Auxiliary field sigma models and Yang-Baxter deformations

A bstract We combine the Yang-Baxter (YB) and bi-Yang-Baxter (bi-YB) deformations with higher-spin auxiliary field deformations to construct multi-parameter families of integrable deformations of the principal chiral model on a Lie group G with semi-simple Lie algebra $$ \mathfrak{g} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> . In the YB case, our construction produces one integrable deformation for each pair ( $$ \mathcal{R} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> , E ), where $$ \mathcal{R} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> is an antisymmetric bilinear operator on $$ \mathfrak{g} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> obeying the modified classical Yang-Baxter equation and E is a function of several variables. In the bi-YB case, the pair becomes a triplet ( $$ \mathcal{R} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> , $$ \overset{\sim }{\mathcal{R}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>R</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> , E ), where $$ \overset{\sim }{\mathcal{R}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>R</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> is another antisymmetric bilinear operator on $$ \mathfrak{g} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> obeying the modified classical Yang-Baxter equation. We show that every model in these families is (weakly) classically integrable by exhibiting a Lax representation for their equations of motion.