Integrability of generalized (matrix) Ernst equations in string theory

TL;DR

This paper explores the integrability of matrix Ernst equations in string theory, revealing their spectral structures and proposing methods for constructing solutions, thus advancing understanding of low-energy string dynamics.

Contribution

It introduces the spectral problem framework for matrix Ernst equations in string theory and analyzes their integrability and solution structures.

Findings

Spectral problems based on 2d×2d linear systems are constructed.

Universal Jordan form structures of matrix coefficients are identified.

Conditions for matrix integrals lead to specific coset structures.

Abstract

The integrability structures of the matrix generalizations of the Ernst equation for Hermitian or complex symmetric $d\times d$-matrix Ernst potentials are elucidated. These equations arise in the string theory as the equations of motion for a truncated bosonic parts of the low-energy effective action respectively for a dilaton and $d\times d$ - matrix of moduli fields or for a string gravity model with a scalar (dilaton) field, U(1) gauge vector field and an antisymmetric 3-form field, all depending on two space-time coordinates only. 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 conditions of existence for each of these systems of two matrix integrals with appropriate symmetries provide a specific (coset) structures of the related matrix variables. An equivalence of these spectral problems to the original field equations is proved and some approach for construction of multiparametric families of their solutions is envisaged.