
Modern Mathematical
Physics:

Leaders: 
A.P. Isaev 
Scientific leader: 
A.T. Filippov 
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 superbranes, 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 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 semiclassicall quantization of the
pulsating string.
Obtaining of quasiclassical string
solutions of different types in 5d KerrAdS 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 oneloop correction to energy of pulsating string solutions.
High precision calculations of quasinormal 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 PastiSorokinTonin action of the abelian selfdual 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 twodimensional gravity. Investigation of a case involving conical singularities in gravitational theory.
Wilson
loops calculation for the supersymmetric case of the 5d KerrAdS
background; the analysis of the leading contributions and comparison
with the results of calculations in the dual conformal theory on
R×S^{3}.
Study
of the dynamics of a quasiclassical pulsating (bosonic) string in
the 5d KerrAdS 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 oneloop 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 infinitedimensional 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 wellproven 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 YangBaxter equation.
Construction of the trigonometric and hyperbolic Calogero and RuijsenaarsSchneider 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 quaternionkahler N=4 mechanics on inhomogeneous target manifolds, including those with WessZumino 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
onepoint matrix of the modular transformations on a torus in the N=1
supersymmetric 2D Liouville field theory.
Study of black
holes and reqular particlelike 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 SchwingerDeWitt 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 nonminimal interactions induced by loop corrections in effective scalartensor 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 nonminimal 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 nonminimal coupling and with the results obtained by other methods.
List of Activities: 



Activity or experiment 
Leaders 


Laboratory
or other 
Main researchers 

1. 
Quantum
groups 
A.P.
Isaev 


BLTP 
M. Buresh, P. Fiziev, A.A. Golubtsova, N.Yu. Kozyrev, D.R. Petrosyan, M. Podoinitsyn, G.S. Pogosyan, A.V. Silantyev


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


BLTP 
S.A. Fedoruk, A. Nersessian, M. Pientek, A. Pietrikovsky, I.B. Samsonov, G. Sarkissyan, S.S. Sidorov, Ya.M. Shnir, A.O. Sutulin

3. 
Quantum
gravity, 
A.T.
Filippov 


BLTP 
B.M. Barbashov, I. Bormotova, E.A. Davydov, V.V. Nesterenko, A.B. Pestov, A.A. Provarov, G.I. Sharygin, E.A. Tagirov, P.V. Tretyakov, P. Yaluvkova, A.F. Zakharov, 3 students


LIT 
I.L. Bogoliubsky, A.M. Chervyakov 

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


YSU 
Australia 
Perth 
UWA 

Sydney 
Univ. 
Brazil 
Juiz de Fora, MG 
UFJF 

Sao Paulo, SP 
USP 

Vitoria, ES 
UFES 
Bulgaria 
Sofia 
INRNE BAS 
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 

Madrid 
ETSIAE 

Santiago de Compostela 
USC 

Valencia 
IFIC 
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 
▲ 