On the Monodromies of N=2 Supersymmetric Yang-Mills Theory with Gauge Group SO(2n)

We present families of algebraic curves describing the moduli-space of $N\!=\!2$ supersymmetric Yang-Mills theory with gauge group $SO(2n)$. We test our curves by computing the weak coupling monodromies and the number of $N\!=\!1$ vacua.Comment: 14 pages, 5 Postscript figures, LaTeX file, uses epsf.st

Stable and unstable attractors in Boolean networks

Boolean networks at the critical point have been a matter of debate for many years as, e.g., scaling of number of attractor with system size. Recently it was found that this number scales superpolynomially with system size, contrary to a common earlier expectation of sublinear scaling. We here point to the fact that these results are obtained using deterministic parallel update, where a large fraction of attractors in fact are an artifact of the updating scheme. This limits the significance of these results for biological systems where noise is omnipresent. We here take a fresh look at attractors in Boolean networks with the original motivation of simplified models for biological systems in mind. We test stability of attractors w.r.t. infinitesimal deviations from synchronous update and find that most attractors found under parallel update are artifacts arising from the synchronous clocking mode. The remaining fraction of attractors are stable against fluctuating response delays. For this subset of stable attractors we observe sublinear scaling of the number of attractors with system size.Comment: extended version, additional figur

Elliptic blowup equations for 6d SCFTs. Part II: Exceptional cases

The building blocks of 6d $(1,0)$ SCFTs include certain rank one theories with gauge group $G=SU(3),SO(8),F_4,E_{6,7,8}$. In this paper, we propose a universal recursion formula for the elliptic genera of all such theories. This formula is solved from the elliptic blowup equations introduced in our previous paper. We explicitly compute the elliptic genera and refined BPS invariants, which recover all previous results from topological string theory, modular bootstrap, Hilbert series, 2d quiver gauge theories and 4d $\mathcal{N}=2$ superconformal $H_{G}$ theories. We also observe an intriguing relation between the $k$-string elliptic genus and the Schur indices of rank $k$ $H_{G}$ SCFTs, as a generalization of Lockhart-Zotto's conjecture at the rank one cases. In a subsequent paper, we deal with all other non-Higgsable clusters with matters

Comment on "c-axis Josephson tunneling in Dx2−y2D_{x^2-y^2}-wave superconductors''

This comment points out that the recent paper by Maki and Haas [Phys. Rev. B {\bf 67}, 020510 (2003)] is completely wrong.Comment: 1 page, submittted to Phys. Rev.

Seiberg-Witten Curve for the E-String Theory

We construct the Seiberg-Witten curve for the E-string theory in six-dimensions. The curve is expressed in terms of affine E_8 characters up to level 6 and is determined by using the mirror-type transformation so that it reproduces the number of holomorphic curves in the Calabi-Yau manifold and the amplitudes of N=4 U(n) Yang-Mills theory on 1/2 K3. We also show that our curve flows to known five- and four-dimensional Seiberg-Witten curves in suitable limits.Comment: 18 pages, 1 figure; appendix C adde

Divergent Time Scale in Axelrod Model Dynamics

We study the evolution of the Axelrod model for cultural diversity. We consider a simple version of the model in which each individual is characterized by two features, each of which can assume q possibilities. Within a mean-field description, we find a transition at a critical value q_c between an active state of diversity and a frozen state. For q just below q_c, the density of active links between interaction partners is non-monotonic in time and the asymptotic approach to the steady state is controlled by a time scale that diverges as (q-q_c)^{-1/2}.Comment: 4 pages, 5 figures, 2-column revtex4 forma

Holomorphic Anomaly in Gauge Theories and Matrix Models

We use the holomorphic anomaly equation to solve the gravitational corrections to Seiberg-Witten theory and a two-cut matrix model, which is related by the Dijkgraaf-Vafa conjecture to the topological B-model on a local Calabi-Yau manifold. In both cases we construct propagators that give a recursive solution in the genus modulo a holomorphic ambiguity. In the case of Seiberg-Witten theory the gravitational corrections can be expressed in closed form as quasimodular functions of Gamma(2). In the matrix model we fix the holomorphic ambiguity up to genus two. The latter result establishes the Dijkgraaf-Vafa conjecture at that genus and yields a new method for solving the matrix model at fixed genus in closed form in terms of generalized hypergeometric functions.Comment: 34 pages, 2 eps figures, expansion at the monopole point corrected and interpreted, and references adde

Symmetry-surfing the moduli space of Kummer K3s.

A maximal subgroup of the Mathieu group M24 arises as the combined holomorphic symplectic automorphism group of all Kummer surfaces whose Kaehler class is induced from the underlying complex torus. As a subgroup of M24, this group is the stabilizer group of an octad in the Golay code. To meaningfully combine the symmetry groups of distinct Kummer surfaces, we introduce the concepts of Niemeier markings and overarching maps between pairs of Kummer surfaces. The latter induce a prescription for symmetry-surfing the moduli space, while the former can be seen as a first step towards constructing a vertex algebra that governs the elliptic genus of K3 in an M24-compatible fashion. We thus argue that a geometric approach from K3 to Mathieu Moonshine may bear fruit.Comment: 20 pages; minor changes; accepted for publication in the Proceedings Volume of String-Math 201

Does dynamics reflect topology in directed networks?

We present and analyze a topologically induced transition from ordered, synchronized to disordered dynamics in directed networks of oscillators. The analysis reveals where in the space of networks this transition occurs and its underlying mechanisms. If disordered, the dynamics of the units is precisely determined by the topology of the network and thus characteristic for it. We develop a method to predict the disordered dynamics from topology. The results suggest a new route towards understanding how the precise dynamics of the units of a directed network may encode information about its topology.Comment: 7 pages, 4 figures, Europhysics Letters, accepte