On the integrable gravity coupled to fermions

TL;DR

This paper extends the inverse scattering method to integrable supergravity in two dimensions, incorporating fermions via supersymmetry, and develops a linear problem framework for solving the coupled equations.

Contribution

It introduces a novel extension of the inverse scattering technique to supergravity with fermions, enabling the construction of super-solitonic solutions.

Findings

Developed a superspace-based spectral linear problem for supergravity.

Extended integrability techniques to include fermionic fields.

Outlined a method for constructing exact super-solitonic solutions.

Abstract

In the present letter we indicate an extension of the pure gravity inverse scattering integration technique to the case when fermions (introduced on the base of supersymmetry) are present. In this way the integrability technique for simple ($N=1$) supergravity in two space-time dimensions coupled to the matter fields taking values in the Lie algebra of $E_{8\left( +8\right) }$ group is developed. This theory contains matter living only in one Weyl representation of $SO\left( 16\right) $ and represents the reduction to two dimensions of the three-dimensional simple supergravity constructed in [1]. Our spectral linear problem use superspace and covers the complete set of principal bosonic and fermionic equations of motion. This linear system, as in pure gravity, contains only the first order poles with respect to the spectral parameter. The procedure of constructing the exact super-solitonic solutions is outlined.