Coeffective and de Rham cohomologies of symplectic manifolds

We discuss the relation of the coeffective cohomology of a symplectic manifold with the topology of the manifold. A bound for the coeffective numbers is obtained. The lower bound is got for compact Kähler manifolds, and the upper one for non-compact exact symplectic manifolds. A Nomizu's type theorem for the coeffective cohomology is proved. Finally, the behaviour of the coeffective cohomology under deformations is studied.     

Walker 4-manifolds with proper almost complex structures

It is shown that a Walker 4-manifold, endowed with a canonical neutral metric depending on three arbitrary functions, admits a specific almost complex structure (called proper) and an associated opposite almost complex structure. We study when these two almost complex structures are integrable and when the corresponding Kähler forms are symplectic. The conditions for the canonical neutral metric to be Kähler imply that the three arbitrary functions in the metric are all harmonic with respect to two coordinate variables, and we obtain a useful method of constructing indefinite Kähler 4-manifolds. Petean’s example of a nonflat indefinite Kähler–Einstein 4-manifold is a special case of this construction.     

Supersymmetry breaking with vanishing F-terms in supergravity theories

In conventional supergravity theories, supersymmetry is broken by a non-zero F-term, and the cosmological constant is fine tuned to zero by a constant in the superpotential W. We discuss a class of supergravity theories with vanishing F-terms but W ≠ 0 being generated dynamically. The cosmological constant is assumed to be cancelled by a non-zero D-term. In this scenario the gravity-mediated soft masses depend only on a single parameter, the gravitino mass. They are automatically universal, independently of the Kähler metric, and real. Thus, dangerous flavor or CP violating interactions are suppressed. Unlike in conventional supergravity models, the Polonyi problem does not arise.     

Quaternionic complexes

Each regular or semi-regular integral affine orbit of the Weyl group of gl(2n + 2, ) invariantly determines a locally exact differential complex on a 4n dimensional quaternionic manifold. This gives quaternionic analogues of Dolbeault cohomology on complex manifolds. We compute the index of such complexes in the hyper-Kähler case, showing that quaternionic cohomology is not trivial.     

Charge susceptibility and charge correlation function in the d---p model

The properties of the charge fluctuation are investigated in the d---p model with the repulsion U_pd between holes on the nearest-neighbor Cu and O sites and the infinite on-site repulsion U_d at the Cu site. We calculate the charge susceptibility χ^c(q, iω_n) and the charge correlation function S^c(q) = TΣ_ωn χ^c(q, iω_n). It is found that S^c(q) has a peak at the Γ point and a maximum in a ring around the Γ point. The former is due to Tχ^c(q, 0). Its intensity is proportional to temperature T and strongly enhanced by U_pd. The latter is due to TΣ_{ωn ≠ 0} χ^c(q, iω_n) and shows a weak T and U_pd dependence. The intensity of the diffuse X-ray scattering on taking the charge fluctuation into account is also calculated. The result is consistent with the experiments in La_2−δSr_δCuO₄.     

The Arnol''d cat: Failure of the correspondence principle

The classical Hamiltonian H = p²/2m + ε(q²/2)Σδ[s-(t/T)] has an integrable mapping of the plane, [q_n+1, p_n+1]= [q_n+1+p_n, q_n+2p_n], as its equations of motion. But then by introducing periodic boundary conditions via (mod 1) applied to both q and p variables, the equations of motion become the Arnol'd cat map, [q_n+1, p_n+1] = [q_n + p_n, q_n + 2p_n], (mod 1), revealing it to be one of the simplest fully chaotic systems which can be derived from a Hamiltonian and analyzed. Consequently, we here quantize the Arnol'd cat and examine its quantum motion for signs of chaos using algorithmic complexity as the litmus. Our analysis reveals that the quantum cat is not chaotic in the deep quantum domain nor does it become chaotic in the classical limit as required by the correspondence principle. We therefore conclude that the correspondence principle, as defined herein, fails for the quantum Arnol'd cat.     

Characteristic polynomials of random Hermitian matrices and Duistermaat–Heckman localisation on non-compact Kähler manifolds

We reconsider the problem of calculating a general spectral correlation function containing an arbitrary number of products and ratios of characteristic polynomials for a N×N random matrix taken from the Gaussian Unitary Ensemble (GUE). Deviating from the standard “supersymmetry” approach, we integrate out Grassmann variables at the early stage and circumvent the use of the Hubbard–Stratonovich transformation in the “bosonic” sector. The method, suggested recently by J.V. Fyodorov [Nucl. Phys. B 621 [PM] (2002) 643], is shown to be capable of calculation when reinforced with a generalisation of the Itzykson–Zuber integral to a non-compact integration manifold. We arrive to such a generalisation by discussing the Duistermaat–Heckman localisation principle for integrals over non-compact homogeneous Kähler manifolds. In the limit of large-N the asymptotic expression for the correlation function reproduces the result outlined earlier by A.V. Andreev and B.D. Simons [Phys. Rev. Lett. 75 (1995) 2304].     

Phase slippage at the interface: normal metal/sliding charge-density wave

Phase slippage is required at the current electrodes of quasi-one-dimensional conductors with a charge density wave (CDW) ground state for the conversion from free to condensed carriers. We have performed at the ESRF high-resolution X-ray measurements of the spatially varying shift q(x) of the CDW satellite wave vector between current contacts on a thin NbSe₃ whisker in the sliding state. Applying direct currents, we observe at 90 K a steep exponential decrease of the shift within a few hundred microns from the contact. The CDW strain profile q(x) reflects the carrier conversion process, via nucleation and growth of phase-dislocation loops. Pulsed current measurements of the shift q show important differences between pulsed and dc current data, revealing a spatially dependent relaxational behaviour of the CDW strain. Using time-resolved high spatial resolution X-ray we observe at 300 μm from the electrode a stretched exponential-type decay of the shift q(t) upon switching off the current (T=75 K): q(t)=q₀[exp(−t/τ)^μ] with τ=23 ms and μ=0.36.     

On a conjecture of Guillemin and Sternberg in geometric quantization of multiplicity-free symplectic spaces

We study the relation between multiplicity-free symplectic manifolds and the multiplicities of group representations obtained by geometric quantization. With the help of a general equivalence theorem we can prove a conjecture of Guillemin and Sternberg in the compact Kähler case.     

The angular intensity correlation functions C(1) and C(10) for the scattering of light from randomly rough dielectric and metal surfaces

We study the statistical properties of the scattering matrix S(q|k) for the problem of the scattering of light of frequency ω from a randomly rough one-dimensional surface, defined by the equation x₃=ζ(x₁), where the surface profile function ζ(x₁) constitutes a zero-mean, stationary, Gaussian random process. This is done by studying the effects of S(q|k) on the angular intensity correlation function C(q,k|q',k')=〈I(q|k)I(q'|k')〉-〈I(q|k)〉〈I(q'|k')〉, where the intensity I(q|k) is defined in terms of S(q|k) by I(q|k)=L^-1₁(ω/c)|S(q|k)|², with L₁ the length of the x₁ axis covered by the random surface. We focus our attention on the C⁽¹⁾ and C⁽¹⁰⁾ correlation functions, which are the contributions to C(q,k|q',k') proportional to δ(q-k-q'+k') and δ(q-k+q'-k'), respectively. The existence of both of these correlation functions is consistent with the amplitude of the scattered field obeying complex Gaussian statistics in the limit of a long surface and in the presence of weak surface roughness. We show that the deviation of the statistics of the scattering matrix from complex circular Gaussian statistics and the C⁽¹⁰⁾ correlation function are determined by exactly the same statistical moment of S(q|k). As the random surface becomes rougher, the amplitude of the scattered field no longer obeys complex Gaussian statistics but obeys complex circular Gaussian statistics instead. In this case the C⁽¹⁰⁾ correlation function should therefore vanish. This result is confirmed by numerical simulation calculations.     

The Casimir operator of a metric connection with skew-symmetric torsion

For any triple (Mⁿ,g,) consisting of a Riemannian manifold and a metric connection with skew-symmetric torsion we introduce an elliptic, second-order operator Ω acting on spinor fields. In case of a naturally reductive space and its canonical connection, our construction yields the Casimir operator of the isometry group. Several non-homogeneous geometries (Sasakian, nearly Kähler, cocalibrated G₂-structures) admit unique connections with skew-symmetric torsion. We study the corresponding Casimir operator and compare its kernel with the space of -parallel spinors.     

Ising spin glass on a D = 3 hierarchical lattice: Effects of tailed coupling distributions

Within a real-space renormalization-group framework, we approach the cubic lattice through a D = 3 diamond-like hierarchical lattice. The model is a standard, nearest-neighbor, Ising spin glass with coupling constants {J_ij} distributed according to the family of continuous probability distributions P_q(J_ij) ∝ 1/[1 + (q − 1)J_ij²/2J²]^{1/(q − 1)} (if 1 + (q − 1) J_ij²/2J² > 0, and zero otherwise; q ). Such distributions, which arise naturally in the treatment, within the recently proposed nonextensive thermostatistics, of anomalous diffusion, reproduce the usual, Gaussian case, for q → 1. Moreover, they present a second moment J_ij² proportional to (5 − 3q)⁻¹ for q < 5/3, diverging for q ≥ 5/3, but keeping a finite width at midheight. In the limit q → 3, P_q(J_ij) collapses with the abscissa, and so the width at midheight diverges. We compute the q-dependence of the spin-glass critical temperature T_c. We show numerically that T_c does not scale with J_ij^21/2 (contrary to the usual belief), but rather with the width at midheight of P_q(J_ij). Our results suggest that T_c vanishes as −1/q when q → −∞; furthermore, we verified that T_c diverges exponentially when q approaches 3 from below.     

The Equivariant Cohomology Theory of Twisted Generalized Complex Manifolds

It has been shown recently by Kapustin and Tomasiello that the mathematical notion of Hamiltonian actions on twisted generalized Kähler manifolds is in perfect agreement with the physical notion of general (2, 2) gauged sigma models with three-form fluxes. In this article, we study the twisted equivariant cohomology theory of Hamiltonian actions on H-twisted generalized complex manifolds. If the manifold satisfies the ${\overline{\partial} \partial}It has been shown recently by Kapustin and Tomasiello that the mathematical notion of Hamiltonian actions on twisted generalized K?hler manifolds is in perfect agreement with the physical notion of general (2, 2) gauged sigma models with three-form fluxes. In this article, we study the twisted equivariant cohomology theory of Hamiltonian actions on H-twisted generalized complex manifolds. If the manifold satisfies the -lemma, we establish the equivariant formality theorem. If in addition, the manifold satisfies the generalized K?hler condition, we prove the Kirwan injectivity in this setting. We then consider the Hamiltonian action of a torus on an H-twisted generalized Calabi-Yau manifold and extend to this case the Duistermaat-Heckman theorem for the push-forward measure. As a side result, we show in this paper that the generalized K?hler quotient of a generalized K?hler vector space can never have a (cohomologically) non-trivial twisting. This gives a negative answer to a question asked by physicists whether one can construct (2, 2) gauged linear sigma models with non-trivial fluxes.     

The geometry of the SU(2) Kepler problem

The SU(2) Kepler problem is defined and analyzed, which is a Hamiltonian system reduced from the conformal Kepler problem on T^*( ⁸ − {0}) by the use of the symplectic SU(2) action lifted from the SU(2) left action on the SU(2) bundle ⁸ − {0} → ⁵ − {0}. This reduced system has a parameter μ ε su(2) coming from the value of the moment map associated with the symplectic SU(2) action. If μ ≠ 0, the phase space of this system have a bundle structure with base space T^*( ⁵ − {0}) and fibre S². The fibre, a (co)adjoint orbit through μ for SU(2), represents the internal degrees of freedom, called the isospin, of the particle of this system. The SU(2) Kepler problem with μ ≠ 0 is then interpreted as describing the motion of a classical particle with isospin in the Newtonian potential plus a specific repulsive potential together with a Yang-Mills field. This Yang-Mills field is to be referred to as BPST Yang's monopole field in ⁵ − {0};, since it becomes the Belavin-Polyakov-Schwartz-Tyupkin instanton, restricted on S⁴. If μ = 0, the SU(2) Kepler problem reduces to the ordinary Kepler problem. Like the ordinary Kepler problem, the Hamiltonian flows of the SU(2) Kepler problem of negative energy are all closed. It is shown in an explicit manner that the energy manifolds and isoenergetic orbit spaces for the SU(2) Kepler problem of negative energy are both homogeneous manifolds on which SU(4) acts transitively to the right; those homogeneous manifold are classified into two, according as the parameter μ is zero or not. For a certain value of μ, however, they contracts to the manifold which represents the set of all the equilibrium states. The isoenergetic orbit spaces are finally shown to be symplectomorphic to certain Kirillov-Konstant-Souriau coadjoint orbits for U(4), if μ is not the exceptional value mentioned above.     

Renormalization of N = 2, 4 Yang-Mills theories with soft breaking of an extended supersymmetry by N = 1 mass term

N = 2, 4 Yang-Mills theories with soft breaking of an extended supersymmetry by mass terms are considered. It is proved that for N = 4 there are no ultraviolet divergences in the mass renormalization constants to all orders of perturbation theory. For N = 2 our two-loop calculations show that the charge and mass renormalization constants contain only one-loop divergences and are the same in this order. It is shown by direct calculation that mass terms can acquire finite quantum corrections starting from the two-loop approximation. The renormalization scheme dependence of N = 4 renormalization group functions is investigated. We have found that unlike renormalization schemes with minimal subtractions of divergences other renormalization schemes give a nonzero β-function. At nonzero masses the β-function in MOM schemes is not zero even at the one-loop level. In the massless case β≠0 beginning from the two-loop approximation.     

Landau damping in degenerate solid state plasmas

Helicon waves are found useful for studying Landau damping in degenerate plasmas. The damping is analyzed as the phase velocity of the wave is increased from ω/q v_F to ω/q v_F. There is no first-orderlike transition at ω/q = V_F. In the collisionless limit, the damping tends to zero as ω/q → v_F. For finite collision times τ it does not vanish for ω/q > v_F. Nonlocal corrections to the wavelength exhibit a peak at ω/q = V_F, which degenerates into a shoulder for ωτ 100.     

Non-Linear realization of supersymmetry algebra from supersymmetric constraint

We discuss spontaneous symmetry breaking of global supersymmetry for a single scalar superfield in an arbitrary Kähler manifold. We show that when the curvature of the manifold goes to infinity (or, equivalently, the masses of the scalar partners of the goldstino go to infinity) a non-linear realization of supersymmetry is obtained. The model can be described, in perfect analogy to the ordinary σ-models, by means of a supersymmetric constraint on the superfield Φ, of the form Φ²=0. The non-linear realization we obtain is different from that of Volkov and Akulov. The differences among the two realizations are discussed.     

Dolbeault cohomology groups of compact pseudo-Kähler homogeneous manifolds

It is well known that a pseudo-Kähler structure is one of the natural generalizations of a Kähler structure. In this paper, we consider the Dolbeault cohomology groups of compact pseudo-Kähler homogeneous manifolds.