Linear Perturbations of Quaternionic Metrics

Abstract

Abstract: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 manifold M are 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 Z S , with a suitable homogeneity condition that ensures that the hyperkähler cone property is preserved. Equivalently, we show that the deformations of M can be encoded in variations of the complex contact transformations which relate different locally flat patches of the twistor space Z M 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.

D3-instantons, mock theta series and twistors

“…solving the gluing conditions (2.13) for the Darboux coordinates as functions of coordinates on the base M and the P 1 coordinate t, and expanding the contact one-form in the vicinity of any fixed point on P 1 . A useful construct in this procedure is the contact potential e Φ , which determines a Kähler potential on Z (see [20] for more details on this procedure).…”

“…As explained in [20,27], the QK metric (2.3) can be cast in this twistorial framework by choosing a covering of P 1 consisting of two patches U + , U − around the north and south poles, t = 0 and t = ∞, and a third patch U 0 which covers the equator. The transition functions between complex Darboux coordinates on each patch are given by 14) whereas the corresponding Darboux coordinates in the patch U 0 read as [20,47] 15) where W cl denotes the 'superpotential'…”

“…The transition functions between complex Darboux coordinates on each patch are given by 14) whereas the corresponding Darboux coordinates in the patch U 0 read as [20,47] 15) where W cl denotes the 'superpotential'…”

“…As a result, the elliptic genus has a theta function decomposition: 19) where θ p,µ is the Siegel-Narain theta series 11 associated to the lattice Λ equipped with the quadratic form κ ab of signature (1, b 2 − 1), 20) and the y-independent coefficients are given by…”

“…18 In fact, the integral over z ′ (4.18) can be rewritten as an integral overw without ever evaluating it in terms of the error function, upon representing both factors in the integrand in terms of their Fourier transforms 20) where η is the Heaviside step function, and carrying out the integral over z ′ , ω, ω ′ . Thus, the integration variablew in (4.21) is Fourier dual to the twistor coordinate z ′ in (4.16).…”