Message passing on the QCDSP supercomputer

The QCDSP machines were designed for lattice gauge calculations. For planning it is crucial to explore this architecture for other computationally intensive tasks. Here I describe an implementation of a simple message passing scheme. With the objective being simplicity, I introduce a small number of generic functions for manipulating a large data set spread over the machine. I test the scheme on three applications: a fast Fourier transform, arbitrary dimension SU(N) pure lattice gauge theory, and the manipulation of Fermionic Fock states through a distributed hash table. These routines compile both on QCDSP and a Unix workstation.Comment: LATTICE99(Algorithms and Machines) - 3 page

Xtoys: cellular automata on xwindows

Xtoys is a collection of xwindow programs for demonstrating simulations of various statistical models. Included are xising, for the two dimensional Ising model, xpotts, for the $q$-state Potts model, xautomalab, for a fairly general class of totalistic cellular automata, xsand, for the Bak-Tang-Wiesenfield model of self organized criticality, and xfires, a simple forest fire simulation. The programs should compile on any machine supporting xwindows.Comment: 4 pages, one figure, uuencoded compressed postscript Contribution to Lattice '95 Also available at http://penguin.phy.bnl.gov/www/papers/BNL-62123.ps.Z Programs available at http://penguin.phy.bnl.gov/www/xtoys/xtoys.htm

Playing with sandpiles

The Bak-Tang-Wiesenfeld sandpile model provdes a simple and elegant system with which to demonstate self-organized criticality. This model has rather remarkable mathematical properties first elucidated by Dhar. I demonstrate some of these properties graphically with a simple computer simulation.Comment: Contribution to the Niels Bohr Summer Institute on Complexity and Criticality; to appear in a Per Bak Memorial Issue of PHYSICA A; 6 pages 3 figure

Surface modes and parity violation in Schwinger model on the lattice

The phase diagram of the Schwinger model on the lattice with Wilson fermions is investigated in the Hartree-Fock approximation. In case of single flavour (not directly amenable to simulations), the calculation indicates the existence of the parity violating phase at both weak and intermediate-to-strong couplings. Hartree-Fock vacuum sustains a nonzero electric field in this broken phase. The phase structure of the model with two flavours is also discussed.Comment: 4 pages, uuencoded compressed PostScript (using uufiles), contribution to LATTICE 9

Chiral anomalies and rooted staggered fermions

A popular approximation in lattice gauge theory is an extrapolation in the number of fermion species away from the four fold degeneracy natural with the staggered fermion formulation. I show that the extrapolation procedure mutilates the expected continuum holomorphic behavior in the quark masses. The conventional resolution proposes canceling the unphysical singularities with a plethora of extra states appearing at finite lattice spacing. This unproven conjecture requires an explicit loss of unitarity and locality. Even if correct, the approach implies large cutoff effects in the low-energy flavor-neutral sector.Comment: 10 pages, no figures; revision includes various clarifications, a changed title, and an additional reference; version to appear in Physics Letters

Contractive Spaces and Relatively Contractive Maps

We present an exposition of contractive spaces and of relatively contractive maps. Contractive spaces are the natural opposite of measure-preserving actions and relatively contractive maps the natural opposite of relatively measure-preserving maps. These concepts play a central role in the work of the author and J.~Peterson on the rigidity of actions of semisimple groups and their lattices and have also appeared in recent work of various other authors. We present detailed definitions and explore the relationship of these phenomena with other aspects of the ergodic theory of group actions, proving along the way several new results, with an eye towards explaining how contractiveness is intimately connected with rigidity phenomena.Comment: arXiv admin note: substantial text overlap with arXiv:1303.394

Wilson Fermions at finite temperature

I conjecture on the phase structure expected for lattice gauge theory with two flavors of Wilson fermions, concentrating on large values of the hopping parameter. Numerous phases are expected, including the conventional confinement and deconfinement phases, as well as an Aoki phase with spontaneous breaking of flavor and parity and a large hopping phase corresponding to negative quark masses.Comment: 9 pages, 4 figures. Talk at Brookhaven Theory Workshop on Relativistic Heavy Ions, July 1996 Replacement contains added reference and acknowledgemen

Quark Masses and Chiral Symmetry

I discuss the global structure of the strongly interacting gauge theory of quarks and gluons as a function of the quark masses and the CP violating parameter $\theta$. I concentrate on whether a first order phase transition occurs at $\theta=\pi.$ I show why this is expected when multiple flavors have a small degenerate mass. This transition can be removed by sufficient flavor-breaking. I speculate on the implications of this structure for Wilson's lattice fermions.Comment: compressed postscript file, 20 pages with 10 figures. Also available at http://penguin.phy.bnl.gov/www/papers/BNL-61796.ps.

Generalized Jacobians and explicit descents

We develop a cohomological description of various explicit descents in terms of generalized Jacobians, generalizing the known description for hyperelliptic curves. Specifically, given an integer $n$ dividing the degree of some reduced effective divisor $\mathfrak{m}$ on a curve $C$, we show that multiplication by $n$ on the generalized Jacobian $J_\frak{m}$ factors through an isogeny $\varphi:A_{\mathfrak{m}} \rightarrow J_{\mathfrak{m}}$ whose kernel is naturally the dual of the Galois module $(\operatorname{Pic}(C_{\overline{k}})/\mathfrak{m})[n]$. By geometric class field theory, this corresponds to an abelian covering of $C_{\overline{k}} := C \times_{\operatorname{Spec}{k}} \operatorname{Spec}(\overline{k})$ of exponent $n$ unramified outside $\mathfrak{m}$. The $n$-coverings of $C$ parameterized by explicit descents are the maximal unramified subcoverings of the $k$-forms of this ramified covering. We present applications of this to the computation of Mordell-Weil groups of Jacobians.Comment: to appear in Math. Com