Final Report Summary - QUANTUMSUPERSYMMETRY (Quantum aspects of extended supersymmetric gauge theories) "Quantum aspects of extended supersymmetric gauge theories"Researcher: Igor SamsonovThe project is aimed at the study of quantum aspects of extended supersymmetric fieldtheories with various numbers of supersymmetries in three-dimensional space-time. The study of various aspects of supersymmetry is one of the central problems in modern high-energy theoretical physics. Originally, the supersymmetry in quantum field theory was proposed as a special kind of symmetry between elementary particles of bosonic and fermionic types. The idea of this symmetry is to provide a universal description of both elementary particles and fundamental interactions of these particles. This idea was inspired by the success of quantum field theory in unified description of three out of four fundamental forces in Nature. Soon after its discovery it was realized that the supersymmetry can play a crucial role at early stages of evolution of our Universe. The study of phenomenological aspects of supersymmetric field theories is an important problem in modern theoretical physics. In particular, the experimental observation of the supersymmetry of elementary particles is one of the main issues which is under the investigation at Large Hadron Collider in CERN, Geneva. On the other hand, purely theoretical aspects of supersymmetric field theories attract constant attention in high energy theoretical physics due to many fascinating properties of such models both on the classical and quantum levels.Modern interest to quantum aspects of supersymmetric field theories is related mainly to the recent developments in the superstring theory which is considered as one of the most realistic models providing a unified description of all fundamental interactions including gravity. It should be noted that supersymmetric field theories in space-time of diverse dimensions (not only four-dimensional) are equally interesting from this point of view. In particular, quantum aspects of various four-dimensional supersymmetric gauge theories are studied at resent in details. However, similar models in three-dimensional space-time are investigated relatively weakly. The main goal of the resent project is to fill this gap by studying various three-dimensional gauge theories with diverse numbers of supersymmetries. In the paper [I.L. Buchbinder, N.G. Pletnev, I.B. Samsonov, Background field formalism and construction of effective action for N=2, d=3 supersymmetric gauge theories, Physics of Particles and Nuclei 44 (2013) 234–249] the superfield method were developed for computing low-energy contributions to the effective action of gauge theories in N=2, d=3 superspace. This approach is a superfield generalization of the well-known background field method which can be applied to the three-dimensional gauge theories of Yang-Mills and Chern-Simons type with N=2,4,6 and 8 supersymmetries. In the paper [I.L. Buchbinder, B.S. Merzlikin, I.B. Samsonov, Two-loop low-energy effective actions in N=2 and N=4 three-dimensional SQED, JHEP 07 (2013) 012] the structure of low-energy effective action in three-dimensional N=2 and N=4 supersymmetric electrodynamics is studied. In particular, two-loop quantum corrections to the low-energy Euler-Heisenberg effective action in these models are calculated. These results were applied to derive two-loop quantum correction to the moduli space metric which attracts high attention at present. In particular, by direct computations it was shown that in the N=4 supersymmetric electrodynamics this metric receives only one-loop quantum corrections.In the paper [I.L. Buchbinder, B.S. Merzlikin, I.B. Samsonov, Two-loop low-energy effective action in Abelian supersymmetric Chern-Simons matter models, Nucl. Phys. B881 (2014) 42–70] two-loop quantum corrections to the low-energy effective action in supersymmetric electrodynamics with Chen-Simons rather than Maxwell kinetic term for the gauge superfield were calculated. These results are compared with the similar ones in the corresponding models with the Maxwell kinetic term for the gauge superfield. It is demonstrated that the structure of low-energy effective in the former is highly restricted by the superconformal symmetry which forbids many superfield structures which are present in the latter. The obtained contributions to the effective action are expressed in terms of superconformal invariants in the N=2 d=3 superspace found previously.In the paper [I.B. Samsonov, D. Sorokin, Superfield theories on S3 and their localization, JHEP 04 (2014) 102] classical and quantum aspects of various supersymmetric field theories on three-dimensional sphere were examined. A superfield approach for construction of classical action of super-Yang-Mills models on S3 with N=2,4 and 8 supersymmetry as well as for the Gaiotto-Witten and Aharony-Bergman-Jafferis-Maldacena theories was developed. A novel feature of such a construction is the possibility to have extended supersymmetry with Killing spinors of different types. One-loop partition functions of such theories using superfield method were computed. It is demonstrated that the superfield approach provides a powerful tool for studying partition functions of such models as it guarantees the automatic cancellation of bosonic and fermionic contributions.To summarize, the results obtained within the frames of the present project make a significant contribution to the study of classical and quantum aspects of three-dimensional supersymmetric gauge theories with various numbers of supersymmetries. The research has been conducted in collaboration with the group of theoretical physics in Tomsk, Russia as well as with the scientists from Istituto Nazionale di Fisica Nucleare, Sezione di Padova, Italy, where the main phase of this project was done. As a result, the present project laid a solid ground for further scientific collaborations between the scientists from these two groups. The results have been disseminated in four papers published in leading international peer-reviewed journals and have been reported on one international conference. They were also reported on three seminars in different research centers in Europe, thus promoting the role of Marie Curie Actions in the support and development of science.