Integrable Models: from Dynamical Solutions to String Theory

We review the status of integrable models from the point of view of their dynamics and integrability conditions. A few integrable models are discussed in detail. We comment on the use it is made of them in string theory. We also discuss the SO(6) symmetric Hamiltonian with SO(6) boundary. This work is especially prepared for the 70th anniversaries of Andr, Swieca (in memoriam) and Roland Koberle.

Chronic VEGF Blockade Worsens Glomerular Injury in the Remnant Kidney Model

VEGF inhibition can promote renal vascular and parenchymal injury, causing proteinuria, hypertension and thrombotic microangiopathy. The mechanisms underlying these side effects are unclear. We investigated the renal effects of the administration, during 45 days, of sunitinib (Su), a VEGF receptor inhibitor, to rats with 5/6 renal ablation (Nx). Adult male Munich-Wistar rats were distributed among groups S+V, sham-operated rats receiving vehicle only; S+Su, S rats given Su, 4 mg/kg/day; Nx+V, Nx rats receiving V; and Nx+Su, Nx rats receiving Su. Su caused no change in Group S. Seven and 45 days after renal ablation, renal cortical interstitium was expanded, in association with rarefaction of peritubular capillaries. Su did not worsen hypertension, proteinuria or interstitial expansion, nor did it affect capillary rarefaction, suggesting little angiogenic activity in this model. Nx animals exhibited glomerulosclerosis (GS), which was aggravated by Su. This effect could not be explained by podocyte damage, nor could it be ascribed to tuft hypertrophy or hyperplasia. GS may have derived from organization of capillary microthrombi, frequently observed in Group Nx+Su. Treatment with Su did not reduce the fractional glomerular endothelial area, suggesting functional rather than structural cell injury. Chronic VEGF inhibition has little effect on normal rats, but can affect glomerular endothelium when renal damage is already present.

A Skyrme-like model with an exact BPS bound

We propose a new Skyrme-like model with fields taking values on the sphere S3 or, equivalently, on the group SU(2). The action of the model contains a quadratic kinetic term plus a quartic term which is the same as that of the Skyrme-Faddeev model. The novelty of the model is that it possess a first order Bogomolny type equation whose solutions automatically satisfy the second order Euler-Lagrange equations. It also possesses a lower bound on the static energy which is saturated by the Bogomolny solutions. Such Bogomolny equation is equivalent to the so-called force free equation used in plasma and solar Physics, and which possesses large classes of solutions. An old result due to Chandrasekhar prevents the existence of finite energy solutions for the force free equation on the entire three- dimensional space R3. We construct new exact finite energy solutions to the Bogomolny equations for the case where the space is the three-sphere S3, using toroidal like coordinates.