Reconsiderations on the formulation of general relativity based on Riemannian structures

We prove that some basic aspects of gravity commonly attributed to the modern connection-based approaches, can be reached naturally within the usual Riemannian geometry-based approach, by assuming the independence between the metric and the connection of the background manifold. These aspects are: 1) the BF-like field theory structure of the Einstein–Hilbert action, of the cosmological term, and of the corresponding equations of motion; 2) the formulation of Maxwellian field theories using only the Riemannian connection and its corresponding curvature tensor, and the subsequent unification of gravity and gauge interactions in a four dimensional field theory; 3) the construction of four and three dimensional geometrical invariants in terms of the Riemann tensor and its traces, particularly the formulation of an anomalous Chern–Simons topological model where the action of diffeomorphisms is identified with the action of a gauge symmetry, close to Witten’s formulation of three-dimensional gravity as a Chern–Simon gauge theory. 4) Tordions as propagating and non-propagating fields are also formulated in this approach. This new formulation collapses to the usual one when the metric connection is invoked, and certain geometrical structures very known in the traditional literature can be identified as remanent structures in this collapse.     

A fundamental quadratic variational principle underlying general relativity

The fundamental result of Lanczos is used in a new type of quadratic variational principle whose field equations are the Einstein field equations together with the Yang-Mills type equations for the Riemann curvature. Additionally, a spin-2 theory of gravity for the special case of the Einstein vacuum is discussed.     

Chern-Simons topological Lagrangians in odd dimensions and their Kaluza-Klein reduction

By clarifying the behavior of generic Chern-Simons secondary invariants under infinitesimal variation and finite gauge transformation, it is proved that they are eligible to be a candidate term in the Lagrangian in odd dimensions (2k ? 1 for gauge theories and 4k ? 1 for gravity). The coefficients in front of these terms may be quantized because of topological reasons. As a possible application, the dimensional reduction of such actions in Kaluza-Klein theory is discussed. The difficulty in defining the Chern-Simons action for topologically nontrivial field configurations is pointed out and resolved.     

The Hamiltonian description of incompressible fluid ellipsoids

We construct the noncanonical Poisson bracket associated with the phase space of first order moments of the velocity field and quadratic moments of the density of a fluid with a free-boundary, constrained by the condition of incompressibility. Two methods are used to obtain the bracket, both based on Dirac’s procedure for incorporating constraints. First, the Poisson bracket of moments of the unconstrained Euler equations is used to construct a Dirac bracket, with Casimir invariants corresponding to volume preservation and incompressibility. Second, the Dirac procedure is applied directly to the continuum, noncanonical Poisson bracket that describes the compressible Euler equations, and the moment reduction is applied to this bracket. When the Hamiltonian can be expressed exactly in terms of these moments, a closure is achieved and the resulting finite-dimensional Hamiltonian system provides exact solutions of Euler’s equations. This is shown to be the case for the classical, incompressible Riemann ellipsoids, which have velocities that vary linearly with position and have constant density within an ellipsoidal boundary. The incompressible, noncanonical Poisson bracket differs from its counterpart for the compressible case in that it is not of Lie-Poisson form.     

The Toda Equations and the Gromov–Witten Theory of the Riemann Sphere

Consequences of the Toda equations arising from the conjectural matrix model for the Riemann sphere are investigated. The Toda equations determine the Gromov–Witten descendent potential (including all genera) of the Riemann sphere from the degree 0 part. Degree 0 series computations via Hodge integrals then lead to higher-degree predictions by the Toda equations. First, closed series forms for all 1-point invariants of all genera and degrees are given. Second, degree 1 invariants are investigated with new applications to Hodge integrals. Third, a differential equation for the generating function of the classical simple Hurwitz numbers (in all genera and degrees) is found – the first such equation. All these results depend upon the conjectural Toda equations. Finally, proofs of the Toda equations in genus 0 and 1 are given.     

Off-shell closed string amplitudes: Towards a computation of the tachyon potential

We derive an explicit formula for the evaluation of the classical closed string action for any off-shell string field, and for the calculation of arbitrary off-shell amplitudes. The formulae require a parametrization, in terms of some moduli space coordinates, of the family of local coordinates needed to insert the off-shell states on Riemann surfaces. We discuss in detail the evaluation of the tachyon potential as a power series in the tachyon field. The expansion coefficients in this series are shown to be geometrical invariants of Strebel quadratic differentials whose variational properties imply that closed string polyhedra, among all possible choices of string vertices, yield a tachyon potential which is as small as possible order by order in the string coupling constant. Our discussion emphasizes the geometrical meaning of off-shell amplitudes.     

Topological Invariants in Braid Theory

Many invariants of knots and links have their counterparts in braid theory. Often, these invariants are most easily calculated using braids. A braid is a set of n strings stretching between two parallel planes. This review demonstrates how integrals over the braid path can yield topological invariants. The simplest such invariant is the winding number – the net number of times two strings in a braid wrap about each other. But other, higher-order invariants exist. The mathematical literature on these invariants usually employs techniques from algebraic topology that may be unfamiliar to physicists and mathematicians in other disciplines. The primary goal of this paper is to introduce higher-order invariants using only elementary differential geometry.Some of the higher-order quantities can be found directly by searching for closed one-forms. However, the Kontsevich integral provides a more general route. This integral gives a formal sum of all finite order topological invariants. We describe the Kontsevich integral, and prove that it is invariant to deformations of the braid.Some of the higher-order invariants can be used to generate Hamiltonian dynamics of n particles in the plane. The invariants are expressed as complex numbers; but only the real part gives interesting topological information. Rather than ignoring the imaginary part, we can use it as a Hamiltonian. For n = 2, this will be the Hamiltonian for point vortex motion in the plane. The Hamiltonian for n = 3 generates more complicated motions.     

Self-dual solutions to euclidean gravity

Recent work on Euclidean self-dual gravitational fields is reviewed. We discuss various solutions to the Einstein equations and treat asymptotically locally Euclidean self-dual metrics in detail. These latter solutions have vanishing classical action and nontrivial topological invariants, and so may play a role in quantum gravity resembling that of the Yang-Mills instantons.     

Geometry and integrability of topological-antitopological fusion

Integrability of equations of topological-antitopological fusion (being proposed by Cecotti and Vafa) describing the ground state metric on a given 2D topological field theory (TFT) model, is proved. For massive TFT models these equations are reduced to a universal form (being independent on the given TFT model) by gauge transformations. For massive perturbations of topological conformal field theory models the separatrix solutions of the equations bounded at infinity are found by the isomonodromy deformations method. Also it is shown that the ground state metric together with some part of the underlined TFT structure can be parametrized by pluriharmonic maps of the coupling space to the symmetric space of real positive definite quadratic forms.     

Topological evolution near the turbulent/non-turbulent interface in turbulent mixing layer

The topological evolution near the turbulent/non-turbulent interface (TNTI) in turbulent mixing layer is studied by means of statistical analysis of the invariants of velocity gradient tensor (VGT) based on direct numerical simulation data. The dynamics of topological evolution is investigated in terms of the source terms of the evolution equations for the invariants, including the pressure effect term, viscous effect term and interaction term among the invariants. It is found that the local topology of fluid particles at the TNTI evolves from non-focal region to focal region in the plane of the second (Q) and the third (R) invariants of the VGT. The topological evolution is mainly associated with the pressure effect term in the TNTI region. According to the analysis of the evolution of enstrophy and dissipation, the enstrophy increase and the dissipation decrease are revealed in the TNTI region, which are caused by viscous vorticity diffusion near the TNTI. A weak correlation between the strain rate and the rotation rate is found in the TNTI region which is related to the reduction of enstrophy production. The alignments between vorticity and strain near the TNTI are investigated and a strong alignment of the vorticity with the extensive eigenvector direction is identified in the TNTI region.     

Variational theory of an ideal spin liquid in the quadratic theory of gravitation in a Riemann-Cartan space

A variational theory of an ideal Weyssenhoff-Raabe spin liquid in a Riemann-Cartan space is constructed in the metric theory of gravitation (taking into account the quadratic invariants in the Lagrangian). The couplings which arise in the theory and are imposed on the dynamical variables are taken into account in the action integral with the help of the method of undetermined Lagrange multipliers. The canonical energy-momentum tensor of an ideal spin liquid arises in a natural manner as a source on the right side of the gravitational field equations.Translated from Izvestiya Vysshykh Uchebnykh Zavedenii, Fizika, No. 10, pp. 101–105, October, 1989.In conclusion I thank Professor V. N. Ponomarev and B. N. Frolov for their constant interest in this work and for a discussion of the results.     

Conformally invariant gauge fixed actions for 2-D topological gravity

We show that invariants of Mumford for moduli spaces of curves are obtainable from a gauge fixed action of a topological quantum field theory in two dimensions. The method is completely analogous to the relation of Donaldson invariants with the topological quantum field theory for gauge theories in four dimensions.Supported by D.O.E. Grant DE-FG02-88ER 25066     

Variational principle in canonical variables,Weber transformation,and complete set of the local integrals of motion for dissipation-free magnetohydrodynamics

The intriguing problem of the “missing” MHD integrals of motion is solved in this paper; i.e., analogues of the Ertel, helicity, and vorticity invariants are obtained. The two latter have been discussed earlier in the literature only for specific cases, and the Ertel invariant is presented for the first time. The set of ideal MHD invariants obtained appears to be complete: to each hydrodynamic invariant corresponds its MHD generalization. These additional invariants are found by means of the fluid velocity decomposition based on its representation in terms of generalized potentials. This representation follows from the discussed variational principle in Hamiltonian (canonical) variables, and it naturally decomposes the velocity field into the sum of “hydrodynamic” and “ magnetic” parts. The “missing” local invariants are expressed in terms of the “ hydrodynamic” part of the velocity and therefore depend on the (nonunique) velocity decomposition; i.e., they are gauge-dependent. Nevertheless, the corresponding conserved integral quantities can be made decomposition-independent by the appropriate choice of the initial conditions for the generalized potentials. It is also shown that the Weber transformation of MHD equations (partial integration of the MHD equations) leads to the velocity representation coinciding with that following from the variational principle with constraints. The necessity of exploiting the complete form of the velocity representation in order to deal with general-type MHD flows (nonbarotropic, rotational, and with all possible types of breaks as well) in terms of single-valued potentials is also under discussion. The new basic invariants found allow one to widen the set of the local invariants on the basis of the well-known recursion procedure.     

关于纯引力共形反常

本文采用几何和拓扑的方法,讨论弯曲时空中的纯引力共形反常,并得到了纯引力共形反常的新的表达式αε^abcdΩ_abΛΩ_cd+βε^abcd H_abΛH_cd,其中α,β是任意常数,Ω_ab与H_ab分别是Riemann曲率2-形式与Thomas引入的共形不变曲率2-形式。这里,第一项正比于Euler类,第二项除了包含通常熟知的Wegl张量平方项以外,还含有其它共形不变的不变量。 关键词：     

Bihamiltonian Hierarchies in 2D Topological Field Theory At One-Loop Approximation

We compute the genus one correction to the integrable hierarchy describing coupling to gravity of a 2D topological field theory. The bihamiltonian structure of the hierarchy is given by a classical W-algebra; we compute the central charge of this algebra. We also express the generating function of elliptic Gromov–Witten invariants via tau-function of the isomonodromy deformation problem arising in the theory of WDVV equations of associativity. Received: 16 January 1998 / Accepted: 2 April 1998     

Polynomial invariants for SU(2) monopoles

We present an explicit expression for the topological invariants associated to SU(2) monopoles in the fundamental representation on spin four-manifolds. The computation of these invariants is based on the analysis of their corresponding topological quantum field theory, and it turns out that they can be expressed in terms of Seiberg-Witten invariants. In this analysis we use recent exact results on the moduli space of vacua of the untwisted N = 1 and N = 2 supersymmetric counterparts of the topological quantum field theory under consideration, as well as on electric-magnetic duality for N = 2 supersymmetric gauge theories.     

Exploiting the 1, 440-fold symmetry of the master two-loop diagram

The 1, 440-element symmetry group of the generic two-loop diagram of massless scalar field theory in 4-2ω dimensions is computed, using tetrahedral symmetry and star-triangle duality. Constructing all quadratic and quartic polynomial invariants, we expand the diagram throughO(ω ⁵), where one first encounters a coefficient that does not appear to be expressible in terms of the Riemann zeta function, thereby strengthening previous suspicions that genuinely new calculational comoplexity arises at the level of 6-loop renormalization.     

Perturbation of pulsating strings

We discuss semiclassical quantization of circular pulsating strings in \( \text {AdS}_3 \times \text {S}^3 \) background with and without the Neveu-Schwarz–Neveu-Schwarz (NS–NS) flux. We find the equations of motion corresponding to the quadratic action in bosonic sector in terms of scalar quantities and invariants of the geometry. The general equations for studying physical perturbations along the string in an arbitrary curved spacetime are written down using covariant formalism. We discuss the stability of these string configurations by studying the solutions of the linearized perturbed equations of motion.     

Functional integral equation for the complete effective action in quantum field theory

Based on a methodological analysis of the effective action approach, certain conceptual foundations of quantum field theory are reconsidered to establish a quest for an equation for the effective action. Relying on the functional integral formulation of Lagrangian quantum field theory, we propose a functional integral equation for the complete effective action which can be understood as a certain fixed-point condition. This is motivated by a critical attitude toward the distinction, artificial from an experimental point of view, between classical and effective action. While for free field theories nothing new is accomplished, for interacting theories the concept differs from the established paradigm. The analysis of this new concept concentrates on gauge field theories, treating QED as the prototype model. An approximative approach to the functional integral equation for the complete effective action of QED is exploited to obtain certain nonperturbative information about the quadratic kernels of the action. As a particular application the approximate calculation of the QED coupling constant α is explicitly studied. It is understood as one of the characteristics of a fixed point given as a solution of the functional integral equation proposed. Finally, within the present approach the vacuum energy problem is considered, as are possible implications for the concept of induced gravity.     

Yang—Mills instantons over Riemann surfaces

Exact solutions to the self-dual Yang—Mills equations over Riemann surfaces of arbitrary genus are constructed. They are characterized by the conformal class of the Riemann surface. They correspond to U(1) instantonic solutions for an Abelian-Higgs system. A functional action of a genus g Riemann surface is constructed, with minimal points being the two-dimensional self-dual connections. The exact solutions may be interpreted as connecting initial and final nontrivial vacuum states of a conformal theory, in the sense of Segal, with a Feynman functor constructed from the functional integral of the action.