Integrable deformation of ℂP ⁿ and generalised Kähler geometry

We build on the results of [1] for generalised frame fields on generalised quotient spaces and study integrable deformations for ℂPⁿ. In particular we show how, when the target space of the Principal Chiral Model is a complex projective space, a two-parameter deformation can be introduced in principle. The second parameter can however be removed via a diffeomorphism, which we construct explicitly, in accordance with the results stemming from a thorough integrability analysis we carry out. We also elucidate how the deformed target space can be seen as an instance of generalised Kähler, or equivalently bi-Hermitian, geometry. In this respect, we find the generic form of the pure spinors for ℂPⁿ and the explicit expression for the generalised Kähler potential for n = 1, 2.