Riemann surfaces and Schrödinger potentials of gauged supergravity

Supersymmetric domain-wall solutions of maximal gauged supergravity areclassified in 4, 5 and 7 dimensions in the presence of non-trivial scalarfields taking values in the coset SL(N, R)/SO(N) for N=8, 6 and 5 respectively.We use an algebro-geometric method based on the Christoffel-Schwarztransformation, which allows for the characterization of the solutions in termsof Riemann surfaces whose genus depends on the isometry group. Theuniformization of the curves can be carried out explicitly for models of lowgenus and results into trigonometric and elliptic solutions for the scalarfields and the conformal factor of the metric. The Schrodinger potentials forthe quantum fluctuations of the graviton and scalar fields are derived on thesebackgrounds and enjoy all properties of supersymmetric quantum mechanics.Special attention is given to a class of elliptic models whose quantumfluctuations are commonly described by the generalized Lame potential\mu(\mu+1)P(z) + \nu(\nu+1)P(z+\omega_1)+ \kappa(\kappa+1)P(z+\omega_2) +\lambda(\lambda+1)P(z+\omega_1 +\omega_2) for the Weierstrass function P(z) ofthe underlying Riemann surfaces with periods 2\omega_1 and 2\omega_2, fordifferent half-integer values of the coupling constants \mu, \nu, \kappa,\lambda.