in Abelian gauge theory revisited

Definition of the partition function of U ( 1 ) gauge theory is extended to a class of four-manifolds containing all compact spaces and certain asymptotically locally flat (ALF) ones including the multi-Taub–NUT spaces. The partition function is calculated via zeta-function regularization and heat kernel techniques with special attention to its modular properties. In the compact case, compared with the purely topological result of Witten, we find a non-trivial curvature correction to the modular weights of the partition function. But the S -duality can be restored by adding gravitational counter terms to the Lagrangian in the usual way. In the ALF case however we encounter non-trivial difficulties stemming from original non-compact ALF phenomena. Fortunately our careful definition of the partition function makes it possible to circumnavigate them and conclude that the partition function has the same modular properties as in the compact case.

Exact gravitational dual of a plasma ball

We present an exact solution for a black hole localized near an infrared wall in four-dimensional anti-deSitter space. By computing the holographic stress tensor we show that the CFT dual of the black hole is a 2+1-dimensional ball (i.e., a disk) of plasma at finite temperature, surrounded by vacuum. This confirms some earlier conjectures about plasma balls in AdS/CFT. We also estimate the value of the surface tension for the ball. The solution displays a number of peculiarities, most notably a non-trivial curvature of the boundary geometry, as well as other properties associated to the vanishing deconfinement temperature of the set up. We discuss how these features are related to specific physics at the infrared and ultraviolet boundaries for this solution, and should not be generic properties of plasma balls.

Skyrmions with holography and hidden local symmetry

We study baryons as Skyrmions in holographic QCD with $\mathrm{D}4/\mathrm{D}8/\overline{\mathrm{D}8}$ multi-D brane system in type IIA superstring theory, and also in the nonlinear sigma model with hidden local symmetry. Comparing these two models, we find that the extra dimension and its nontrivial curvature can largely change the role of (axial) vector mesons for baryons in four-dimensional space-time. In the hidden local symmetry approach, the $\ensuremath{\rho}$-meson field as a massive Yang-Mills field has a singular configuration in Skyrmion, which gives a strong repulsion for the baryon as a stabilizer. When the ${a}_{1}$ meson is added in this approach, the stability of Skyrmion is lost by the cancellation of $\ensuremath{\rho}$ and ${a}_{1}$ contributions. On the contrary, in holographic QCD, the $\ensuremath{\rho}$-meson field does not appear as a massive Yang-Mills field due to the extra dimension and its nontrivial curvature. We show that the $\ensuremath{\rho}$-meson field has a regular configuration in Skyrmion, which gives a weak attraction for the baryon in holographic QCD. We argue that Skyrmion with $\ensuremath{\pi}$, $\ensuremath{\rho}$, and ${a}_{1}$ mesons become stable due to the curved extra dimension and also the presence of the Skyrme term in holographic QCD. From this result, we also discuss the features of our truncated-resonance analysis on baryon properties with $\ensuremath{\pi}$ and $\ensuremath{\rho}$ mesons below the cutoff scale ${M}_{\mathrm{KK}}\ensuremath{\sim}1\text{ }\text{ }\mathrm{GeV}$ in holographic QCD, which is compared with other 5D instanton analysis.

Comment on “Nontrivial geometries: Bounds on the curvature of the universe”

The paper “Nontrivial geometries: Bounds on the curvature of the universe” [L. Mersini-Houghton, E. Kafexhiu, Nontrivial geometries: bounds on the curvature of the universe, Astropart. Phys. 29 (2008) 167 [0705.0332v1 [astro-ph]]] seeks to study the effect of non-trivial spatial curvature in homogeneous and isotropic models. We note that the space considered is not homogeneous, and that the equations of motion used are inconsistent with the metric. Also, we explain why the spatial curvature of homogeneous and isotropic spacetimes always evolves like 1 / a 2 , contrary to the central assumption of [1].

Fermions in three-dimensional spinfoam quantum gravity

We study the coupling of massive fermions to the quantum mechanical dynamics of spacetime emerging from the spinfoam approach in three dimensions. We first recall the classical theory before constructing a spinfoam model of quantum gravity coupled to spinors. The technique used is based on a finite expansion in inverse fermion masses leading to the computation of the vacuum to vacuum transition amplitude of the theory. The path integral is derived as a sum over closed fermionic loops wrapping around the spinfoam. The effects of quantum torsion are realised as a modification of the intertwining operators assigned to the edges of the two-complex, in accordance with loop quantum gravity. The creation of non-trivial curvature is modelled by a modification of the pure gravity vertex amplitudes. The appendix contains a review of the geometrical and algebraic structures underlying the classical coupling of fermions to three dimensional gravity.

Canonical multi-symplectic structure on the total exterior algebra bundle

The aim of this paper is to construct multi-symplectic structures starting with the geometry of an oriented Riemannian manifold, independent of a Lagrangian or a particular partial differential equation (PDE). The principal observation is that on an n -dimensional orientable manifold M there is a canonical quadratic form Θ associated with the total exterior algebra bundle on M . On the fibre, which has dimension 2 n , the form Θ can be locally decomposed into n classical symplectic structures. When concatenated, these n -symplectic structures define a partial differential operator, J ∂ , which turns out to be a Dirac operator with multi-symplectic structure. The operator J ∂ generalizes the product operator J (d/d t ) in classical symplectic geometry, and M is a generalization of the base manifold (i.e. time) in classical Hamiltonian dynamics. The structure generated by Θ provides a natural setting for analysing a class of covariant nonlinear gradient elliptic operators. The operator J ∂ is elliptic, and the generalization of Hamiltonian systems, J ∂ Z =∇ S ( Z ), for a section Z of the total exterior algebra bundle, is also an elliptic PDE. The inverse problem—find S ( Z ) for a given elliptic PDE—is shown to be related to a variant of the Legendre transform on k -forms. The theory is developed for flat base manifolds, but the constructions are coordinate free and generalize to Riemannian manifolds with non-trivial curvature. Some applications and implications of the theory are also discussed.

POINCARÉ GAUGE GRAVITY: SELECTED TOPICS

In the gauge theory of gravity based on the Poincaré group (the semidirect product of the Lorentz group and the spacetime translations) the mass (energy–momentum) and the spin are treated on an equal footing as the sources of the gravitational field. The corresponding spacetime manifold carries the Riemann–Cartan geometric structure with the nontrivial curvature and torsion. We describe some aspects of the classical Poincaré gauge theory of gravity. Namely, the Lagrange–Noether formalism is presented in full generality, and the family of quadratic (in the curvature and the torsion) models is analyzed in detail. We discuss the special case of the spinless matter and demonstrate that Einstein's theory arises as a degenerate model in the class of the quadratic Poincaré theories. Another central point is the overview of the so-called double duality method for constructing of the exact solutions of the classical field equations.

Monte Carlo based modeling of carbon nanostructured surfaces

We have developed Monte Carlo based algorithms to produce realistic models of complex carbon nanostructures with nontrivial curvature, including fullerene junctions between nanotubes. The models are constructed by first analytically defining curved surfaces and then optimizing the configuration of carbon atoms on these surfaces using a realistic interatomic potential. We illustrate our method by generating several previously proposed and also new types of structures, which all show realistic physical properties. Our method is not limited to these structures and can be used to generate large models of nanostructured materials with complex surface geometries and porous structure on the nano-scale.

Cartan–Weyl Dirac and Laplacian Operators, Brownian Motions: The Quantum Potential and Scalar Curvature, Maxwell’s and Dirac-Hestenes Equations, and Supersymmetric Systems

We present the Dirac and Laplacian operators on Clifford bundles over space–time, associated to metric compatible linear connections of Cartan–Weyl, with trace-torsion, Q. In the case of nondegenerate metrics, we obtain a theory of generalized Brownian motions whose drift is the metric conjugate of Q. We give the constitutive equations for Q. We find that it contains Maxwell’s equations, characterized by two potentials, an harmonic one which has a zero field (Bohm-Aharonov potential) and a coexact term that generalizes the Hertz potential of Maxwell’s equations in Minkowski space.We develop the theory of the Hertz potential for a general Riemannian manifold. We study the invariant state for the theory, and determine the decomposition of Q in this state which has an invariant Born measure. In addition to the logarithmic potential derivative term, we have the previous Maxwellian potentials normalized by the invariant density. We characterize the time-evolution irreversibility of the Brownian motions generated by the Cartan–Weyl laplacians, in terms of these normalized Maxwell’s potentials. We prove the equivalence of the sourceless Maxwell equation on Minkowski space, and the Dirac-Hestenes equation for a Dirac-Hestenes spinor field written on Minkowski space provided with a Cartan–Weyl connection. If Q is characterized by the invariant state of the diffusion process generated on Euclidean space, then the Maxwell’s potentials appearing in Q can be seen alternatively as derived from the internal rotational degrees of freedom of the Dirac-Hestenes spinor field, yet the equivalence between Maxwell’s equation and Dirac-Hestenes equations is valid if we have that these potentials have only two components corresponding to the spin-plane. We present Lorentz-invariant diffusion representations for the Cartan–Weyl connections that sustain the equivalence of these equations, and furthermore, the diffusion of differential forms along these Brownian motions. We prove that the construction of the relativistic Brownian motion theory for the flat Minkowski metric, follows from the choices of the degenerate Clifford structure and the Oron and Horwitz relativistic Gaussian, instead of the Euclidean structure and the orthogonal invariant Gaussian. We further indicate the random Poincare–Cartan invariants of phase-space provided with the canonical symplectic structure. We introduce the energy-form of the exact terms of Q and derive the relativistic quantum potential from the groundstate representation. We derive the field equations corresponding to these exact terms from an average on the invariant state Cartan scalar curvature, and find that the quantum potential can be identified with 1 / 12R(g), where R(g) is the metric scalar curvature. We establish a link between an anisotropic noise tensor and the genesis of a gravitational field in terms of the generalized Brownian motions. Thus, when we have a nontrivial curvature, we can identify the quantum nonlocal correlations with the gravitational field. We discuss the relations of this work with the heat kernel approach in quantum gravity. We finally present for the case of Q restricted to this exact term a supersymmetric system, in the classical sense due to E.Witten, and discuss the possible extensions to include the electromagnetic potential terms of Q

Quantum magnetic algebra and magnetic curvature

The symplectic geometry of the phase space associated with a charged particle is determined by the addition of the Faraday 2-form to the standard structure on the Euclidean phase space. In this paper we describe the corresponding algebra of Weyl-symmetrized functions in coordinate and momentum operators satisfying nonlinear commutation relations. The multiplication in this algebra generates an associative product of functions on the phase space. This product is given by an integral kernel whose phase is the symplectic area of a groupoid-consistent membrane. A symplectic phase space connection with non-trivial curvature is extracted from the magnetic reflections associated with the Stratonovich quantizer. Zero and constant curvature cases are considered as examples. The quantization with both static and time dependent electromagnetic fields is obtained. The expansion of the product by the deformation parameter, written in the covariant form, is compared with the known deformation quantization formulas.

QUANTUM HOLONOMIES FOR QUANTUM COMPUTING

Holonomic Quantum Computation (HQC) is an all-geometrical approach to quantum information processing. In the HQC strategy information is encoded in degenerate eigen-spaces of a parametric family of Hamiltonians. The computational network of unitary quantum gates is realized by driving adiabatically the Hamiltonian parameters along loops in a control manifold. By properly designing such loops the nontrivial curvature of the underlying bundle geometry gives rise to unitary transformations i.e., holonomies that implement the desired unitary transformations. Conditions necessary for universal QC are stated in terms of the curvature associated to the non-abelian gauge potential (connection) over the control manifold. In view of their geometrical nature the holonomic gates are robust against several kind of perturbations and imperfections. This fact along with the adiabatic fashion in which gates are performed makes in principle HQC an appealing way towards universal fault-tolerant QC.

Non-linear inplane deformation and buckling of rings and high arches

A non-linear theory is presented for stretching and inplane-bending of isotropic beams which have constant initial curvature and lie in their plane of symmetry. For the kinematics, the geometrically exact one-dimensional (1-D) measures of deformation are specialized for small strain. The 1-D constitutive law is developed in terms of these measures via an asymptotically correct dimensional reduction of the geometrically non-linear 3-D elasticity under the assumptions of comparable magnitudes of initial radius of curvature and wavelength of deformation, small strain, and small ratio of cross-sectional diameter to initial radius of curvature ( h/R). The 1-D constitutive law contains an asymptotically correct refinement of O(h/R) beyond the usual stretching and bending strain energies which, for doubly symmetric cross sections, reduces to a stretch–bending elastic coupling term that depends on the initial radius of curvature and Poisson’s ratio. As illustrations, the theory is applied to inplane deformation and buckling of rings and high arches. In spite of a very simple final expression for the second variation of the total potential, it is shown that the only restriction on the validity of the buckling analysis is that the prebuckling strain remains small. Although the term added in the refined theory does not affect the buckling loads, it is shown that non-trivial prebuckling displacements, curvature, and bending moment of high arches are impossible to calculate accurately without this term.

Generalized supergravity in two dimensions

Among the usual constraints of (1,1) supergravity in d = 2 the condition of vanishing bosonic torsion is dropped. Using the inverse supervierbein and the superconnection considerably simplifies the formidable computational problems. It allows us to solve the constraints for those fields before taking into account the (identically fulfilled) Bianchi identities. The relation of arbitrary functions in the seminal paper of Howe to supergravity multiplets is clarified. The local supersymmetry transformations remain the same, but, somewhat surprisingly, the transformations of zweibein and Rarita-Schwinger field decouple from those of the superconnection multiplet. A method emerges naturally of how to construct ‘non-Einsteinian’ supergravity theories with non-trivial curvature and torsion in d = 2 which, apart from their intrinsic interest, may be relevant for models of super black holes and for novel generalizations in superstring theories. Several explicit examples of such models are presented, some of which immediately allow for a dilatonic formulation for the bosonic part of the action.

Grassmanian and chiral anomaly

We discuss the Grassmanian of self-adjoint global elliptic boundary conditions with γ 5- and gauge-invariance of the domain for the Dirac operator over the 4-ball coupled to a gauge configuration with non-trivial curvature form. We show that this space contains a variety of boundary conditions in addition to the spectral Atiyah-Patodi-Singer projection and that some of them, like the Calderón projector, imply the vanishing of the index of the Dirac operator and therefore the invariance of the fermion determinant under global (i.e. rigid) chiral transformations.

Curved-space quantization: Toward a resolution of the Dirac versus reduced quantization question.

It is well known that Dirac quantization of gauge theories is not, in general, equivalent to reduced quantization. When both approaches are self-consistent some additional criterion must be found in order to decide which approach is more natural, or correct. Now, in many cases quantization on the physical degrees of freedom is properly curved-space quantization, with a highly nontrivial curvature: neither constant nor Ricci flat. On the other hand, the configuration space of the unreduced gauge theory is often (Ricci) flat, which makes Dirac quantization considerably simpler. We show that the natural minimal'' Dirac quantization scheme, together with certain restrictions we impose on the observables, is sufficient to make the quantum commutator of quadratic observables free of van Hove anomalies. This means the minimal'' Dirac quantization (acting in the physical Hilbert space) is actually a curved-space quantization scheme suitable for the type of curvature mentioned above, at least within a restricted (but still interesting) class of observables. In fact, we demonstrate that this curved-space quantization scheme, unlike minimal'' reduced quantization, has remarkable similarities with other curved-space quantization schemes proposed elsewhere. However, unlike these other schemes, it contains a piece which depends in an essential way on the gauge structure ofmore » the unreduced theory, and so could not have been guessed working strictly from within the classical reduced theory.« less

A classification of Riemannian 3-manifolds with constant principal ricci curvaturesρ1= ρ2≠ ρ3

This paper has been motivated by various problems and results in differential geometry. The main motivation is the study of curvature homogeneous Riemannian spaces initiated in 1960 by I.M. Singer (see Section 9—Appendix for the precise definitions and references). Up to recently, only sporadic classes of examples have been known of curvature homogeneous spaces which are not locally homogeneous. For instance, isoparametric hypersurfaces in space forms give nice examples of nontrivial curvature homogeneous spaces (see [FKM]). To study the topography of curvature homogeneous spaces more systematically, it is natural to start with the dimension n = 3. The following results and problems have been particularly inspiring.