Linear perturbations of quaternionic metrics

We extend the twistor methods developed in our earlier work on linear deformations of hyperkähler manifolds [1] to the case of quaternionic-Kähler manifolds. Via Swann’s construction, deformations of a 4d-dimensional quaternionic-Kähler manifoldMare in one-to-one correspondence with deformations of its 4d + 4-dimensional hyperkähler cone S. The latter can be encoded in variations of the complex symplectomorphisms which relate different locally flat patches of the twistor space ZS, with a suitable homogeneity condition that ensures that the hyperkähler cone property is preserved. Equivalently, we show that the deformations ofMcan be encoded in variations of the complex contact transformations which relate different locally flat patches of the twistor space ZM of M, by-passing the Swann bundle and its twistor space. We specialize these general results to the case of quaternionic-Kähler metrics with d + 1 commuting isometries, obtainable by the Legendre transform method, and linear deformations thereof.We illustrate our methods for the hypermultiplet moduli space in string theory compactifications at tree- and one-loop level.