From conformal invariance towards dynamical symmetries of the collisionless Boltzmann equation

Dynamical symmetries of the collisionless Boltzmann transport equation, or Vlasov equation, but under the influence of an external driving force, are derived from non-standard representations of the $2D$ conformal algebra. In the case without external forces, the symmetry of the conformally invariant transport equation is first generalised by considering the particle momentum as an independent variables. This new conformal representation can be further extended to include an external force. The construction and possible physical applications are outlined.Comment: Latex2e, 18 pages, no figure

Non-local meta-conformal invariance, diffusion-limited erosion and the XXZ chain

Diffusion-limited erosion is a distinct universality class of fluctuating interfaces. Although its dynamical exponent $z=1$, none of the known variants of conformal invariance can act as its dynamical symmetry. In $d=1$ spatial dimensions, its infinite-dimensional dynamic symmetry is constructed and shown to be isomorphic to the direct sum of three loop-Virasoro algebras, with the maximal finite-dimensional sub-algebra $\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})$. The infinitesimal generators are spatially non-local and use the Riesz-Feller fractional derivative. Co-variant two-time response functions are derived and reproduce the exact solution of diffusion-limited erosion. The relationship with the terrace-step-kind model of vicinal surfaces and the integrable XXZ chain are discussed.Comment: Latex 2e, 28 pp, 4 figures (revised, with 2 new figures

Quantum critical transport, duality, and M-theory

We consider charge transport properties of 2+1 dimensional conformal field theories at non-zero temperature. For theories with only Abelian U(1) charges, we describe the action of particle-vortex duality on the hydrodynamic-to-collisionless crossover function: this leads to powerful functional constraints for self-dual theories. For the n=8 supersymmetric, SU(N) Yang-Mills theory at the conformal fixed point, exact hydrodynamic-to-collisionless crossover functions of the SO(8) R-currents can be obtained in the large N limit by applying the AdS/CFT correspondence to M-theory. In the gravity theory, fluctuating currents are mapped to fluctuating gauge fields in the background of a black hole in 3+1 dimensional anti-de Sitter space. The electromagnetic self-duality of the 3+1 dimensional theory implies that the correlators of the R-currents obey a functional constraint similar to that found from particle-vortex duality in 2+1 dimensional Abelian theories. Thus the 2+1 dimensional, superconformal Yang Mills theory obeys a "holographic self duality" in the large N limit, and perhaps more generally.Comment: 35 pages, 4 figures; (v2) New appendix on CFT2, corrected normalization of gauge field action, added ref

Dynamical symmetries and causality in non-equilibrium phase transitions

Dynamical symmetries are of considerable importance in elucidating the complex behaviour of strongly interacting systems with many degrees of freedom. Paradigmatic examples are cooperative phenomena as they arise in phase transitions, where conformal invariance has led to enormous progress in equilibrium phase transitions, especially in two dimensions. Non-equilibrium phase transitions can arise in much larger portions of the parameter space than equilibrium phase transitions. The state of the art of recent attempts to generalise conformal invariance to a new generic symmetry, taking into account the different scaling behaviour of space and time, will be reviewed. Particular attention will be given to the causality properties as they follow for co-variant $n$-point functions. These are important for the physical identification of n-point functions as responses or correlators.Comment: Latex2e, 26 pages, 1 figure. Final form, a new example added & typos correcte

The landscape of the Hubbard model

I present a pedagogical survey of a variety of quantum phases of the Hubbard model. The honeycomb lattice model has a conformal field theory connecting the semi-metal to the insulator with Neel order. States with fractionalized excitations are linked to the deconfined phases of gauge theories. I also consider the confining phases of such gauge theories, and show how Berry phases of monopoles induce valence bond solid order. The triangular lattice model can display a metal-insulator transition from a Fermi liquid to a deconfined spin liquid, and I describe the theory of this transition. The bilayer triangular lattice is used to illustrate another compressible metallic phase, the `fractionalized Fermi liquid'. I make numerous connections of these phases and critical points to the AdS/CFT correspondence. In particular, I argue that two recent holographic constructions connect respectively to the Fermi liquid and fractionalized Fermi liquid phases.Comment: 56 pages, 16 figures; TASI and Chandrasekhar lectures; (v3) expanded discussion of phases with Fermi surfaces; (v5) added section on Mott transition on the triangular lattic

Stable Clustering Ansatz, Consistency Relations and Gravity Dual of Large-Scale Structure

Gravitational clustering in the nonlinear regime remains poorly understood. Gravity dual of gravitational clustering has recently been proposed as a means to study the nonlinear regime. The stable clustering ansatz remains a key ingredient to our understanding of gravitational clustering in the highly nonlinear regime. We study certain aspects of violation of the stable clustering ansatz in the gravity dual of Large Scale Structure (LSS). We extend the recent studies of gravitational clustering using AdS gravity dual to take into account possible departure from the stable clustering ansatz and to arbitrary dimensions. Next, we extend the recently introduced consistency relations to arbitrary dimensions. We use the consistency relations to test the commonly used models of gravitational clustering including the halo models and hierarchical ans\"atze. In particular we establish a tower of consistency relations for the hierarchical amplitudes: $Q, R_a, R_b, S_a,S_b,S_c$ etc. as a functions of the scaled peculiar velocity $h$. We also study the variants of popular halo models in this context. In contrast to recent claims, none of these models, in their simplest incarnation, seem to satisfy the consistency relations in the soft limit.Comment: 21 pages, 4 figure

Advances in perturbative thermal field theory

The progress of the last decade in perturbative quantum field theory at high temperature and density made possible by the use of effective field theories and hard-thermal/dense-loop resummations in ultrarelativistic gauge theories is reviewed. The relevant methods are discussed in field theoretical models from simple scalar theories to non-Abelian gauge theories including gravity. In the simpler models, the aim is to give a pedagogical account of some of the relevant problems and their resolution, while in the more complicated but also more interesting models such as quantum chromodynamics, a summary of the results obtained so far are given together with references to a few most recent developments and open problems.Comment: 84 pages, 18 figues, review article submitted to Reports on Progress in Physics; v2, v3: minor additions and corrections, more reference

Scale Symmetry in the Universe

Scale symmetry is a fundamental symmetry of physics that seems however not to be fully realized in the universe. Here, we focus on the astronomical scales ruled by gravity, where scale symmetry holds and gives rise to a truly scale invariant distribution of matter, namely it gives rise to a fractal geometry. A suitable explanation of the features of the fractal cosmic mass distribution is provided by the nonlinear Poisson--Boltzmann--Emden equation. An alternative interpretation of this equation is connected with theories of quantum gravity. We study the fractal solutions of the equation and connect them with the statistical theory of random multiplicative cascades, which originated in the theory of fluid turbulence. The type of multifractal mass distributions so obtained agrees with results from the analysis of cosmological simulations and of observations of the galaxy distribution.Comment: 19 pages, 5 figure