Spin-chain with PSU(2|2)⊗U(1)3 and non-linear σ-model with D(2,1;γ)

• We give a proper account for a non-linear σ  -model on PSU(2|2)⊗U(1)3/{H⊗U(1)}PSU(2|2)⊗U(1)3/{H⊗U(1)}.
• It is identified with the spin-chain with the PSU(2|2)⊗U(1)3PSU(2|2)⊗U(1)3 symmetry.
• There exists a quantity in the non-linear σ-model, which we may identify with the spin-variable.

We propose that the spin-chain with the PSU(2|2)⊗U(1)3PSU(2|2)⊗U(1)3 symmetry is equivalent to the non-linear σ  -model on PSU(2|2)⊗U(1)3/{H⊗U(1)}PSU(2|2)⊗U(1)3/{H⊗U(1)} with a certain subgroup H. To this end we show that the spin-variable of the former theory is identified as the Killing scalar of the latter and their correlation functions can have the same integrability. It is crucial to think that the respective theory gets the PSU(2|2)⊗U(1)3PSU(2|2)⊗U(1)3 symmetry by a symmetry reduction of the exceptional supergroup D(2,1;γ)D(2,1;γ), rather than by an extension of PSU(2|2)PSU(2|2).