Modern Mathematical Physics: Integrability, Gravity and Supersymmetry

Participating Countries and International organizations:

Armenia, Australia, Brazil, Bulgaria, CERN, China, Czech Republic, France, Germany, Greece, Iran, Ireland, Israel, Italy, Japan, Poland, Portugal, Russia, Serbia, Spain, United Kingdom, USA.

The problem under study and the main purpose of the research:

The main task of the Theme is the development of mathematical methods for solving the most important problems of modern theoretical physics, namely: development of new mathematical methods for studying and describing a wide class of classical and quantum integrable systems and their exact solutions; analyzing and searching for solutions to a wide range of problems of supersymmetric theories, including models of strings and other extended objects; study of nonperturbative regimes in supersymmetric gauge theories; development of cosmological models of the early Universe, gravitational waves and black holes.

Brief annotation and scientific rationale:

Our project is devoted to important problems of modern mathematical physics. The three most important investigation directions of the project are the study of holographic duality, construction of supersymmetric theories and description of unitary irreducible representations of Poincare groups in higher dimensions. Each of these directions can be regarded separately but in our project we concentrate our attention on the problems which lie on the borders of these major directions. As byproducts, we study certain applied problems, including those that arise in connection with booster thematic.

Expected results upon completion of the project:

The first problem of the project is the study of algebraic and differential structures in holographical systems, which belongs to the subject of modern mathematical physics considered in the context of holographic duality. This part of the project is focused on the study of properties of integrable systems appearing in holographical models. 

The second problem is devoted to the construction of an action of the non-Abelian N=(1,0), d=6 tensor multiplet possessing as many as possible numbers of properties of six dimensional superconformal theories. This problem is related to the first one since it is devoted to field theories with extended supersymmetry which are very important in the mathematical physics studies since they help to describe common properties of quantum fields theories and many aspects of the string theory.

The third problem of our project arises in the context of studies of the models with higher spin fields requiring a certain description of unitary irreducible representations of Poincare groups and symmetry groups of AdS spaces. According to Wigner, each irreducible representation of the four-dimensional Poincare group is assotiated with an elementary particle (field). This conception is generalized to the case of arbitrary dimension and to the case of other groups including supergroups. Therefore, when studying different field models, one first of all asks the question of classification and explicit construction of unitary irreducible representations of the symmetry group of the studied theory.

Expected results of the project in the current year:
Universal formulas will be constructed for the projectors onto invariant subspaces and eigenvalues of the split Casimir operator in the tensor product of four adjoint representations of simple Lie algebras and superalgebras. Universal formulas will be found for the eigenvalues of higher Casimir operators in the representations included in the tensor cube of the adjoint representation of simple Lie algebras and superalgebras. The color factors of Feynman diagrams in quantum gauge field theories will be found. 

The superfield action of the N=(1,0), d=6 tensor multiplet in harmonic superspace will be constructed, which includes interaction with a non-abelian gauge field in the framework of tensor hierarchy, and it will be found what expected properties of six-dimensional superconformal field theory are satisfied. 

For a 3d supergravity model, a holographic RG flow at finite temperature corresponding to а black hole solution will be constructed. Using the mapping onto the Poincare sphere, the stability of holographic RG flows will be investigated. It is also planned to calculate thermodynamic quantities and investigate the phase diagram corresponding to the RG flow. 

Using the method of the generalized Wigner operator, local relativistic fields will be constructed, on which unitary irreducible massless helicity representations of the 4D Poincaré group are realized. It is planned to build objects corresponding to gauge potentials (using an auxiliary “index” 4-vector variable) and objects corresponding to field strengths (using auxiliary commuting Weyl spinors).

Brief annotation and scientific rationale:

The project is aimed at solving fundamental problems of modern theoretical physics associated with the development of superfield methods in gauge theories with extended supersymmetry in various dimensions, including extended supersymmetric mechanics. The implementation of the project includes the construction of new field and quantum-mechanical models with global and gauge symmetries, the development of new, including geometric, methods for studying the structure of these models at the classical and quantum levels, the study of the structure of the corresponding quantum effective actions and classical solutions of these models, including black holes. All tasks of the project are set by the modern development of theoretical physics and are organically joined by the unity of methods and approaches.

Expected results upon completion of the project:

Calculating all leading and subleading in the dimensional regularization parameter two-loop counterterms in 6D, N=(1,0) and N=(1,1) supersymmetric gauge theories. 

Constructing a one-loop induced effective action in the theory of hypermultiplet interacting with N=2 supergravity in the harmonic superspace approach. 

Development of the methods of calculation of the one-loop induced effective action in the theory of hypermultiplet coupled to external fields of N=2 harmonic gauge superfields. 

Derivation of 4D, N=2 harmonic superfield formulation for N=2 supersymmetric gauge fermionic higher spin fields. 

Working out 4D, N=2 superfield gauge theory of higher spin fields in the AdS space. 

Development of effective methods for describing gauge fields and superfields of an infinite spin in an arbitrary space-time dimension. 

Finding Lagrangians describing the interactions of infinite spin fields and higher spin fields with fields of a fixed spin. 

Finding out superfield harmonic Lagrangians of sigma models obtained by T-duality from 2D, N=(4,4) supersymmetric hyperkahler and quaternion-kahler sigma models. 

Building a superfield matrix formulation of new N=4 and N=8 supersymmetric extensions of integrable many-particle systems and their quantization. 

Construction of new models of N-extended supersymmetric quantum mechanics by using the superfield  gauging  method, which describe the interaction of dynamic and semidynamic multiplets of various types.

Construction and study of N=4 models of supersymmetric mechanics based on the interaction of linear and nonlinear supermultiplets with the component content (4,4,0), (3,4,1) and (2,4,2). 

Constructing the Hamiltonian formulation and performing quantization of the generalizations of systems with the nonlinear (2.4.2) supermultiplet. 

Constructing an extension of N=4 supersymmetric mechanics with (3.4.1) supermultiplet to the class of systems parametrized by an arbitrary holomorphic function. 

Construction and study of many-particle systems with nonlinear supermultiplets. 

Construction of a superfield description of Calogero-type models with extended N≥4 supersymmetries.

Analysis of the integrability of N-extended supersymmetric systems of the Euler–Calogero–Moser and Calogero–Moser–Sutherland types for the A(n-1) series of the Coxeter group. 

Finding an explicit form of the functionally independent conserved Liouville currents in N=2 supersymmetric Calogero models for all root systems of Coxeter groups. 

Construction of two new exactly calculated rarefied elliptic beta integrals associated with special lens spaces and a special subgroup of the modular transformations group SL(2,Z). 

Computation of a matrix of modular transformations of one-point conformal blocks on a torus in the Neveu-Schwarz sector of the N=1 superconformal Liouville field theory, based on the expression of this matrix as an integral of the product of certain elements of the fusion matrix. 

Obtaining the difference equations for the fusion matrix in the Neveu-Schwarz sector of the N=1 superconformal Liouville field theory.   

Finding a new class of solutions of GR with gauge multicomponent matter fields in models with spontaneous symmetry breaking. 

Constructing and exploring a new class of solutions of extenged Einstein gravity with the Chern-Simons term that represents stationary rotating black holes.

Expected results of the project in the current year: 

Construction of a theory of N=2 higher spin superfields against the AdS₄ superbackground. 

Calculation of the quantum effective action of the hypermultiplet induced by interaction with higher-spin N=2 gauge superfields. 

Construction and study of a new class of solutions of GR with multicomponent gauge matter fields in models with spontaneous symmetry breaking. 

Finding the Lagrangians describing free infinite spin(super)fields using the twistor approach and BRST methods
Construction of the minimal interaction of infinite spin fields and higher spin fields with the fixed spin fields.

Construction and study at the classical and quantum levels of new N=4 and N=8 supersymmetric matrix systems with extended deformed supersymmetry and matrix systems of superconformal mechanics.

Hamiltonian formulation and quantization of systems with the generalization of (2.4.2) nonlinear chiral supermultiplet suggested in the article by S. Bellucci, A. Nersessian, Phys. Rev. D 73, 107701 (2006). 

Compution of the matrix of modular transformations of one-point conformal blocks on the torus in the Neveu-Schwarz sector of 2D N=1 superconformal Liouville field theory.

Derivation of difference equations for the fusion matrix in the Neveu-Schwartz sector of the 2D N=1 superconformal Liouville field theory.

Obtaining a superconformal index of the 4D N=1 superconformal field theory on general lens space and calculation of its rarefied beta integral.

 Сonstruction of models of the N=4 supersymmetric mechanics with spin degrees of freedom based on the interaction of linear and nonlinear supermultiplets.

 Construction of a supersymmetric generalization of the Euler-Calogero-Moser model for an arbitrary number of particles and study of its integrability and superintegrability in the case of N=2 supersymmetry. 

Brief annotation and scientific rationale:

The project is aimed at solving the fundamental problems of classical and quantum gravity and conducting advanced theoretical research at the national and world level in this area at BLTP JINR. In classical gravity, the project is focused on studying all kinds of gravitational wave phenomena, including shock waves in General Relativity, as well as various sources of gravitational wave background such as cosmic strings. One of the directions of the project is the elaboration of cosmological models that explain the properties of the observable Universe based on field theory methods and modified gravity. In the field of quantum gravity, it is planned to develop an apparatus of quantum field theory in an external classical gravitational background and new methods for an approximate estimation of the effective gravitational action in various regimes. Asymptotic symmetries in gravity, the relationship between gravity, thermodynamics and quantum entanglement, the holographic properties of gravity, and the AdS/CFT correspondence will also be explored.

Expected results upon completion of the project:
Development of field theory methods against the background of shock gravitational waves using the method of supertranslations at the wave front; study of classical field effects induced by shock waves, including the astrophysical applications.

Study of classical effects in the gravitational field of shock gravitational waves, including the case of the gravitational field induced by null cosmic strings (cosmic strings moving at the speed of light); study of gravitational (electromagnetic) radiation induced by the motion of null cosmic strings near massive (charged) sources, estimation of the parameters of these objects corresponding to the observed characteristics of induced radiation.

Study of physical effects associated with the formation of caustics and other defects on the world sheet of the null cosmic string as possible sources of gravitational bursts; development of the holonomy method proposed for describing free classical fields against the background of a gravitational shock wave.

Quantization and study of quantum effects in the gravitational field of shock gravitational waves, calculation of the expectation value of the renormalized energy-momentum tensor.

Derivation and study of the properties of exact solutions of the Einstein equations related to the subject of this project, for example, the search for non-trivial solutions that have global hyperbolic isometry and allow the introduction of holonomy associated with these transformations.

Study of the gravitational entropy associated with various surfaces in Riemannian geometry, in particular, study of the entropy formed when the light cones of the past and future (causal diamonds) intersect, as well as study of quantum corrections and renormalization of this quantity.

Further development of spectral geometry methods applied to nonlinear spectral problems; using these methods to study the finite-temperature QFT on stationary manifolds of a general form, as well as the application of this theory to calculate the effects of quark-gluon matter taking into account rotation and acceleration.

Study of cosmological models of modified gravity, an attempt to explain on their basis the key characteristics of the observed cosmology such as the accelerated expansion of the Universe; the study of cosmological perturbations in a teleparallel theory with a non-minimal scalar-tensor coupling, where the main object is the torsion scalar, in contrast to general relativity, where the main object is the Ricci scalar.

Construction of integrable cosmological potentials for spatially flat cosmologies with one scalar field for searching and constructing realistic completely integrable inflationary models with a phase transition; study of phase transitions in quantum theory, including gravity, and the formation dynamics of walls separating regions with different field values, the development of the thick-wall approximation method taking into account gravity, as well as the construction and study of exactly solvable inflationary models with phase transitions.

Development of methods in the framework of the Picard-Lefschetz theory and their application for calculating Lorentz path integrals in problems of quantum field theory, gravity and cosmology, and, in particular, in problems of describing the lensing of gravitational waves.

Expected results of the project in the current year:

Calculation of an aproximate stress-energy tensor for weak high frequency waves in f(R) gravity against an arbitrary curved space-time background, which is not an obligatory solution of this vacuum f(R) gravity, using Aizekson method for high frequency waves in Einstien gravity. A possible application of this result is to take into account the back reaction from previously borned scalarons on evolution of dark energy in the Starobinsky and Hu-Sawicki gravity models.

Investigation of classical effects against the background of shock gravitational waves, the special cases of which are the gravitational fields of massless ultrarelativistic particles and null cosmic strings. Quantitative estimates of these effects will be made and the possibility of their observation in gravitational experiments will be studied.

The method of soldering two metrics together across null hypersurfaces will be used to find new solutions in the general relativity, which can be interpreted as gravitational shock waves. The geometry and physical properties of these null surfaces will be investigated. The results will be formulated in the language of Carrollian symmetry groups. The asymptotic BMS supertranslation charges will be found.

Revision of quantum field theory against the classical backgound of a gravitational shock wave and study of quantum effects in this spacetime; calculation of the expectation value of renormalized stress-energy tensor.