We show that the hydrodynamics of the perfect fluid admits a natural modification that incorporates a chiral gravitational anomaly (also known as a mixed gauge-gravity anomaly) alongside the chiral current anomaly. The modification introduces the gravitational and gauge Chern-Simons in a manner analogous to the Jackiw-Pi modification of gravity [1], and features an analog of the axion in fluid mechanics. As a result, flows of a spinless perfect fluid acquire spin through motion, with the spin being equal to the fluid helicity. Furthermore, spacetime curvature induces an analogue of the Mathisson-Papapetrou force and a geometric counterpart of the chiral magnetic effect. a geometric counterpart of the chiral magnetic effect.

We present a hydrodynamic framework derived from the action of a perfect fluid, modified by the hydrodynamic analog of Novikov's multivalued functional. This modification introduces spin degrees of freedom into the fluid. The structure closely resembles the Abelian version of the Wess-Zumino functional, commonly applied in field theories with chiral anomalies. The deformation incorporates the transport properties of Weyl fermions and exhibits the chiral anomaly in the case of a charged fluid. It is also consistent with Onsager's semiclassical quantization of circulation. Additionally, we discuss the hydrodynamic analog of instantons and related topological invariants.

The two-dimensional Gross-Neveu model is anticipated to undergo a crystalline phase transition at high baryon charge densities. This conclusion is drawn from the mean-field approximation, which closely resembles models of Peierls instability. We demonstrate that this transition indeed occurs when both the rank of the symmetry group and the dimension of the particle representation contributing to the baryon density are large (the large-N limit). We derive this result through the exact solution of the model, developing the large-N limit of the Bethe Ansatz. Our analytical construction of the large-N solution of the Bethe Ansatz equations aligns perfectly with the periodic (finite-gap) solution of the Korteweg-de Vries (KdV) of the mean-field analysis.

We show that the chiral anomaly of quantum field theories with Dirac fermions subject to an axial background field is an inherent property of kinematics of a perfect classical fluid. Celebrated Beltrami flows (stationary solutions of Euler equations with extensive helicity) exhibit the chiral anomaly equivalent to that known for Dirac fermions. A prominent effect of the chiral anomaly is the transport electric current at equilibrium. We show that it is also a property of Beltrami flows.

We argue that a close analog of the axial-current anomaly of quantum field theories with fermions occurs in the classical Euler fluid. The conservation of the axial current (closely related to the helicity of inviscid barotropic flow) is anomalously broken by the external electromagnetic field as ∂_{μ}j_{A}^{μ}=2E·B, similar to that of the axial current of a quantum field theory with Dirac fermions, such as QED.

We developed the spacetime-covariant Hamilton principle for barotropic flows of a perfect fluid in the external axial-vector potential conjugate to the helicity current. Such flows carry helicity, a chiral imbalance, controlled by the axial potential. The interest in such a setting is motivated by the recent observation that the axial-current anomaly of quantum field theories with Dirac fermions appears as a kinematic property of classical hydrodynamics. Especially interesting effects occur under the simultaneous actions of the electromagnetic field and the axial-vector potential. With the help of the Hamilton principle, we obtain the extension of the Euler equations by the axial potential and derive anomalies in the divergence of the axial and vector current. Our approach provides a hydrodynamic expression for vector and axial currents and lays down a platform for studying flows with a chiral imbalance and their anomalies.

We introduce and study a model of a logarithmic gas with inverse temperature β on an arbitrary smooth closed contour in the plane. This model generalizes Dyson’s gas (the β-ensemble) on the unit circle. We compute the non-vanishing terms of the large N expansion of the free energy (N is the number of particles) by iterating the ‘loop equation’ that is the Ward identity with respect to reparametrizations and dilatation of the contour. We show that the main contribution to the free energy is expressed through the spectral determinant of the Neumann jump operator associated with the contour, or equivalently through the Fredholm determinant of the Neumann–Poincare (double layer) operator. This result connects the statistical mechanics of the Dyson gas to the spectral geometry of the interior and exterior domains of the supporting contour.

We study flows of barotropic perfect fluid under the simultaneous action of the electromagnetic field and the axial-vector potential, the external field conjugate to the fluid helicity. We obtain the deformation of the Euler equation by the axial-vector potential and the deformations of various currents by two external fields. We show that the divergence of the vector and axial currents are controlled by the chiral anomaly known in quantum field theories with Dirac fermions. We obtain these results by extending the variational principle for barotropic flows of a perfect fluid by coupling with the external axial-vector potential.

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 β this system is equivalent to the eigenvalue version of certain random matrix models, where it is referred 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. Institute of Biochemical Physics, Kosygina str. 4, 119991 Moscow, Russia and ITEP, Bol. Cheremushkinskaya str. 25, 117259 Moscow, Russia James Franck Institute and Enrico Fermi Institute of the University of Chicago, 5640 S.Ellis Avenue, Chicago, IL 60637, USA and Landau Institute for Theoretical Physics, Moscow, Russia 1

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 show that 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 signlike 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 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 the odd viscosity of 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.

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.

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-Soulé curvature formula for the Quillen metric and the adiabatic curvature for the electromagnetic and geometric adiabatic transport of the integer Quantum Hall state. We then obtain the geometric part of the adiabatic phase in QHE, given by the Chern–Simons functional.

We develop quantum hydrodynamics of inner waves in the bulk of fractional quantum Hall states. We show that the optical absorption of inner waves is a sole effect of the gravitational anomaly. We obtain the formula for the `oscillator-strengthu0027, which is expressed in terms of static structure factor and does not explicitly depend on a model interaction potential.

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.

In Spring 2015, the Galileo Galilei Institute for Theoretical Physics hosted an eight-week Workshop on “Statistical Mechanics, Integrability and Combinatorics”. The Workshop addressed a series of questions in the realm of exactly solvable models of statistical mechanics, featuring numerous ties and overlaps with various problems in modern combinatorics, probability theory, and representation theory. Much recent progress in these areas exploits the underlying notion of quantum integrability. We report here on the scientific motivations and background for this activity and on its main outputs.

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.