Poisson-Lie T-duality of String Effective Actions: A New Approach to the Dilaton Puzzle

For a particular class of backgrounds, equations of motion for string sigma models targeted in mutually dual Poisson-Lie groups are equivalent. This phenomenon is called the Poisson-Lie T-duality. On the level of the corresponding string effective actions, the situation becomes more complicated due to the presence of the dilaton field. A novel approach to this problem using Levi-Civita connections on Courant algebroids is presented. After the introduction of necessary geometrical tools, formulas for the Poisson-Lie T-dual dilaton fields are derived. This provides a version of Poisson-Lie T-duality for string effective actions.Comment: One subsection added, several typos and minor mistakes correcte

A Characterization of Reduced Forms of Linear Differential Systems

A differential system $[A] : \; Y'=AY$, with $A\in \mathrm{Mat}(n, \bar{k})$ is said to be in reduced form if $A\in \mathfrak{g}(\bar{k})$ where $\mathfrak{g}$ is the Lie algebra of the differential Galois group $G$ of $[A]$. In this article, we give a constructive criterion for a system to be in reduced form. When $G$ is reductive and unimodular, the system $[A]$ is in reduced form if and only if all of its invariants (rational solutions of appropriate symmetric powers) have constant coefficients (instead of rational functions). When $G$ is non-reductive, we give a similar characterization via the semi-invariants of $G$. In the reductive case, we propose a decision procedure for putting the system into reduced form which, in turn, gives a constructive proof of the classical Kolchin-Kovacic reduction theorem.Comment: To appear in : Journal of Pure and Applied Algebr

Sub-quadratic Decoding of One-point Hermitian Codes

We present the first two sub-quadratic complexity decoding algorithms for one-point Hermitian codes. The first is based on a fast realisation of the Guruswami-Sudan algorithm by using state-of-the-art algorithms from computer algebra for polynomial-ring matrix minimisation. The second is a Power decoding algorithm: an extension of classical key equation decoding which gives a probabilistic decoding algorithm up to the Sudan radius. We show how the resulting key equations can be solved by the same methods from computer algebra, yielding similar asymptotic complexities.Comment: New version includes simulation results, improves some complexity results, as well as a number of reviewer corrections. 20 page

c=2 Rational Toroidal Conformal Field Theories via the Gauss Product

We find a concise relation between the moduli $\tau, \rho$ of a rational Narain lattice $\Gamma(\tau,\rho)$ and the corresponding momentum lattices of left and right chiral algebras via the Gauss product. As a byproduct, we find an identity which counts the cardinality of a certain double coset space defined for isometries between the discriminant forms of rank two lattices.Comment: AMS-TeX, 45 pages; title changed, minor errors corrected, acknowledgement adde