SQCD: A Geometric Apercu

We take new algebraic and geometric perspectives on the old subject of SQCD. We count chiral gauge invariant operators using generating functions, or Hilbert series, derived from the plethystic programme and the Molien-Weyl formula. Using the character expansion technique, we also see how the global symmetries are encoded in the generating functions. Equipped with these methods and techniques of algorithmic algebraic geometry, we obtain the character expansions for theories with arbitrary numbers of colours and flavours. Moreover, computational algebraic geometry allows us to systematically study the classical vacuum moduli space of SQCD and investigate such structures as its irreducible components, degree and syzygies. We find the vacuum manifolds of SQCD to be affine Calabi-Yau cones over weighted projective varieties.Comment: 49 pages, 1 figur

A new 2D static hand gesture colour image dataset for ASL gestures

It usually takes a fusion of image processing and machine learning algorithms in order to build a fully-functioning computer vision system for hand gesture recognition. Fortunately, the complexity of developing such a system could be alleviated by treating the system as a collection of multiple sub-systems working together, in such a way that they can be dealt with in isolation. Machine learning need to feed on thousands of exemplars (e.g. images, features) to automatically establish some recognisable patterns for all possible classes (e.g. hand gestures) that applies to the problem domain. A good number of exemplars helps, but it is also important to note that the efficacy of these exemplars depends on the variability of illumination conditions, hand postures, angles of rotation, scaling and on the number of volunteers from whom the hand gesture images were taken. These exemplars are usually subjected to image processing first, to reduce the presence of noise and extract the important features from the images. These features serve as inputs to the machine learning system. Different sub-systems are integrated together to form a complete computer vision system for gesture recognition. The main contribution of this work is on the production of the exemplars. We discuss how a dataset of standard American Sign Language (ASL) hand gestures containing 2425 images from 5 individuals, with variations in lighting conditions and hand postures is generated with the aid of image processing techniques. A minor contribution is given in the form of a specific feature extraction method called moment invariants, for which the computation method and the values are furnished with the dataset

Functional integration and abelian link invariants

The functional integral computation of the various topological invariants, which are associated with the Chern-Simons field theory, is considered. The standard perturbative setting in quantum field theory is rewieved and new developments in the path-integral approach, based on the Deligne-Beilinson cohomology, are described in the case of the abelian U(1) Chern-Simons field theory formulated in S^1 x S^2.Comment: 20 pages, 4 figures, Contribution to the Proceedings of the workshop "Chern-Simons Gauge theory: 20 years after", Bonn, August 200

Curve counting, instantons and McKay correspondences

We survey some features of equivariant instanton partition functions of topological gauge theories on four and six dimensional toric Kahler varieties, and their geometric and algebraic counterparts in the enumerative problem of counting holomorphic curves. We discuss the relations of instanton counting to representations of affine Lie algebras in the four-dimensional case, and to Donaldson-Thomas theory for ideal sheaves on Calabi-Yau threefolds. For resolutions of toric singularities, an algebraic structure induced by a quiver determines the instanton moduli space through the McKay correspondence and its generalizations. The correspondence elucidates the realization of gauge theory partition functions as quasi-modular forms, and reformulates the computation of noncommutative Donaldson-Thomas invariants in terms of the enumeration of generalized instantons. New results include a general presentation of the partition functions on ALE spaces as affine characters, a rigorous treatment of equivariant partition functions on Hirzebruch surfaces, and a putative connection between the special McKay correspondence and instanton counting on Hirzebruch-Jung spaces.Comment: 79 pages, 3 figures; v2: typos corrected, reference added, new summary section included; Final version to appear in Journal of Geometry and Physic

Multi-Colour Braid-Monoid Algebras

We define multi-colour generalizations of braid-monoid algebras and present explicit matrix representations which are related to two-dimensional exactly solvable lattice models of statistical mechanics. In particular, we show that the two-colour braid-monoid algebra describes the Yang-Baxter algebra of the critical dilute A-D-E models which were recently introduced by Warnaar, Nienhuis, and Seaton as well as by Roche. These and other solvable models related to dense and dilute loop models are discussed in detail and it is shown that the solvability is a direct consequence of the algebraic structure. It is conjectured that the Yang-Baxterization of general multi-colour braid-monoid algebras will lead to the construction of further solvable lattice models.Comment: 32 page