Diophantine approximation on Veech surfaces

We show that Y. Cheung's general $Z$-continued fractions can be adapted to give approximation by saddle connection vectors for any compact translation surface. That is, we show the finiteness of his Minkowski constant for any compact translation surface. Furthermore, we show that for a Veech surface in standard form, each component of any saddle connection vector dominates its conjugates. The saddle connection continued fractions then allow one to recognize certain transcendental directions by their developments

Dynamics of a family of piecewise-linear area-preserving plane maps II. Invariant circles

This paper studies the behavior under iteration of the maps T_{ab}(x,y)=(F_{ab}(x)-y,x) of the plane R^2, in which F_{ab}(x)=ax if x>=0 and bx if x<0. The orbits under iteration correspond to solutions of the nonlinear difference equation x_{n+2}= 1/2(a-b)|x_{n+1}| + 1/2(a+b)x_{n+1} - x_n. This family of maps has the parameter space (a,b)\in R^2. These maps are area-preserving homeomorphisms of R^s that map rays from the origin into rays from the origin. This paper shows the existence of special parameter values where T_{ab} has every nonzero orbit an invariant circle with irrational rotation number, and these invariant circles are piecewise unions of arcs of conic sections. Numerical experiments suggest the possible existence of many other parameter values having invariant circles.Comment: v2 corresponds to second half of old part I; 27 pages latex, 9 ps figure files. Current part I is math.DS/0301294, part III is math.DS/0505103; v3 reflects prior work of Beardon, Bullett and Rippo

Noneuclidean Tessellations and their relation to Reggie Trajectories

The coefficients in the confluent hypergeometric equation specify the Regge trajectories and the degeneracy of the angular momentum states. Bound states are associated with real angular momenta while resonances are characterized by complex angular momenta. With a centrifugal potential, the half-plane is tessellated by crescents. The addition of an electrostatic potential converts it into a hydrogen atom, and the crescents into triangles which may have complex conjugate angles; the angle through which a rotation takes place is accompanied by a stretching. Rather than studying the properties of the wave functions themselves, we study their symmetry groups. A complex angle indicates that the group contains loxodromic elements. Since the domain of such groups is not the disc, hyperbolic plane geometry cannot be used. Rather, the theory of the isometric circle is adapted since it treats all groups symmetrically. The pairing of circles and their inverses is likened to pairing particles with their antiparticles which then go one to produce nested circles, or a proliferation of particles. A corollary to Laguerre's theorem, which states that the euclidean angle is represented by a pure imaginary projective invariant, represents the imaginary angle in the form of a real projective invariant.Comment: 27 pages, 4 figure

Quantisation of Monopoles with Non-abelian Magnetic Charge

Magnetic monopoles in Yang-Mills-Higgs theory with a non-abelian unbroken gauge group are classified by holomorphic charges in addition to the topological charges familiar from the abelian case. As a result the moduli spaces of monopoles of given topological charge are stratified according to the holomorphic charges. Here the physical consequences of the stratification are explored in the case where the gauge group SU(3) is broken to U(2). The description due to A. Dancer of the moduli space of charge two monopoles is reviewed and interpreted physically in terms of non-abelian magnetic dipole moments. Semi-classical quantisation leads to dyonic states which are labelled by a magnetic charge and a representation of the subgroup of U(2) which leaves the magnetic charge invariant (centraliser subgroup). A key result of this paper is that these states fall into representations of the semi-direct product U(2) \semidir R^4. The combination rules (Clebsch-Gordan coefficients) of dyonic states can thus be deduced. Electric-magnetic duality properties of the theory are discussed in the light of our results, and supersymmetric dyonic BPS states which fill the SL(2,Z)-orbit of the basic massive W-bosons are found.Comment: 57 pages, harvmac, amssym, two eps figures, minor mistakes and typos corrected, references added; to appear in Nucl. Phys.

Local statistics for random domino tilings of the Aztec diamond

We prove an asymptotic formula for the probability that, if one chooses a domino tiling of a large Aztec diamond at random according to the uniform distribution on such tilings, the tiling will contain a domino covering a given pair of adjacent lattice squares. This formula quantifies the effect of the diamond's boundary conditions on the behavior of typical tilings; in addition, it yields a new proof of the arctic circle theorem of Jockusch, Propp, and Shor. Our approach is to use the saddle point method to estimate certain weighted sums of squares of Krawtchouk polynomials (whose relevance to domino tilings is demonstrated elsewhere), and to combine these estimates with some exponential sum bounds to deduce our final result. This approach generalizes straightforwardly to the case in which the probability distribution on the set of tilings incorporates bias favoring horizontal over vertical tiles or vice versa. We also prove a fairly general large deviation estimate for domino tilings of simply-connected planar regions that implies that some of our results on Aztec diamonds apply to many other similar regions as well.Comment: 42 pages, 7 figure

Taub-NUT Dynamics with a Magnetic Field

We study classical and quantum dynamics on the Euclidean Taub-NUT geometry coupled to an abelian gauge field with self-dual curvature and show that, even though Taub-NUT has neither bounded orbits nor quantum bound states, the magnetic binding via the gauge field produces both. The conserved Runge-Lenz vector of Taub-NUT dynamics survives, in a modified form, in the gauged model and allows for an essentially algebraic computation of classical trajectories and energies of quantum bound states. We also compute scattering cross sections and find a surprising electric-magnetic duality. Finally, we exhibit the dynamical symmetry behind the conserved Runge-Lenz and angular momentum vectors in terms of a twistorial formulation of phase space.Comment: 36 pages, three figure