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Project submitted for the base funding of Artem Alikhanyan National Laboratory (ANL) Principal Investigator: Ara Sedrakyan TITLE:

Investigations in low dimensional physics(d=1,2,3,4): Applications to non-critical strings and condensed matter physics

Division, group: DURATION:

Theoretical physics

3 year

Estimated Project Costs (± 20%) (эта часть заполняется по согласованию с дирекцией Национальной Лаборатории) Estimated total cost of the project (US $) 177900 $ Including: Payments to Individual Participants

4800$ per month

Equipment

$1,700.00

Materials Other Direct Costs

$400.00

Travel

$3,000.00

Salaries: 1.

Ara Sedrakyan – Doctor of Science ---full stuff--800 $

2.

Hrachia Babujyan- Doctor of Science—full stuff--800 $

3.

David Karakhanyan-Doctor of Science—full stuff—800 $

4

Shahane Khachatryan-Candidate of Science—full stuff--650 $

5

Mkhitar Mirumyan-Candidate of Science—full stuff-- 100 $

6

Ara Martirosyan- researcher- full stuff--100 $

7

Tigran Hakobyan—Candidate of Science- 05 stuff--300 $

8

Tigran Sedrakyan-Candidate of Science- 0.5 stuff- 100$

9

Vagharsh Mkhitaryan-Candidate of Science-0.5 stuff—100 $

10

Hayk Minassyan-Doctor of Science-full stuff- 650 $

11

Armen Melikyan-Doctor of Science-0.5 stuff—300 $

12.

K. Madoyan---aspirant--100 $

PROBLEM:

The main goal of this research project is to investigate certain aspects of the relation between gauge theories and strings, which follows the line of the modern development of theoretical high energy physics. It is planned to further study the string/gauge correspondence, supersymmetric gauge theories and large -N matrix models which are associated with them. The present project proposes to carry out an extensive research of non -critical string theories and to gain new insight into the structure of integrable systems, their mathematical aspects and to apply it on revealing of new opportunities to construct a string theory in real (d=2,3,4) dimensions. The essential ingredient of our investigations is the string representation of the 3D Ising model considered on a regular and BCC (body center cubic) lattices. We believe that the obtained results will give new insight into quantum chromodynamics and statistical physics, in particular, to extend the existing results about the string/gauge correspondence to the theories with less supersymmetries. The main tasks of the investigation include: - mathematical aspects of string and gauge theories; - matrix models and supersymmetric gauge theories; - numerical study of $Z_2$ gauge theory on a lattice; - investigation of the string/gauge correspondence; - non-critical string and integrable models. The investigation will use the methods of Quantum Field Theory, General Relativity, Superstring Theory and Statistical Mechanics. OBJECTIVES:

The main aim of the present project is twofold. First we want to formulate of a new type of matrix models (motivated by the algebraic Bethe ansatz of integrable systems) and investigate the possible new aspects of integrability inherent i n these models. We want to use these models in the study of non-critical strings in real (d=2,3,4) dimensions and in the study of plateau-plateau transitions in the quantum Hall effect. One of the most important trends in theoretical physics in the last decades is the development of exact methods which are completely different from perturbation theory. Resolution of the strong coupling problem would give us a full understanding of the structure of interactions in non-abelian gauge theory. One promising possibility of overcoming the limitations of perturbation theory is the application of exact integrability. From this point of view the two dimensional integrable quantum field theories are in a sense a laboratory for investigations of those properties of quantum field theories, which cannot be described via perturbation theory. Therefore, second line of our investigations have intention to apply the methods of integrable systems to the problems of condensed matter physics. Particular emphasize will be made on investigations of transport problem, metal-insulator transitions and problem of graphene. These investigations will hopefully result in building new links between different fields of mathematical physics, and will connect string theory, quantum criticality in disordered systems and transport problems in condensed matter physics. The approach will lead to analytic calculations of observable’s in the above mentioned fields.

All investigations are grouped into 5 Tasks, which are essentially interrelated. Task 1: The string theories corresponding to integrable models on random surfaces. The technique of representing integrable models as lattice quantum field theories allows a generalization from a regular lattice to various kinds of random lattices and thus offers a unique possibility to consider the integrable models on random surfaces. Moreover, there are some preliminary results demonstrating how to construct the corresponding matrix models, which will allow to advance the technique of calculation of physical observables even further. One of the major goals of the present project is the development of this approach. In particular we intend to: a) Formulate and investigate matrix models corresponding to integrable models on random surfaces. The main difference compared to the conventional approaches is that the kinetic (quadratic) part of the matrix action is defined by the R-matrix of the integrable model. b) Define and calculate the new Itzykson-Zuber type integrals which appears in a). c) The investigation of the string theory corresponding to the XXZ model on random surfaces. d) The investigation of the string theory corresponding to the Hubbard model on random surfaces.

Task 2, Investigation of the string theoretical approach to 3D Ising model. The signfactor of the 3d Ising model is an example of a network model on random surfaces with a particular R-matrix based on the SU(2) group. Here we intend to:

a) Investigate integrability of the corresponding $R-$matrix from the Yang-Baxter equation point of view.

b) Find a topological quantity corresponding to the sign-factor on the basis of conformal field theories with twisted N=2 supersymmetry.

c) Formulate the three particle R-matrix which will reproduce the partition function of 3d-Ising model in its string representation and check if it will satisfy Zamolodchikov's tetrahedron equations, the sufficient conditions for commutativity of the corresponding two dimensional transfer matrices. This work is in progress.

Task 3. Investigations of form-factors. We use the algebraic Bethe Ansatz and his generalization-off shell Bethe Ansatz, inside of ‘bootstrap program’ construct the form factors for most essential operators.

a) Using off-shell Bethe ansatz equations for Belavin’s Z N ⊗Z N -symmetric solution [43] of the Yang-Baxter equations we are going to construct the solutions of the sl(n) invariant KZ equations. In the degenerated case we will get the solutions of trigonometric and elliptic Calogero models.

b) We will also construct O(N) sigma models form factors using the “bootstrap program” and off-shell Bethe Ansatz. We will provide some examples, and calculate the form factors exactly and compare the results with field theoretical 1/N expansion.

c) Generalizing the methods of Tarasov we will construct the off-shell O(N) Bethe Ansatz and apply the results for obtaining the form-factors of the O(N) σ - and Gross-Neveu models.

d) We also would like to focus our attention to possible relation newely established AGT relation between CFT and Nekrasov partition function in N=2 super Yang-Mils, With the off-shell Betha ansatz technique..

Task 4. We propose that the relaxation of electrons in metal exited by light above Fermi level is conditioned not only by electron-phonon interactions (which are the case in conductivity measurements) but also by electron-electron-phonon three particle scattering processes. The investigation of this mechanism is especially important in low dimensional structures - clusters and nanoparticles since in these new objects only the optical measurements are possible. It will also help in identifying the structural peculiarities of MCs and MNPs synthesized by various methods. a) Calculation of electron-electron, electron-phonon and electron-surface scattering rates in metal during the optical excitation. b) Investigation of the contribution of the optical interband transitions and radiation damping mechanism into the surface plasmon linewidth with taking into account the retardation effects in noble metal nanoparticles.

c)

Development of the method for the identification of cluster symmetry based on spectroscopic measurements with small and medium transition and noble metal clusters of various geometries.

Task 5. Investigations in Calogero Model. a) Construct the complete set of the integrals of motion for the system described by the spherical part of the Calogero Hamiltonian. To extract from this set the Liouville integrals of motion.

b) Investigate the ordering of energy levels for the spin and fermionic quantum chains with $SO(n)$ and $SP(n)$ symmetries. To find the quantum numbers and degeneracies for the ground state.

c) To construct an analogue of the Temperley-Lieb basis for the SU(N) singlet states and to apply it for the investigation of resonating-valence-bond ground states.

Task 6. Investigation of new solutions of Yang-Baxter equation Consideration of the YB equations with the R-matrices having the given quantum symmetry at roots of unity will give essentially new solutions due to the existence of the indecomposable representations. We will observe the cases sl q (2) / osp q (1|2). Earlier we have analyzed in details the finite dimensional non-reducible representations (and their fusion rules) of the quantum super-algebra osp q (1|2) at roots of unity and confirmed their equivalence with the corresponding representations of the algebra sl q (2).

Description of deliverables. The results of all investigations will be published in well known international journals: Nuclear Physics B, Phys. Rev. D, B, Journal of Phys. A. C, JHEP, Phys. Rev. Lett. Phys. Lett. A. B. IMPACT: The relevance of a project trying to define string theory in non-critical dimensions is hard to underestimate. The excitement of the superstring theory when it first entered in a serious way as a ``theory of everything'' was tremendous. Not much of this excitement is left today and many people (non-string theorists in particular) talk ironically about a ``theory of nothing''. While it might not be so bad, it is clear that a relation to the world as we observe it has not yet been established. While this does not disprove the superstring approach it is clear that it calls for alternative approaches to be taken seriously. At the present stage diversity should be encouraged, and the present approach represents an alternative. Non-critical string theory in 4d has the potential of being a theory of quantum gravity, which is what we desperately are looking for. It does not present a unification of gravity with the rest of the forces, but who are we to dictate that there has to be such a unification. Again, experiments should hopefully show us whether or not there is such a unification. Considering the present state of superstring theory we consider the proposed project as highly relevant and very timely. Before moving to four-dimensional non-critical string theory it is important to establish that one has at all the possibility of studying non-critical strings in dimensions larger than one, and a major part of this proposal deals with this aspect. Here it shouldbe noted that the string representation of the 3d Ising model serves as an important

inspiration, as often emphasized by Polyakov. And this model indeed plays an important role in our research proposal. Brief survey of the worldwide researches made on the project topic, the competitiveness of the project, and achievements of the group (not more than 2 pages): New type of Matrix model is developed in a paper [1] which allows to consider integrable models on a random surface. This gives possibility to formulate new type of non-critical strings. Correct continuum limit of Chalker-Coddington network model was constructed in a paper [2]. In the paper [3] the complete conformal spectrum of the network model and the logarithmic corrections were derived by representation theoretical tools. Asymptotically, the scaling dimensions show a degeneracy growing exponentially with one of the quantum numbers. The physical relevance of these results is that any correlation function (of either bosonic or fermionic local fields) and the response to perturbations away from criticality can be written down. The CC-model was investigated in various works of A. Sedrakyan. In [54, 55] an exact action (Lagrangian) formulation of the original CC transfer matrix model as a field theory on the 2D lattice was given. First of all it was shown, that before taking into account the disorder over U(1) phases, the CC-model is equivalent to some inhomogeneous modification of the XX model in the background of U(1) fields. The basic element of this construction is the fermionized version of the standard R-operator of the XX model, where the transfer matrix is a staggered product of π /2 -rotated Roperators. The Landauer resistance of the CC-model based on the average over random U(1) phases of the direct product of the transfer matrix and its complex conjugate was analyzed. By use of the powerful technique of the algebraic Bethe ansatz the eigenstates of the model and the corresponding spectral parameters (from Bethe equations) were obtained. A network model of CC-type based on SU(2) group (instead of U(1)), developed in [57] in connection with 3d Ising model, can be applied to SQHE. The Bethe equations derived by A.S. are a good starting point for the further application of the non-linear integral equation approach developed by Klümper for the investigation of the critical behavior of the plateau transitions. Some preliminary work was done also on investigations of random network models. In the papers [7,8], a technique was developed for the formulation of the action of any random network model. This form gives an outlook for generalizations of the CC-model on random networks, to understand the deviations of the numerical values of the localization length obtained in recent calculations from the experiments. The new numerical technique [5] for the calculation of the Lyapunov exponent in the CC-model is

expected to be applicable to random networks and other problems like the Anderson localization. In the form-factor program the corresponding quantities in sine-Gordon model are calculated in [17], and it is established the quantum field equation of motion in [19]. The indecomposable representations of a quantum super-group at roots of unity and their fusions are analysed in [58]. References: List of important publications of the group members

1 J. Ambjørn, A. Sedrakyan, Matrix model for anizotropic Heisenberg chain interacting with 2d gravity, article in preparation. 2 I. Gruzberg, A. Sedrakyan, Continuum limit of Chalker-Godington network model, article in preparation. 3 B. Aufgebauer, M. Brockman, W. Nuding, A. Kl¨umper, A. Sedrakyan,- The complete conformal spectrum of a gl(2 j 1) invariant network model and logarithmic corrections, Arxiv: 1008.2653v1, Accepted in JSTAT 4 A. Sinner, A. Sedrakyan, K. Ziegler, Optical conductivity of graphene in the presence of random lattice deformations, Arxiv:1010.4510v1, submitted to Phys.Rev.B 5 M. Amado, A. Malyshev, A. Sedrakyan, F. Dominguez-Adame, Numerical study of the localization length critical index in a network model of plateau-plateau transitions in the quantum Hall effect, arxiv: 0912.4403v1, submitted to Phys. Rev. Lett. 6 Sh. Khachatryan and A. Sedrakyan, Characteristics of two-dimensional lattice models from a fermionic realization: Ising and XYZ models, Phys. Rev. B 80, 125128 (2009). 7 Sh. Khachatryan, A. Sedrakyan, P. Sorba, Network models: Action formulation, Nucl. Phys. B 825 (2010), 444. 8 Sh. Khachatryan, R. Schrader , A. Sedrakyan, Grassmann-Gaussian integrals and generalized star products, J. Phys. A: Math. Theor. 42, (2009) 304019. 9 V. Mkhitaryan, A. Sedrakyan, On the next to nearest neighbor and chiral-spin correlation fubctions in generalized XXX chain, Phys. Rev. B.77, (2007) 035111. 10 E. Diaz, A. Sedrakyan, D. Sedrakyan, and F. Domnguez-Adame, Absence of extended states in a ladder model of DNA, Phys. Rev. B 75, (2007) 014201. 11 M. Kohmoto and A. Sedrakyan, Hofshtadter problem on the honeycomb and triangular lattices: Bethe Ansatz solution, cond-mat/0603285, Phys.Rev. B 73 (2006) 235118. 12 A.Sedrakyan and F.Dom´ınguez-Adame, Comment on ”Sequencing-Independent delocalization in a DNA-like double chain with base pairing”, Phys. Rev. Lett 96 (2006) 059703. 13 J.Ambjorn, Sh.Khachatryan, A.Sedrakyan, Simplified tetrahedron equations: Fermionic realization, cond- .mat/0508148, Nucl. Phys. B 734 (2006) 287. 14 R.Schrader, H.Schulz-Balde and A.Sedrakyan, Perturbative test of single parameter scaling for 1D random media, mat-phys/0405019, Annales Henri Poincare 5 (2004) 1159118. 15 J.Ambjorn, Sh.Khachatryan, A.Sedrakyan, An integrable model with non-reducible three particle R-matrix, cond-mat/0403513, J.Physics A: Math.Gen. 37 (2004) 7397. 16 A.Sedrakyan, Action formulation of the network model of plateau-plateau transitions in quantum Hall effect, 17. H.Babujian,A.Foerster and M.Karowski ,J.Phys.A,v.41(2008) 18. H.Babujian,A.Foerster and M.Karowski, Theor.Math.Phys.v.155,N1,2008 19. H.Babujian,A.Foerster and M.Karowski,SIGMA,V.2,2006 20. H.Babujian,A.Foerster and M.Karowski, Nucl.Phys.B736:169-198,2006 21. T. Hakobyan, The ordering of energy levels for SU(n) symmetric antiferromagnetic chains, Nucl. Phys. B 699, 575-594 (2004)

22. G. Balint-Kurti, A. Bogdanov, A. Gevorkyan, Yu. Gorbachev, T. Hakobyan, G. Nyman, I. Shoshmina, and E. Stankova, Grid-Technology for Chemical Reactions Calculation, Lect. Notes Comp. Sc. 3516, 933-936 (2005) 23. T. Hakobyan, Energy-level ordering and ground-state quantum numbers for a frustrated two-leg spin-1/2 ladder, Phys. Rev. B 75, 214421 (2007) 24. T. Hakobyan, Antiferromagnetic ordering of energy levels for a spin ladder with fourspin cyclic exchange: Generalization of the Lieb-Mattis theorem, Phys. Rev. B 78, 012407 (2008) 25. T. Hakobyan, Re°ection Symmetry and energy-level ordering of frustrated ladder models, in Proc. of Int. Workshop on "Sypersymmetries and Quantum Symmetries", Dubna, July-August 2007, Ed. E. Ivanov and S. Fedoruk, pp. 383-386, JINR 2008 26. T. Hakobyan and A. Nersessian, Lobachevsky geometry of (super)conformal mechanics, Phys. Lett. A 373, 1001-1004 (2009) 27. T. Hakobyan, A. Nersessian and V. Yeghikyan, Cuboctahedric Higgs oscillator from the Calogero model, arXiv:0808.0430 28. T. Hakobyan, Energy-level ordering for generalized frustrated spin ladder models, in Proc. of XVII Int. Colloquium on "Group Theoretical Methods in Physics", Yerevan, August 2008, Accepted to Phys. Atom. Nuclei 29. T. Sedrakyan, Phys. Rev. B 69, 085109 (2004); 30. T. Sedrakyan, A. Ossipov, Phys. Rev. B 70, 214206 (2004). 31. T. A. Sedrakyan, Nucl. Phys. B 729 [FS], 526 (2005). 32. T. Sedrakyan, A. V. Chubukov, Arxiv:0901.1459v1 . 33. T. A. Sedrakyan, E. G. Mishchenko, and M. E. Raikh, Phys. Rev. Lett. 99, 036401 (2007). 34. T. A. Sedrakyan, M. E. Raikh, Phys. Rev. Lett. 100, 086808 (2008). 35. T. A. Sedrakyan, M. E. Raikh, Phys. Rev. Lett. 100, 106806 (2008). 36. T. A. Sedrakyan, E. G. Mishchenko, and M. E. Raikh, Phys. Rev. Lett. 99, 206405 (2007). 37. T. A. Sedrakyan, M. E. Raikh, Phys. Rev. B 77, 115353 (2008). 38. T. A. Sedrakyan, E. G. Mishchenko, and M. E. Raikh, Phys. Rev. B 73, 245325 (2006). 39. T. A. Sedrakyan, E. G. Mishchenko, and M. E. Raikh, Phys. Rev. B 74, 235423 (2006). 40. V. V. Mkhitaryan, T. A. Sedrakyan, Annales Henri Poincare 7, (2006) 1579-1590. 41. V.V. Mkhitaryan and M.E. Raikh, Phys. Rev. B 77, 195329 (2008), "Supergap anomalies in cotunneling between N-S and between S-S leads via a small quantum dot". 42. V.V. Mkhitaryan and M.E. Raikh, Phys. Rev. B 77, 245428 (2008), "2D skew scattering in the vicinity and away from resonant scattering condition". 43. V.V. Mkhitaryan and M.E. Raikh, Phys. Rev. B 78, 195409 (2008), "Disorder-induced tail states in a gapped bilayer graphene". 44. V. V. Mkhitaryan, Y. Fang, J. Gerton, E. G. Mishchenko, and M. E. Raikh, Phys. Rev. Lett. 101, 256401 (2008), "Scattering of plasmons at the inter- section of two metallic nanotubes: Im- plications for tunnelling". 45. V.V. Mkhitaryan and M.E. Raikh, ( Accepted in Phys. Rev. B), "Quantum site percolation on triangular lattice and the integer quantum Hall e®ect". 12 46. A.Melikyan, H.Minassian, On Surface Plasmon Damping in Metallic Nanoparticles. Applied Physics B, Lasers and Optics, 78(3-4), pp. 455- 457(2004). 47. T.Makaryan, A.Melikyan, H.Minassian, Surface Plasmon Frequency Spectrum in a System of Two Spherical Dielectric Coated Metallic Nanoparticles. Acta Physica Polonica A, v. 112, n5, pp.1031-1035 (2007). 48. T.Makaryan, A.Melikyan, H.Minassian, Impact of Interface on Plasmon Frequencies of Metallic Nanosphere. Proceedings of International Conference on "Laser Physics 2007", Ashtarak. 49. A. Melikyan and H. Minassian, Calculation of Longitudinal Surface Plasmon Frequencies in Noble Metal Nanorods. Chem. Phys. Lett., v. 452, pp. 139-143 (2008). 50. M. Chergui, A. Melikyan and H. Minassian. Calculation of Surface Plasmon Frequencies of Two, Three and Four Strongly Interacting Nanospheres. Journal of Physical Chemistry C 2009 (in press).

51. K.Madoyan, A.Melikyan, H.Minassian. Plasma Oscillation Frequencies of a System of Two Nearly Touching Nanospheres. Submitted to the Chemical 52.T.Makaryan, Influence of interface on surface plasmon frequencies of metallic nanosphere. Physica E: Low-dimensional Systems and Nanostructures V43, Issue 1, pp. 134-137 (2010). 53. T. Makaryan, K. Madoyan, A. Melikyan and H. Minassian, "Theoretical study of surface plasmon frequencies in a system of two coupled spheres and comparison with experimental data," Nanophotonics III, David L. Andrews, Jean-Michel Nunzi, Andreas Ostendorf, Editors, Proc. SPIE 7212, 77121I (2010). 54. A. Sedrakyan, Phys. Rev. B 68, 235329 (2003). 55. A. Sedrakyan, Nucl. Phys. B 554, 514 (1999). 56.

A.

G.

Sedrakyan, `Integrable chain models with staggered R-matrices', in

Statistical Field Theories , (A. Cappelli and G. Mussardo, eds.) 67, cond-mat/0112077. 57.A. Kavalov, A. Sedrakyan, B 285, (1987) 264. 58. D. Karakhanyan, Sh. Khachatryan, J. Phys. A: Math. Theor .42 (2009) 375205 (28pp). 59. H. Babujian, A. Foerster, M. Karowski-Nucl.Phys.B825:396-425,2010. 60. S. Derkachov, D. Karakhanyan, R. Kirschner, Nucl.Phys.B785:263-285, 2007. 61. D. Karakhanyan, Sh. Khachatryan, Nuclear Physics B 808 [FS] (2009) 525–545.

Personnel Commitments (chart, total number of project participants, responsibilities of each). The list of researchers: 1

Ara Sedrakyan – responsible for Task 1-Task 2

2

Hrachia Babujyan- responsible for Task 3

3

David Karakhanyan- responsible for Task 6

4

Shahane Khachatryan-responsible for Task 2a, 2c and partly Task 6

5

Mkhitar Mirumyan-responsible for Task 2a, 2c

6

Ara Martirosyan- responsible for Task 1c-1d

7

Tigran Hakobyan—responsible for Task.5

8

Tigran Sedrakyan-responsible for Task 1

9

Vagharsh Mkhitaryan-reponsible for Task 5

10

Hayk Minassyan-responsible for Task 4

11

Armen Melikyan- responsible for Task 4

12

K. Madoyan-aspirant- responsible partly for Task 4

Equipment Equipment description

Two computers

Cost (US $)

1700 US$

Total

Подробное описание заказываемого оборудования, указание сайта фирмы.

Materials Materials description

Cost (US $)

Direct cost description

Cost (US $)

Other Direct Costs

Two desks Two chairs

200 200

Travel costs (US $) CIS travel

International travel

6

Total

3000

Technical Approach and Methodology The most challenging problem in theoretical high energy physics is the unification of gauge theories (electroweak theory and quantum chromodynamics) with the theory of gravitation. The most prominent feature of the new theory would be the construction of a quantum theory of gravity and the understanding of confinement in the theory of strong interactions.

At present the only known candidate which can possibly unify all the types of fundamental interactions and solve the problems of gravity and confinement is the superstring theory. This theory was originally proposed in the early 1970's as a model describing hadrons, the dynamics of which is governed by the strong interactions. Shortly after that it was realized that in its simplest form the string theory could not describe strong interactions and, in fact, it was much more suitable as a model that contains quantum gravity. This idea obtained significant impetus in the middle of the 1980's after discovery of anomaly cancelation and the emergence of internally consistent perturbative superstring theories.

In parallel to this development there were extensive studies of string theory in the so-called Polyakov's approach which is based on the Feynman quantization scheme. The hope was to go away from critical dimensions (26 for bosonic strings and 10 for

superstrings) where quantization can be performed while preserving all the symmetries of the model. One essential ingredient of this approach is the study conformal field theories on two-dimensional random surfaces, i.e.\ the combined study of the dynamics of the conformal field theories and the dynamics of the random surfaces. These theories are non-critical string theories and apart from being models of string theory and of two-dimensional quantum gravity coupled to matter they also reveal amazing new aspects of conformal field theory. One of the beautiful aspects of non-critical string theory is that it allows an explicit (lattice) regularization of the field theory via the use of so-called dynamical triangulations, and that this regularization turned into a virtue via the use of random matrix theory which allows one solve analytically the regularized theory and then take the continuum limit. Whenever results can be compared with continuum calculations using the machinery of conformal field theory the results agree.

However the theory of non-critical strings is successful only

for effective

dimensions (or, in other words, central charge of matter fields) less or equal to one. The problem of the conformal anomaly can be solved only for these values of the central charge. Therefore, quantization of strings in ``physical'' dimensions (d=2,3,4) is still an open and important problem; it is especially attractive to try to use some kind of string picture in the theory of strong interaction (where it originated back in the seventies). All evidence suggests that there should exist an effective string theory of strong interactions because of confinement: due to the nontrivial structure of the ground state of non-Abelian gauge theories, the gauge fields form a flux tube between quarks, which can be approximated by strings. Moreover, the string picture naturally appears in the strong coupling limit of nonAbelian gauge theories on a lattice, where partition function can be represented as a functional integral over random surfaces with some additional structure on them.

A clear example of such representation is provided by the 3d Ising model (3DIM), which is the simplest gauge theory in 3d with the group $Z_2$. One can formulate it as a model of 2d random surfaces imbedded into three dimensional Euclidean space with fermionic degrees living on the random surface. On the other hand it can be formulated as an ordinary statistical model with ``field degrees of freedom'' living on the lattice sites and at the critical point it is believed to correspond to a non-trivial continuum field theory. The theory cannot be solved analytically, but critical indices etc can be calculated with standard approximation techniques. This fact that there exists such a continuum quantum field theory and that it has a fermionic random surface representation indicates the possibility of quantization of strings in 3d (a point of view which have been advocated repeatedly by Polyakov) and the study of this problem constitutes an important part of the research proposal.

One of the possible ways to revitalize Polyakov's approach to non-critical string

theory is to consideration lattice regularized integrable models with interactions on random surfaces and correctly analyze their continuum limit. In the vicinity of the critical point integrable models with interaction are also conformally invariant and therefore their behavior on random surfaces will extend our understanding both of strings and of conformal theories. Moreover, even when the models have a mass gap the interaction of excitations with the surface fluctuation (with 2d gravity) may create new, non-trivial quantum critical points. Our motivation is that this approach might present a new mechanism to cross the d=1 barrier since the problem in the older models was that the interaction between matter and the surface became too strong and thus led to a phase transition where the random surface degenerated into so-called branched polymers. A non-trivial matter interaction or a mass gap might cure this problem and the study of interacting integrable models on random surfaces constitutes a major part of this proposal.

We see two directions of research. First one can try to generalize the Lagrangian formulation of integrable models such that one can implement the integrable models on any surface and then finally analyze the continuum limit of the combined system of matter and surface fluctuations near the critical point. The algebraic Bethe ansatz (ABA) (or quantum inverse scattering method) is a Hamiltonian approach which falls within this line of attack. Investigations in this direction have already been started \footnote{A. Sedrakyan, Phys.Rev.B. 68, 235329 (2003), and in contribution to the proceedings of Advanced NATO Workshop on Statistical Field Theories, Editors: A.Capelli, G.Mussardo, Como, June 18-23, 2001.}. However, the main aspects like the role of the Yang- Baxter equation for the models on random surfaces and the technique of summation over random surfaces need to be elaborated upon. In order to proceed one can try to formulate a new type of random matrix model, where the kinetic (quadratic) part of the matrix action depends on the R-matrix of the integrable model under consideration. We consider this line of research as potential very promising, based on some preliminary (unpublished) calculations. Random matrices have been used not only in the non-critical string theory as described above, but also in a wide variety of physical systems including quantum chromodynamics, complex nuclei, chaotic systems and disordered mesoscopic conductors. The theory deals with the statistical properties of large matrices with randomly distributed elements. The probability distribution of the matrix elements is taken as input, from which the correlation functions of eigenvalues and eigenvectors are derived as output. From the correlation functions one can obtain the physical properties of the system. Of particular interest for us is the way random matrix have been used in the string representation of $SU(N)$ gauge theories for large N proposed by t'Hooft. Representing the gauge field propagators in Feynman diagrams by double lines of opposite orientation, one

can easily see that we obtain an orientable surface. It can be shown that this surface can be identified with the so-called random Manhattan lattices. Then the Feynman diagram can be regarded as a partition function for some kind of a model of statistical physics. (The elaboration of these ideas is part of Task 1 of the research proposal.)

As mentioned above matrix models constitute a versatile tool. The second direction, and an independent motivation for the investigation and development of the new type of matrix model in Task 1 is its connection with a class of random network models which have important applications in solid state physics, in particular in the quantum Hall effect. The Chalker-Coddington network model is a phenomelogical model developed in order to describe the plateau-plateau transitions in the quantum Hall effect. The model shows critical behavior with the critical index being quite close to the corresponding experimental value. This network model has been investigated by Zirnbauer using so-called supersymmetric random matrix models. It has been shown, that the continuum limit of the Chalker-Coddington network model corresponds to a supersymmetric non-linear sigma-model with a topological term, formulated earlier by Pruisken et al. However, first of all analytical calculation of the critical exponent in the Chalker and Coddington network model still remains an open problem, even having at hand the supersymmetric non-linear sigma-model. Secondly, recent numerical investigations of the multicritical behavior in Chalker-coddington model\footnote{F.Evers,A.Mildenberger,A.D.MirlinPhys.Rev.Lett.101,116803(2008); H.Obuse,A.R.Subramaniam, A.Furusaki,I.A.Gruz\-berg,A.W.W.Ludwig-Phys.Rev.Lett.101,116802(2008).} indicate that sigma model approach to the plateau-plateau transitions is too simplified. Moreover, recent high-precision numerical studies \footnote{K.Slevin,T.Ohtsuki-Phys.Rev.B80,041304(2009); M.Amado,A.Malyshev,A.Sedrakyan,F.Dominguez-Adame-arxiv:0912.4403v1.} show that the localization index in flat CC model nevertheless is different from experimental value. Clearly there is a gap between the understanding of the physics and the technical capacities. In this project we intend to fill this gap by developing an alternative analytic technique by using the random matrix models related to 2d-gravity (developed in Task 1) as a (hopefully better) model of the randomness of the network. Some work already in progress and there are preliminary results \footnote{J. Ambjorn, A. Sedrakyan, article in preparation}.

Quite remarkable it appears \footnote{A. Sedrakyan, Nucl.Phys. B554 (1999) 514; and Contribution to the proceedings of Advanced NATO Workshop on Statistical Field Theories, Editors: A.Capelli, G.Mussardo, Como, June 18-23, 2001.} that the Chalker-Coddington network model is the $U(1)$-group version of the so called

sign-factor model of three dimensional Ising model in flat space\footnote{A. Kavalov, A. Sedrakyan, Nucl. Phys. B 285, (1987) 264-278.}, in this way bringing together all the aspects mentioned above in a quite unexpected way. Our preliminary investigations show that the extension of the Chalker-Coddington model which includes {\em randomness of the network} is necessary as a result result of a hidden reparametrization invariance of the model. This fact is a strong indication that the Chalker-Coddington model is related to non-critical string theory. It is necessary to carry out some numerical investigations of the localization length in Chalker-Coddington model on random networks in order to check the above statements. This investigation are forming our second main direction of investigation (and is described in Task 2), but as described, quite surprisingly it might be closely related to the non-critical string theory research we have advocated above.

We strongly believe that this link between string theory and condensed matter physics has a great potential leading to progress in quantum Hall physics as well as in the field of disordered conductors.