# Riemann surfaces and Schrodinger potentials of gauged supergravity

Research paper by **I. Bakas, A. Brandhuber, K. Sfetsos**

Indexed on: **11 Feb '00**Published on: **11 Feb '00**Published in: **High Energy Physics - Theory**

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#### Abstract

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.