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Modern Mathematical Physics:
Strings and Gravity, Supersymmetry, Integrability

• In quantum integrable models with high symmetry the closed expression for the scalar product of the nested Bethe vectors in terms of the sums over partitions of the sets of the Bethe parameters will be obtained. The cases when these sums can be reduced to the product of determinants will be investigated. The determinant formulas for the scalar product of the Bethe vectors in the nested quantum integrable models with higher symmetries will be used to calculate the form-factors of the local operators and correlation function.
  
  The dilaton field as a model of dark matter will be taken into account to construct the new self-consistent models of compact relativistic stars. This class of models avoids the well-known difficulties of the f(R) theories of gravity. The models of compact stars with different equations of state will be studied, and the predictions of the models will be compared with astrophysical data.
  
  New relations for the confluent Heun functions will be derived by using the corresponding monodromy group and the asymptotics of these functions around the irregular singular point will be studied. New numerical algorithms for calculations with the confluent Heun functions will be developed.
  
  Existence of bound states of spinor-particles and their physical spectra in the metrics of Schwarzschild, Kerr, Reisner-Nordstrem, Kerr-Newman, Veidja etc., will be investigated by using new boundary conditions. Nonquantum motion of test particles in these metrics will be studied.
  
  The ingredients of real Lagrangian geometry will be extended to the case of complex variety with holomorphic symplectic form with the aim to find a complex version of the Bohr-Sommerfeld condition for complex Lagrangian submanifolds.
  
  The properties of gluon propagator in the 3d gauge theory on the lattice with SU(2) symmetry will be studied numerically.
  
  The density, correlators and parameters of the interaction of Abelian monopoles near the critical temperature will be calculated within the lattice QCD with gauge group SU(3).
  
  
• The off-shell superfield formulations for the AdS₃ S ³ and AdS₅ S ⁵ superstrings within the Pohlmeyer reduction procedure will be constructed. The corresponding quantization procedures will be developed.
  
  The new models of the N=4 and N=8 supersymmetric mechanics which includes semi-dynamical degrees of freedom interacting with the background non-Abelian gauge fields will be constructed.
  
  The new models of Landau type with the extended world-line supersymmetry will be developed. The possibility of their applications in the theory of quantum Hall, Yang-Mills theory and superstrings will be analyzed.
  
  A class of self-dual models of nonlinear electrodynamics and its supesymmetric extensions will be constructed.
  
  
• The effective scalar theory, which has cosmological solutions and black hole solutions, will be constructed within the affine generalizations of the Einstein theory of gravity. The integrability of this model will be studied and an iterative procedure for finding the general solutions will be developed.
  
  The new type of phenomenological cosmological Lagrangians will be constructed. Their general properties and methods of their covariant generalization will be developed. The relations between Galilean theory and gravity with torsion will be analyzed in the linear approximation.
  
    The phase transition diagrams for the complete symmetric Toda system will be analyzed. The extreme multi-center black hole solutions of the D=4 N=2 supergravity will be constructed and classified. The integrable supergravity cosmologies will be found and investigated.
  
  

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