Noncommutative Schur polynomials and the crystal limit of the U_q sl(2)-vertex model

Starting from the Verma module of U_q sl(2) we consider the evaluation module for affine U_q sl(2) and discuss its crystal limit (q=0). There exists an associated integrable statistical mechanics model on a square lattice defined in terms of vertex configurations. Its transfer matrix is the generating function for noncommutative complete symmetric polynomials in the generators of the affine plactic algebra, an extension of the finite plactic algebra first discussed by Lascoux and Sch\"{u}tzenberger. The corresponding noncommutative elementary symmetric polynomials were recently shown to be generated by the transfer matrix of the so-called phase model discussed by Bogoliubov, Izergin and Kitanine. Here we establish that both generating functions satisfy Baxter's TQ-equation in the crystal limit by tying them to special U_q sl(2) solutions of the Yang-Baxter equation. The TQ-equation amounts to the well-known Jacobi-Trudy formula leading naturally to the definition of noncommutative Schur polynomials. The latter can be employed to define a ring which has applications in conformal field theory and enumerative geometry: it is isomorphic to the fusion ring of the sl(n)_k -WZNW model whose structure constants are the dimensions of spaces of generalized theta-functions over the Riemann sphere with three punctures.Comment: 24 pages, 6 figures; v2: several typos fixe

Bott periodicity and stable quantum classes

We use Bott periodicity to relate previously defined quantum classes to certain "exotic Chern classes" on $BU$. This provides an interesting computational and theoretical framework for some Gromov-Witten invariants connected with cohomological field theories. This framework has applications to study of higher dimensional, Hamiltonian rigidity aspects of Hofer geometry of $\mathbb{CP} ^{n}$, one of which we discuss here.Comment: prepublication 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

Open-closed string correspondence: D-brane decay in curved space

This paper analyzes the effect of curved closed string backgrounds on the stability of D-branes within boundary string field theory. We identify the non-local open string background that implements shifts in the closed string background and analyze the tachyonic sector off-shell. The renormalization group flow reveals some characteristic properties, which are expected for a curved background, like the absence of a stable space-filling brane. In 3-dimensions we describe tachyon condensation processes to lower-dimensional branes, including a curved 2-dimensional brane. We argue that this 2-brane is perturbatively stable. This is in agreement with the known maximally symmetric WZW-branes and provides further support to the bulk-boundary factorization approach to open-closed string correspondence.Comment: 23 pages, harvma

An algebraic approach to logarithmic conformal field theory

A comprehensive introduction to logarithmic conformal field theory, using an algebraic point of view, is given. A number of examples are explained in detail, including the c=-2 triplet theory and the k=-4/3 affine su(2) theory. We also give some brief introduction to the work of Zhu.Comment: 55 pages, LaTeX, 2 figures; notes of lectures delivered at the school on Logarithmic Conformal Field Theory and its Applications, IPM Tehran, September 200

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

Loop Groups and Discrete KdV Equations

A study is presented of fully discretized lattice equations associated with the KdV hierarchy. Loop group methods give a systematic way of constructing discretizations of the equations in the hierarchy. The lattice KdV system of Nijhoff et al. arises from the lowest order discretization of the trivial, lowest order equation in the hierarchy, b_t=b_x. Two new discretizations are also given, the lowest order discretization of the first nontrivial equation in the hierarchy, and a "second order" discretization of b_t=b_x. The former, which is given the name "full lattice KdV" has the (potential) KdV equation as a standard continuum limit. For each discretization a Backlund transformation is given and soliton content analyzed. The full lattice KdV system has, like KdV itself, solitons of all speeds, whereas both other discretizations studied have a limited range of speeds, being discretizations of an equation with solutions only of a fixed speed.Comment: LaTeX, 23 pages, 1 figur

Gauged diffeomorphisms and hidden symmetries in Kaluza-Klein theories

We analyze the symmetries that are realized on the massive Kaluza-Klein modes in generic D-dimensional backgrounds with three non-compact directions. For this we construct the unbroken phase given by the decompactification limit, in which the higher Kaluza-Klein modes are massless. The latter admits an infinite-dimensional extension of the three-dimensional diffeomorphism group as local symmetry and, moreover, a current algebra associated to SL(D-2,R) together with the diffeomorphism algebra of the internal manifold as global symmetries. It is shown that the `broken phase' can be reconstructed by gauging a certain subgroup of the global symmetries. This deforms the three-dimensional diffeomorphisms to a gauged version, and it is shown that they can be governed by a Chern-Simons theory, which unifies the spin-2 modes with the Kaluza-Klein vectors. This provides a reformulation of D-dimensional Einstein gravity, in which the physical degrees of freedom are described by the scalars of a gauged non-linear sigma model based on SL(D-2,R)/SO(D-2), while the metric appears in a purely topological Chern-Simons form.Comment: 23 pages, minor changes, v3: published versio

Loop Groups, Kaluza-Klein Reduction and M-Theory

We show that the data of a principal G-bundle over a principal circle bundle is equivalent to that of a \hat{LG} = U(1) |x LG bundle over the base of the circle bundle. We apply this to the Kaluza-Klein reduction of M-theory to IIA and show that certain generalized characteristic classes of the loop group bundle encode the Bianchi identities of the antisymmetric tensor fields of IIA supergravity. We further show that the low dimensional characteristic classes of the central extension of the loop group encode the Bianchi identities of massive IIA, thereby adding support to the conjectures of hep-th/0203218.Comment: 26 pages, LaTeX, utarticle.cls, v2:clarifications and refs adde

Some computations in the cyclic permutations of completely rational nets

In this paper we calculate certain chiral quantities from the cyclic permutation orbifold of a general completely rational net. We determine the fusion of a fundamental soliton, and by suitably modified arguments of A. Coste , T. Gannon and especially P. Bantay to our setting we are able to prove a number of arithmetic properties including congruence subgroup properties for $S, T$ matrices of a completely rational net defined by K.-H. Rehren .Comment: 30 Pages Late