Witt Equation Research Articles

On the Riemannian geometry of Seiberg–Witten moduli spaces

We construct a natural L 2 -metric on the perturbed Seiberg–Witten moduli spaces M μ + of a compact 4-manifold M , and we study the resulting Riemannian geometry of M μ + . We derive a formula which expresses the sectional curvature of M μ + in terms of the Green operators of the deformation complex of the Seiberg–Witten equations. In case M is simply connected, we construct a Riemannian metric on the Seiberg–Witten principal U ( 1 ) bundle P → M μ + such that the bundle projection becomes a Riemannian submersion. On a Kähler surface M , the L 2 -metric on M μ + coincides with the natural Kähler metric on moduli spaces of vortices.

Cosmological Perturbations in Quantum Backgrounds and Confrontation with Inflationary Predictions

We show that a contracting universe which bounces due to quantum cosmological effects and connects to the hot big-bang expansion phase, can produce an almost scale invariant spectrum of perturbations provided the perturbations are produced during an almost matter dominated era in the contraction phase. This is achieved using Bohmian solutions of the canonical Wheeler-de Witt equation, thus treating both the background and the perturbations in a fully quantum manner. We find an almost scale invariant spectrum. Taking into account the spectral index constraint as well as the CMB normalization measure yields an equation of state that should be less than ω . 8×10 −4 , implying nS −1 ∼O ` 10 −4 ´ ,a nd that the characteristic size of the Universe at the bounce is L0 ∼ 10 3 LPlanck, a region where one expects that the Wheeler-DeWitt equation should be valid without being spoiled by string or loop quantum gravity effects. We have also obtained a consistency relation between the tensor to scalar ratio T/ S and the scalar spectral index as T/ S ∼ 4 × 10 −2 √ n S − 1, leading to potentially measurable differences with inflationary predictions.

The Seiberg–Witten equations and the Weinstein conjecture

Let M denote a compact, oriented 3-manifold and let a denote a contact 1-form on M. This article proves that the vector field that generates the kernel of the 2-form da has at least one closed, integral curve.

DEFINING AN SU(3)-CASSON/U(2)-SEIBERG–WITTEN INTEGER INVARIANT FOR INTEGRAL HOMOLOGY 3-SPHERES

An open question is the possibility of defining an integer valued SU(3)-Casson invariant for integral homology 3-spheres which involves counting the irreducible portion of the non-degenerate (perturbed) moduli space of flat SU(3)-connections plus counter-terms associated to only the non-degenerate (perturbed) reducible portion of the moduli space. The obstruction to this is the non-trivial spectral flow of a family of twisted signature operators in 3-dimensions. The parallel U(2)-Seiberg–Witten theory also has a similiar obstruction but arising from the non-trivial spectral flow of a family of twisted Dirac operators. By taking the SU(3)-flat and U(2)-Seiberg–Witten equations simultaneously the obstructions can be made to cancel and an integer invariant is obtained.

Loop quantum cosmology and the Wheeler–De Witt equation

We present some results concerning the large volume limit of loop quantum cosmology in the flat homogeneous and isotropic case. We derive the Wheeler–De Witt equation in this limit. Looking for the action from which this equation can also be obtained, we then address the problem of the modifications to be brought to the Friedman’s equation and to the equation of motion of the scalar field, in the classical limit.

Tomography of quantum states of the universe and cosmological dynamics

It has been proven that conventional quantum mechanics can be reformulated without using the concept of wave function or matrix density. Quantum states can be described in terms of a standard positive probability distribution. Such probability distributions (tomograms) satisfy classical-like equations, which substitute the Schrödinger equation. We consider quantum cosmology in the probability distribution approach. We derive the tomogram corresponding to theWheeler-de Witt equation. We also use the circumstance that cosmological evolution equations for a homogeneous and isotropic universe can be converted into very simple classical equations to derive a quantum cosmological model which links the tomogram reconstructed from observations to the initial conditions of the universe.

Solution of Wheeler–De Witt Equation, Potential Well and TunnelEffect

This paper uses the relation of the cosmic scale factorand scalar field tosolve Wheeler–De Witt equation, gives the tunnel effect of the cosmic scalefactor a and quantum potential well of scalar field, and makes it fit withthe physics of cosmic quantum birth. By solving Wheeler–De Witt equation weachieve a general probability distribution of the cosmic birth, and give theanalysis of cosmic quantum birth.

Quantum Origin of a Rotating Universe of the Bianchi IX Type

A closed cosmological model with rotation of the Bianchi IX type is constructed. Λ-term and anisotropic liquid are the sources of gravitational field for the model. A quantum origin of the Universe is examined. The Wheeler — De Witt equation is derived. A tunneling coefficient for the Universe is calculated. It is found for a particular case that the probability of quantum birth of a rotating Universe is higher than for the model without rotation.

On a system of Seiberg–Witten equations

We introduce and study a system of Seiberg–Witten equations. These are r copies of the usual Seiberg–Witten equations coupled to each other involving r connections on r Spin C structures as well as r positive spinors and are Abelian generalizations of the Seiberg–Witten equations. For r = 2 , we show that the moduli space of solutions is a compact, orientable and smooth manifold. For minimal surfaces of general type, we are able to determine the basic classes.

Existence and nonexistence results for higher-order semilinear evolution inequalities with critical potential

We prove nonexistence results for higher-order semilinear evolution equations and inequalities of the form ∂ k u ∂ t k − Δ u + λ | x | 2 u ⩾ | u | q in R N × ( 0 , ∞ ) , where λ ⩾ − ( N − 2 2 ) 2 . This problem can be seen as a higher-order evolution version of the nonlinear Wheeler–De Witt equation which appears in the theory of quantum cosmology. In order to show that our result is sharp in the parabolic case, we establish the existence of positive solutions to the semilinear equation ∂ u ∂ t − Δ u + λ | x | 2 u ⩾ u q in R N × ( 0 , ∞ ) , for λ ⩾ 0 . The nonexistence results are based on the test function method, developed by Mitidieri, Pohozaev, Tesei and Véron. The existence result is established by the construction of an explicit global solution of the semilinear parabolic inequality.

Radon transform of the Wheeler-De Witt equation and tomography of quantum states of the universe

The notion of standard positive probability distribution function (tomogram) which describes the quantum state of universe alternatively to wave function or to density matrix is introduced. Connection of the tomographic probability distribution with the Wigner function of the universe and with the star-product (deformation) quantization procedure is established. Using the Radon transform the Wheeler-De Witt generic equation for the probability function is written in tomographic form. Some examples of the Wheeler-DeWitt equation in the minisuperspace are elaborated explicitly for a homogeneous isotropic cosmological models. Some interpretational aspects of the probability description of the quantum state are discussed.

Witten triples and the Seiberg–Witten equations on a complex surface

AbstractWe study the solutions of the Seiberg–Witten equations on complex surfaces. We show that for a large class of parameters, the gauge equivalence classes of irreducible solutions of the twisted Seiberg–Witten equations correspond to stable Witten triples. We prove that on Kähler surfaces this correspondence is the set‐theoretical support of an isomorphism of real‐analytic spaces. This makes it possible to take multiplicities into account and generalizes and unifies results previously obtained by Witten. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Adiabatic Limit for Some Nonlinear Equations of Gauge Field Theory

We consider the adiabatic limit for nonlinear dynamic equations of gauge field theory. Our main example of such equations is given by the Abelian (2+1)-dimensional Higgs model. We show next that the Taubes correspondence, which assigns pseudoholomorphic curves to solutions of Seiberg--Witten equations on symplectic 4-manifolds, may be interpreted as a complex analogue of the adiabatic limit construction in the (2+1)-dimensional case.

A Quantum Cosmological Bianchi I Model with Interacting Spinor and Scalar Fields

A Bianchi I model of the Universe filled with interacting nonlinear spinor and scalar fields is studied within quantum geometrodynamics. Three types of interaction are considered: gradient, Yukawa, and axion ones. For massless fermion fields, the variables in the Wheeler – de Witt equation will separate. The solution can be interpreted using a two-component perfect liquid. One component corresponds to a massless scalar field, while the other – to a nonlinear spinor field. The interaction between the spinor and scalar fields can lead to elimination of singularity of the wave function. There is a possibility of existence of a discrete spectrum of the quantum Universe, as well as tunneling from the region with a rigorous equation of state to the region of the de Sitter vacuum.

NONSINGULAR COSMOLOGIES FROM BRANES

We analyze possible cosmological scenarios on a brane where the brane acts as a dynamical boundary of various black holes with anti-de Sitter or de Sitter asymptotic. In many cases, the brane is found to describe the completely nonsingular universe. In some cases, quantum gravity era of the brane-universe can also be avoided by properly tuning bulk parameters. We further discuss the creation of a brane-universe by studying its wave function. This is done by employing Wheeler–De Witt equation in the mini superspace formalism.

TIME AND TACHYON

Recent analysis suggests that the classical dynamics of a tachyon on an unstable D-brane is described by a scalar Born–Infeld type action with a runaway potential. The classical configurations in this theory at late time are in one to one correspondence with the configuration of a system of noninteracting (incoherent), nonrotating dust. We discuss some aspects of canonical quantization of this field theory coupled to gravity, and explore, following an earlier work on this subject, the possibility of using the scalar field (tachyon) as the definition of time in quantum cosmology. At late "time" we can identify a subsector in which the scalar field decouples from gravity and we recover the usual Wheeler–de Witt equation of quantum gravity.

Quantum Birth of the Rotating Universe

Quantum birth of the Universe of the Bianchi type IX filled by a rotating anisotropic liquid is studied. The Wheeler–De Witt equation is derived for the model under consideration. The tunneling wave function as a solution to the equation is found using the WKB method. The tunneling coefficient of the Universe is calculated. The probabilities of quantum birth of the Universe with and without rotation are compared for different formulations of the problem.

Self-dual teleparallel formulation of general relativity and the positive energy theorem

A self-dual and anti-self-dual decomposition of the teleparallel formulation of Einstein's general relativity is carried out and the self-dual Lagrangian of the teleparallel formulation of Einstein's general relativity, which is equivalent to the Ashtekar Lagrangian in vacuum, is obtained. Its Hamiltonian formulation and the constraint analysis are developed. Starting from Witten's equation the gauge condition of Nester and co-workers is derived directly and a new expression of the boundary term is obtained. Using this expression and Witten's identity the proof of the positive energy theorem by Nester and co-workers is extended to a case including momentum.

Born-Infeld Action from Supergravity

We show that the Born-Infeld action with the Wess-Zumino terms for the Ramond-Ramond fields, which is the D3-brane effective action, is a solution to the Hamilton-Jacobi (H-J) equation of type IIB supergravity. Adopting the radial coordinate as time, we develop the ADM formalism for type IIB supergravity reduced on $S^5$ and derive the H-J equation, which is the classical limit of the Wheeler-De Witt equation and whose solutions are classical on-shell actions. The solution to the H-J equation reproduces the on-shell actions for the supergravity solution of a stack of D3-branes in a $B_2$ field and the near-horizon limit of this supergravity solution, which is conjectured to be dual to noncommutative Yang Mills and reduces to $AdS_5 \times S^5$ in the commutative limit. Our D3-brane effective action is that of a probe D3-brane, and the radial time corresponds to the vacuum expectation value of the Higgs field in the dual Yang Mills. Our findings can be applied to the study of the holographic renormalization group.

THE GEOMETRIC TRIANGLE FOR 3-DIMENSIONAL SEIBERG–WITTEN MONOPOLES

In this paper we prove the surgery formula relating the moduli spaces of solutions of suitably perturbed 3-dimensional Seiberg–Witten equations on a homology 3-sphere Y and on the 3-manifolds Y1and Y0obtained, respectively, by +1 and 0-surgery on a knot K in Y.