Coulomb Gas Partition Function of a Layered Loop Model

We consider a two-dimensional bi-layered loop model with a certain interlayer coupling and study its spectrum on a torus. Each layer consists of an $O(n)$ model on a honeycomb lattice with periodic boundary conditions; these layers are stacked such that the links of the lattice intersect each other. A complex Boltzmann weight $\lambda$ with unit modulus is assigned to each intersection of two loops each from each layer. The model is reduced to an inhomogeneous vertex model at a special point of parameters. The continuum partition function is represented, based on the idea of the Coulomb gas, by a path integral over two compact bosonic fields. The modular invariance of the partition function follows naturally. Further, because of the topological nature of the interlayer coupling, the fluctuation of loops decomposes into a local and a global part. The existence of the latter leads to a sum over all the pairs of torus knots, which can be Poisson ressummed by the M\"{o}bius inversion formula. This reveals the operator content of the theory. The multiplicity of each operator is explicitly given by a combination of two Ramanujan sums. We calculate each scaling dimension as a function of $\lambda$. We present the flow of dimensions which connects the decoupled-$O(1)$ models at $\lambda=1$ and the layered-$O(1)$ model with the non-trivial coupling $\lambda=-1$. The lower spectrum in the latter model is related to that of a known coset model.Comment: 22 pages, 7 figures, uses iopart.cl

Laser energy converted into electric power

Apparatus verifies concepts of converting laser energy directly into electric energy. Mirror, placed in beam and inclined at angle to it, directs small amount of incident radiation to monitor which establishes precise power levels and other beam characteristics. Second mirror and condensing lens direct bulk of laser energy into laser plasmadynamic converter

Conformal amplitude hierarchy and the Poincare disk

The amplitude for the singlet channels in the 4-point function of the fundamental field in the conformal field theory of the 2d $O(n)$ model is studied as a function of $n$. For a generic value of $n$, the 4-point function has infinitely many amplitudes, whose landscape can be very spiky as the higher amplitude changes its sign many times at the simple poles, which generalize the unique pole of the energy operator amplitude at $n=0$. In the stadard parameterization of $n$ by angle in unit of $\pi$, we find that the zeros and poles happen at the rational angles, forming a hierarchical tree structure inherent in the Poincar\'{e} disk. Some relation between the amplitude and the Farey path, a piecewise geodesic that visits these zeros and poles, is suggested. In this hierarchy, the symmetry of the congruence subgroup $\Gamma(2)$ of $SL(2,\mathbb{Z})$ naturally arises from the two clearly distinct even/odd classes of the rational angles, in which one respectively gets the truncated operator algebras and the logarithmic 4-point functions.Comment: 13 pages, 2 figures. see this version; corrections made; references adde

Transcendental lattices and supersingular reduction lattices of a singular K3K3 surface

A (smooth) K3 surface X defined over a field k of characteristic 0 is called singular if the N\'eron-Severi lattice NS (X) of X over the algebraic closure of k is of rank 20. Let X be a singular K3 surface defined over a number field F. For each embedding \sigma of F into the complex number field, we denote by T(X^\sigma) the transcendental lattice of the complex K3 surface X^\sigma obtained from X by \sigma. For each prime ideal P of F at which X has a supersingular reduction X_P, we define L(X, P) to be the orthogonal complement of NS(X) in NS(X_P). We investigate the relation between these lattices T(X^\sigma) and L(X, P). As an application, we give a lower bound of the degree of a number field over which a singular K3 surface with a given transcendental lattice can be defined.Comment: 40 pages, revised version, to appear in Transactions of the American Mathematical Societ

Worker Participation in Management Decision Making

Draft Presented to International Evidence: Worker-Management Institutions and Economic Performance Conference, U.S. Commission on the Future of Worker-Management Relations Suggested Citation Shimada, H. (1994).Paper_Shimada_020694.pdf: 10729 downloads, before Oct. 1, 2020