Ramification points of Seiberg-Witten curves

When the Seiberg-Witten curve of a four-dimensional N = 2 supersymmetric gauge theory wraps a Riemann surface as a multi-sheeted cover, a topological constraint requires that in general the curve should develop ramification points. We show that, while some of the branch points of the covering map can be identified with the punctures that appear in the work of Gaiotto, the ramification points give us additional branch points whose locations on the Riemann surface can have dependence not only on gauge coupling parameters but on Coulomb branch parameters and mass parameters of the theory. We describe how these branch points can help us to understand interesting physics in various limits of the parameters, including Argyres-Seiberg duality and Argyres-Douglas fixed points

ADE Spectral Networks

We introduce a new perspective and a generalization of spectral networks for 4d $\mathcal{N}=2$ theories of class $\mathcal{S}$ associated to Lie algebras $\mathfrak{g} = \textrm{A}_n$, $\textrm{D}_n$, $\textrm{E}_{6}$, and $\textrm{E}_{7}$. Spectral networks directly compute the BPS spectra of 2d theories on surface defects coupled to the 4d theories. A Lie algebraic interpretation of these spectra emerges naturally from our construction, leading to a new description of 2d-4d wall-crossing phenomena. Our construction also provides an efficient framework for the study of BPS spectra of the 4d theories. In addition, we consider novel types of surface defects associated with minuscule representations of $\mathfrak{g}$.Comment: 68 pages plus appendices; visit http://het-math2.physics.rutgers.edu/loom/ to use 'loom,' a program that generates spectral networks; v2: version published in JHEP plus minor correction

BPS Graphs: From Spectral Networks to BPS Quivers

We define "BPS graphs" on punctured Riemann surfaces associated with $A_{N-1}$ theories of class $\mathcal{S}$. BPS graphs provide a bridge between two powerful frameworks for studying the spectrum of BPS states: spectral networks and BPS quivers. They arise from degenerate spectral networks at maximal intersections of walls of marginal stability on the Coulomb branch. While the BPS spectrum is ill-defined at such intersections, a BPS graph captures a useful basis of elementary BPS states. The topology of a BPS graph encodes a BPS quiver, even for higher-rank theories and for theories with certain partial punctures. BPS graphs lead to a geometric realization of the combinatorics of Fock-Goncharov $N$-triangulations and generalize them in several ways.Comment: 48 pages, 44 figure

Axion as a cold dark matter candidate: low-mass case

Axion as a coherently oscillating scalar field is known to behave as a cold dark matter in all cosmologically relevant scales. For conventional axion mass with 10^{-5} eV, the axion reveals a characteristic damping behavior in the evolution of density perturbations on scales smaller than the solar system size. The damping scale is inversely proportional to the square-root of the axion mass. We show that the axion mass smaller than 10^{-24} eV induces a significant damping in the baryonic density power spectrum in cosmologically relevant scales, thus deviating from the cold dark matter in the scale smaller than the axion Jeans scale. With such a small mass, however, our basic assumption about the coherently oscillating scalar field is broken in the early universe. This problem is shared by other dark matter models based on the Bose-Einstein condensate and the ultra-light scalar field. We introduce a simple model to avoid this problem by introducing evolving axion mass in the early universe, and present observational effects of present-day low-mass axion on the baryon density power spectrum, the cosmic microwave background radiation (CMB) temperature power spectrum, and the growth rate of baryon density perturbation. In our low-mass axion model we have a characteristic small-scale cutoff in the baryon density power spectrum below the axion Jeans scale. The small-scale deviations from the cold dark matter model in both matter and CMB power spectra clearly differ from the ones expected in the cold dark matter model mixed with the massive neutrinos as a hot dark matter component.Comment: 9 pages, 8 figure

Universality class of the restricted solid-on-solid model with hopping

We study the restricted solid-on-solid (RSOS) model with finite hopping distance $l_{0}$, using both analytical and numerical methods. Analytically, we use the hard-core bosonic field theory developed by the authors [Phys. Rev. E {\bf 62}, 7642 (2000)] and derive the Villain-Lai-Das Sarma (VLD) equation for the $l_{0}=\infty$ case which corresponds to the conserved RSOS (CRSOS) model and the Kardar-Parisi-Zhang (KPZ) equation for all finite values of $l_{0}$. Consequently, we find that the CRSOS model belongs to the VLD universality class and the RSOS models with any finite hopping distance belong to the KPZ universality class. There is no phase transition at a certain finite hopping distance contrary to the previous result. We confirm the analytic results using the Monte Carlo simulations for several values of the finite hopping distance.Comment: 13 pages, 3 figure

Derivation of continuum stochastic equations for discrete growth models

We present a formalism to derive the stochastic differential equations (SDEs) for several solid-on-solid growth models. Our formalism begins with a mapping of the microscopic dynamics of growth models onto the particle systems with reactions and diffusion. We then write the master equations for these corresponding particle systems and find the SDEs for the particle densities. Finally, by connecting the particle densities with the growth heights, we derive the SDEs for the height variables. Applying this formalism to discrete growth models, we find the Edwards-Wilkinson equation for the symmetric body-centered solid-on-solid (BCSOS) model, the Kardar-Parisi-Zhang equation for the asymmetric BCSOS model and the generalized restricted solid-on-solid (RSOS) model, and the Villain--Lai--Das Sarma equation for the conserved RSOS model. In addition to the consistent forms of equations for growth models, we also obtain the coefficients associated with the SDEs.Comment: 5 pages, no figur