Kac--Moody groups and automorphic forms in low dimensional supergravity theories

Kac--Moody groups $G$ over $\mathbb{R}$ have been conjectured to occur as symmetry groups of supergravity theories dimensionally reduced to dimensions less than 3, and their integral forms $G(\mathbb{Z})$ conjecturally encode quantized symmetries. In this review paper, we briefly introduce the conjectural symmetries of Kac--Moody groups in supergravity as well as the known evidence for these conjectures. We describe constructions of Kac--Moody groups over $\R$ and $\Z$ using certain choices of fundamental modules that are considered to have physical relevance. Eisenstein series on certain finite dimensional algebraic groups are known to encode quantum corrections in the low energy limit of superstring theories. We describe briefly how the construction of Eisenstein series extends to Kac--Moody groups. The constant terms of Eisenstein series on $E_9$, $E_{10}$ and $E_{11}$ are predicted to encode perturbative string theory corrections.Comment: arXiv admin note: text overlap with arXiv:1308.619

Integral forms of Kac-Moody groups and Eisenstein series in low dimensional supergravity theories

Kac-Moody groups $G$ over $\mathbb{R}$ have been conjectured to occur as symmetry groups of supergravities in dimensions less than 3, and their integer forms $G(\mathbb{Z})$ are conjecturally U-duality groups. Mathematical descriptions of $G(\mathbb{Z})$, due to Tits, are functorial and not amenable to computation or applications. We construct Kac-Moody groups over $\mathbb{R}$ and $\mathbb{Z}$ using an analog of Chevalley's constructions in finite dimensions and Garland's constructions in the affine case. We extend a construction of Eisenstein series on finite dimensional semisimple algebraic groups using representation theory, which appeared in the context of superstring theory, to general Kac-Moody groups. This coincides with a generalization of Garland's Eisenstein series on affine Kac-Moody groups to general Kac-Moody groups and includes Eisenstein series on $E_{10}$ and $E_{11}$. For finite dimensional groups, Eisenstein series encode the quantum corrections in string theory and supergravity theories. Their Kac-Moody analogs will likely also play an important part in string theory, though their roles are not yet understood

A Note on Topological M5-branes and String-Fivebrane Duality

We derive the stability conditions for the M5-brane in topological M-theory using kappa-symmetry. The non-linearly self-dual 3-form on the world-volume is necessarily non-vanishing, as is the case also for the 2-form field strengths on coisotropic branes in topological string theory. It is demonstrated that the self-duality is consistent with the stability conditions, which are solved locally in terms of a tensor in the representation 6 of SU(3) in G_2. The double dimensional reduction of the M5-brane is the D4-brane, and its direct reduction is an NS5-brane. We show that the equation of motion for the 3-form on the NS5-brane wrapping a Calabi-Yau space is exactly the Kodaira-Spencer equation, providing support for a string-fivebrane duality in topological string theory.Comment: 11 pp, plain te

Multi-breathers and high order rogue waves for the nonlinear Schr\"odinger equation on the elliptic function background

We construct the multi-breather solutions of the focusing nonlinear Schr\"odinger equation (NLSE) on the background of elliptic functions by the Darboux transformation, and express them in terms of the determinant of theta functions. The dynamics of the breathers in the presence of various kinds of backgrounds such as dn, cn, and non-trivial phase-modulating elliptic solutions are presented, and their behaviors dependent on the effect of backgrounds are elucidated. We also determine the asymptotic behaviors for the multi-breather solutions with different velocities in the limit $t\to\pm\infty$, where the solution in the neighborhood of each breather tends to the simple one-breather solution. Furthermore, we exactly solve the linearized NLSE using the squared eigenfunction and determine the unstable spectra for elliptic function background. By using them, the Akhmediev breathers arising from these modulational instabilities are plotted and their dynamics are revealed. Finally, we provide the rogue-wave and higher-order rogue-wave solutions by taking the special limit of the breather solutions at branch points and the generalized Darboux transformation. The resulting dynamics of the rogue waves involves rich phenomena: depending on the choice of the background and possessing different velocities relative to the background. We also provide an example of the multi- and higher-order rogue wave solution.Comment: 45 pages, 16 figure

The Role of Small Peptides in Cancer Physiology and Chemotherapy

The targeting of proven anticancer drugs specifically to cancer cells would provide a unique opportunity to restrict neoplasms without damaging the cancer patient. The present research utilizes the phenomenon of illicit transport, i.e. the coupling of normally impermeant metabolites to permeant metabolites, in targeting the drug melphalan to mouse Ehrlich ascites tumor cells. The dipeptide beta-alanyl-melphalan was synthesized and tested in vitro for toxicity towards mouse Ehrlich ascites tumor cells, mouse liver cells, and mouse 3T3 embryonic cells. The parent compound, melphalan, was used as a control treatment. In addition, both melphalan and beta-alanyl-melphalan were utilized in in vivo chemotherapeutic assays to assess the efficacy of both drugs to restrict tumor cell growth in a mouse model system. The dipeptide, beta-alanyl-melphalan, was synthesized using standard liquid synthesis procedures and assayed for purity and stability by high performance liquid chromatography. The peptide was shown to be greater than 85% pure and was significantly more stable at 37\sp\circC than melphalan, exhibiting a half-life in solution of 607.71 minutes. The half-life of melphalan under similar conditions was 105.21 minutes. The inclusion of proteins in solutions of melphalan increased the stability of this drug, providing for a half-life of 176.72 minutes. Both melphalan and beta-alanyl-melphalan were stable at 0\sp\circC. In in vitro toxicity assays, melphalan was shown to be toxic to all three cell systems studied, whereas beta-alanyl-melphalan was toxic only towards the Ehrlich ascites tumor cells and the 3T3 fibroblast cells. The dipeptide containing melphalan was not toxic to the mouse liver cells at concentrations up to 0.1 mM. Toxicity assays included assessment of both plasma membrane permeability and cell proliferation after drug treatment. Morphological studies, using scanning and transmission electron microscopy as well as light microscopy, of treated cells corroborated the toxicity assays, revealing reduced cell numbers, aberrant cell morphologies, and cell destruction where drug treatment had been demonstrated to alter membrane integrity and/or cell proliferation was observed, i.e. beta-alanyl-melphalan treatment of mouse liver cells, cellular morphologies were demonstrated to be similar to nontreated liver cells. In vivo chemotherapy assays, using Ehrlich ascites tumor cells injected into the abdominal cavity of mice, revealed that melphalan, at concentrations of 5 and 10 mg/kg, was an effective anticancer drug providing for T/C rations of 179 and 193 respectively. The dipeptide, beta-alanyl-melphalan, was also an effective anticancer drug, exhibiting reduced toxicity towards the tumor bearing animal when compared to the parent drug melphalan, providing for T/C rations of 152 at a drug concentration of 40 mg/kg. Neither drug had observable effects on animal activities, i.e. food and water consumption, yet significantly restricted tumor cell growth as assessed by increasing body weights and survival times of tumor bearing mice