Participating Countries and International organizations:
Armenia, Australia, Brazil, Bulgaria, Canada, CERN, Czech Republic, Estonia, France, Germany, Greece, ICTP, India, Israel, Iran, Ireland, Italy, Japan, Lithuania, Luxembourg, Norway, Poland, Portugal, Republic of Korea, Russia, Spain, Taiwan, Ukraine, United Kingdom, USA.

Issues addressed and main goals of research:
The main purpose of research in modern mathematical physics is the development of mathematical methods for solving the most important problems of modern theoretical physics: clarifying the nature of fundamental interactions and their symmetries, construction and study of effective field models arising in the theory of strings and other extended objects, uncovering of the geometric description of quantum symmetries and their spontaneous breaking in the framework of search for a unified theory of all fundamental interactions, including quantum gravity. Mathematical physics in recent years has been characterized by increasing interest in identifying and effective use of integrability in various areas, and in applying powerful mathematical methods of quantum groups, supersymmetry and non-commutative geometry to quantum theories of fundamental interactions as well as to classical models.
 The main goals and tasks of the research within the theme include: development of new mathematical methods for investigation and description of a variety of classical and quantum integrable models and their exact solutions; analysis of a wide range of problems in supersymmetric theories including models of superstrings and superbranes, study of non-perturbative regimes in supersymmetric gauge theories; development of cosmological models of the early Universe, primordial gravitational waves and black holes. The decisive factor in solving the above problems will be the crucial use of the mathematical methods of the theory of integrable systems, quantum groups and noncommutative geometry as well as superspace techniques.

Expected main results in the current year:

• Study of the interior of a black hole using random matrix ensembles holographically dual to dilaton gravity. Calculation of spectral correlators for 2d dilaton gravity, analysis using random matrix ensembles.
   
  Investigation of the phase diagram for the thermal ensemble of N=4 super-Yang-Mills theory on RxS³ in the framework of the holographic approach. Calculation of the confinement-deconfinement phase transitiontemperature, calculation of the circular Wilson loop in Kerr-AdS₅, and the corresponding quark-antiquarkinteraction potential.
   
  Construction of a solution describing aclosed bosonic (pulsating) string in the 5d space of the Kerr-AdS black hole. Calculation of the energy spectrum of a string, using the Bohr-Sommerfeld analysis, that is dual to the dispersion relations for the operators of the thermal N=4 SYM on RxS³.
   
  Construction of the N=(1,0), d=6 non-abelian tensor hierarchy off the mass shell and the action of non-abelian tensormultiplet that is  invariant with respect to the obtained gauge transformations.
   
  Analysis of the spectrum of the Casimir operators for the six-dimensional Poincaré group on the subspace of states corresponding to massive particles. Construction of the field realization of massive representations of thesix-dimensional Poincaré group.
   
  Derivation of 4D and 6D covariant equations for the wave functions of particles with an infinite (continuous) spin within the framework of the generalized Wigner scheme.
   
  Application of the Manin matrices to the theory of representations of quantum algebras on quantum linear spaces. Interpretation of the Manin matrices as generalized (co)points of some quadratic algebras. Generalization of the tensor product of representationsto the quantum case. Generalization of the theory of quantum linear spaces to the infinite-dimensional case and to the case ofsuper-algebras.
   
  New method for construction of lagrangian cycles in algebraic varieties in the framework of Mirror Symmetry: construction of the generalized Mironov cycles in Grassmannians; germs of lagrangian cycles on divisors and their growth by the inverse flows ofLiouville fields.
   
  Construction of the trigonometric and hyperbolicRuijsenaars-Schneider models with extended supersymmetry andanalysis of their integrability.
   
• Computation of the two-loop divergences in the 6D, N=(1,1) supersymmetric Yang-Mills theory in the quantumharmonic 6D, N=(1,0) superfield approach in the general background for checking the hypothesis that the relevant expression possesses  hidden 6D, N=(0,1) supersymmetry and  vanishes on the full set of the equations of motion for 6D, N=(1,1) theory.
   
  Study of the unitary representations of the Poincaré group in six-dimensional space-time, including massless infinite spin representations, in space-time and twistor formulations.
  Construction and study by various methods of the supersymmetric generalizations of many-particle integrable systems of diverse kinds, including both the nonrelativistic Calogero-Moser-Sutherland systems and their relativistic analogs – the Ruijsenaars-Schneider models.  
   
  Construction and study of new multicomponent solutions of the CP2 Skyrme model with the SU(3) symmetry breaking potential.
  Construction and study of new types of boson stars and hairy black holes in the U(1) gauged Einstein-Friedberg-Lee-Sirlin model. 
   
  Construction and study of compactified spin-Ruijsenaars-Schneider model as a system on phase space given bycomplex Grassmannian, and the study of its supersymmetric extensions.
   
  Construction and study of an isotropic optical profiles that are dual to the two-center Coulomb problem generalizing the classical "Maxwell fish eye" refraction indices.
   
  Construction of a new model of the N=4 supersymmetric mechanics with the coordinate (1,4,3) and (2,4,2) multiplets interacting with the spin (3,4,1) multiplet, consideration of its SU(2|1)deformation and quantizationin a few simple cases.
   
  Study of various limits of the hyperbolic hypergeometric integral related to the fusion matrix of two-dimensional Liouville conformal field theory and study of limiting forms of the difference equations and symmetry relations for rarefied elliptic and hyperbolic hypergeometric integrals paying special attention to the supersymmetric case.

• Generalization of the approach to the decay of a false vacuum in 4-dimensional scalar field theory to a space with an arbitrary number of dimensions. Consideration of a wide class of unbounded potentials for which Coleman instantons do not exist. Obtaining of universal formulas for basic physical quantities related to the problem of the decay of a false vacuum in any number of dimensions. For any number of dimensions, construction of integrable potentials for which instanton equations have exact solutions.
   
  Investigation of scalar-tensor gravity models by effective field theory methods. Derivation of ascalar field effective potential generated in a scalar-tensor gravity model with a massive scalar field admitting quartic self-interaction up to the leading gravitational corrections. Study of the effect of quantum corrections on low-energy phenomenology, especially on the ability of this model to describe slow-roll inflationary expansion.
   
  Comprehensive study of the dynamics of null cosmic strings on physically interesting manifolds (geometries with black holes, cosmological models, and others) making use of the optical equationthat takes into account the general characteristics of the motion of a string in an arbitrary gravitational field in a coordinate-invariant and reparameterization-invariant form.
   
  Formulation and study of quantum field theory on manifolds with holonomy, the elements of which are related to parabolic transformations of the Lorentz group (the so-called zero rotations), obtaining of Green's functions, calculation of the expectation values of the energy-momentum tensor, study of the behavior of the heat kernel, etc.
   
  Investigation of spaces with global parabolic isometries in order to accurately describe the gravitational field of null cosmic strings. Study of the influence of null cosmic strings on the spectrum of inhomogeneities of the microwave background radiation - an analogue of the Kaiser-Stebbins effect.
   
  Investigation of the main scenarios of the test particle and photon motion for the spherically symmetric Stephani cosmological model with  accelerated expansion.
   
  Construction of a model of a cosmological black hole in the dust-filled universe on the basis of the exact solution to the Einstein equations of the Lemaitre-Tolman-Bondi class for different types of spatial curvature.Analysis of the cosmological horizon in the Lemaitre-Tolman-Bondi metric with nonzero pressure.
   
  Consideration of cosmological anisotropic models of Bianchi type I in the theories of teleparallel gravity f(T). Study of the possibility of the existence in such theories of solutions with bounce and recollapse, as well as the possibility of dynamic isotropization during the expansion of the Universe. Examination ofthe structure of the cosmological singularity.
   
  Development of the universal effective method for solving typical problems in the classical nonrelativistic theory of gravity, in particular, the calculation of perturbations of Kepler’s ellipses.
   
  Construction of solutions for massless cosmic strings moving in space-times containing non-trivial objects such as singularities, black holes, matter flows, fluctuations of matter density. Analysis of the possibility of extracting information from observational data related to the cosmic string about objects with which the string previously interacted andfinding out whether it is possible to obtain in this way data on the physics of processes that took place on the Planck scale during the Big Bang.
   
  Comprehensiveinvestigationof the properties of the previously constructed solution for a dyon black hole with a dilaton field with asymptotic Minkowski space in the context of the holographic approach, which allows one to relate the parameters of solutions, such as temperature and free energy, with the parameters of dual field models.
   
  Investigation of the properties of dynamical systems arising in models of a gravitating scalar field in five-dimensional space-time with a specific potential arising in the context of holographic duality. Interpretation of the obtained properties of dynamical systems in the context of dual field theories.

 Collaboration