On the universal Representation of the Scattering Matrix of Affine Toda Field Theory

By exploiting the properties of q-deformed Coxeter elements, the scattering matrices of affine Toda field theories with real coupling constant related to any dual pair of simple Lie algebras may be expressed in a completely generic way. We discuss the governing equations for the existence of bound states, i.e. the fusing rules, in terms of q-deformed Coxeter elements, twisted q-deformed Coxeter elements and undeformed Coxeter elements. We establish the precise relation between these different formulations and study their solutions. The generalized S-matrix bootstrap equations are shown to be equivalent to the fusing rules. The relation between different versions of fusing rules and quantum conserved quantities, which result as nullvectors of a doubly q-deformed Cartan like matrix, is presented. The properties of this matrix together with the so-called combined bootstrap equations are utilised in order to derive generic integral representations for the scattering matrix in terms of quantities of either of the two dual algebras. We present extensive case-by-case data, in particular on the orbits generated by the various Coxeter elements.Comment: 57 page

Quantum cohomology via vicious and osculating walkers

We relate the counting of rational curves intersecting Schubert varieties of the Grassmannian to the counting of certain non-intersecting lattice paths on the cylinder, so-called vicious and osculating walkers. These lattice paths form exactly solvable statistical mechanics models and are obtained from solutions to the Yang–Baxter equation. The eigenvectors of the transfer matrices of these models yield the idempotents of the Verlinde algebra of the gauged u^(n)k -WZNW model. The latter is known to be closely related to the small quantum cohomology ring of the Grassmannian. We establish further that the partition functions of the vicious and osculating walker model are given in terms of Postnikov’s toric Schur functions and can be interpreted as generating functions for Gromov–Witten invariants. We reveal an underlying quantum group structure in terms of Yang–Baxter algebras and use it to give a generating formula for toric Schur functions in terms of divided difference operators which appear in known representations of the nil-Hecke algebra

Atmospheric neutrons

Contributions to fast neutron measurements in the atmosphere are outlined. The results of a calculation to determine the production, distribution and final disappearance of atmospheric neutrons over the entire spectrum are presented. An attempt is made to answer questions that relate to processes such as neutron escape from the atmosphere and C-14 production. In addition, since variations of secondary neutrons can be related to variations in the primary radiation, comment on the modulation of both radiation components is made

Auxiliary matrices for the six-vertex model and the algebraic Bethe ansatz

We connect two alternative concepts of solving integrable models, Baxter's method of auxiliary matrices (or Q-operators) and the algebraic Bethe ansatz. The main steps of the calculation are performed in a general setting and a formula for the Bethe eigenvalues of the Q-operator is derived. A proof is given for states which contain up to three Bethe roots. Further evidence is provided by relating the findings to the six-vertex fusion hierarchy. For the XXZ spin-chain we analyze the cases when the deformation parameter of the underlying quantum group is evaluated both at and away from a root of unity.Comment: 32 page

Exactly solvable potentials of Calogero type for q-deformed Coxeter groups

We establish that by parameterizing the configuration space of a one-dimensional quantum system by polynomial invariants of q-deformed Coxeter groups it is possible to construct exactly solvable models of Calogero type. We adopt the previously introduced notion of solvability which consists of relating the Hamiltonian to finite dimensional representation spaces of a Lie algebra. We present explicitly the $G_2^q$-case for which we construct the potentials by means of suitable gauge transformations.Comment: 22 pages Late

Auxiliary matrices for the six-vertex model at roots of 1 and a geometric interpretation of its symmetries

The construction of auxiliary matrices for the six-vertex model at a root of unity is investigated from a quantum group theoretic point of view. Employing the concept of intertwiners associated with the quantum loop algebra $U_q(\tilde{sl}_2)$ at $q^N=1$ a three parameter family of auxiliary matrices is constructed. The elements of this family satisfy a functional relation with the transfer matrix allowing one to solve the eigenvalue problem of the model and to derive the Bethe ansatz equations. This functional relation is obtained from the decomposition of a tensor product of evaluation representations and involves auxiliary matrices with different parameters. Because of this dependence on additional parameters the auxiliary matrices break in general the finite symmetries of the six-vertex model, such as spin-reversal or spin conservation. More importantly, they also lift the extra degeneracies of the transfer matrix due to the loop symmetry present at rational coupling values. The extra parameters in the auxiliary matrices are shown to be directly related to the elements in the enlarged center of the quantum loop algebra $U_q(\tilde{sl}_2)$ at $q^N=1$. This connection provides a geometric interpretation of the enhanced symmetry of the six-vertex model at rational coupling. The parameters labelling the auxiliary matrices can be interpreted as coordinates on a three-dimensional complex hypersurface which remains invariant under the action of an infinite-dimensional group of analytic transformations, called the quantum coadjoint action.Comment: 52 pages, TCI LaTex, v2: equation (167) corrected, two references adde

Factorization of the transfer matrices for the quantum sl(2) spin chains and Baxter equation

It is shown that the transfer matrices of homogeneous sl(2) invariant spin chains with generic spin, both closed and open, are factorized into the product of two operators. The latter satisfy the Baxter equation that follows from the structure of the reducible representations of the sl(2) algebra.Comment: 14 pages, 9 figures, typos correcte

BMPix and PEAK tools: New methods for automated laminae recognition and counting — Application to glacial varves from Antarctic marine sediment

We present tools for rapid and quantitative detection of sediment lamination. The BMPix tool extracts color and gray-scale curves from images at pixel resolution. The PEAK tool uses the gray-scale curve and performs, for the first time, fully automated counting of laminae based on three methods. The maximum count algorithm counts every bright peak of a couplet of two laminae (annual resolution) in a smoothed curve. The zero-crossing algorithm counts every positive and negative halfway-passage of the curve through a wide moving average, separating the record into bright and dark intervals (seasonal resolution). The same is true for the frequency truncation method, which uses Fourier transformation to decompose the curve into its frequency components before counting positive and negative passages. We applied the new methods successfully to tree rings, to well-dated and already manually counted marine varves from Saanich Inlet, and to marine laminae from the Antarctic continental margin. In combination with AMS14C dating, we found convincing evidence that laminations in Weddell Sea sites represent varves, deposited continuously over several millennia during the last glacial maximum. The new tools offer several advantages over previous methods. The counting procedures are based on a moving average generated from gray-scale curves instead of manual counting. Hence, results are highly objective and rely on reproducible mathematical criteria. Also, the PEAK tool measures the thickness of each year or season. Since all information required is displayed graphically, interactive optimization of the counting algorithms can be achieved quickly and conveniently

Multi-Color Imaging of Magnetic Co/Pt Multilayers

We demonstrate for the first time the realization of a spatial resolved two color, element-specific imaging experiment at the free-electron laser facility FERMI. Coherent imaging using Fourier transform holography was used to achieve direct real space access to the nanometer length scale of magnetic domains of Co/Pt heterostructures via the element-specific magnetic dichroism in the extreme ultraviolet spectral range. As a first step to implement this technique for studies of ultrafast phenomena we present the spatially resolved response of magnetic domains upon femtosecond laser excitation