
Modern
Mathematical Physics: 
Leaders: 
A.P. Isaev 
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 noncommutative 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 super
branes, study of nonperturbative 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 super space 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 superYangMills theory on RxS^{3} in the framework of the holographic approach. Calculation of the confinementdeconfinement phase transition temperature, calculation of the circular Wilson loop in KerrAdS_{5}, and the corresponding quarkantiquark interaction potential.
Construction of a solution describing a closed bosonic (pulsating) string in the 5d space of the KerrAdS black hole. Calculation of the energy spectrum of a string, using the BohrSommerfeld analysis, that is dual to the dispersion relations for the operators of the thermal N=4 SYM on RxS^{3}.
Construction of the N=(1,0), d=6 nonabelian tensor hierarchy off the mass shell and the action of nonabelian tensormultiplet that is invariant with respect to the obtained gauge transformations.
Analysis of the spectrum of the Casimir operators for the sixdimensional Poincaré group on the subspace of states corresponding to massive particles. Construction of the field realization of massive representations of the sixdimensional 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 representations to the quantum case. Generalization of the theory of quantum linear spaces to the infinitedimensional case and to the case of superalgebras.
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 of Liouville fields.
Construction of the trigonometric and hyperbolic RuijsenaarsSchneider models with extended supersymmetry andanalysis of their integrability.
Computation of the twoloop divergences in the 6D, N=(1,1) supersymmetric YangMills theory in the quantum harmonic 6D, N=(1,0) super field 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 sixdimensional spacetime, including massless infinite spin representations, in spacetime and twistor formulations.
Construction and study by various methods of the supersymmetric generalizations of manyparticle integrable systems of diverse kinds, including both the nonrelativistic CalogeroMoserSutherland systems and their relativistic analogs – the RuijsenaarsSchneider 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 EinsteinFriedbergLeeSirlin model.
Construction and study of compactified spinRuijsenaarsSchneider 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 twocenter 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(21)deformation and quantizationin a few simple cases.
Study of various limits of the hyperbolic hypergeometric integral related to the fusion matrix of twodimensional 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 4dimensional 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 scalartensor gravity models by effective field theory methods. Derivation of ascalar field effective potential generated in a scalartensor gravity model with a massive scalar field admitting quartic selfinteraction up to the leading gravitational corrections. Study of the effect of quantum corrections on lowenergy phenomenology, especially on the ability of this model to describe slowroll 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 coordinateinvariant and reparameterizationinvariant 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 socalled zero rotations), obtaining of Green's functions, calculation of the expectation values of the energymomentum 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 KaiserStebbins 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 dustfilled universe on the basis of the exact solution to the Einstein equations of the LemaitreTolmanBondi class for different types of spatial curvature. Analysis of the cosmological horizon in the LemaitreTolmanBondi 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 of the 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 spacetimes containing nontrivial 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.
Comprehensivei nvestigation of 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 fivedimensional spacetime 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.




Activity or experiment 
Leaders 



Laboratory
or other 
Main researchers 

1. 
Quantum
groups 
A.P.
Isaev 


BLTP 
Ch. Burdik, H. Dimov, P. Fiziev, A.A. Golubtsova, N.Yu. Kozyrev, M. Podoinitsyn, G.S. Pogosyan, A.A. Provorov, A.V. Silantyev


UC 
S.Z. Pakuliak 
2. 
Supersymmetry 
E.A. Ivanov 


BLTP 
S.A.
Fedoruk, A. Nersessian, G. Sarkissyan, 
3. 
Quantum
gravity, 
I.G.
Pirozhenko 


BLTP 
I.
Bormotova, E.A. Davydov, D.V. Fursaev, B. Latosh, 

MLIT 
A.M. Chervyakov 

VBLHEP 
E.E. Donets 
Collaboration
Country or International Organization 
City 
Institute or laboratory 
Armenia 
Yerevan 
Foundation ANSL 


YSU 
Australia 
Perth, WA 
UWA 

Sydney, NSW 
Univ. 
Brazil 
Juiz de Fora, MG 
UFJF 

Sao Paulo, SP 
USP 

Vitoria, ES 
UFES 
Bulgaria 
Sofia 
INRNE BAS 


SU 
Canada 
Edmonton 
U of A 

Montreal 
Concordia 
CERN 
Geneva 
CERN 
Czech Republic 
Opava 
SlU 

Prague 
CTU 

Rez 
NPI CAS 
Estonia 
Tartu 
UT 
France 
AnnecyleVieux 
LAPP 

Lyon 
ENS Lyon 

Marseille 
CPT 

Nantes 
SUBATECH 

Paris 
ENS 


LUTH 

Tours 
Univ. 
Germany 
Bonn 
UniBonn 

Hannover 
LUH 

Leipzig 
UoC 

Oldenburg 
IPO 

Potsdam 
AEI 
Greece 
Athens 
UoA 

Thessaloniki 
AUTH 
ICTP 
Trieste 
ICTP 
India 
Chennai 
IMSc 

Kolkata 
BNC 


IACS 
Iran 
Tehran 
IPM 
Ireland 
Dublin 
DIAS 
Israel 
Tel Aviv 
TAU 
Italy 
Frascati 
INFN LNF 

Padua 
UniPd 

Pisa 
INFN 

Trieste 
SISSA/ISAS 

Turin 
UniTo 
Japan 
Tokyo 
Keio Univ. 


UT 
Lithuania 
Vilnius 
VU 
Luxembourg 
Luxembourg 
Univ. 
Norway 
Trondheim 
NTNU 
Poland 
Bialystok 
UwB 

Lodz 
UL 

Wroclaw 
UW 
Portugal 
Aveiro 
UA 
Republic of Korea 
Seoul 
SKKU 
Russia 
Chernogolovka 
LITP RAS 

Kazan 
KFU 

Moscow 
ITEP 


LPI RAS 


MI RAS 


MSU 


SAI MSU 

Moscow, Troitsk 
INR RAS 

Novosibirsk 
NSU 

Protvino 
IHEP 

St. Petersburg 
PDMI RAS 

Tomsk 
TPU 


TSPU 
Spain 
Barcelona 
IEECCSIC 

Bilbao 
UPV/EHU 

Santiago de Compostela 
USC 

Valencia 
IFIC 

Valladolid 
UVa 
Taiwan 
Taoyuan City 
NCU 
Ukraine 
Kharkov 
KhNU 


NSC KIPT 

Kiev 
BITP NASU 
United Kingdom 
Cambridge 
Univ. 

Canterbury 
Univ. 

Durham 
Univ. 

Glasgow 
U of G 

Leeds 
UL 

London 
Imperial College 

Nottingham 
Univ. 
USA 
Amherst, MA 
UMass 

College Park, MD 
UMD 

Coral Gables, FL 
UM 

New York, NY 
CUNY 


SUNY 

Norman, OK 
OU 

Piscataway, NJ 
Rutgers 

Rochester, NY 
UR 

Tempe, AZ 
ASU 
▲ 