2. Modern Mathematical Physics: gravity, supersymmetry and integrability Report for 2003-2007 (theme 1047) Scientific leader: A.T.Filippov Leaders: A.S.Sorin, A.P.Isaev Programme Advisory Committee for Particle Physics 29 meeting, June 10-11 2008

3. MATHEMATICAL PHYSICS IS “KNOW-HOW” IN THEORETICAL PHYSICS

4. Modern Mathematical Physics (projects) Quantum Groups and Integrable Systems Leaders: A.P.Isaev, A.A.Vladimirov Supersymmetry Leader: E.A.Ivanov Gravity, Cosmology and Strings Leaders: A.T.Filippov, V.V.Nesterenko, A.S.Sorin

5. I Isaev А.P. Burdik C. Мir-Каsimov R.М. Мitryushkin V.К. Оs’kin А.F. Pakuliak S.Z. Pogosyan G.S. Sissakian A.N. Тyurin N.А. II Ivanov Е.А. Zupnik B.М. Кrivonos S.О. Sirilo D.H. Sutulin А.О. Fedoruk S.А. Pentek M. III Filippov A.T. Nesterenko V.V. Sorin A.S. Barbashov B.M. Dimitrov B. Ivanova T.A. Tret’yakov P.V. Pestov A.B. Pirozhenko I.G. Popov A.D. Tagirov E.A. Fursaev D.V. Shavokhina N. Taniyldyzy Sh.F. Egorova I.А. Кozlovsky S.А. Теlezhnikov S. Doctor of science – 15 PhD – 11 Without degree – 7 + 2 (including - 1 postdocs, 1 PhD student, 4 students, 1 postgraduate) ___________________ Total: 33 + 2 Arnaeva А. Belev S.

6. I. Quantum groups and integrable systems II. Supersymmetry III. Gravity, cosmology and (super)strings Basic directions for investigations

7. Investigations were performed in the following directions: supergravity and superstring theories development of the methods of modern mathematical physics and their application to the fundamental problems of theoretical physics study of the new types of (super)symmetries Вasic results

8. In the framework of integrable gravity models the physics of black holes was investigated and the relationship was discovered between the static states, cosmological scenarios, and wave-like solutions. A.T.Filippov, “A New Integrable Model of (1+1)-Dimensional Dilaton Gravity Coupled to Toda Matter”, arXiv:0801.1312; "Integrable Models of 1+1 Dimensional Dilaton Gravity Coupled to Scalar Matter", Theor. Math. Phys. 146 (2006) 95- 107; "Dimensional Reduction of Gravity and Relation between Static States, Cosmologies and Waves",hep-th/0612258.

9. The properties of entanglement entropy, generated by the spatial separation of fundamental degrees of freedom, were investigated. The results can be used for elucidating the microscopic origin of the Bekenstein- Hawking entropy in the black hole physics. D.V.Fursaev, “Entanglement entropy in quantum gravity and the Plateau groblem, arXiv:0711.1221

10. Pietro Fre and Alexander Sorin Nucl.Phys. B733 (2006) 334, arXiv:0710.1059 (2007) FUTUR E PAST Cosmic billiard Cosmic evolution can be seen as a smooth billiard scattering with quantized angles defined by the Weyl group of the U-duality of supersting theory UNIVERSE EVOLUTION: Bounces  Smooth Weyl reflections Walls  Dynamical hyperplanes Billiard table  Weyl chamber Time arrow  Weyl group ordering (entropy)

11. Integrable open (spin) chain models of gl(n|m)- and osp(n|m)-type were formulated in algebraic terms. It helps to investigate these models (which generalize Heisenberg spin chains) from unified point of view. In particular the energy spectrum for these spin chains were explicitly found in special cases of chains. These investigations are important in statistical mechanics and (as it was shown recently) in (super) Yang-Mills field theories. E.g., energy spectrum for special spin chains = = anomaly dimensions in QCD for high energies !!! A.P. Isaev and O.V. Ogievetsky, "On Baxterized Solutions of Reflection Equation and Integrable Chain Models", Nucl. Phys. B 760 [PM] (2007) 167-183; A.P. Isaev, O.V. Ogievetsky and A.F. Os'kin, "Chain models on Hecke algebra for corner type representations", Rep. on Math. Phys. 61, No.2 (2008) 309 – 315.

12. Once again about spin chain models: The projection method for constructing the states of quantum integrable models was formulated and compared to the nested Bethe ansatz. Universal Bethe equations were obtained allowing to unify hierarchical and analytical Bethe ansatzes in one frame. This theory describes the universal properties of the space of states in the quantum integrable models associated with the algebras gl(N). B. Enriquez, S. Khoroshkin, S. Pakuliak, ``Weight functions and Drinfeld currents``, Comm. Math. Phys. 276 (2007) 691-725

13. The generalization of the BRST approach to the quantization of models with quadratic constraints was developed. The method gives the possibility to describe the whole spectrum of massive and massless higher spin fields including the models on the curved anti-de Sitter spaces. X. Bekaert, I.L. Buchbinder, A. Pashnev, M. Tsulaia,”On higher spin theory: Strings, BRST, dimensional reductions'', Class. Quant. Grav. 21 (2004) S1457-1464. A.P. Isaev and O.V. Ogievetsky, "BRST Operator for Quantum Lie Algebras: Explicit Formula", Int. J. Mod. Phys. A., Vol. 19, Supplement (2004) 240-247.

14. Boson Fermion Fermion Boson

15. Theories with deformed supersymmetry were investigated. Theories with the “nonanticommutative” deformations were constructed and analysed in the case of extended (N=2 and N=4) supersymmetry in four dimensions. Such deformations can be treated as an alternative mechanism of supersymmetry breaking. E. Ivanov, O. Lechtenfeld, B. Zupnik, ``Nilpotent deformations of N=2 superspace'',JHEP 0402 (2004) 012. S. Ferrara, E. Ivanov, O. Lechtenfeld, E. Sokatchev, B. Zupnik,``Non-anticommutative chiral singlet deformation of N=(1,1) gauge theory'',Nucl. Phys. B 704 (2005) 154-180

16. The properties of supersymmetric field theopries in diverse dimensions were studied on the basis of supersymmetric quantum mechanics. Some of these models are the supersymmetric extensions of some important integrable one-dimensional systems (Calogero-Moser models, various versions of the Landau problem, etc). A new class of mechanics models with extended N=4 and N=8 supersymmetries was proposed and studied. S. Bellucci, E. Ivanov, S. Krivonos, O. Lechtenfeld,``ABC of N = 8, D = 1 supermultiplets'',Nucl. Phys. B 699 (2004) 226- 252 E. Ivanov, O. Lechtenfeld, A. Sutulin,``Hierarchy of N=8 Mechanics Models'', Nucl. Phys. B 790 (2008) 493-523

17. Lagrangian tori for the projective plane and other toric Fano varieties were studied in the context of the Mirror Symmetry Conjecture for Fano varieties given by Hori and Vafa. The Hori - Vafa conjecture can be formulated in terms of the Bohr - Sommerfeld conditions, which implies a relationship between Mirror Symmetry and Geometric Quantization in terms of Lagrangian geometry. N.A.Tyurin, “Geometric quantization in algebraic Lagrangian geometry”, London Mathematical Society, Lecture Notes 338 (2007)

18. Publications Моnographs – 2 1.) A.Galperin, E.Ivanov, V.Ogievetsky, E.Sokatchev “Harminic Superspace”, Cambrige University Press (2007), Second Edition 2.) N.A.Tyurin, “Geometric quantization in algebraic lagrangian geometry”, London Mathematical Society, Lecture Notes 338 (2007) Journals ~ 210 Conference contributions ~ 60 _____________________________________ Total ~ 272

19. Published in: Nucl. Phys. B Phys.Rev. A Phys. Lett. B Lett.Math.Phys. JHEP Mod.Phys.Lett. Phys. Rev. D Russian Math. Surveys Тheor.Мat.Phys. Algebra and Analysis Comm.Math.Phys. Int.Journ.Math.Phys. Class.Quant.Grav. J.Math.Phys. J.Phys. A J.Geom.Phys. Found.Phys. Contemp.Math. Amer.Math.Soc. London Math.Soc. Yadernaya Fizika Phys.Elem.Part. & Atom.Nucl. Letters in PEPAN Europhys.Lett.

20. International collaboration Reports at conferences and lectures at schools ~ 130 Collaboration visits ~ 70 Visitors to BLTP ~ 60 Countries: JINR Member States, CERN, Australia, France, Germany, Great Britain, Italy, USA, ….

21. GRANTS RFBR DFG CNRS INTAS IN2P3 Helmholtz Assosiation UNESCO Dynasty FP6-NEST NANOCASE PROGRAMMES: Heisenberg-Landau Votruba-Blokhincev Bogoliubov-Infeld

22. Courses of lectures in “Dubna” University, Moscow Phys.Tech.Inst., UNC JINR – 7 (every year) Courses of lectures on International Schools ~ 15 (every year) Presentation of Theses: Doctor of Science – 5 PhD – 4 We supervise diploma theses of students and postgraduate students Teaching Activity

23. Proposal for the new theme ”Modern mathematical physics: gravity, supersymmetry, integrability” Period of execution 2009–2013 Direction: Theoretical Physics Leaders: Sorin A.S., Isaev A.P. Scientific leader: Filippov A.T. The superstring theory is generally accepted as the unique candidate for the unified theory of all fundamental interactions, including the quantum gravity. This theory as well as the calculations beyond the perturbation theory in QFT are the key subjects of modern mathematical physics. Development of the methods appropriate for studying the whole complex of these problems is in the focus of the proposal.

24. The main topics of the Theme include: study of superstring theory: different regimes, vacuums, classical and quantum solutions; application of modern methods of mathematical physics to the fundamental problems of theoretical physics: nonperturbative regime of supersymmetric gauge theories, microscopic description of black holes, cosmological models of early universe, investigation of classical and quantum gravity; investigation of the new types of symmetries of physical models, such as supersymmetries, quantum, nonlinear and infinite dimensional symmetries. The use of mathematical methods of the theory of integrable systems, quantum groups, and non- commutative geometry is the decisive factor in investigation of these problems.

25. Expected results Construction and investigation of new supersymmetric models, unified theories and models of fundamental interactions. Elaboration of new approaches to investigation of nonperturbative regimes in quantum field theories, quantum gravity and cosmology. Investigation of noncommutative gauge and supersymmetric field theories on the base of the noncommutative geometry. Development of mathematical methods for the theory of fundamental interactions, in particular, the methods related to the integrable systems, theory of symmetries, and noncommutative geometry. Application of these methods to the construction and investigation of modern cosmological models. The collaboration: Australia, Armenia, Belgium, Bulgaria, Brazil, Great Britain, Hungary, Germany, Greece, Georgia, India, Italy, Canada, Mexico, Poland, Russia, Rumania, USA, Turkey, Ukraine, France, Czech Republic, Switzerland, Serbia, Croatia, Japan, CERN, ICTP, SISSA.

26. Rector: A.T. Filippov Leaders: A.S. Sorin, V.V. Voronov

27. DIAS-TH at JINR : Standing Activity  DIAS-TH is a substructure of BLTP supported by the JINR’s budget as one of priority activities of the Institute.  DIAS-TH organizes and controls all educational programmes for young scientists, graduates, and students.  DIAS-TH programme includes both the standing activity through the year and the standard short schools (about 4 - 5 in a year). The main goals of DIAS-TH:  training courses for young scientists, graduates and undergraduates from the JINR MS and other countries  looking for and supporting gifted young theorists in the JINR Member States  creating databases of students and young researchers;  organization of schools of different level in Dubna  coordination with similar schools in the JINR MS, Germany, and other cntrs.  support of the JINR and CERN experimental programmes by organizing lecture courses and lectures with a special emphasis on the LHC program  coordination with the schools and workshops supported by CERN, UNESCO, ICTP, and other organizations  publication and distribution of lectures and discussions in different forms, in particular, with the use of modern electronic means, etc.

28. Supported by In 2008, we have received the grant in the framework of the UNESCO International Basic Science Programme Supporting up to 2005

29. In 2004-2008, the following activities in the framework of DIAS-TH were:  Winter School on Theoretical Physics (2004--2008);  International Student School on Selected Topics in Nuclear Theory (2004);  Advanced Summer School on Modern Mathematical Physics (2004--2008);  Research Workshop "Nucleation Theory and Applications" (2004--2008);  School on Modern Cosmology and Astrophysics (2004);  School on Heavy Quark Physics (2005, 2008);  International School on Nuclear Theory and Astrophysical Applications (2005, 2007);  School and Workshop on Calculations for Modern and Future Colliders (2006);  School on Few-Body Problems in Physics (2006);  School on Dense Matter in Heavy Ion Collisions and Astrophysics (2006, 2008)

30. Other activities:  Lecture courses and review lectures on new trends in modern physics were given for students and post-graduates.  Computer processing of video records of lectures was carried out.  A new lecture hall for DIAS-TH was opened.  Web-site of DIAS-TH was opened and supported http://theor.jinr.ru/~diastp/diasth

31. Prominent scientists from JINR, Russia, Ukraine, Czechia, and Germany were invited as lecturers. 60 students and post- graduates from JINR, Belarus, Germany, Kazakhstan, Poland, Russia, Romania, Slovakia, Ukraine, and Czechia took part in the School.

32. The School was attended by last-year students of universities, post-graduate students, and young scientists: 64 participants from Russia, Belorus, Belgium, Bolgaria, Venezuela, Germany, Spain, Poland, Ukraine, and Chechia. Lectures were given by the leading scientists actively working on modern problems of theoretical and mathematical physics: I. Buchbinder, L. Cardoso, A. Filippov, E. Ivanov, V. Rubakov, A. Starobinsky, M. Vasiliev. 13 cycles of lectures were supplemented with seminars.

33. The program of three schools and research workshop covered such fields of theoretical physics as particle astrophysics, gravity and cosmology, sypersymmetry and string theory, conformal field theory, structure of an atomic nucleus, nuclear reaction theory, halo nuclei, properties of neutron stars, nucleosynthesis in stars, properties of nuclear matter, critical phenomena and nucleation theory. The leading scientists from Canada, China, Finland, Germany, Romania, Russia, Spain and JINR gave more than 40 lecture courses for about 170 PhD students and young scientists from Armenia, Belarus, China,Croatia, Czechia, Germany, Italy, Japan, Poland, Russia,Turkey, Ukraine, Uzbekistan, and JINR University Centre. Activities were supported by the Helmholtz Foundation, Russian Foundation for Basic Research, Dinastiya Foundation and JINR.

34. 2008 DIAS-TH Activity The first school in this year: Physics at the LHC, new ideas and perspectives. Special additional lectures on modern technics, accelerators, CERN experimental facilities etc. 70 participants from Russia and the JINR MS. Two weeks of introductory lectures and review talks on new trends in modern physics

35. You are welcome to the School!

36. Thank you for attention!