An Algebraic Classification of Solution Generating Techniques

TL;DR

This paper develops an algebraic framework to classify solution-generating techniques in (modified) supergravity and integrable-preserving transformations of 2D sigma-models, revealing their underlying double Lie algebra structure.

Contribution

It introduces a unified algebraic classification based on double Lie algebras, connecting supergravity solution techniques with integrable model transformations.

Findings

Classifies continuous deformations via Lie algebra cohomologies.

Establishes the algebraic equivalence of supergravity and sigma-model transformations.

Provides a detailed example with the bi-Yang-Baxter-Wess-Zumino model.

Abstract

We consider a two-fold problem: on the one hand, the classification of a family of solution-generating techniques in (modified) supergravity and, on the other hand, the classification of a family of canonical transformations of 2-dimensional $\sigma$-models giving rise to integrable-preserving transformations. Assuming a generalised Scherk-Schwarz ansatz, in fact, the two problems admit essentially the same algebraic formulation, emerging from an underlying double Lie algebra $\mathfrak d$. After presenting our derivation of the classification, we discuss in detail the relation to modified supergravity and the additional conditions to recover the standard (unmodified) supergravity. Starting from our master equation - that encodes all the possible continuous deformations allowed in the family of solution-generating techniques - we show that these are classified by the Lie algebra cohomologies $H^2(\mathfrak h,\mathbb R)$ and $H^3(\mathfrak h,\mathbb R)$ of the maximally isotropic subalgebra $\mathfrak h$ of the double Lie algebra $\mathfrak d$. {We illustrate our results with a non-trivial example, the bi-Yang-Baxter-Wess-Zumino model.