Solutions to Yang-Mills field equations in eight dimensions and the last Hopf map

We will show that the Hopf map admits a sourceless, topologically non-trivial gauge field. This result will be cast in the form of a solution to eight dimensional Euclidean Yang-Mills field equations with topological chargeQ=1. This solution is Spin (9) symmetric and leads to a new generalized duality conditionFF=±(FF)*.     

Reduction of Vaisman structures in complex and quaternionic geometry

We consider locally conformal Kähler geometry as an equivariant (homothetic) Kähler geometry: a locally conformal Kähler manifold is, up to equivalence, a pair $(K,\Gamma )$, where $K$ is a Kähler manifold and $\Gamma$ is a discrete Lie group of biholomorphic homotheties acting freely and properly discontinuously. We define a new invariant of a locally conformal Kähler manifold $(K,\Gamma )$ as the rank of a natural quotient of $\Gamma$, and prove its invariance under reduction. This equivariant point of view leads to a proof that locally conformal Kähler reduction of compact Vaisman manifolds produces Vaisman manifolds and is equivalent to a Sasakian reduction. Moreover, we define locally conformal hyperKähler reduction as an equivariant version of hyperKähler reduction and in the compact case we show its equivalence with 3-Sasakian reduction. Finally, we show that locally conformal hyperKähler reduction induces hyperKähler with torsion (HKT) reduction of the associated HKT structure and the two reductions are compatible, even though not every HKT reduction comes from a locally conformal hyperKähler reduction.     

The Einstein–Maxwell Equations and Conformally Kähler Geometry

Page’s Einstein metric on \({{\mathbb{CP}}_2\#\overline{\mathbb{CP}}_2}\) is conformally related to an extremal Kähler metric. Here we construct a family of conformally Kähler solutions of the Einstein–Maxwell equations that deforms the Page metric, while sweeping out the entire Kähler cone of \({{\mathbb{CP}}_2\#\overline{\mathbb{CP}}_2}\). The same method also yields analogous solutions on every Hirzebruch surface. This allows us to display infinitely many geometrically distinct families of solutions of the Einstein–Maxwell equations on the smooth 4-manifolds \({S^2 \times S^2}\) and \({{\mathbb{CP}}_2\#\overline{\mathbb{CP}}_2}\).     

Harmonic maps and stability on locally conformal Kähler manifolds

We study harmonic and pluriharmonic maps on locally conformal Kähler manifolds. We prove that there are no nonconstant holomorphic pluriharmonic maps from a locally conformal Kähler manifold to a Kähler manifold and that any holomorphic harmonic map from a compact locally conformal Kähler manifold to a Kähler manifold is stable.     

Conformal Invariance and Stochastic Loewner Evolution Predictions for the 2D Self-Avoiding Walk—Monte Carlo Tests

Simulations of the two-dimensional self-avoiding walk (SAW) are performed in a half-plane and a cut-plane (the complex plane with the positive real axis removed) using the pivot algorithm. We test the conjecture of Lawler, Schramm, and Werner that the scaling limit of the two-dimensional SAW is given by Schramm's stochastic Loewner evolution (SLE). The agreement is found to be excellent. The simulations also test the conformal invariance of the SAW since conformal invariance implies that if we map infinite length walks in the cut-plane into the half plane using the conformal map $z \to \sqrt z$ , then the resulting walks will have the same distribution as the SAW in the half plane. The simulations show excellent agreement between the distributions.     

Teleparallel Kähler Calculus for Spacetime

In a recent paper [J. G. Vargas and D. G. Torr, Found. Phys. 27, 599 (1997)], we have shown that a subset of the differential invariants that define teleparallel connections in spacetime generates a teleparallel Kaluza-Klein space (KKS) endowed with a very rich Clifford structure. A canonical Dirac equation hidden in this structure might be uncovered with the help of a teleparallel Kähler calculus in KKS. To bridge the gap to such a calculus from the existing Riemannian Kähler calculus in spacetime, we commence the construction of a teleparallel Kähler calculus in spacetime. In the process, we notice: (a) Unknown to him, one of Einstein's equations in his attempt at unification with teleparallelism states that the interior covariant derivative of the torsion is zero. (b) A mechanism exists in the tangent bundle of teleparallel spaces for producing confinement (in the applicable cases, one would have to show why nonconfinement also occurs, rather than the other way around). (c) When the torsion is not zero, the interior covariant derivative in the sense of Kähler, F, does not coincide with *d*F. The system (dF = 0, F = j) rather than (dF = 0, *d*F = j) should then be used for generalizations of Maxwell's electrodynamics.     

Quantization as analysis in partial differential varieties

Replacing positive-energy considerations by considerations of invariance under theS-operator, and applying Paneitz' extension of the stability theory of the school of M. G. Krein, a long-sought canonical positive symplectic complex structure in the stable phase space of infinite-dimensional classical field-theoretic systems can be mathematically determined. This almost-Kählerization of the phase space then yields a (positive-definite) infinite-dimensional Riemannian structure that serves to specify formally, and convergently in finite-mode approximation, the physical vacuum measure for functional integrals involved in the associated quantized field. The method applies to a general class of nonlinear wave equations including that of Yang-Mills.Invited talk at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–21, 1981.     

Eingenvalues of the Dirac operator on compact Kähler manifolds

Kählerian twistor operators are introduced to get lower bounds for the eigenvalues of the Dirac operator on compact spin Kähler manifolds. In odd complex dimensions, manifolds with the smallest eigenvalues are characterized by an over determined system of differential equations similar to the Riemannian case. In these dimensions, we show the existence of a unique natural Kählerian twistor operator. It is also proved that, on a Kähler manifold with nonzero scalar curvature, the space of Riemannian twistor-spinors is trivial.This work has been partially supported by the EEC programme GADGET Contract Nr. SC1-0105     

The Map Between Conformal Hypercomplex/ Hyper-Kähler and Quaternionic(-Kähler) Geometry

We review the general properties of target spaces of hypermultiplets, which are quaternionic-like manifolds, and discuss the relations between these manifolds and their symmetry generators. We explicitly construct a one-to-one map between conformal hypercomplex manifolds (i.e. those that have a closed homothetic Killing vector) and quaternionic manifolds of one quaternionic dimension less. An important role is played by `ξ-transformations', relating complex structures on conformal hypercomplex manifolds and connections on quaternionic manifolds. In this map, the subclass of conformal hyper-Kähler manifolds is mapped to quaternionic-Kähler manifolds. We relate the curvatures of the corresponding manifolds and furthermore map the symmetries of these manifolds to each other.     

Spectra and Eigenstates of Spin Chain Hamiltonians

We prove that translationally invariant Hamiltonians of a chain of n qubits with nearest-neighbour interactions have two seemingly contradictory features. Firstly in the limit \({n \rightarrow \infty}\) we show that any translationally invariant Hamiltonian of a chain of n qubits has an eigenbasis such that almost all eigenstates have maximal entanglement between fixed-size sub-blocks of qubits and the rest of the system; in this sense these eigenstates are like those of completely general Hamiltonians (i.e., Hamiltonians with interactions of all orders between arbitrary groups of qubits). Secondly, in the limit \({n \rightarrow \infty}\) we show that any nearest-neighbour Hamiltonian of a chain of n qubits has a Gaussian density of states; thus as far as the eigenvalues are concerned the system is like a non-interacting one. The comparison applies to chains of qubits with translationally invariant nearest-neighbour interactions, but we show that it is extendible to much more general systems (both in terms of the local dimension and the geometry of interaction). Numerical evidence is also presented that suggests that the translational invariance condition may be dropped in the case of nearest-neighbour chains.     

Holonomy groups,complex structures andD=4 topological Yang-Mills theory

We study conditions for the existence of extended supersymmetry in topological Yang-Mills theory. These conditions are most conveniently formulated in terms of the holonomy group of the underlying manifold, on which the topological Yang-Mills theory is defined. For irreducible manifolds we find that extended supersymmetries are in 1–1 correspondence with covariantly constant complex structures. Therefore, the topological Yang-Mills theory on any Kähler manifold possesses one additional supersymmetry and on any hyper Kähler manifold there are three additional supersymmetries. The Donaldson map, which plays a crucial role in the construction of the topological invariants, is generalized for Kähler manifolds, thus providing candidates for new invariants of complex manifolds.     

Polychromatic Arm Exponents for the Critical Planar FK-Ising Model

Schramm–Loewner evolution (SLE) is a one-parameter family of random planar curves introduced by Schramm in 1999 as the candidates for the scaling limits of the interfaces in the planar critical lattice models. This is the only possible process with conformal invariance and a certain “domain Markov property”. In 2010, Chelkak and Smirnov proved the conformal invariance of the scaling limits of the critial planar FK-Ising model which gave the convergence of the interface to \(\text {SLE}_{16/3}\). We derive the arm exponents of \(\text {SLE}_{\kappa }\) for \(\kappa \in (4,8)\). Combining with the convergence of the interface, we derive the arm exponents of the critical FK-Ising model. We obtain six different patterns of boundary arm exponents and three different patterns of interior arm exponents of the critical planar FK-Ising model on the square lattice.     

Exact solutions for the massless plane symmetric scalar field in general relativity,with cosmological constant

The Klein–Gordon equations are solved for the case of a plane-symmetric static massless scalar field in general relativity with cosmological constant, generalizing the solutions found by Taub, Novotny and Horsky, and Singh. A separate class of solutions is obtained in which the metrics reduce to flat space in the limit that .The static solutions can be used to generate time-dependent cosmological solutions, one of which exhibits rapid inflation followed by continued exponential expansion at all later times.     

On the Yang-Mills effective action with instanton gauge field

The calculation of the determinant for the second order covariant derivative operator is analyzed in the space R⁴ and for Yang-Mills instanton field configurations. The problems inherent to the ultraviolet ζ-function renormalization method and infrared divergencies of this operator are reviewed. A method for estimating the asymptotic coefficients of its determinant has been discussed and the failure of the general conformal invariance induced by any regularization technique has also been considered. A particular solution valid for a Yang-Mills multi instanton configuration, valuable in order to get the non conformally invariant piece of the general solution, is the main result obtained here. We give it as an integral equation, namely in a semi-explicit form.     

Weak field approximation in a model of de Sitter gravity: Schwarzschild solutions and galactic rotation curves

Weak field approximate solutions in the $\Lambda \rightarrow 0$ limit of a model of de Sitter gravity have been presented in the static and spherically symmetric case. Although the model looks different from general relativity, among those solutions, there still exist the weak Schwarzschild fields with the smooth connection to regular internal solutions obeying the Newtonian gravitational law. The existence of such solutions would determine the value of the coupling constant, which is different from that of the previous literature. Moreover, there also exist solutions that could deduce the galactic rotation curves without invoking dark matter.     

Spectral Analysis and Zeta Determinant on the Deformed Spheres

We consider a class of singular Riemannian manifolds, the deformed spheres , defined as the classical spheres with a one parameter family g[k] of singular Riemannian structures, that reduces for k = 1 to the classical metric. After giving explicit formulas for the eigenvalues and eigenfunctions of the metric Laplacian , we study the associated zeta functions . We introduce a general method to deal with some classes of simple and double abstract zeta functions, generalizing the ones appearing in . An application of this method allows to obtain the main zeta invariants for these zeta functions in all dimensions, and in particular and . We give explicit formulas for the zeta regularized determinant in the low dimensional cases, N = 2,3, thus generalizing a result of Dowker [25], and we compute the first coefficients in the expansion of these determinants in powers of the deformation parameter k. Partially supported by FAPESP: 2005/04363-4     

On the Small Mass Limit of Quantum Brownian Motion with Inhomogeneous Damping and Diffusion

We study the small mass limit (or: the Smoluchowski–Kramers limit) of a class of quantum Brownian motions with inhomogeneous damping and diffusion. For Ohmic bath spectral density with a Lorentz–Drude cutoff, we derive the Heisenberg–Langevin equations for the particle’s observables using a quantum stochastic calculus approach. We set the mass of the particle to equal \(m = m_{0} \epsilon \), the reduced Planck constant to equal \(\hbar = \epsilon \) and the cutoff frequency to equal \(\varLambda = E_{\varLambda }/\epsilon \), where \(m_0\) and \(E_{\varLambda }\) are positive constants, so that the particle’s de Broglie wavelength and the largest energy scale of the bath are fixed as \(\epsilon \rightarrow 0\). We study the limit as \(\epsilon \rightarrow 0\) of the rescaled model and derive a limiting equation for the (slow) particle’s position variable. We find that the limiting equation contains several drift correction terms, the quantum noise-induced drifts, including terms of purely quantum nature, with no classical counterparts.     

A special class of holomorphic mappings and the Faddeev-Hopf model

Abstract

Pseudo horizontally weakly conformal maps [16] extend both holomorphic and (semi)conformal maps into an almost Hermitian manifold. We find critical points for the (generalized) Faddeev-Hopf model [28] in this larger class.     

Renormalized Traces and Cocycles on the Algebra of S 1-Pseudo-differential Operators

Using renormalized (or weighted) traces of classical pseudo-differential operators and calculus on formal symbols. We exhibit three cocycles on the Lie algebra of classical pseudo-differential operators $Cl(S^1,\mathbb{C}^n)Using renormalized (or weighted) traces of classical pseudo-differential operators and calculus on formal symbols. We exhibit three cocycles on the Lie algebra of classical pseudo-differential operators acting on . We first show that the Schwinger functional associated to the Dirac operator is a cocycle on , and not only on a restricted algebra Then, we investigate two bilinear functionals and , which satisfies We show that and are two cocycles in , and and have the same nonvanishing cohomology class. We finaly calculate on classical pseudo-differential operators of order 1 and on differential operators of order 1, in terms of partial symbols. By this last computation, we recover the Virasoro cocyle and the K?hler form of the loop group. Mathematics Subject Classification (1991). 47G30, 47N50     

On the curvature of Kähler–Norden manifolds

Using the one-to-one correspondence between Kähler–Norden and holomorphic Riemannian metrics, important relations between various Riemannian invariants of manifolds endowed with such metrics are established. Especially, the holomorphic versions of the recurrence of the Riemann, Ricci, projective are defined and investigated. For four-dimensional Kähler–Norden manifolds, it is proved that they are of holomorphically recurrent curvature on the set where the holomorphic scalar curvature does not vanish. Furthermore, a four-dimensional Kähler–Norden manifold is (locally) conformally flat if and only if its holomorphic scalar curvature is constant pure imaginary. The present paper continues author’s investigations of Kähler–Norden manifolds from the papers [K. Słuka, On Kähler manifolds with Norden metrics, An. Ştiint. Univ. Al.I. Cuza IaşI Ser. Ia Mat. 47 (2001) 105–122; K. Słuka, Properties of the Weyl conformal curvature of Kähler–Norden manifolds, in: Proc. Colloq. Diff. Geom. on Steps in Differential Geometry, July 25–30, 2000, Debrecen, 2001, pp. 317–328].