D-instantons and twistors

Abstract

Finding the exact, quantum corrected metric on the hypermultiplet moduli space in Type II string compactifications on Calabi-Yau threefolds is an outstanding open problem. We address this issue by relating the quaternionic-Kähler metric on the hypermultiplet moduli space to the complex contact geometry on its twistor space. In this framework, Euclidean D-brane instantons are captured by contact transformations between different patches. We derive those by recasting the previously known A-type D2-instanton corrections in the language of contact geometry, covariantizing the result under electro-magnetic duality, and using mirror symmetry. As a result, we are able to express the effects of all D-instantons in Type II compactifications concisely as a sum of dilogarithm functions. We conclude with some comments on the relation to microscopic degeneracies of four-dimensional BPS black holes and to the wall-crossing formula of Kontsevich and Soibelman, and on the form of the yet unknown NS5-brane instanton contributions.

D3-instantons, mock theta series and twistors

“…Although these effects are exponentially suppressed compared to D1-D(-1)-instantons in the large volume limit, they are still exponentially larger than D5 and NS5-brane instantons. Thus, a natural question is whether the twistorial construction of [27,28] produces a modular invariant quaternionKähler metric, in the large volume limit where D5-NS5-instantons are ignored but the effects of D3-D1-D(-1)-instantons with non-zero D3-brane charge are still retained. 1 In this work, we shall answer this question in the affirmative, at least in the one-instanton approximation.…”

“…We mainly follow [27], incorporating further improvements introduced in [28,30,31], as well as some new insights on the action of S-duality on the tree level geometry.…”

“…Using these techniques, combining earlier results on the twistorial description of the perturbative metric and of D1-D(-1)-instantons [22][23][24][25] and taking inspiration from a similar construction in the context of gauge theories [26], a general construction of the Dinstanton corrected HM moduli space was laid out in [27,28], in terms of the generalized Donaldson-Thomas (DT) invariants Ω(γ; z) which count D-instantons. For reasons that will become clear below, we refer to the construction of [27,28] as the 'type IIA' construction.…”

“…For reasons that will become clear below, we refer to the construction of [27,28] as the 'type IIA' construction. In type IIA string theory compactified on a Calabi-Yau (CY) threefold Y, D-instantons come from D2-branes wrapping special Lagrangian cycles, while in type IIB string theory compactified on the mirror threefoldŶ, they come from D5-D3-D1-D(-1)-branes wrapping even-dimensional cycles, or more generally coherent sheaves onŶ.…”

“…Although the DT invariants typically depend on the complex structure moduli (in type IIA) or Kähler moduli (in type IIB), they are piecewise constant away from walls of marginal stability, and their discontinuity is such that the resulting contact structure on Z (and hence, the QK metric on M H ) is smooth [26]. In fact, the type IIA construction of [27,28] is essentially dictated by consistency with wall-crossing, and is isomorphic to the gauge theory construction of [26] under the QK/HK correspondence [29,30].…”