Fractal Spectrum of a Quasi_periodically Driven Spin System

We numerically perform a spectral analysis of a quasi-periodically driven spin 1/2 system, the spectrum of which is Singular Continuous. We compute fractal dimensions of spectral measures and discuss their connections with the time behaviour of various dynamical quantities, such as the moments of the distribution of the wave packet. Our data suggest a close similarity between the information dimension of the spectrum and the exponent ruling the algebraic growth of the 'entropic width' of wavepackets.Comment: 17 pages, RevTex, 5 figs. available on request from [email protected]

Evaluating Sums over the Matsubara Frequencies

Perturbative calculations in field theory at finite temperature involve sums over the Matsubara frequencies. Besides the usual difficulties that appear in perturbative computations, these sums give rise to some new obstacles that are carefully analized here. I present a fast and realible recipe to work out sums over the Matsubara frequencies. As this algorithm leads to deal with very cumbersome algebraic expressions, it has been written for computers by using the symbolic manipulation program Mathematica. It is also shown this algorithm to be self-consistent when it is applied to more than one loop computations.Comment: 10 page

Towards Classification of 5d SCFTs: Single Gauge Node

We propose a number of apparently equivalent criteria necessary for the consistency of a 5d SCFT in its Coulomb phase and use these criteria to classify 5d SCFTs arising from a gauge theory with simple gauge group. These criteria include the convergence of the 5-sphere partition function; the positivity of particle masses and monopole string tensions; and the positive definiteness of the metric in some region in the Coulomb branch. We find that for large rank classical groups simple classes of SCFTs emerge where the bounds on the matter content and the Chern-Simons level grow linearly with rank. For classical groups of rank less than or equal to 8, our classification leads to additional cases which do not fit in the large rank analysis. We also classify the allowed matter content for all exceptional groups.Comment: 52 pages + appendix, 11 tables, 12 figure

A Canonical Form for Positive Definite Matrices

We exhibit an explicit, deterministic algorithm for finding a canonical form for a positive definite matrix under unimodular integral transformations. We use characteristic sets of short vectors and partition-backtracking graph software. The algorithm runs in a number of arithmetic operations that is exponential in the dimension $n$, but it is practical and more efficient than canonical forms based on Minkowski reduction

Scalar curvature in conformal geometry of Connes-Landi noncommutative manifolds

We first propose a conformal geometry for Connes-Landi noncommutative manifolds and study the associated scalar curvature. The new scalar curvature contains its Riemannian counterpart as the commutative limit. Similar to the results on noncommutative two tori, the quantum part of the curvature consists of actions of the modular derivation through two local curvature functions. Explicit expressions for those functions are obtained for all even dimensions (greater than two). In dimension four, the one variable function shows striking similarity to the analytic functions of the characteristic classes appeared in the Atiyah-Singer local index formula, namely, it is roughly a product of the $j$-function (which defines the $\hat A$-class of a manifold) and an exponential function (which defines the Chern character of a bundle). By performing two different computations for the variation of the Einstein-Hilbert action, we obtain a deep internal relations between two local curvature functions. Straightforward verification for those relations gives a strong conceptual confirmation for the whole computational machinery we have developed so far, especially the Mathematica code hidden behind the paper.Comment: 44 pages, 11 figures, some minor updates from the previous versio

A Combinatorial Formula for Principal Minors of a Matrix with Tree-metric Exponents and Its Applications

Let $T$ be a tree with a vertex set $\{ 1,2,\dots, N \}$. Denote by $d_{ij}$ the distance between vertices $i$ and $j$. In this paper, we present an explicit combinatorial formula of principal minors of the matrix $(t^{d_{ij}})$, and its applications to tropical geometry, study of multivariate stable polynomials, and representation of valuated matroids. We also give an analogous formula for a skew-symmetric matrix associated with $T$.Comment: 16 page

String Consistency for Unified Model Building

We explore the use of real fermionization as a test case for understanding how specific features of phenomenological interest in the low-energy effective superpotential are realized in exact solutions to heterotic superstring theory. We present pedagogic examples of models which realize SO(10) as a level two current algebra on the world-sheet, and discuss in general how higher level current algebras can be realized in the tensor product of simple constituent conformal field theories. We describe formal developments necessary to compute couplings in models built using real fermionization. This allows us to isolate cases of spin structures where the standard prescription for real fermionization may break down.Comment: harvmac (available from xxx.lanl.gov), 30 pages (reduced format), if you are using harvmac for the first time, make sure to adjust the "site dependent options" at the beginning of the harvmac file. Shortened introduction and added table 3, listing the complete massless spectrum with U(1) charges of Model A. Version to appear in journa

Instantons, Topological Strings and Enumerative Geometry

We review and elaborate on certain aspects of the connections between instanton counting in maximally supersymmetric gauge theories and the computation of enumerative invariants of smooth varieties. We study in detail three instances of gauge theories in six, four and two dimensions which naturally arise in the context of topological string theory on certain non-compact threefolds. We describe how the instanton counting in these gauge theories are related to the computation of the entropy of supersymmetric black holes, and how these results are related to wall-crossing properties of enumerative invariants such as Donaldson-Thomas and Gromov-Witten invariants. Some features of moduli spaces of torsion-free sheaves and the computation of their Euler characteristics are also elucidated.Comment: 61 pages; v2: Typos corrected, reference added; v3: References added and updated; Invited article for the special issue "Nonlinear and Noncommutative Mathematics: New Developments and Applications in Quantum Physics" of Advances in Mathematical Physic