Lax pairs on Yang-Baxter deformed backgrounds

Understand

We explicitly derive Lax pairs for string theories on Yang-Baxter deformed backgrounds, 1) gravity duals for noncommutative gauge theories, 2) $\gamma$-deformations of S$^5$, 3) Schr\"odinger spacetimes and 4) abelian twists of the global AdS$_5$\,.

• Then we can find out a concise derivation of Lax pairs based on simple replacement rules.
• Furthermore, each of the above deformations can be reinterpreted as a twisted periodic boundary conditions with the undeformed background by using the rules.
• As another derivation, the Lax pair for gravity duals for noncommutative gauge theories is reproduced from the one for a $q$-deformed AdS$_5\times$S$^5$ by taking a scaling limit.