Density of Spherically-Embedded Stiefel and Grassmann Codes

The density of a code is the fraction of the coding space covered by packing balls centered around the codewords. This paper investigates the density of codes in the complex Stiefel and Grassmann manifolds equipped with the chordal distance. The choice of distance enables the treatment of the manifolds as subspaces of Euclidean hyperspheres. In this geometry, the densest packings are not necessarily equivalent to maximum-minimum-distance codes. Computing a code's density follows from computing: i) the normalized volume of a metric ball and ii) the kissing radius, the radius of the largest balls one can pack around the codewords without overlapping. First, the normalized volume of a metric ball is evaluated by asymptotic approximations. The volume of a small ball can be well-approximated by the volume of a locally-equivalent tangential ball. In order to properly normalize this approximation, the precise volumes of the manifolds induced by their spherical embedding are computed. For larger balls, a hyperspherical cap approximation is used, which is justified by a volume comparison theorem showing that the normalized volume of a ball in the Stiefel or Grassmann manifold is asymptotically equal to the normalized volume of a ball in its embedding sphere as the dimension grows to infinity. Then, bounds on the kissing radius are derived alongside corresponding bounds on the density. Unlike spherical codes or codes in flat spaces, the kissing radius of Grassmann or Stiefel codes cannot be exactly determined from its minimum distance. It is nonetheless possible to derive bounds on density as functions of the minimum distance. Stiefel and Grassmann codes have larger density than their image spherical codes when dimensions tend to infinity. Finally, the bounds on density lead to refinements of the standard Hamming bounds for Stiefel and Grassmann codes.Comment: Two-column version (24 pages, 6 figures, 4 tables). To appear in IEEE Transactions on Information Theor

Loops, Surfaces and Grassmann Representation in Two- and Three-Dimensional Ising Models

Starting from the known representation of the partition function of the 2- and 3-D Ising models as an integral over Grassmann variables, we perform a hopping expansion of the corresponding Pfaffian. We show that this expansion is an exact, algebraic representation of the loop- and surface expansions (with intrinsic geometry) of the 2- and 3-D Ising models. Such an algebraic calculus is much simpler to deal with than working with the geometrical objects. For the 2-D case we show that the algebra of hopping generators allows a simple algebraic treatment of the geometry factors and counting problems, and as a result we obtain the corrected loop expansion of the free energy. We compute the radius of convergence of this expansion and show that it is determined by the critical temperature. In 3-D the hopping expansion leads to the surface representation of the Ising model in terms of surfaces with intrinsic geometry. Based on a representation of the 3-D model as a product of 2-D models coupled to an auxiliary field, we give a simple derivation of the geometry factor which prevents overcounting of surfaces and provide a classification of possible sets of surfaces to be summed over. For 2- and 3-D we derive a compact formula for 2n-point functions in loop (surface) representation.Comment: 31 pages, 9 figure

Moduli of vortices and Grassmann manifolds

We use the framework of Quot schemes to give a novel description of the moduli spaces of stable n-pairs, also interpreted as gauged vortices on a closed Riemann surface with target Mat(r x n, C), where n >= r. We then show that these moduli spaces embed canonically into certain Grassmann manifolds, and thus obtain natural Kaehler metrics of Fubini-Study type; these spaces are smooth at least in the local case r=n. For abelian local vortices we prove that, if a certain "quantization" condition is satisfied, the embedding can be chosen in such a way that the induced Fubini-Study structure realizes the Kaehler class of the usual L^2 metric of gauged vortices.Comment: 22 pages, LaTeX. Final version: last section removed, typos corrected, two references added; to appear in Commun. Math. Phy

Black Hole Superpartners and Fixed Scalars

Some bosonic solutions of supergravities admit Killing spinors of unbroken supersymmetry. The anti-Killing spinors of broken supersymmetry can be used to generate the superpartners of stringy black holes. This has a consequent feedback on the metric and the graviphoton. We have found however that the fixed scalars for the black hole superpartners remain the same as for the original black holes. Possible phenomenological implications of this result are discussed.Comment: 6 pages, Late