Integrable deformation of ℂPn and generalised Kähler geometry

Abstract

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.

Deformed $$\sigma $$-models, Ricci flow and Toda field theories

“…We derive, in explicit form, the metric of the dual model, which is the subject of proposition 2. The second result, formulated as proposition 3, is that the dual manifold is Kähler (which is also a statement that N = (2, 2) supersymmetry [46] is preserved after T -duality). We obtain an explicit and compact expression for the Kähler potential, the main ingredient in this formula being the dilogarithm Li 2 (z).…”

“…For higher values of n, one could presumably follow the same route, but we leave this generalization for the future, and for the moment, we utilize the machinery of [22] to construct the η-deformed version of the CP n−1 model. We note that the analysis of this geometry has been undertaken in the recent papers [44][45][46]. In particular, it was noticed in [44] that it is useful to perform T -duality on all of the angles (the projective space, as well as its deformation, are toric manifolds), in order to get rid of the B-field.…”

“…One can integrate out the fields V , W (the quadratic part is a Hermitian form in the variables (V , W ), (V , W )) and get a deformed CP n−1 η Lagrangian that depends on the U -fields. The calculation along these lines seems rather formidable and leads to a generalized Kähler manifold [46]. As it has already been observed in [44], a simpler answer can be obtained if one performs T -duality in all the angular directions of the deformed space.…”

“…where we have introduced the notation ρ = 1 − η R. As elaborated in [46], this is really an example of generalized Kähler geometry. In particular, the quadratic form in the last two lines of (3.11) encodes the generalized Kähler metric [66].…”

“…Before starting this section, we point out that the η-deformation of the CP n−1 -model has been recently considered in the works [44][45][46]. We will comment below on the relation of their results to our discussion.…”