Castelnuovo-Mumford Regularity in Biprojective Spaces

We define the concept of regularity for bigraded modules and bigraded polynomial ring. In this setting we prove analogs of some of the classical results on $m$-regularity for graded modules over polynomial algebras.Comment: 17 Pages, 2 figure

Curvilinear Base Points, Local Complete Intersection and Kozsul Syzygies in Biprojective Spaces

We prove analogs of results of Cox/Schenck on the structure of certain ideals in the bigraded polynomial ring k[s,u;t,v].Comment: 12 page

Edge States and Broken Symmetry Phases of Laterally Confined 3^3He Films

Broken symmetries in topological condensed matter systems have implications for the spectrum of Fermionic excitations confined on surfaces or topological defects. The Fermionic spectrum of confined (quasi-2D) $^3$He-A consists of branches of chiral edge states. The negative energy states are related to the ground-state angular momentum, $L_z = (N/2) \hbar$, for $N/2$ Cooper pairs. The power law suppression of the angular momentum, $L_z(T) \simeq (N/2)\,\hbar\,[1 - \frac{2}{3}(\pi T/\Delta)^2 ]$ for $0 \le T \ll T_c$, in the fully gapped 2D chiral A-phase reflects the thermal excitation of the chiral edge Fermions. We discuss the effects of wave function overlap, and hybridization between edge states confined near opposing surfaces on the edge currents, ground-state angular momentum and ground-state order parameter. Under strong lateral confinement, the chiral A phase undergoes a sequence of phase transitions, first to a pair density wave (PDW) phase with broken translational symmetry at $D_{c2} \approx 16 \xi_0$. The PDW phase is described by a periodic array of chiral domains with alternating chirality, separated by domain walls. The period of PDW phase diverges as the confinement length $D\rightarrow D_{c_2}$. The PDW phase breaks time-reversal symmetry, translation invariance, but is invariant under the combination of time-reversal and translation by a one-half period of the PDW. The mass current distribution of the PDW phase reflects this combined symmetry, and orignates from the spectra of edge Fermions and the chiral branches bound to the domain walls. Under sufficiently strong confinement a second-order transition occurs to the non-chiral "polar phase" at $D_{c1} \approx 9\xi_0$, in which a single p-wave orbital state of Cooper pairs is aligned along the channel.Comment: 16 pages, 16 figure

Zonal Flow Magnetic Field Interaction in the Semi-Conducting Region of Giant Planets

All four giant planets in the Solar System feature zonal flows on the order of 100 m/s in the cloud deck, and large-scale intrinsic magnetic fields on the order of 1 Gauss near the surface. The vertical structure of the zonal flows remains obscure. The end-member scenarios are shallow flows confined in the radiative atmosphere and deep flows throughout the entire planet. The electrical conductivity increases rapidly yet smoothly as a function of depth inside Jupiter and Saturn. Deep zonal flows will inevitably interact with the magnetic field, at depth with even modest electrical conductivity. Here we investigate the interaction between zonal flows and magnetic fields in the semi-conducting region of giant planets. Employing mean-field electrodynamics, we show that the interaction will generate detectable poloidal magnetic field perturbations spatially correlated with the deep zonal flows. Assuming the peak amplitude of the dynamo alpha-effect to be 0.1 mm/s, deep zonal flows on the order of 0.1 - 1 m/s in the semi-conducting region of Jupiter and Saturn would generate poloidal magnetic perturbations on the order of 0.01% - 1% of the background dipole field. These poloidal perturbations should be detectable with the in-situ magnetic field measurements from the Juno mission and the Cassini Grand Finale. This implies that magnetic field measurements can be employed to constrain the properties of deep zonal flows in the semi-conducting region of giant planets.Comment: 38 pages, 12 figures, revised submission to Icaru