Quiver varieties and a noncommutative P²

To any finite group Γ ⊂ SL₂(ℂ) and each element t in the center of the group algebra Of Γ we associate a category, Coh(ℙ²_(Γ, τ),ℙ¹). It is defined as a suitable quotient of the category of graded modules over (a graded version of) the deformed preprojective algebra introduced by Crawley-Boevey and Holland. The category Coh(ℙ²_(Γ, τ),ℙ¹) should be thought of as the category of coherent sheaves on a ‘noncommutative projective space’, ℙ²_(Γ, τ), equipped with a framing at ℙ¹, the line at infinity. Our first result establishes an isomorphism between the moduli space of torsion free objects of Coh(ℙ²_(Γ, τ),ℙ¹) and the Nakajima quiver variety arising from G via the McKay correspondence. We apply the above isomorphism to deduce a generalization of the Crawley-Boevey and Holland conjecture, saying that the moduli space of ‘rank 1’ projective modules over the deformed preprojective algebra is isomorphic to a particular quiver variety. This reduces, for Γ = {1}, to the recently obtained parametrisation of the isomorphism classes of right ideals in the first Weyl algebra, A₁, by points of the Calogero– Moser space, due to Cannings and Holland and Berest and Wilson. Our approach is algebraic and is based on a monadic description of torsion free sheaves on ℙ²_(Γ, τ). It is totally different from the one used by Berest and Wilson, involving τ-functions

Current in narrow channels of anisotropic superconductors

We argue that in channels cut out of anisotropic single crystal superconductors and narrow on the scale of London penetration depth, the persistent current must cause the transverse phase difference provided the current does not point in any of the principal crystal directions. The difference is proportional to the current value and depends on the anisotropy parameter, on the current direction relative to the crystal, and on the transverse channel dimension. An experimental set up to measure the transverse phase is proposed.Comment: 3 pages, 1 figur

The role of cosmic rays in magnetic hydrodynamics of interstellar medium

Cosmic ray (CR) propagation in the Galaxy and generally in the cosmic plasma is usually considered in the diffusion approximation. The diffusion is regarded to result from CR scattering due to their interaction with a magnetic and an electric field. In most cases the fields are assumed to be given. Meanwhile, in the Galaxy the CR energy density w sub cr is similar to I eV/cm, i.e., it is comparable with the energy densities of the magnetic field and turbulent motions in the interstellar gas. Therefore, for the Galaxy it becomes necessary to take into account the influence of CR on the gas dynamics and on the magnetic fields in this gas. The simplest way to this is to use the hydrodynamic approximation, but this is possible only on scales greatly exceeding the CR free path lambda before scattering and only for times larger than lambda/v approx. equals lambda/c. One should thus obtain corresponding MHD equations and establish the limits of their applicability

On the powerful X-ray emission of radiogalaxies

Radio galactic emission and cosmic X-radiatio

Laser cooling of electron beams for linear colliders

A novel method of electron beam cooling is considered which can be used for linear colliders. The electron beam is cooled during collision with focused powerful laser pulse. With reasonable laser parameters (laser flash energy about 10 J) one can decrease transverse beam emittances by a factor about 10 per one stage. The ultimate transverse emittances are much below those achievable by other methods. Beam depolarization during cooling is about 5--15 % for one stage. This method is especially useful for photon colliders and opens new possibilities for e+e- colliders.Comment: 4 pages, Latex, v2 corresponds to the PRL paper with erratum (in 1998) include

Finite-Size Effects in the ϕ4\phi^{4} Field Theory Above the Upper Critical Dimension

We demonstrate that the standard O(n) symmetric $\phi^{4}$ field theory does not correctly describe the leading finite-size effects near the critical point of spin systems on a $d$-dimensional lattice with $d > 4$. We show that these finite-size effects require a description in terms of a lattice Hamiltonian. For $n \to \infty$ and $n=1$ explicit results are given for the susceptibility and for the Binder cumulant. They imply that recent analyses of Monte-Carlo results for the five-dimensional Ising model are not conclusive.Comment: 4 pages, latex, 1 figur

The role of energy-momentum conservation in emission of Cherenkov gluons

The famous formula for the emission angle of Cherenkov radiation should be modified when applied to hadronic reactions because of recoil effects. They impose the upper limit on the energy of the gluon emitted at a given angle. Also, it leads to essential corrections to the nuclear refractive index value as determined from the angular position of Cherenkov rings.Comment: 6

On some problems of gamma-astronomy

Gamma ray emissions from young supernova remnants are discussed and calculated. The positron annihilation line is also calculated. Decay of charged pions in remnants cause generation of high energy neutrinos. This emission of neutrinos is reviewed. The CR origin and gamma emission from Magellanic clouds help to establish the intensity gradient in the galaxy. This gamma astronomical data is briefly discussed

Thinning of superfluid films below the critical point

Experiments on $^4$He films reveal an attractive Casimir-like force at the bulk $\lambda$-point, and in the superfluid regime. Previous work has explained the magnitude of this force at the $\lambda$ transition and deep in the superfluid region but not the substantial attractive force immediately below the $\lambda$-point. Utilizing a simple mean-field calculation renormalized by critical fluctuations we obtain an effective Casimir force that is qualitatively consistent with the scaling function $\vartheta$ obtained by collapse of experimental data.Comment: 4 page