Lax pairs for deformed Minkowski spacetimes

TL;DR

This paper develops a general method to construct Lax pairs for Yang-Baxter deformations of 4D Minkowski spacetime, including various backgrounds, advancing integrability techniques in deformed spacetime models.

Contribution

It introduces a unified approach to derive Lax pairs for deformed Minkowski spacetimes using a simple replacement law, applicable to arbitrary classical r-matrices.

Findings

Derived a general Lax pair expression for deformed Minkowski spacetime.

Constructed explicit Lax pairs for multiple backgrounds including pp-waves and Hashimoto-Sethi.

Validated the Lax pair's applicability to various classical r-matrices with Poincaré generators.

Abstract

We proceed to study Yang-Baxter deformations of 4D Minkowski spacetime based on a conformal embedding. We first revisit a Melvin background and argue a Lax pair by adopting a simple replacement law invented in 1509.00173. This argument enables us to deduce a general expression of Lax pair. Then the anticipated Lax pair is shown to work for arbitrary classical $r$-matrices with Poinca\'e generators. As other examples, we present Lax pairs for pp-wave backgrounds, the Hashimoto-Sethi background, the Spradlin-Takayanagi-Volovich background.