Local $\beta$-deformations and Yang-Baxter sigma model

Understand

Homogeneous Yang-Baxter (YB) deformation of AdS$_5\times$S$^5$ superstring is revisited.

• We calculate the YB sigma model action up to quadratic order in fermions and show that homogeneous YB deformations are equivalent to $\beta$-deformations of the AdS$_5\times$S$^5$ background when the classical $r$-matrices consist of bosonic generators.
• In order to make our discussion clearer, we discuss YB deformations in terms of the double-vielbein formalism of double field theory.
• We further provide an O(10,10)-invariant string action that reproduces the Green-Schwarz type II superstring action up to quadratic order in fermions.