Central extensions of current groups in two dimensions

In this paper we generalize some of these results for loop algebras and groups as well as for the Virasoro algebra to the two-dimensional case. We define and study a class of infinite dimensional complex Lie groups which are central extensions of the group of smooth maps from a two dimensional orientable surface without boundary to a simple complex Lie group G. These extensions naturally correspond to complex curves. The kernel of such an extension is the Jacobian of the curve. The study of the coadjoint action shows that its orbits are labelled by moduli of holomorphic principal G-bundles over the curve and can be described in the language of partial differential equations. In genus one it is also possible to describe the orbits as conjugacy classes of the twisted loop group, which leads to consideration of difference equations for holomorphic functions. This gives rise to a hope that the described groups should possess a counterpart of the rich representation theory that has been developed for loop groups. We also define a two-dimensional analogue of the Virasoro algebra associated with a complex curve. In genus one, a study of a complex analogue of Hill's operator yields a description of invariants of the coadjoint action of this Lie algebra. The answer turns out to be the same as in dimension one: the invariants coincide with those for the extended algebra of currents in sl(2).Comment: 17 page

Chern-Simons Reduction and non-Abelian Fluid Mechanics

We propose a non-Abelian generalization of the Clebsch parameterization for a vector in three dimensions. The construction is based on a group-theoretical reduction of the Chern-Simons form on a symmetric space. The formalism is then used to give a canonical (symplectic) discussion of non-Abelian fluid mechanics, analogous to the way the Abelian Clebsch parameterization allows a canonical description of conventional fluid mechanics.Comment: 12 pages, REVTeX; revised for publication in Phys Rev D; email to [email protected]

Decomposable representations and Lagrangian submanifolds of moduli spaces associated to surface groups

In this paper, we construct a Lagrangian submanifold of the moduli space associated to the fundamental group of a punctured Riemann surface (the space of representations of this fundamental group into a compact connected Lie group). This Lagrangian submanifold is obtained as the fixed-point set of an anti-symplectic involution defined on the moduli space. The notion of decomposable representation provides a geometric interpretation of this Lagrangian submanifold

Long range facial image acquisition and quality

Abstract This chapter introduces issues in long range facial image acquisition and measures for image quality and their usage. Section 1, on image acquisition for face recognition discusses issues in lighting, sensor, lens, blur issues, which impact short-range biometrics, but are more pronounced in long-range biometrics. Section 2 introduces the design of controlled experiments for long range face, and why they are needed. Section 3 introduces some of the weather and atmospheric effects that occur for long-range imaging, with numerous of examples. Section 4 addresses measurements of “system quality”, including image-quality measures and their use in prediction of face recognition algorithm. That section introduces the concept of failure prediction and techniques for analyzing different “quality ” measures. The section ends with a discussion of post-recognition ”failure prediction ” and its potential role as a feedback mechanism in acquisition. Each section includes a collection of open-ended questions to challenge the reader to think about the concepts more deeply. For some of the questions we answer them after they are introduced; others are left as an exercise for the reader. 1 Image Acquisition Before any recognition can even be attempted, they system must acquire an image of the subject with sufficient quality and resolution to detect and recognize the face. The issues examined in this section are the sensor-issues in lighting, image/sensor resolution issues, the field-of view, the depth of field, and effects of motion blur

Gauge-Invariant Coordinates on Gauge-Theory Orbit Space

A gauge-invariant field is found which describes physical configurations, i.e. gauge orbits, of non-Abelian gauge theories. This is accomplished with non-Abelian generalizations of the Poincare'-Hodge formula for one-forms. In a particular sense, the new field is dual to the gauge field. Using this field as a coordinate, the metric and intrinsic curvature are discussed for Yang-Mills orbit space for the (2+1)- and (3+1)-dimensional cases. The sectional, Ricci and scalar curvatures are all formally non-negative. An expression for the new field in terms of the Yang-Mills connection is found in 2+1 dimensions. The measure on Schroedinger wave functionals is found in both 2+1 and 3+1 dimensions; in the former case, it resembles Karabali, Kim and Nair's measure. We briefly discuss the form of the Hamiltonian in terms of the dual field and comment on how this is relevant to the mass gap for both the (2+1)- and (3+1)-dimensional cases.Comment: Typos corrected, more about the non-Abelian decomposition and inner products, more discussion of the mass gap in 3+1 dimensions. Now 23 page

Moduli of Abelian varieties, Vinberg theta-groups, and free resolutions

We present a systematic approach to studying the geometric aspects of Vinberg theta-representations. The main idea is to use the Borel-Weil construction for representations of reductive groups as sections of homogeneous bundles on homogeneous spaces, and then to study degeneracy loci of these vector bundles. Our main technical tool is to use free resolutions as an "enhanced" version of degeneracy loci formulas. We illustrate our approach on several examples and show how they are connected to moduli spaces of Abelian varieties. To make the article accessible to both algebraists and geometers, we also include background material on free resolutions and representation theory.Comment: 41 pages, uses tabmac.sty, Dedicated to David Eisenbud on the occasion of his 65th birthday; v2: fixed some typos and added reference

SU(2) WZW Theory at Higher Genera

We compute, by free field techniques, the scalar product of the SU(2) Chern-Simons states on genus > 1 surfaces. The result is a finite-dimensional integral over positions of ``screening charges'' and one complex modular parameter. It uses an effective description of the CS states closely related to the one worked out by Bertram. The scalar product formula allows to express the higher genus partition functions of the WZW conformal field theory by finite-dimensional integrals. It should provide the hermitian metric preserved by the Knizhnik-Zamolodchikov-Bernard connection describing the variations of the CS states under the change of the complex structure of the surface.Comment: 44 pages, IHES/P/94/10, Latex fil

Riemannian Gauge Theory and Charge Quantization

In a traditional gauge theory, the matter fields \phi^a and the gauge fields A^c_\mu are fundamental objects of the theory. The traditional gauge field is similar to the connection coefficient in the Riemannian geometry covariant derivative, and the field-strength tensor is similar to the curvature tensor. In contrast, the connection in Riemannian geometry is derived from the metric or an embedding space. Guided by the physical principal of increasing symmetry among the four forces, we propose a different construction. Instead of defining the transformation properties of a fundamental gauge field, we derive the gauge theory from an embedding of a gauge fiber F=R^n or F=C^n into a trivial, embedding vector bundle F=R^N or F=C^N where N>n. Our new action is symmetric between the gauge theory and the Riemannian geometry. By expressing gauge-covariant fields in terms of the orthonormal gauge basis vectors, we recover a traditional, SO(n) or U(n) gauge theory. In contrast, the new theory has all matter fields on a particular fiber couple with the same coupling constant. Even the matter fields on a C^1 fiber, which have a U(1) symmetry group, couple with the same charge of +/- q. The physical origin of this unique coupling constant is a generalization of the general relativity equivalence principle. Because our action is independent of the choice of basis, its natural invariance group is GL(n,R) or GL(n,C). Last, the new action also requires a small correction to the general-relativity action proportional to the square of the curvature tensor.Comment: Improved the explanations, added references, added 3 figures and an appendix, corrected a sign error in the old figure 4 (now figure 5). Now 33 pages, 7 figures and 2 tables. E-mail Serna for annimation

N=2 Topological Yang-Mills Theory on Compact K\"{a}hler Surfaces

We study a topological Yang-Mills theory with $N=2$ fermionic symmetry. Our formalism is a field theoretical interpretation of the Donaldson polynomial invariants on compact K\"{a}hler surfaces. We also study an analogous theory on compact oriented Riemann surfaces and briefly discuss a possible application of the Witten's non-Abelian localization formula to the problems in the case of compact K\"{a}hler surfaces.Comment: ESENAT-93-01 & YUMS-93-10, 34pages: [Final Version] to appear in Comm. Math. Phy

Gauge Orbit Types for Theories with Classical Compact Gauge Group

We determine the orbit types of the action of the group of local gauge transformations on the space of connections in a principal bundle with structure group O(n), SO(n) or $Sp(n)$ over a closed, simply connected manifold of dimension 4. Complemented with earlier results on U(n) and SU(n) this completes the classification of the orbit types for all classical compact gauge groups over such space-time manifolds. On the way we derive the classification of principal bundles with structure group SO(n) over these manifolds and the Howe subgroups of SO(n).Comment: 57 page