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 results in the current year:

1. Investigation of holographic renormalization group flows in 3d supergravity at zero and finite temperatures using the theory of dynamical systems. Construction of asymptotic gravity solutions corresponding to holographic RG flows in 3d supergravity. Investigation of fixed-point deformation using the trace of the energy-momentum tensor (TTbar-deformation) in the framework of the holographic approach.

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 and analysis using random matrix ensembles.

In the context of holographic correspondence, integrable structures on Sasaki-Einstein-type manifolds Y^p,q and L^p,q,r following the chain Fuchsian equations, Heun equations and Painleve-type equations will be studied. Test of the holographic duality hypothesis using string dynamics on these manifolds. The focus will be on Y^p,q spaces which can be used as internal manifolds for supersymmetric AdS5 or Schrodinger invariant IIB supergravity solutions.

Construction of analogs of super-Schwarzians and Schwarzian mechanincs associated with d=1 superconformal algebras with extended supersymmetry, in particular, osp(N|2), su(1,1|N), osp(4*|4), F(4). Study of what properties of super-Schwarzians can be extended to the case of N>4 supersymmetries.

Construction of a twistor description of massless fields with continuous spin in four-dimensional Minkowski space. Investigation of the transition from massless fields with continuous spin to massless fields with helicities in this description. Study of the dynamics of massless fields with continuous spin.

Derivation of universal formulas for the projectors onto invariant subspaces and the corresponding eigenvalues of the split Casimir operator in the tensor product of four adjoint representations of simple Lie algebras and Lie superalgebras. Construction of a matrix model that defines the interpretation of the diagrams corresponding to the split Casimir operator in the defining and adjoint representations as Feynman diagrams. Derivation of group factors of the diagrams of this model.

New methods in Geometric Quantization of synthetic type, unified vector and lagrangian approaches, based on the programme of Special Bohr-Sommerfeld geometry.

Inversion of operators related to generalizations of V.P. Maslov’s quasiclassical approximations and topological properties of Liouville vector fields on open symplectic manifolds.

2. Construction and investigation of new types of static Q-cloudy black holes in the Einstein-Maxwell-Fridberg-Lee-Sirlin model.
   Construction of N=4,8 supersymmetric extensions of systems with generic Kähler phase space.

Exploration of the integrability issues in the supersymmetric Euler-Calogero-Moser model and construction of the relevant set of integrals of motion.

Development of the BRST formalism for describing massless infinite spin fields and superfields in 6D space.

Working out the manifestly N=(4,4) supersymmetric harmonic superfield approach to T-duality in the hyper-Kähler and quaterrnion-Kähler 2D sigma models.

Construction of the harmonic superspace formulation of N=2 superconformal higher spins and its reduction to AdS background.
Construction and investigation of new multi-soliton solution of the Skyrme-Maxwell theory.

Study of N=4, d=1 non-linear mirror multiplets of supersymmetric quantum mechanics and construction for them of Wess-Zumino-type Lagrangians and couplings to other N=4, d=1 mirror multiplets.

Construction of generalized lens elliptic gamma functions and proof that they describe the superconformal index of 4D N=1 supersymmetric theories on a product of a circle and a generalized squashed lens space. 

3. Analysis of inflationary scenarios in scalar-tensor models of gravity, calculation of observable parameters such the slope of the primordial perturbation spectrum and the tensor-to-scalar ratio.

Investigation of photon orbits in the regime of a strong gravitational field in modified gravity theories and setting limits on the parameters of modified theories based on current observational data.

Study of quantum effects in scalar-tensor models of gravity and opportunities for their empirical verification.

Development of the FeynGrav package and its application to the computation of one-loop amplitudes in scalar-tensor gravity models. Investigation of the structure of divergencies in these models and the possibility of their ultraviolet extension.

Study of phenomenological theories of gravity containing higher derivatives of the Ricci scalar and the trace of the energy-stress tensor. Analysis of instabilities in this kind of theories and possibility to construct ghost-free and phantom-free subclasses. Investigation of possible cosmological consequences of these theories.

Investigation of gravitational bursts generated by null strings and setting limits on the parameters of such strings based on current observational data.

Analysis of the influence of the gravitational-wave background on physical processes available for observation.

Study of diffraction and interference of electromagnetic and gravitational waves aginst the background of null cosmic strings. Application of the Picard-Lefschetz theory for estimating the diffraction integrals arising from these problems. Investigation of the caustics of world surfaces of null cosmic strings by methods of the Arnold’s theory of singularities of differentiable mappings.
Investigation of quantum fluctuations of an electromagnetic field against the background of anisotropic integrable optical profiles generalizing the classical “Maxwell’s fisheye”.

Development of a quantum field theory approach to the description of topological insulators.

Collaboration