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Integrable deformation of \(\mathbb{CP}^n\) and generalised Kähler geometry. (English) Zbl 1456.81229

Summary: We build on the results of [the authors, ibid. 2020, No. 9, Paper No. 44, 25 p. (2020; Zbl 1454.81103)] for generalised frame fields on generalised quotient spaces and study integrable deformations for \(\mathbb{CP}^n\). 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 \(\mathbb{CP}^n\) and the explicit expression for the generalised Kähler potential for \(n = 1, 2\).

MSC:

81R12 Groups and algebras in quantum theory and relations with integrable systems
83E30 String and superstring theories in gravitational theory
83C60 Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism

Citations:

Zbl 1454.81103
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References:

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