Deformations of special geometry: in search of the topological string

The topological string captures certain superstring amplitudes which are also encoded in the underlying string effective action. However, unlike the topological string free energy, the effective action that comprises higher-order derivative couplings is not defined in terms of duality covariant variables. This puzzle is resolved in the context of real special geometry by introducing the so-called Hesse potential, which is defined in terms of duality covariant variables and is related by a Legendre transformation to the function that encodes the effective action. It is demonstrated that the Hesse potential contains a unique subsector that possesses all the characteristic properties of a topological string free energy. Genus $gleq3$ contributions are constructed explicitly for a general class of effective actions associated with a special-Kahler target space and are shown to satisfy the holomorphic anomaly equation of perturbative type-II topological string theory. This identification of a topological string free energy from an effective action is primarily based on conceptual arguments and does not involve any of its more specific properties. It is fully consistent with known results. A general theorem is presented that captures some characteristic features of the equivalence, which demonstrates at the same time that non-holomorphic deformations of special geometry can be dealt with consistently.

Spontaneous breaking of time reversal symmetry in strongly interacting two dimensional electron layers in silicon and germanium

We report experimental evidence of a remarkable spontaneous time reversal symmetry breaking in two dimensional electron systems formed by atomically confined doping of phosphorus (P) atoms inside bulk crystalline silicon (Si) and germanium (Ge). Weak localization corrections to the conductivity and the universal conductance fluctuations were both found to decrease rapidly with decreasing doping in the Si:P and Ge:P $delta-$layers, suggesting an effect driven by Coulomb interactions. In-plane magnetotransport measurements indicate the presence of intrinsic local spin fluctuations at low doping, providing a microscopic mechanism for spontaneous lifting of the time reversal symmetry. Our experiments suggest the emergence of a new many-body quantum state when two dimensional electrons are confined to narrow half-filled impurity bands.

BPS black holes, the Hesse potential, and the topological string

The Hesse potential is constructed for a class of four-dimensional N=2 supersymmetric effective actions with S- and T-duality by performing the relevant Legendre transform by iteration. It is a function of fields that transform under duality according to an arithmetic subgroup of the classical dualities reflecting the monodromies of the underlying string compactification. These transformations are not subject to corrections, unlike the transformations of the fields that appear in the effective action which are affected by the presence of higher-derivative couplings. The class of actions that are considered includes those of the FHSV and the STU model. We also consider heterotic N=4 supersymmetric compactifications. The Hesse potential, which is equal to the free energy function for BPS black holes, is manifestly duality invariant. Generically it can be expanded in terms of powers of the modulus that represents the inverse topological string coupling constant, $g_s$, and its complex conjugate. The terms depending holomorphically on $g_s$ are expected to correspond to the topological string partition function and this expectation is explicitly verified in two cases. Terms proportional to mixed powers of $g_s$ and $bar g_s$ are in principle present.

Subleading and non-holomorphic corrections to N=2 BPS black hole entropy

BPS black hole degeneracies can be expressed in terms of an inverse Laplace transform of a partition function based on a mixed electric/magnetic ensemble, which involves a non-trivial integration measure. This measure has been evaluated for black holes with various degrees of supersymmetry and for N=4 supersymmetric black holes all results agree. It generally receives contributions from non-holomorphic corrections. An explicit evaluation of these corrections in the context of the effective action of the FHSV model reveals that these are related to, but quantitatively different from, the non-holomorphic corrections to the topological string, indicating that the relation between the twisted partition functions of the latter and the effective action is more subtle than has so far been envisaged. The effective action result leads to a duality invariant BPS free energy and arguments are presented for the existence of consistent non-holomorphic deformations of special geometry that can account for these effects. A prediction is given for the measure based on semiclassical arguments for a class of N=2 black holes. Furthermore an attempt is made to confront some of the results of this paper to a recent proposal for the microstate degeneracies of the STU model.