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We present a consistent scheme of quantization of chiral flows (flows with extensive vorticity) in ideal hydrodynamics in two dimensions. Chiral flows occur in rotating superfluid, rotating turbulence, and also in electronic systems in magnetic field in the regime of a fractional Hall effect. The quantization is based on a geometric relation of chiral flows to two-dimensional quantum gravity and is implemented by the gravitational anomaly. The effect of the gravitational anomaly changes the major property of classical hydrodynamics, the Helmholtz law: vortices are no longer frozen into the flow. Effects of quantization could be cast in the form of quantum stress. We show that the quantum stress is a generator of Virasoro algebra, the centrally extended algebra of holomorphic diffeomorphisms.

We present a consistent scheme of quantization of chiral flows (flows with extensive vorticity) in ideal hydrodynamics in two dimensions. Chiral flows occur in rotating superfluid, rotating turbulence and also in electronic systems in the magnetic field in the regime of a fractional Hall effect. The quantization is based on a geometric relation of chiral flows to two-dimensional quantum gravity and is implemented by the gravitational anomaly. The effect of the gravitational anomaly violates the major property of classical hydrodynamics, the Helmholtz law: vortices are no longer frozen into the flow. Effects of quantization could be cast in the form of quantum stress. We show that the quantum stress generates Virasoro algebra, the centrally extended algebra of holomorphic diffeomorphisms., Comment: Contribution for the JETP special issue in honor of I.M. Khalatnikov centenary

We develop the quantum hydrodynamics of inner waves in the bulk of fractional quantum Hall states. We show that the inelastic light scattering by inner waves is a sole effect of the gravitational anomaly. We obtain the formula for the oscillator-strength or mean energy of optical absorption expressed solely in terms of an independently measurable static structure factor. The formula does not explicitly depend on a model interaction potential., Comment: 6 pages, v.2 minor changes in introduction and display of equations

We show that a vortex matter, that is a dense assembly of vortices in an incompressible two-dimensional flow, such as a fast rotating superfluid or turbulent flows with sign-like eddies, exhibits (i) a boundary layer of vorticity (vorticity layer), and (ii) a nonlinear wave localized within the vorticity layer, the edge wave. Both are solely an effect of the topological nature of vortices. Both are lost if the vortex matter is approximated as a continuous vorticity patch. The edge wave is governed by the integrable Benjamin-Davis-Ono equation exhibiting solitons with a quantized total vorticity. Quantized solitons reveal the topological nature of the vortices through their dynamics. The edge wave and the vorticity layer are due to odd viscosity of the vortex matter. We also identify the dynamics with the action of the Virasoro-Bott group of diffeomorphisms of the circle, where odd viscosity parametrizes the central extension. Our edge wave is a hydrodynamic analog of the edge states of the fractional quantum Hall effect., Comment: 6 pages, 1 figure

There is a well known analogy between the Laughlin trial wave function for the fractional quantum Hall effect, and the Boltzmann factor for the two-dimensional one-component plasma. The latter requires analytic continuation beyond the finite geometry used in its derivation. We consider both disk and cylinder geometry, and focus attention on the exact and asymptotic features of the edge density. At the special coupling \Gamma := q^2/k_BT=2 the system is exactly solvable. In particular the k-point correlation can be written as a k \times k determinant, allowing the edge density to be computed to first order in \Gamma - 2. A double layer structure is found, which in turn implies an overshoot of the density as the edge of the leading support is approached from inside the plasma. Asymptotic analysis shows that the deviation from the leading order (step function) value is different for into the plasma than for outside. For general \Gamma, a Gaussian fluctuation formula is used to study the large deviation form of the density for N large but finite. This asymptotic form involves thermodynamic quantities which we independently study, and moreover an appropriate scaling gives the asymptotic decay of the limiting edge density outside of the plasma., Comment: 36 pages, 6 figures

We study the generating functional, the adiabatic curvature and the adiabatic phase for the integer quantum Hall effect (QHE) on a compact Riemann surface. For the generating functional we derive its asymptotic expansion for the large flux of the magnetic field, i.e., for the large degree $k$ of the positive Hermitian line bundle $L^k$. The expansion consists of the anomalous and exact terms. The anomalous terms are the leading terms of the expansion. This part is responsible for the quantization of the adiabatic transport coefficients in QHE. We then identify the non-local (anomalous) part of the expansion with the Quillen metric on the determinant line bundle, and the subleading exact part with the asymptotics of the regularized spectral determinant of the Laplacian for the line bundle $L^k$, at large $k$. Finally, we show how the generating functional of the integer QHE is related to the gauge and gravitational (2+1)d Chern-Simons functionals. We observe the relation between the Bismut-Gillet-Soul\'e curvature formula for the Quillen metric and the adiabatic curvature for the electromagnetic and geometric adiabatic transport of the integer Quantum Hall state. Then we relate the adiabatic phase in QHE to the eta invariant and show that the geometric part of the adiabatic phase is given by the Chern-Simons functional., Comment: 36 pages, v4: greatly expanded version, added: references, Sec. 1.1 and Appendix A with background material, examples in Sec. 2.3 and Sec. 4, Thm. 3 and Sec. 5 expanded with more details on the relation between the adiabatic phase, eta invariant and Chern-Simons functional. To appear in Commun. Math. Phys

We study quantum Hall states on surfaces with conical singularities. We show that the electronic fluid at the cone tip possesses an intrinsic angular momentum, which is due solely to the gravitational anomaly. We also show that quantum Hall states behave as conformal primaries near singular points, with a conformal dimension equal to the angular momentum. Finally, we argue that the gravitational anomaly and conformal dimension determine the fine structure of the electronic density at the conical point. The singularities emerge as quasiparticles with spin and exchange statistics arising from adiabatically braiding conical singularities. Thus, the gravitational anomaly, which appears as a finite size correction on smooth surfaces, dominates geometric transport on singular surfaces.

In Hele-Shaw flows at vanishing surface tension, the boundary of a viscous fluid develops cusp-like singularities. In recent papers [1, 2] we have showed that singularities trigger viscous shocks propagating through the viscous fluid. Here we show that the weak solution of the Hele-Shaw problem describing viscous shocks is equivalent to a semiclassical approximation of a special real solution of the Painleve I equation. We argue that the Painleve I equation provides an integrable deformation of the Hele-Shaw problem which describes flow passing through singularities. In this interpretation shocks appear as Stokes level-lines of the Painleve linear problem., A more detailed derivation is included

In [Can et al. 2016], quantum Hall states on singular surfaces were shown to possess an emergent conformal symmetry. In this paper, we develop this idea further and flesh out details on the emergent conformal symmetry in holomorphic adiabatic states, which we define in the paper. We highlight the connection between the universal features of geometric transport of quantum Hall states and holomorphic dimension of primary fields in conformal field theory. In parallel we compute the universal finite-size corrections to the free energy of a critical system on a hyperbolic sphere with conical and cusp singularities, thus extending the result of Cardy and Peschel for critical systems on a flat cone [Cardy, Peschel 1988], and the known results for critical systems on polyhedra and flat branched Riemann surfaces., Comment: v1: 55 pages, 2 figures, v2: minor edits - invited contribution to a special issue of J. Phys. A for John Cardy's 70th birthday

Hele-Shaw flow at vanishing surface tension is ill-defined. In finite time, the flow develops cusp-like singularities. We show that the ill-defined problem admits a weak {\it dispersive} solution when singularities give rise to a graph of shock waves propagating in the viscous fluid. The graph of shocks grows and branches. Velocity and pressure jump across the shock. We formulate a few simple physical principles which single out the dispersive solution and interpret shocks as lines of decompressed fluid. We also formulate the dispersive weak solution in algebro-geometrical terms as an evolution of the Krichever-Boutroux complex curve. We study in detail the most generic (2,3) cusp singularity, which gives rise to an elementary branching event. This solution is self-similar and expressed in terms of elliptic functions., Comment: 24 pages, 11 figures; references added; figures changed

We argue that in addition to the Hall conductance and the nondissipative component of the viscous tensor, there exists a third independent transport coefficient, which is precisely quantized. It takes constant values along quantum Hall plateaus. We show that the new coefficient is the Chern number of a vector bundle over moduli space of surfaces of genus 2 or higher and therefore cannot change continuously along the plateau. As such, it does not transpire on a sphere or a torus. In the linear response theory, this coefficient determines intensive forces exerted on electronic fluid by adiabatic deformations of geometry and represents the effect of the gravitational anomaly. We also present the method of computing the transport coefficients for quantum Hall states., Comment: 6 pages, discussion of angular momentum formulas in sec. 7 is amended

Bubbles of inviscid fluid surrounded by a viscous fluid in a Hele-Shaw cell can merge and break-off. During the process of break-off, a thinning neck pinches off to a universal self-similar singularity. We describe this process and reveal its integrable structure: it is a solution of the dispersionless limit of the AKNS hierarchy. The singular break-off patterns are universal, not sensitive to details of the process and can be seen experimentally. We briefly discuss the dispersive regularization of the Hele-Shaw problem and the emergence of the Painlev\'e II equation at the break-off., Comment: 27 pages, 9 figures; typo corrected

We show that universal transport coefficients of the fractional quantum Hall effect (FQHE) can be understood as a response to variations of spatial geometry. Some transport properties are essentially governed by the gravitational anomaly. We develop a general method to compute correlation functions of FQH states in a curved space, where local transformation properties of these states are examined through local geometric variations. We introduce the notion of a generating functional and relate it to geometric invariant functionals recently studied in geometry. We develop two complementary methods to study the geometry of the FQHE. One method is based on iterating a Ward identity, while the other is based on a field theoretical formulation of the FQHE through a path integral formalism., v1: 60+21 pages; v2: 63 pages, 8 appendices, minor changes to improve clarity and format, fix typos, and expand references; v3: 65 pages, 8 appendices, revised and expanded discussions in the introduction and sections 9-11, major revisions to appendix E, minor changes in notation to improve clarity, and further expanded references

In general or normal random matrix ensembles, the support of eigenvalues of large size matrices is a planar domain (or several domains) with a sharp boundary. This domain evolves under a change of parameters of the potential and of the size of matrices. The boundary of the support of eigenvalues is a real section of a complex curve. Algebro-geometrical properties of this curve encode physical properties of random matrix ensembles. This curve can be treated as a limit of a spectral curve which is canonically defined for models of finite matrices. We interpret the evolution of the eigenvalue distribution as a growth problem, and describe the growth in terms of evolution of the spectral curve. We discuss algebro-geometrical properties of the spectral curve and describe the wave functions (normalized characteristic polynomials) in terms of differentials on the curve. General formulae and emergence of the spectral curve are illustrated by three meaningful examples., Comment: 44 pages, 14 figures; contains the first part of the original file. The second part will be submitted separately

We study how the solution of the two-dimensional Dirichlet boundary problem for smooth simply connected domains depends upon variations of the data of the problem. We show that the Hadamard formula for the variation of the Dirichlet Green function under deformations of the domain reveals an integrable structure. The independent variables corresponding to the infinite set of commuting flows are identified with harmonic moments of the domain. The solution to the Dirichlet boundary problem is expressed through the tau-function of the dispersionless Toda hierarchy. We also discuss a degenerate case of the Dirichlet problem on the plane with a gap. In this case the tau-function is identical to the partition function of the planar large $N$ limit of the Hermitean one-matrix model., Comment: 25 pages, 2 figures, LaTeX

A distinguishing feature of fractional quantum Hall (FQH) states is a singular behavior of equilibrium densities at boundaries. In contrast to states at integer filling fraction, such quantum liquids posses an additional dipole moment localized near edges. It enters observable quantities such as universal dispersion of edge states and Lorentz shear stress. For a Laughlin state, this behavior is seen as a peak, or overshoot, in the single particle density near the edge, reflecting a general tendency of electrons in FQH states to cluster near edges. We compute the singular edge behavior of the one particle density by a perturbative expansion carried out around a completely filled Landau level. This correction is shown to fully capture the dipole moment and the major features of the overshoot observed numerically. Furthermore, it exhibits the Stokes phenomenon with the Stokes line at the boundary of the droplet, decaying like a Gaussian inside and outside the liquid with different decay lengths. In the limit of vanishing magnetic length the shape the overshoot is a singular double layer with a capacity that is a universal function of the filling fraction. Finally, we derive the edge dipole moment of Pfaffian FQH states. The result suggests an explicit connection between the magnitude of the dipole moment and the bulk odd viscosity., Comment: v1: 5 pages, 2 figures; v2: 7 pages, 2 figures, includes an expanded discussion of the singular double layer, and two new sections discussing the edge dipole moment of Pfaffian states. Otherwise, only minor changes to improve clarity

We develop a collective field theory for fractional quantum Hall (FQH) states. We show that in the leading approximation for a large number of particles, the properties of Laughlin states are captured by a Gaussian free field theory with a (filling fraction dependent) background charge. Gradient corrections to the Gaussian field theory arise from ultraviolet regularization. They are the origin of the gravitational anomaly and are described by the Liouville theory of quantum gravity. The field theory simplifies the computation of correlation functions in FQH states and makes manifest the effect of quantum anomalies., Comment: v1: 20 pages; v2: 6 pages, considerably revised and rewritten for the sake of clarity and brevity, v3: 7 pages, updated to reflect the published version which includes a discussion of the effects spin

We discuss the origin of the associativity (WDVV) equations in the context of quasiclassical or Whitham hierarchies. The associativity equations are shown to be encoded in the dispersionless limit of the Hirota equations for KP and Toda hierarchies. We show, therefore, that any tau-function of dispersionless KP or Toda hierarchy provides a solution to associativity equations. In general, they depend on infinitely many variables. We also discuss the particular solution to the dispersionless Toda hierarchy that describes conformal mappings and construct a family of new solutions to the WDVV equations depending on finite number of variables., 16 pages, LaTeX

We show that conformal maps of simply connected domains with an analytic boundary to a unit disk have an intimate relation to the dispersionless 2D Toda integrable hierarchy. The maps are determined by a particular solution to the hierarchy singled out by the conditions known as “string equations”. The same hierarchy locally solves the 2D inverse potential problem, i.e., reconstruction of the domain out of a set of its harmonic moments. This is the same solution which is known to describe 2D gravity coupled to c= matter. We also introduce a concept of the τ-function for analytic curves.

We present asymptotically exact wave functions of an incommensurate Harper equation---one-dimensional Schr\"odinger equation of one particle on a lattice in a cosine potential. The wave functions can be written as an infinite product of string polynomials. The roots of these polynomials are solutions of Bethe equations. They are classified according to the string hypothesis. The string hypothesis gives asymptotically exact values of roots and reveals the hierarchical structure of the spectrum of the Harper equation.

We present numerical evidence that solutions of the Bethe Ansatz equations for a Bloch particle in an incommensurate magnetic field (Azbel-Hofstadter or AH model), consist of complexes-"strings". String solutions are well-known from integrable field theories. They become asymptotically exact in the thermodynamic limit. The string solutions for the AH model are exact in the incommensurate limit, where the flux through the unit cell is an irrational number in units of the elementary flux quantum. We introduce the notion of the integral spectral flow and conjecture a hierarchical tree for the problem. The hierarchical tree describes the topology of the singular continuous spectrum of the problem. We show that the string content of a state is determined uniquely by the rate of the spectral flow (Hall conductance) along the tree. We identify the Hall conductances with the set of Takahashi-Suzuki numbers (the set of dimensions of the irreducible representations of $U_q(sl_2)$ with definite parity). In this paper we consider the approximation of noninteracting strings. It provides the gap distribution function, the mean scaling dimension for the bandwidths and gives a very good approximation for some wave functions which even captures their multifractal properties. However, it misses the multifractal character of the spectrum., Comment: revtex, 30 pages, 6 Figs, 8 postscript files are enclosed, important references are added

We study algebro-geometric (finite-gap) and elliptic solutions of fully discretized KP or 2D Toda equations. In bilinear form they are Hirota's difference equation for $\tau$-functions. Starting from a given algebraic curve, we express the $\tau$-function and the Baker-Akhiezer function in terms of the Riemann theta function. We show that the elliptic solutions, when the $\tau$-function is an elliptic polynomial, form a subclass of the general algebro-geometric solutions. We construct the algebraic curves of the elliptic solutions. The evolution of zeros of the elliptic solutions is governed by the discrete time generalization of the Ruijsenaars-Schneider many body system. The zeros obey equations which have the form of nested Bethe-Ansatz equations, known from integrable quantum field theories. We discuss the Lax representation and the action-angle-type variables for the many body system. We also discuss elliptic solutions to discrete analogues of KdV, sine-Gordon and 1D Toda equations and describe the loci of the zeros., Comment: 22 pages, Latex with emlines2.sty

Functional relation for commuting quantum transfer matrices of quantum integrable models is identified with classical Hirota's bilinear difference equation. This equation is equivalent to the completely discretized classical 2D Toda lattice with open boundaries. The standard objects of quantum integrable models are identified with elements of classical nonlinear integrable difference equation. In particular, elliptic solutions of Hirota's equation give complete set of eigenvalues of the quantum transfer matrices. Eigenvalues of Baxter's $Q$-operator are solutions to the auxiliary linear problems for classical Hirota's equation. The elliptic solutions relevant to Bethe ansatz are studied. The nested Bethe ansatz equations for $A_{k-1}$-type models appear as discrete time equations of motions for zeros of classical $\tau$-functions and Baker-Akhiezer functions. Determinant representations of the general solution to bilinear discrete Hirota's equation and a new determinant formula for eigenvalues of the quantum transfer matrices are obtained.

In this paper we propose a comprehensive framework for the quantum hydrodynamics of the Fractional Quantum Hall (FQH) states. We suggest that the electronic fluid in the FQH regime could be phenomenologically described by the quantized hydrodynamics of vortices in an incompressible rotating liquid. We demonstrate that such hydrodynamics captures all major features of FQH states including the subtle effect of Lorentz shear stress. We present a consistent quantization of hydrodynamics of an incompressible fluid providing a powerful framework to study FQHE and superfluid. We obtain the quantum hydrodynamics of the vortex flow by quantizing the Kirchhoff equations for vortex dynamics., Comment: The paper is written for the special issue of JETP dedicated to Anatoly Larkin

Turbulent flows of incompressible liquid in two dimensions are comprised of dense systems of vortices. Such system of vortices can be treated as a fluid and itself could be described in terms of hydrodynamics. We develop the hydrodynamics of the vortex fluid. This hydrodynamics captures characteristics of fluid flows averaged over fast circulations in the inter-vortex space. The hydrodynamics of the vortex fluid features the anomalous stress absent in Euler's hydrodynamics. The anomalous stress yields a number of interesting effects. Some of them are: a deflection of stream lines, a correction to the Bernoulli law, accumulation of vortices in regions with high curvature in the curved space. The origin of the anomalous stresses is a divergence of inter-vortex interactions at the micro-scale which manifest at the macro-scale. We obtain the hydrodynamics of the vortex fluid from the Kirchhoff equations for dynamics of point-like vortices, Comment: 6 pages

We argue that dynamics of gapless Fractional Quantum Hall Edge states is essentially non-linear and that it features fractionally quantized solitons propagating along the edge. Observation of solitons would be a direct evidence of fractional charges. We show that the non-linear dynamics of the Laughlin's FQH state is governed by the quantum Benjamin-Ono equation. Non-linear dynamics of gapless edge states is determined by gapped modes in the bulk of FQH liquid. The dispersion of edge modes is traced to the double boundary layer of FQH states., 6 pages, 2 Figures

We present the exact and explicit solution of the principle chiral field model in two dimensions for infinitely large rank group. In particular we show that in the large-[ital N] limit the spectrum of the theory does not contain pointlike particles. The energy of the ground state as a function of external Noether'' field and the beta function are explicitly found. The nonperturbative threshold behavior near the mass gap [ital m] is [ital f]([ital h])[similar to]([ital h][minus][ital m])/ln([ital h][minus][ital m]), exhibiting a similarity with the 1D bosonic string theory.

We present a theory of large shift Raman scattering in Mott insulators and show that inelastic light scattering provides information about electronic current algebra. We argue that the recent experiment where a new excitation below the optical absorption threshold was observed in crossed polarizations gives evidence of anomalous terms in the current alegebra. We show that it suggests there exists an exciton bound state with a topological magnetic excitation with odd parity with respect to a spatial reflection.

We present a new approach to the problem of Bloch electrons in magnetic field, by making explicit a natural relation between magnetic translations and the quantum group ${\mathit{U}}_{\mathit{q}}$(${\mathit{sl}}_{2}$). The approach allows us to express the spectrum and the Bloch function as solutions of the Bethe-ansatz equations typical for completely integrable quantum systems.

We present the Bethe ansatz solution for the problem of Bloch electrons in magnetic field described by the Harper’s equation [Formula: see text] We made explicit a natural relation between magnetic translations and the quantum group Uq(sl2) and express the midband spectrum and the Bloch function in terms of solutions of the Bethe ansatz equations. In this letter we present solution for the critical value λ=1.

When a classical device suddenly perturbs a degenerate Fermi gas, a semiclassical nonequilibrium Fermi state arises. Similar states appear when a charge is transferred into a Fermi gas through a localized contact. Semiclassical Fermi states are characterized by a Fermi energy or Fermi momentum that slowly depends on space and/or time. We show that the Fermi distribution of a semiclassical Fermi state has a universal nature. It is described by Airy functions regardless of the details of the perturbation.

In Hele-Shaw flows, a boundary of a viscous fluid develops unstable fingering patterns. At vanishing surface tension, fingers evolve to cusp-like singularities preventing a smooth flow. We show that the Hele-Shaw problem admits a weak solution where a singularity triggers viscous shocks. Shocks form a growing, branching tree of a line distribution of vorticity where pressure has a finite discontinuity. A condition that the flow remains curl-free at a macroscale uniquely determines the shock graph structure. We present a self-similar solution describing shocks emerging from a generic (2, 3)-cusp singularity—an elementary branching event of a branching shock graph.

Any smooth spatial disturbance of a degenerate Fermi gas inevitably becomes sharp. This phenomenon, called the gradient catastrophe, causes the breakdown of a Fermi sea to multiconnected components characterized by multiple Fermi points. We argue that the gradient catastrophe can be probed through a Fermi-edge singularity measurement. In the regime of the gradient catastrophe the Fermi-edge singularity problem becomes a nonequilibrium and nonstationary phenomenon. We show that the gradient catastrophe transforms the single-peaked Fermi-edge singularity of the tunneling (or absorption) spectrum to a sequence of multiple asymmetric singular resonances. An extension of the bosonic representation of the electronic operator to nonequilibrium states captures the singular behavior of the resonances.

We show that the gauge theory of the two-dimensional doped Mott insulator predicts an anomalous anisotropic negative contribution to the magnetoresistance, which dominates at low temperatures. This anomalous contribution to the magnetoresistivity falls rapidly with increasing temperature, so the total magnetoresistivity is positive at room temperatures. This effect is due to orbital motion and is highly anistropic with respect to the orientation of the magnetic field. We discuss possible applications of these results to the high-{ital T}{sub {ital c}} materials.

We present a new approach to the problem of Bloch electrons in magnetic field, by making explicit a natural relation between magnetic translations and the quantum group U9(sl2). The approach allows to express the spectrum and the Bloch function as solutions of the Bethe-Ansatz equations typical for completely integrable quantum systems

We present new nonlinear differential equations for spacetime correlation functions of Fermi gas in one spatial dimension. The correlation functions we consider describe non-stationary processes out of equilibrium. The equations we obtain are integrable equations. They generalize known nonlinear differential equations for correlation functions at equilibrium and provide vital tools to study non-equilibrium dynamics of electronic systems. The method we developed is based only on Wick's theorem and the hydrodynamic description of the Fermi gas. Differential equations appear directly in bilinear form., Comment: 10+ pages, appendix with some technical details is added

We develop a hydrodynamic description of the classical Calogero-Sutherland liquid: a Calogero-Sutherland model with an infinite number of particles and a non-vanishing density of particles. The hydrodynamic equations, being written for the density and velocity fields of the liquid, are shown to be a bidirectional analogue of Benjamin-Ono equation. The latter is known to describe internal waves of deep stratified fluids. We show that the bidirectional Benjamin-Ono equation appears as a real reduction of the modified KP hierarchy. We derive the Chiral Non-linear Equation which appears as a chiral reduction of the bidirectional equation. The conventional Benjamin-Ono equation is a degeneration of the Chiral Non-Linear Equation at large density. We construct multi-phase solutions of the bidirectional Benjamin-Ono equations and of the Chiral Non-Linear equations., Comment: 28 pages, 1 figure

It is shown that the parity-violated ground state of two-dimensional strongly correlated electronic systems (commensurate flux state) provides a physical realization of the anyon mechanism of superconductivity. The electromagnetic response of the commensurate flux state is calculated. It is found that the superconducting ground state is accompanied by the {ital restoration} of parity and time-reversal symmetry: the parity-violation signal is very small at low energies and momentum, and vanishes in the static limit as well as in the uniform limit.

We examine long wavelength fluctuations in two-dimensional magnetic systems with the symmetry group of a large rank N. The mean field solution is obtained and the existence of the parity-violating ground state is established. On the basis of the 1/N expansion, an effective gauge theory containing the Chern-Simons term is derived, which allows one to obtain a spectrum, spin and statistics of long wavelength excitations.

A semiclassical wave-packet propagating in a dissipationless Fermi gas inevitably enters a "gradient catastrophe" regime, where an initially smooth front develops large gradients and undergoes a dramatic shock wave phenomenon. The non-linear effects in electronic transport are due to the curvature of the electronic spectrum at the Fermi surface. They can be probed by a sudden switching of a local potential. In equilibrium, this process produces a large number of particle-hole pairs, a phenomenon closely related to the Orthogonality Catastrophe. We study a generalization of this phenomenon to the non-equilibrium regime and show how the Orthogonality Catastrophe cures the Gradient Catastrophe, providing a dispersive regularization mechanism. We show that a wave packet overturns and collapses into modulated oscillations with the wave vector determined by the height of the initial wave. The oscillations occupy a growing region extending forward with velocity proportional to the initial height of the packet. We derive a fundamental equation for the transition rates (MKP-equation) and solve it by means of the Whitham modulation theory., Comment: 5 pages, 1 figure, revtex4, prl

Using the Calogero model as an example, we show that the transport in interacting nondissipative electronic systems is essentially nonlinear and unstable. Nonlinear effects are due to the curvature of the electronic spectrum near the Fermi energy. As is typical for nonlinear systems, a propagating semiclassical wave packet develops a shock wave at a finite time. A wave packet collapses into oscillatory features which further evolve into regularly structured localized pulses carrying a fractionally quantized charge. The Calogero model can be used to describe fractional quantum Hall edge states. We discuss perspectives of observation of quantum shock waves and a direct measurement of the fractional charge in fractional quantum Hall edge states.

We consider critical curves -- conformally invariant curves that appear at critical points of two-dimensional statistical mechanical systems. We show how to describe these curves in terms of the Coulomb gas formalism of conformal field theory (CFT). We also provide links between this description and the stochastic (Schramm-) Loewner evolution (SLE). The connection appears in the long-time limit of stochastic evolution of various SLE observables related to CFT primary fields. We show how the multifractal spectrum of harmonic measure and other fractal characteristics of critical curves can be obtained., Published version

Preface.- Random Matrices and Number Theory. 1. Introduction. 2. 3. Characteristic polynomials of random unitary matrices 4. Other compact groups. 5. Families of L-functions and Symmetry. 6. Asymptotic expansions. References J.P. Keating.- 2D Quantum Gravity, Matrix Models and Graph Combinatorics. 1. Introduction. 2. Matrix models for 2D quantum gravity. The one-matrix model I: large N limit and the enumeration of planar graphs. 4. The trees behind the graphs. 5. The one-matrix model II: topological expansions and quantum gravity. 6. The combinatorics beyond matrix models: geodesic distance in planar graphs. 7. Planar graphs as spatial branching processes. 8. Conclusion P. Di Francesco.- Eigenvalue Dynamics, Follytons and Large N Limits of Matrices. References J. Arnlind, J. Hoppe.- Random Matrices and Supersymmetry in Disordered Systems. Supersymmetry method. 2. Wave functions fluctuations in a finite volume. Multifractality. Recent and possible future developments. Summary. Acknowledgements. References K.B. Efetov.- Hydrodynamics of Correlated Systems. 1. Introduction. 2. Instanton or rare fluctuation method. 3. Hydrodynamic approach. 4. Linearized hydrodynamics or bosonization. 5. EFP through an asymptotics of the solution. 6. Free fermions. 7. Calogero-Sutherland model. 8. Free fermions on the lattice. 9. Conclusion. Acknowledgements. Appendix: Hydrodynamic approach to non-Galilean invariant systems. Appendix: Exact results for EFP in some integrable models. References. A.G. Abanov.- QCD, Chiral Random Matrix Theory and Integrability. 1. Summary. 2. Introduction. 3. QCD. 4. The Dirac Spectrum in QCD. 5. Low Energy Limit of QCD. 6. Chiral RMT and the QCD Dirac Spectrum. 7. Integrability and the QCD Partition Function. 8. QCD at Finite Baryon Density. 9. Full QCD at Nonzero Chemical Potential. 10. Conclusions. Acknowledgements. References J.J. M. Verbaarschot.- Euclidan Random Matrices: Solved and Open Problems. 1. Introduction. 2. Basic definitions. 3. Physical motivations. 4. Field theory. 5. The simplest case. 6. Phonnos. References G. Parisi.- Matrix Models and Growth Processes. 1. Introduction. 2. Some ensembles of random matrices with complex eigenvalues. 3. Exact results at finite N. 4. Large N limit. 5. The matrix model as a growth problem. References A. Zabrodin.- Matrix Models and Topological Strings. 1. Introduction. 2. Matrix models. 3. Type B topological strings and matrix models. 4. Type A topological strings, Chern-Simons theory and matrix models M. Marino.- Matrix Models of Moduli Space. 1. Introduction. 2. Moduli Space of Riemann Surfaces and its Topology. 3. Quadratic Differentials and Fatgraphs. 4. The Penner model. 5. Penner Model and Matrix Gamma Function. 6. The Kontsevich Model. 7. Applications to String Theory. 8. Conclusions. References S. Mukhi.- Matrix Models and 2D String Theory. 1. Introduction. 2. An overview of string theory. 3. Strings in D-dimensional spacetime. 4. Discretized surfaces and 2D string theory. 5. An overview of observables. 6. Sample calculation: the disk one-point function. 7. Worldsheet description of matrix eigenvalues. 8. Further results. 9. Open problems. References E.J. Martinec.- Matrix Models as Conformal Field Theories. 1. Introduction and historical notes. 2. Hermitian matrix integral: saddle points and hyperellptic curves. 3. The hermitian matrix model as a chiral CFT. 4. Quasiclassical expansions: CFT on a hyperelliptic Riemann surface. 5. Generalization to chains of random matrices. References I.K. Kostov.

We discuss the 1/N expansion of the free energy of N logarithmically interacting charges in the plane in an external field. For some particular values of the inverse temperature beta this system is equivalent to the eigenvalue version of certain random matrix models, where it is refered to as the "Dyson gas" of eigenvalues. To find the free energy at large N and the structure of 1/N-corrections, we first use the effective action approach and then confirm the results by solving the loop equation. The results obtained give some new representations of the mathematical objects related to the Dirichlet boundary value problem, complex analysis and spectral geometry of exterior domains. They also suggest interesting links with bosonic field theory on Riemann surfaces, gravitational anomalies and topological field theories., 37 pages, 1 figure

We show that unstable fingering patterns of two-dimensional flows of viscous fluids with open boundary are described by a dispersionless limit of the Korteweg-de Vries hierarchy. In this framework, the fingering instability is linked to a known instability leading to regularized shock solutions for nonlinear waves, in dispersive media. The integrable structure of the flow suggests a dispersive regularization of the finite-time singularities.

Fractal geometry of critical curves appearing in 2D critical systems is characterized by their harmonic measure. For systems described by conformal field theories with central charge $c\leqslant 1$, scaling exponents of harmonic measure have been computed by B. Duplantier [Phys. Rev. Lett. {\bf 84}, 1363 (2000)] by relating the problem to boundary two-dimensional gravity. We present a simple argument that allows us to connect harmonic measure of critical curves to operators obtained by fusion of primary fields, and compute characteristics of fractal geometry by means of regular methods of conformal field theory. The method is not limited to theories with $c\leqslant 1$., Some more corrections

We show that singularities developed in the Hele-Shaw problem have a structure identical to shock waves in dissipativeless dispersive media. We propose an experimental setup where the cell is permeable to a nonviscous fluid and study continuation of the flow through singularities. We show that a singular flow in this nontraditional cell is described by the Whitham equations identical to Gurevich-Pitaevski solution for a regularization of shock waves in Korteveg-de Vriez equation. This solution describes regularization of singularities through creation of disconnected bubbles.

Collective field theory for the Calogero model represents particles with fractional statistics in terms of hydrodynamic modes---density and velocity fields. We show that the quantum hydrodynamics of this model can be written as a single evolution equation on a real holomorphic Bose field---the quantum integrable Benjamin-Ono equation. It renders tools of integrable systems to studies of nonlinear dynamics of 1D quantum liquids.