SUSY SUGRA GEOMETRY · Supersymmetry, supergravity and geometry in particle physics and string theory

Supersymmetry, supergravity and geometry in particle physics and string theory

Project context and objectives The Standard Model of particle physics, which has so far provided a successful description of elementary particles, is believed to be an incomplete theory. New experiments are starting to probe physics beyond the Standard Model. Supersymmetry, supergravity, and string theory are the theoretical tools that are expected to describe the new physics proposing solutions to fundamental problems such as the nature of dark matter, the unification of forces and the formulation of quantum gravity. Moreover, a symbiotic relation between string theory and mathematics has proven to be central for both fields. The present project aims at a deeper understanding of challenging features of supersymmetry, supergravity and string theories in the intersection of physics and mathematics. A central aspect of this research project is supersymmetry. A powerful method of studying supersymmetry is to use superspace techniques. As supersymmetry extends the space-time symmetry algebra with a set of fermionic charges, superspaces are an extension of the space-time with a set of fermionic coordinates. If one is interested in the quantum behavior of supersymmetric theories, the superspace formalisms have proved to be extremely powerful. Besides the simplifications that arise in quantum computations, superspace techniques prove to have a powerful and direct connection to topics in complex geometry through the study of supersymmetric non-linear sigma-models (SNLSM). In fact, a direct relation between the target space geometry of supersymmetric sigma-models and the number of supercharges of supersymmetric field theories has been known since the early days of supersymmetry. The off-shell formulation of SNLSM in superspace provides a way to understand and generate new target space geometries. In particular a proper superspace description of the sigma-model provides a procedure to generate and read off the geometrical objects, such as the metric, of the target space in which the fields live. This is highly non-trivial in cases with extended supersymmetry where complicated target spaces - such as hyper-Kahler and quaternionic-Kahler - arise and is peculiar of superspace techniques. The Marie Curie fellow is an expert in this formalism and recently obtained a series of major results in the study of extended supergravity theories. The major objectives of this project were to continue to develop new superspace techniques for supersymmetric and supergravity theories and apply them, in order to understand the geometrical structures of the low-energy dynamics of String Theory which is described by supersymmetric and supergravity models in different dimensions. Project achievements Working on this project we have in fact constructed new superspace techniques for supersymmetric and supergravity theories living in three space-time dimensions (3D). Firstly, we developed superspace techniques to construct general off-shell N = 1, 2, 3, 4 superconformal sigma-models in 3D. Such techniques are applicable to building a general off-shell supersymmetric model and are a natural framework to studying hyperkahler cones. Hyperkahler cones are intimately related to quaternion Kahler manifolds. As a second step, we have constructed the superspace geometry of 3D N-extended conformal supergravity and constructed off-shell supergravity-matter couplings in the cases N = 1, 2, 3, 4. This was a major result of our work that gave us the opportunity to explore properties of 3D supergravities with cosmological terms and their associated anti de-Sitter geometries (AdS) - which are among the simplest, and most important, backgrounds of supergravity models. We have classified the geometrical conditions on the target space of supersymmetric sigma-models possessing N = 2, 3 supersymmetry in the constant curvature AdS geometry. Interestingly, SNLSM in AdS possess new non-trivial geometrical conditions that have never before been fully explored. One main objective was the study of N = 2 sigma-models in four dimensions (4D) possessing global and local supersymmetry. As a first step towards fully understanding general supergravity backgrounds, we analysed properties of 4D N = 2 SNLSM in AdS by using new superspace techniques. It turns out that the sigma-model target space, that in the flat case is constrained to be a hyperkahler manifold, in the AdS case must be a non-compact hyperkahler manifold endowed with a Killing vector field which generates an SO (2) group of rotations on the two-sphere of complex structures. In extending the programme of construction and the study of supergravity theories in various dimensions by using superspace, part of this project dealt with the analysis of the six-dimensional case. The fellow developed a new superspace geometry for six-dimensional N = (1, 0) conformal supergravity suitable to describe the various 'Weyl' multiplets of conformal supergravity. As a next step, the fellow extended to six-dimensions the projective superspace approach to study general supegravity-matter couplings. In the development of the project and the analysis of 3D supersymmetric models, we also used the power of superspace to perform state-of-the-art quantum computations in the context of AdS 4 /CFT 3 dualities. These are mainstream topics in high-energy theoretical physics which, on one hand, are important theoretical tools to study strongly coupled string theory in terms of weakly coupled gauge theories. On the other hand they allow computations in strongly coupled gauge theories to be made by using supergravity. The fellow computed the four-loop spectrum of operators in the SU (2) × SU (2) sector of ABJM and ABJ supersymmetric N = 6 Chern-Simons theories and proved its integrability.

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