Riemann surfaces and Schrodinger potentials of gauged supergravity

Supersymmetric domain-wall solutions of maximal gauged supergravity are classified in 4, 5 and 7 dimensions in the presence of non-trivial scalar fields 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-Schwarz transformation, which allows for the characterization of the solutions in terms of Riemann surfaces whose genus depends on the isometry group. The uniformization of the curves can be carried out explicitly for models of low genus and results into trigonometric and elliptic solutions for the scalar fields and the conformal factor of the metric. The Schrodinger potentials for the quantum fluctuations of the graviton and scalar fields are derived on these backgrounds and enjoy all properties of supersymmetric quantum mechanics. Special attention is given to a class of elliptic models whose quantum fluctuations 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) of the underlying Riemann surfaces with periods 2\omega_1 and 2\omega_2, for different half-integer values of the coupling constants \mu, \nu, \kappa, \lambda