Frobenius manifolds, Integrable Hierarchies and Minimal Liouville Gravity

We use the connection between the Frobrenius manifold and the Douglas string equation to further investigate Minimal Liouville gravity. We search a solution of the Douglas string equation and simultaneously a proper transformation from the KdV to the Liouville frame which ensure the fulfilment of the conformal and fusion selection rules. We find that the desired solution of the string equation has explicit and simple form in the flat coordinates on the Frobenious manifold in the general case of (p,q) Minimal Liouville gravity.Comment: 17 pages; v2: typos removed, some comments added, minor correction

AGT conjecture and Integrable structure of Conformal field theory for c=1

AGT correspondence gives an explicit expressions for the conformal blocks of $d=2$ conformal field theory. Recently an explanation of this representation inside the CFT framework was given through the assumption about the existence of the special orthogonal basis in the module of algebra $\mathcal{A}=Vir\otimes\mathcal{H}$. The basis vectors are the eigenvectors of the infinite set of commuting integrals of motion. It was also proven that some of these vectors take form of Jack polynomials. In this note we conjecture and verify by explicit computations that in the case of the Virasoro central charge $c=1$ all basis vectors are just the products of two Jack polynomials. Each of the commuting integrals of motion becomes the sum of two integrals of motion of two noninteracting Calogero models. We also show that in the case $c\neq1$ it is necessary to use two different Feigin-Fuks bosonizations of the Virasoro algebra for the construction of all basis vectors which take form of one Jack polynomial.Comment: 16 pages, added references, corrected typo

Stripes in thin ferromagnetic films with out-of-plane anisotropy

We examine the T=0 phase diagram of a thin ferromagnetic film with a strong out-of-plane anisotropy in the vicinity of the reorientation phase transition (with Co on Pt as an example). The phase diagram in the anisotropy-applied field plane is universal in the limit where the film thickness is the shortest length scale. It contains uniform fully magnetized and canted phases, as well as periodically nonuniform states: a weakly modulated spin-density wave and strongly modulated stripes. We determine the boundaries of metastability of these phases and point out the existence of a critical point at which the difference between the SDW and stripes vanishes. Out-of-plane magnetization curves exhibit a variety of hysteresis loops caused by the coexistence of one or more phases. Additionally, we study the effect of a system edge on the orientation of stripes. We compare our results with recent experiments.Comment: added references and clarified derivations in response to referee comment

Comment on ``Magnon wave forms in the presence of a soliton in two--dimensional antiferromagnets with a staggered field''

Very recently Fonseca and Pires [Phys. Rev. B 73, 012403(2006)] have studied the soliton--magnon scattering for the isotropic antiferromagnet and calculated ``exact'' phase shifts, which were compared with the ones obtained by the Born approximation. In this Comment we correct both the soliton and magnon solutions and point out the way how to study correctly the scattering problem.Comment: 2 pages (RevTeX

A nested sequence of projectors and corresponding braid matrices R^(θ)\hat R(\theta): (1) Odd dimensions

A basis of $N^2$ projectors, each an ${N^2}\times{N^2}$ matrix with constant elements, is implemented to construct a class of braid matrices $\hat{R}(\theta)$, $\theta$ being the spectral parameter. Only odd values of $N$ are considered here. Our ansatz for the projectors $P_{\alpha}$ appearing in the spectral decomposition of $\hat{R}(\theta)$ leads to exponentials $exp(m_{\alpha}\theta)$ as the coefficient of $P_{\alpha}$. The sums and differences of such exponentials on the diagonal and the antidiagonal respectively provide the $(2N^2 -1)$ nonzero elements of $\hat{R}(\theta)$. One element at the center is normalized to unity. A class of supplementary constraints imposed by the braid equation leaves ${1/2}(N+3)(N-1)$ free parameters $m_{\alpha}$. The diagonalizer of $\hat{R}(\theta)$ is presented for all $N$. Transfer matrices $t(\theta)$ and $L(\theta)$ operators corresponding to our $\hat{R}(\theta)$ are studied. Our diagonalizer signals specific combinations of the components of the operators that lead to a quadratic algebra of $N^2$ constant $N\times N$ matrices. The $\theta$-dependence factors out for such combinations. $\hat R(\theta)$ is developed in a power series in $\theta$. The basic difference arising for even dimensions is made explicit. Some special features of our $\hat{R}(\theta)$ are discussed in a concluding section.Comment: latex file, 32 page

Low energy dynamics of a CP^1 lump on the sphere

Low-energy dynamics in the unit-charge sector of the CP^1 model on spherical space (space-time S^2 x R) is treated in the approximation of geodesic motion on the moduli space of static solutions, a six-dimensional manifold with non-trivial topology and metric. The structure of the induced metric is restricted by consideration of the isometry group inherited from global symmetries of the full field theory. Evaluation of the metric is then reduced to finding five functions of one coordinate, which may be done explicitly. Some totally geodesic submanifolds are found and the qualitative features of motion on these described.Comment: 15 pages, 9 postscript figure

Stability analysis for soliton solutions in a gauged CP(1) theory

We analyze the stability of soliton solutions in a Chern-Simons-CP(1) model. We show a condition for which the soliton solutions are stable. Finally we verified this result numerically.Comment: 13 pages, numerical analysis is added. To be published in Mod. Phys. Lett.