Quantum hypermultiplet moduli spaces in N=2 string vacua: a review

The hypermultiplet moduli space M_H in type II string theories compactified on a Calabi-Yau threefold X is largely constrained by supersymmetry (which demands quaternion-K\"ahlerity), S-duality (which requires an isometric action of SL(2, Z)) and regularity. Mathematically, M_H ought to encode all generalized Donaldson-Thomas invariants on X consistently with wall-crossing, modularity and homological mirror symmetry. We review recent progress towards computing the exact metric on M_H, or rather the exact complex contact structure on its twistor space.Comment: 31 pages; Contribution to the Proceedings of String Math 2012; v2: references added, misprints corrected, published versio

SUSY Quivers, Intersecting Branes and the Modest Hierarchy Problem

We present a class of chiral non-supersymmetric D=4 field theories in which quadratic divergences appear only at two loops. They may be depicted as ``SUSY quivers'' in which the nodes represent a gauge group with extended e.g., N=4 SUSY whereas links represent bifundamental matter fields which transform as chiral multiplets with respect to different N=1 subgroups. One can obtain this type of field theories from simple D6-brane configurations on Type IIA string theory compactified on a six-torus. We discuss the conditions under which this kind of structure is obtained from D6-brane intersections. We also discuss some aspects of the effective low-energy field theory. In particular we compute gauge couplings and Fayet-Iliopoulos terms from the Born-Infeld action and show how they match the field theory results. This class of theories may be of phenomenological interest in order to understand the modest hierarchy problem i.e., the stability of the hierarchy between the weak scale and a fundamental scale of order 10-100 TeV which appears e.g. in low string scale models. Specific D-brane models with the spectrum of the SUSY Standard Model and three generations are presented.Comment: 36 pages, using JHEP3.cls, 8 figures. References update

D3-instantons, Mock Theta Series and Twistors

The D-instanton corrected hypermultiplet moduli space of type II string theory compactified on a Calabi-Yau threefold is known in the type IIA picture to be determined in terms of the generalized Donaldson-Thomas invariants, through a twistorial construction. At the same time, in the mirror type IIB picture, and in the limit where only D3-D1-D(-1)-instanton corrections are retained, it should carry an isometric action of the S-duality group SL(2,Z). We prove that this is the case in the one-instanton approximation, by constructing a holomorphic action of SL(2,Z) on the linearized twistor space. Using the modular invariance of the D4-D2-D0 black hole partition function, we show that the standard Darboux coordinates in twistor space have modular anomalies controlled by period integrals of a Siegel-Narain theta series, which can be canceled by a contact transformation generated by a holomorphic mock theta series.Comment: 42 pages; discussion of isometries is amended; misprints correcte

S-duality in Twistor Space

In type IIB string compactifications on a Calabi-Yau threefold, the hypermultiplet moduli space $M_H$ must carry an isometric action of the modular group SL(2,Z), inherited from the S-duality symmetry of type IIB string theory in ten dimensions. We investigate how this modular symmetry is realized at the level of the twistor space of $M_H$, and construct a general class of SL(2,Z)-invariant quaternion-Kahler metrics with two commuting isometries, parametrized by a suitably covariant family of holomorphic transition functions. This family should include $M_H$ corrected by D3-D1-D(-1)-instantons (with fivebrane corrections ignored) and, after taking a suitable rigid limit, the Coulomb branch of five-dimensional N=2 gauge theories compactified on a torus, including monopole string instantons. These results allow us to considerably simplify the derivation of the mirror map between type IIA and IIB fields in the sector where only D1-D(-1)-instantons are retained.Comment: 29 pages, 1 figur

Type IIA Moduli Stabilization

We demonstrate that flux compactifications of type IIA string theory can classically stabilize all geometric moduli. For a particular orientifold background, we explicitly construct an infinite family of supersymmetric vacua with all moduli stabilized at arbitrarily large volume, weak coupling, and small negative cosmological constant. We obtain these solutions from both ten-dimensional and four-dimensional perspectives. For more general backgrounds, we study the equations for supersymmetric vacua coming from the effective superpotential and show that all geometric moduli can be stabilized by fluxes. We comment on the resulting picture of statistics on the landscape of vacua.Comment: 48 pages, 2 figures, LaTeX. v2: references added. v3: minor comments & references adde

Twistor Approach to String Compactifications: a Review

We review a progress in obtaining the complete non-perturbative effective action of type II string theory compactified on a Calabi-Yau manifold. This problem is equivalent to understanding quantum corrections to the metric on the hypermultiplet moduli space. We show how all these corrections, which include D-brane and NS5-brane instantons, are incorporated in the framework of the twistor approach, which provides a powerful mathematical description of hyperkahler and quaternion-Kahler manifolds. We also present new insights on S-duality, quantum mirror symmetry, connections to integrable models and topological strings.Comment: 99 pages; minor corrections; journal versio

Frontiers in complex dynamics

Rational maps on the Riemann sphere occupy a distinguished niche in the general theory of smooth dynamical systems. First, rational maps are complex-analytic, so a broad spectrum of techniques can contribute to their study (quasiconformal mappings, potential theory, algebraic geometry, etc.). The rational maps of a given degree form a finite-dimensional manifold, so exploration of this {\em parameter space} is especially tractable. Finally, some of the conjectures once proposed for {\em smooth} dynamical systems (and now known to be false) seem to have a definite chance of holding in the arena of rational maps. In this article we survey a small constellation of such conjectures centering around the density of {\em hyperbolic} rational maps --- those which are dynamically the best behaved. We discuss some of the evidence and logic underlying these conjectures, and sketch recent progress towards their resolution.Comment: 18 pages. Abstract added in migration