Modern Mathematical Physics:
Gravity, Supersymmetry and Strings

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:

• Сonstruction of the hierarchy of Mironov lagrangian cycles in Grassmannians of any degree of homogeneity.
  
  Search for minimality and the Hamiltonian minimality conditions for the Mironov lagrangian cycles in the Kahler – Einstein manifolds.

Obtaining of giant magnons and single spike quasiclassical string solutions on the Schr5×T1,1 background. Search for dispersion relations for these classes of string solutions in the above background by using finite combinations of conserved charges.

Investigation of pulsating strings on the Schr5×T1,1 background. Obtaining of energy spectra and perturbative quantum corrections up to first order in the small parameter by semi-classicall quantization of the pulsating string.

Obtaining of quasiclassical string solutions of different types in 5d Kerr-AdS background geometry, mostly of the pulsating string class.

Сalculation of quadratic fluctuations of the world sheet of different types of string configurations and obtaininig of a one-loop correction to energy of pulsating string solutions.

High precision calculations of quasi-normal modes and study of their physical applications.

Development of new methods for solution of quasilinear partial differential equations in the complex domain of variables and their physical applications.

Construction of an N=(1,0) superfield analog of the Pasti-Sorokin-Tonin action of the abelian self-dual tensor field in six dimensional spacetime, search for its generalizations with the nonabelian tensor field and N=(2,0) supersymmetry.

Research and effective construction of state spaces in orthogonal and symplectic quantum integrable systems associated with the Yangians of the classical series B(n), C(n) and D(n) Lie algebras. Investigation of the scalar products of state vectors in these models.

Development of a new model of black hole evaporation in terms of a matrix model for dual two-dimensional gravity. Investigation of a case involving conical singularities in gravitational theory.

Wilson loops calculation for the supersymmetric case of the 5d Kerr-AdS background; the analysis of the leading contributions and comparison with the results of calculations in the dual conformal theory on R×S³.

Study of the dynamics of a quasi-classical pulsating (bosonic) string in the 5d Kerr-AdS space. Search for anomalous dimensions of operators for the dual gauge theory. Сalculation of the quadratic fluctuations of the world sheet for various types of string configurations, obtaining of a one-loop correction to energy of pulsating strings.

Derivation of covariant equations of particles with continuous (infinite) spin within the framework of the generalized Wigner scheme. Search for infinite-dimensional analogs of the Wigner operators that transform a massless test pulse into an arbitrary one.

Construction of projectors onto irreducible representations of symmetry groups of multidimensional (anti) de Sitter spaces using the multidimensional Poincaré group algebraic method well-proven in the theory of representations, the key object of which is the Brauer algebra.

Construction of the split Casimir operator (SCO) for the exceptional Lie algebras in the defining and adjoint representations as well as calculation of the characteristic identity for SCO and finding of the corresponding solutions to the Yang-Baxter equation.

Construction of the trigonometric and hyperbolic Calogero and Ruijsenaars-Schneider models with extended supersymmetry.

• Study of quantum models of N=4 an N=8 supersymmetric mechanics, their integrability, the presence of hidden (super) symmetries and exploring of their relationships with supersymmetric gauge theories.

Construction of new examples of quaternion-kahler N=4 mechanics on inhomogeneous target manifolds, including those with Wess-Zumino terms in the Lagrangian, investigation of their hamiltonian structure and quantization for a few simple cases.

Generalization of the method of gauging isometries in N=4 mechanics to the N=8 case, study of its possble role for constructing new models of N=8 mechanics and finding out interrelations between different models.

Setting up new superextensions of integrable multiparticle systems of the Calogero type and their covariant quantization with application of the 1D harmonic superspace approach and the gauging method and revealing of the relation of these methods to the hamiltonian approach to the same systems.

Analysis of the quantum structure of the superfield effective action in the 6D and 5D supersymmetric gauge theories and supergravities on the basis of the appropriate harmonic superspace formulations.

Study of conformal field theories and their connections with integrable models and N=2 supersymmetric gauge theories as well as applications of these formalisms in condensed matter, statistical, and gravitational physics.

Study of the p>1 и q>1 limits of the difference equations for rarefied elliptic hypergeometric integrals. Obtaining of the corresponding difference equations for the parafermionic hyperbolic hypergeometric integrals in this limit, which will be used to study supersymmetric extensions of the relativistic Calogero models and also for calculation of the one-point matrix of the modular transformations on a torus in the N=1 supersymmetric 2D Liouville field theory.

Study of black holes and reqular particle-like localized solutions of the Einstein equations in asymptotically flat 3+1 dimensional spacetime and in the asymptotically AdS spacetime.

• Derivation of vacuum energy for quantum fields on the background of several crossed or moving cosmic strings. Study of various effects related to cosmic strings, their peculiarities and observability in the massless string limit.
  
  Development of the heat kernel approach and derivation of the Schwinger-DeWitt coefficients for the SU(N) gluodynamics with boundaries and in the external field, specifically, for studying the influence of the boundary conditions on the effective potential and the free energy. Derivation of the uniform asymptotic expansion for the hypergeometric function arising in the free energy computation.

Investigation of non-minimal interactions induced by loop corrections in effective scalar-tensor theories of gravity. Determination of the restrictions on the magnitude of these interactions by the observed form of Newton's law as well as by data on violation of the weak equivalence principle.

Investigation of cosmological perturbations in a covariant formulation of teleparallel gravity with non-minimal coupling. Derivation of equations for scalar perturbations within this approach and obtaining of the spectrum of scalar perturbations during inflation. Comparative analyses with the case of teleparallel gravity without non-minimal coupling and with the results obtained by other methods.

Collaboration