The order of curvature operators on loop groups

For loop groups (free and based), we compute the exact order of the curvature operator of the Levi-Civita connection depending on a Sobolev space parameter. This extends results of Freed and Maeda-Rosenberg-Torres.Comment: to appear in Letters in Mathematical Physic

The use of disjunct eddy sampling methods for the determination of ecosystem level fluxes of trace gases

The concept of disjunct eddy sampling (DES) for use in measuring ecosystem-level micrometeorological fluxes is re-examined. The governing equations are discussed as well as other practical considerations and guidelines concerning this sampling method as it is applied to either the disjunct eddy covariance (DEC) or disjunct eddy accumulation (DEA) techniques. A disjunct eddy sampling system was constructed that could either be combined with relatively slow sensors (response time of 2 to 40 s) to measure fluxes using DEC, or could also be used to accumulate samples in stable reservoirs for later laboratory analysis (DEA technique). Both the DEC and DEA modes of this sampler were tested against conventional eddy covariance (EC) for fluxes of either CO2 (DEC) or isoprene (DEA). Good agreement in both modes was observed relative to the EC systems. However, the uncertainty in a single DEA flux measurement was considerable (40%) due to both the reduced statistical sampling and the analytical precision of the concentration difference measurements. We have also re-investigated the effects of nonzero mean vertical wind velocity on accumulation techniques as it relates to our DEA measurements. Despite the higher uncertainty, disjunct eddy sampling can provide an alternative technique to eddy covariance for determining ecosystem-level fluxes for species where fast sensors do not currently exist

Gauge fixing and Gribov copies in pure Yang-Mills on a circle

%In order to understand how gauge fixing can be affected on the %lattice, we first study a simple model of pure Yang-mills theory on a %cylindrical spacetime [$SU(N)$ on $S^1 \times$ {\bf R}] where the %gauge fixed subspace is explicitly displayed. On the way, we find that %different gauge fixing procedures lead to different Hamiltonians and %spectra, which however coincide under a shift of states. The lattice %version of the model is compared and lattice gauge fixing issues are %discussed. (---TALK GIVEN AT LATTICE 92---AMSTERDAM, 15 SEPT. 92)Comment: 4 pages + 1 PostScript figure (appended), UVA-ITFA-92-34/ETH-IPS-92-22. --just archiving published versio

Fractional Loop Group and Twisted K-Theory

We study the structure of abelian extensions of the group $L_qG$ of $q$-differentiable loops (in the Sobolev sense), generalizing from the case of central extension of the smooth loop group. This is motivated by the aim of understanding the problems with current algebras in higher dimensions. Highest weight modules are constructed for the Lie algebra. The construction is extended to the current algebra of supersymmetric Wess-Zumino-Witten model. An application to the twisted K-theory on $G$ is discussed.Comment: Final version in Commun. Math. Phy

Loop groups and noncommutative geometry

We describe the representation theory of loop groups in terms of K-theory and noncommutative geometry. This is done by constructing suitable spectral triples associated with the level l projective unitary positive-energy representations of any given loop group $LG$. The construction is based on certain supersymmetric conformal field theory models associated with LG in the setting of conformal nets. We then generalize the construction to many other rational chiral conformal field theory models including coset models and the moonshine conformal net.Comment: Revised versio

Non-abelian Harmonic Oscillators and Chiral Theories

We show that a large class of physical theories which has been under intensive investigation recently, share the same geometric features in their Hamiltonian formulation. These dynamical systems range from harmonic oscillations to WZW-like models and to the KdV dynamics on $Diff_oS^1$. To the same class belong also the Hamiltonian systems on groups of maps. The common feature of these models are the 'chiral' equations of motion allowing for so-called chiral decomposition of the phase space.Comment: 1

String Equations for the Unitary Matrix Model and the Periodic Flag Manifold

The periodic flag manifold (in the Sato Grassmannian context) description of the modified Korteweg--de Vries hierarchy is used to analyse the translational and scaling self--similar solutions of this hierarchy. These solutions are characterized by the string equations appearing in the double scaling limit of the symmetric unitary matrix model with boundary terms. The moduli space is a double covering of the moduli space in the Sato Grassmannian for the corresponding self--similar solutions of the Korteweg--de Vries hierarchy, i.e. of stable 2D quantum gravity. The potential modified Korteweg--de Vries hierarchy, which can be described in terms of a line bundle over the periodic flag manifold, and its self--similar solutions corresponds to the symmetric unitary matrix model. Now, the moduli space is in one--to--one correspondence with a subset of codimension one of the moduli space in the Sato Grassmannian corresponding to self--similar solutions of the Korteweg--de Vries hierarchy.Comment: 21 pages in LaTeX-AMSTe

Vector Coherent State Realization of Representations of the Affine Lie Algebra sl^(2)\hat{sl}(2)

The method of vector coherent states is generalized to study representations of the affine Lie algebra $\hat{sl}(2)$. A large class of highest weight irreps is explicitly constructed, which contains the integrable highest weight irreps as special cases.Comment: 8 pages plain latex. To appear in J. Phys.

Going beyond defining: Preschool educators\u27 use of knowledge in their pedagogical reasoning about vocabulary instruction

Previous research investigating both the knowledge of early childhood educators and the support for vocabulary development present in early childhood settings has indicated that both educator knowledge and enacted practice are less than optimal, which has grave implications for children\u27s early vocabulary learning and later reading achievement. Further, the nature of the relationship between educators\u27 knowledge and practice is unclear, making it difficult to discern the best path towards improved knowledge, practice, and children\u27s vocabulary outcomes. The purpose of the present study was to add to the existing literature by using stimulated recall interviews and a grounded approach to examine how 10 preschool educators used their knowledge to made decisions about their moment-to-moment instruction in support of children\u27s vocabulary development. Results indicate that educators were thinking in highly context-specific ways about their goals and strategies for supporting vocabulary learning, taking into account important knowledge of their instructional history with children and of the children themselves to inform their decision making in the moment. In addition, they reported thinking about research-based goals and strategies for supporting vocabulary learning that went beyond simply defining words for children. Implications for research and professional development are discussed

Geometry of W-algebras from the affine Lie algebra point of view

To classify the classical field theories with W-symmetry one has to classify the symplectic leaves of the corresponding W-algebra, which are the intersection of the defining constraint and the coadjoint orbit of the affine Lie algebra if the W-algebra in question is obtained by reducing a WZNW model. The fields that survive the reduction will obey non-linear Poisson bracket (or commutator) relations in general. For example the Toda models are well-known theories which possess such a non-linear W-symmetry and many features of these models can only be understood if one investigates the reduction procedure. In this paper we analyze the SL(n,R) case from which the so-called W_n-algebras can be obtained. One advantage of the reduction viewpoint is that it gives a constructive way to classify the symplectic leaves of the W-algebra which we had done in the n=2 case which will correspond to the coadjoint orbits of the Virasoro algebra and for n=3 which case gives rise to the Zamolodchikov algebra. Our method in principle is capable of constructing explicit representatives on each leaf. Another attractive feature of this approach is the fact that the global nature of the W-transformations can be explicitly described. The reduction method also enables one to determine the ``classical highest weight (h. w.) states'' which are the stable minima of the energy on a W-leaf. These are important as only to those leaves can a highest weight representation space of the W-algebra be associated which contains a ``classical h. w. state''.Comment: 17 pages, LaTeX, revised 1. and 7. chapter