Integrability of Generalized (Matrix) Ernst Equations in String Theory

G. A. Alekseev

Teoretičeskaâ i matematičeskaâ fizika, Tome 144 (2005) no. 2, pp. 214-225

Résumé

We elucidate the integrability structures of the matrix generalizations of the Ernst equation for Hermitian or complex symmetric $(d\times d)$-matrix Ernst potentials. These equations arise in string theory as the equations of motion for the truncated bosonic parts of the low-energy effective action for the respective dilaton and $(d\times d)$-matrix of moduli fields or for a string gravity model with a scalar (dilaton) field, a $U(1)$ gauge vector field, and an antisymmetric 3-form field, all depending on only two space-time coordinates. We construct the corresponding spectral problems based on the overdetermined $(2d\times 2d)$-linear systems with a spectral parameter and the universal (i.e., solution-independent) structures of the canonical Jordan forms of their matrix coefficients. The additionally imposed existence conditions for each of these systems of two matrix integrals with appropriate symmetries provide specific (coset) structures of the related matrix variables. We prove that these spectral problems are equivalent to the original field equations, and we envisage an approach for constructing multiparametric families of their solutions.

Détail

Keywords: Ernst equations, string gravity, integrability, spectral problems
Mots-clés : monodromy.