Final Report Summary - HYPERGRAV (The last piece of the puzzle: Off-shell hypermultiplets in string theory and complex geometry) Supersymmetry has played an important role in the field of theoretical particle physics for the last thirty years. This symmetry, which pairs bosons and fermions into supermultiplets with identical quantum numbers, is especially important in efforts to unify Einstein's theory of general relativity with quantum field theory. These efforts primarily involve string theory, whose ten-dimensional description yields quantum gravity along with an infinite tower of high-mass string modes, which must subsequently be compactified down to the four dimensions we actually observe. The so-called landscape of possible ways this may occur generically possesses (local) supersymmetry (known as supergravity) and so any effort to understand string theory must be informed by our knowledge of supersymmetry.This project concerns the class of N=2 supersymmetric theories in four dimensions. These represent a middle ground between the less constrained N=1 theories (which may yet play a role in low energy particle phenomenology) and the highly constrained N=4 gauge and N=8 supergravity theories. Their rich structure include two different multiplets of matter fields – vector multiplets, which include gauge bosons, and hypermultiplets, which include general charged matter. Both occur in low energy effective actions derived from string theory and so are equally relevant in understanding aspects of quantum gravity. While the vector multiplet sector is largely understood, the hypermultiplet sector remains a more difficult problem largely due to the complexity of its moduli space, which is a quaternion-Kahler (QK) manifold in the presence of supergravity. The objectives of this project are:1. To construct a complete description of N=2 supergravity-matter systems, including the interactions of hypermultiplets and vector multiplets, while keeping supersymmetry both off-shell and manifest.2. To develop a description of quaternion-Kahler geometries based on the off-shell supergravity description of hypermultiplets.3. To analyze the general structure of rigid N=2 supersymmetric systems.A related objective we have also sought is:4. To develop higher-derivative actions and techniques for conformal supergravity in four, five, and six dimensions.N=2 supergravity-matter systems. In order to formulate supersymmetry in a universal way independently from any specific action, it is necessary to introduce non-dynamical degrees of freedom (auxiliary fields). For the case of N=2 hypermultiplets, these turn out to be infinite in number, but this infinity can be tamed using superspace. In harmonic superspace these fields correspond to Fourier modes on an auxiliary two-sphere, while in projective superspace they correspond to components of a Laurent expansion on complex projective space. A few years ago, a certain analytic continuation of harmonic superspace to projective superspace was proposed with the goal of deriving more efficient Feynman supergraph techniques. Building on this idea, we showed how to map any two-derivative action involving supergravity-matter systems in harmonic superspace to projective superspace.Quaternion-Kahler geometry. The difficulty of handling hypermultiplets coupled to supergravity comes ultimately from the complexity of their target space geometry. In the absence of supergravity, hypermultiplets possess a hyperKahler target space geometry with three covariantly constant complex structures. Using projective superspace techniques, the target space geometry can be extended by the additional bosonic manifold CP^1, which defines the so-called twistor space of the hyperKahler manifold. Our existing research in supersymmetric sigma models showed that the twistor space description of the three complex structures could be directly identified with the canonical symplectic two-form associated with the Hamiltonian description of harmonic superspace sigma models. Upon the inclusion of supergravity, the hypermultiplet target space geometry becomes a significantly more complicated quaternion-Kahler (QK) manifold. Our research has hinted how this could be naturally geometrized in superspace. We are now seeking the direct geometric construction of supergravity-matter QK systems.Rigid supersymmetric systems. Recent research involving supersymmetric systems on rigid curved manifolds has led to impressive computations of certain quantities such as Wilson loops using the technique of supersymmetric localization. This has spawned exhaustive classifications of the types of rigid curved manifolds that support N=1 supersymmetry, but not much work had been done on N=2 systems. We examined general N=2 rigid supersymmetric systems with the following goals:* The classification of all rigid geometries in both Lorentzian and Euclidean signatures admitting full N=2 SUSY, up to discrete identifications (such as orbifolds). In addition to the round spheres, hyperboloids, and their Lorentzian cousins, we found a number of surprising possibilities including classes of deformed three-spheres, three-hyperboloids and variants with differing signature (e.g. a constant radial slicing of the Taub-NUT geometry). For each case, we completely specified the geometry and rigid supersymmetry algebra and gave the explicit form of the Killing spinors.* The classification of all possible two-derivative actions involving vector and hypermultiplets on these spaces in either Lorentzian or Euclidean signature. This exhausts the possibilities for the low energy effective N=2 supersymmetric Lagrangians with eight supercharges on rigid 4D spacetimes.Higher-derivative actions. Armed with general techniques for the construction of supergravity-matter systems, it is natural to consider possible classes of higher-derivative actions. A natural sector to first characterize is the pure supergravity sector. Much work has already been done in four dimensions, so we have worked to extend existing techniques to the cases of 5D N=1 and 6D (1,0) systems. The 5D case has led to a fairly exhaustive analysis in superspace of possible supergravity-matter systems. The 6D case is particularly interesting as it is the largest number of dimensions where hypermultiplets may appear, and recently, knowledge of its structure (as well as that of its (2,0) cousin) has become important in the study of general 6D CFTs. Heat kernel arguments imply two possibilities for (1,0) actions and we are nearing completion on both of them. The final results are expected in mid-2016.It has proven an interesting question to also complete the construction of all conformal supergravity actions for 4D N=4: these can be interpreted in a certain sector as a subclass of N=2 theories, but even there, their analysis has remained elusive. It is known from string theory arguments (e.g. IIA compactifications on K3xT^2) that these schematically are of the form f(T, T*) C^2 where C is the Weyl tensor and f is a function of the scalar field parametrizing a SU(1,1) / U(1) coset space. The supersymmetrization of this action is expected to be computationally quite involved, but recent developments have brought them within reach. We hope to finish their classification in late 2016.Future applications. Possible future application of the results of this project involve black hole mechanics. Hawking showed that in the semi-classical limit, black holes radiate as a black body with a temperature inversely proportional to their mass. Using arguments from thermodynamics, one can relate that temperature to the so-called Bekenstein-Hawking entropy, which is proportional to the area of the black hole's event horizon. But just as classical thermodynamics relates macroscopic descriptions of a physical system (e.g. the pressure, volume and temperature of a gas) to its microscopic degrees of freedom (e.g. the masses, velocities, and interactions of the constituent molecules), the black hole should possess a microscopic quantum gravity description, with the Bekenstein-Hawking entropy related to the degeneracy of the microstates. In 1996, Strominger and Vafa demonstrated this explicitly for a certain class of black holes in string theory in the thermodynamic limit. Now researchers have begun to explore subleading corrections to the entropy, which correspond to higher derivative terms in the effective supergravity action. We are interested in evaluating the effects of certain higher-derivative terms, including those involving hypermultiplet, to the black hole entropy.