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Modern Mathematical Physics:
Gravity, Supersymmetry and Strings

• Study of the symmetries of the space of states or Bethe vectors for different quantum integrable systems. These symmetries will be used to obtain effective formulas for the scalar products of the vectors from these spaces. The effective formulas for the scalar products allow investigation of nontrivial physical models solvable by the hierarchical Bethe Ansatz method.
  
  The construction and studies of the generalized (deformed) Calogero- Moser systems will be continued. In particular, the relations of the generalized KP hierarchies with the Calogero - Moser systems and their spin versions as well as the construction of classical integrable systems on quiver varieties and their quantization will be investigated.
  
  The development of special Bohr - Sommerfeld geometry of algebraic varieties needs to solve the main problem - construction of finite dimensional moduli spaces of stable special Bohr - Sommerfeld cycles. The main conjecture says that these moduli spaces are algebraic. The construction of the Landau - Ginzburg models on the moduli spaces of the special Bohr - Sommerfeld cycles over Fano varieties will be provided.
  
  Investigations of the confinement-deconfinement transportation, using exact solutions of the holographic flow of renormgroup with SL(2,C)-symmetry and AdS-fixed point will be continued including - the construction of the holographic RG flows with a couple of effective charges. Interpretation of the flows as a collection of branes in the corresponding supergravity theory; - studies of the transport coefficients of quark-gluon plasma using holographic approach in 5 dimensional Kerr - AdS solution.
  
  Study the relation between n-dimensional N=4 supersymmetric mechanics and the WDVV equation, generalization of the latter to curved spaces, i.e. to arbitrary Riemannian. In this curved WDVV equation, the third derivative of the pre-potential is replaced by the third-rank Codazzi tensor, while the WDVV equation itself acquires a non-trivial right-hand side given by the Riemann curvature tensor. The solutions of the curved WDVV equation have been found for metrics with a potential and on arbitrary isotropic spaces. The latter solution is built on an arbitrary solution of the flat WDVV equation. Thus, any such flat solution can be lifted to a curved solution on an isotropic space. It is planned to construct the corresponding N = 4 supersymmetric mechanics with non-trivial potentials.
  
  Study of the boundary three-point function in the 2D conformal Liouville field theory in the semi-classical limits. In particular, we are going to address the light and heavy asymptotic limits. Since the boundary three-point function is related to the fusion matrix, full understanding of these limits gives us information on the corresponding behavior of the fusion matrix. Analysis of the boundary three-point function in the heavy asymptotic limit. This can be done by estimating the action of the Liouville theory with a boundary on solutions with the three boundary singularities. Remembering that, as was mentioned above, the boundary three-point function is related to the fusion matrix and that heavy asymptotic limit of the conformal blocks is related to the solutions of the Heun and Painleve VI equations, one can obtain in this way information on the monodromy properties of their solutions.
  
  
• Study of the structure of superfield counterterms and other invariants in N=(1,0), N=(1,1) and N=(2,0) supersymmetric gauge theories in 6 dimensions by the harmonic superspace methods. The analysis of these models in the framework of generic AdS/CFT correspondence, quantization of such theories, computing their quantum effective action and learning the full structure of admissible counter-terms. For such an analysis, of high importance is the formalism of harmonic superfields with the maximal number of manifestly realized supersymmetries.
  
  Investigations of multiparticle systems with extended Poincare d=1 and superconformal supersymmetries and various SU(m|n) deformed supersymmetries. These studies will be based on the construction of new models of supersymmetric mechanics by using gauging isometries of matrix superfield systems.
  
  Construction of new models of multiparticle mechanics with extended supersymmetry on curved spaces, investigation of the quantum properties of the constructed models, their integrability and connection with matrix models of the string theory as well as their application in nuclear physics models, elementary particle physics and high energy.
  
  Construction on the complex / quaternionic Euclidean and projective spaces of superintegrable analogs of known oscillator-like systems, allowing the interaction with constant magnetic / instanton field, and further supersymmetrization of them.
  
  Construction and study of superintegrable generalizations of generalized oscillator models (with additional Calogero-like potentials) on complex / quaternion projective spaces that interact with external magnetic constant / instanton fields, and then performing their "weak N=4 supersymmetrization". We plan to build analogues of the Smorodinsky-Winternitz and Rosokhatius systems and their "weak" N=4 supersymmetric extensions, study their symmetry algebra and classical and quantum-mechanical solutions, and extend this analysis to Calogero type systems.
  
  Construction of the twistor formulations of particles and superparticles of fixed spin (helicity), as well as higher spin particles.
  
  Investigation the properties of topological solitons in classical and quantum field theory in flat and curved space-time as well as the investigation of black holes and regular localized field configurations in the extended models of gravity coupled to the matter fields, including non-Abelian fields, will be continued.
  
  
• Owing to the birth of gravitational-wave astronomy and the acquisition of new observational data (LIGO, VIRGO, etc.), it became possible to test both various theories of modified gravity and effective models of black holes and other compact highly gravitating objects. In this regard, the following research directions are outlined:
  - the study of the cosmological consequences of various theories of modified gravity;
  - the development and study of new modified gravity theories, capable of explaining inflation and modern dark energy in a single approach;
  - the construction of effective models such as rotating single and double black holes and other compact objects (such as NUT solutions) in various theories of gravity;
  -the development of new approaches and methods of mathematical physics to the study of effective models of compact objects in various theories of modified gravity;
  -the study of various boundary effects in conformal theories, such as Casimir effect, and their possible holographic description in dual gravity theories in order to comprehend the behavior of these effects in the strong coupling regime;
  - the calculation of the Casimir effect due to the interaction of the quantum field with another quantum field confined in the spatially non-connected regions (two half spaces, for instance) and elaboration of the methods explicitly taking this interaction into account without replacing it by effective boundary conditions;
  - the elaboration of spectral geometry methods (zeta functions, heat kernel expansions) for differential operators on the singular background or with singular potential, along with the development of the spectral summation method with the goal to employ it in boundary problems with matching conditions on the interfaces between different material media.
  
  

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