Gross-Neveu models, nonlinear Dirac equations, surfaces and strings

Recent studies of the thermodynamic phase diagrams of the Gross-Neveu model (GN2), and its chiral cousin, the NJL2 model, have shown that there are phases with inhomogeneous crystalline condensates. These (static) condensates can be found analytically because the relevant Hartree-Fock and gap equations can be reduced to the nonlinear Schrödinger equation, whose deformations are governed by the mKdV and AKNS integrable hierarchies, respectively. Recently, Thies et al. have shown that time-dependent Hartree-Fock solutions describing baryon scattering in the massless GN2 model satisfy the Sinh-Gordon equation, and can be mapped directly to classical string solutions in AdS3. Here we propose a geometric perspective for this result, based on the generalized Weierstrass spinor representation for the embedding of 2d surfaces into 3d spaces, which explains why these well-known integrable systems underlie these various Gross-Neveu gap equations, and why there should be a connection to classical string theory solutions. This geometric viewpoint may be useful for higher dimensional models, where the relevant integrable hierarchies include the Davey-Stewartson and Novikov-Veselov systems

Chern-Simons diffusion rate in strongly coupled N=4 SYM plasma in an external magnetic field

We calculate the Chern-Simons diffusion rate in a strongly coupled N=4 super Yang-Mills plasma in the presence of a constant external U(1) R magnetic flux via the holographic correspondence. Because of the strong interactions between the charged fields and non-Abelian gauge fields, the external Abelian magnetic field affects the thermal Yang-Mills dynamics and increases the diffusion rate, regardless of its strength. We obtain the analytic results for the Chern-Simons diffusion rate both in the weak and strong magnetic field limits. In the latter limit, we show that the diffusion rate scales as B×T2 and this can be understood as a result of a dynamical dimensional reduction

Resurgence and the Nekrasov-Shatashvili limit: connecting weak and strong coupling in the Mathieu and Lamé systems

Abstract: The Nekrasov-Shatashvili limit for the low-energy behavior of N=2 and N=2* supersymmetric SU(2) gauge theories is encoded in the spectrum of the Mathieu and Lamé equations, respectively. This correspondence is usually expressed via an all-orders Bohr-Sommerfeld relation, but this neglects non-perturbative effects, the nature of which is very different in the electric, magnetic and dyonic regions. In the gauge theory dyonic region the spectral expansions are divergent, and indeed are not Borel-summable, so they are more properly described by resurgent trans-series in which perturbative and non-perturbative effects are deeply entwined. In the gauge theory electric region the spectral expansions are convergent, but nevertheless there are non-perturbative effects due to poles in the expansion coefficients, and which we associate with worldline instantons. This provides a concrete analog of a phenomenon found recently by Drukker, Mariño and Putrov in the large N expansion of the ABJM matrix model, in which non-perturbative effects are related to complex space-time instantons. In this paper we study how these very different regimes arise from an exact WKB analysis, and join smoothly through the magnetic region. This approach also leads to a simple proof of a resurgence relation found recently by Dunne and Ünsal, showing that for these spectral systems all non-perturbative effects are subtly encoded in perturbation theory, and identifies this with the Picard-Fuchs equation for the quantized elliptic curve

Self-consistent crystalline condensate in chiral Gross-Neveu and bogoliubov-de gennes systems

We derive a new exact self-consistent crystalline condensate in the (1+1)-dimensional chiral Gross-Neveu model. This also yields a new exact crystalline solution for the one dimensional Bogoliubov-de Gennes equations and the Eilenberger equation of semiclassical superconductivity. We show that the functional gap equation can be reduced to a solvable nonlinear equation and discuss implications for the temperature-chemical potential phase diagram

Hydrodynamics, resurgence, and transasymptotics

The second order hydrodynamical description of a homogeneous conformal plasma that undergoes a boost-invariant expansion is given by a single nonlinear ordinary differential equation, whose resurgent asymptotic properties we study, developing further the recent work of Heller and Spalinski [Phys. Rev. Lett. 115, 072501 (2015)]. Resurgence clearly identifies the nonhydrodynamic modes that are exponentially suppressed at late times, analogous to the quasinormal modes in gravitational language, organizing these modes in terms of a trans-series expansion. These modes are analogs of instantons in semiclassical expansions, where the damping rate plays the role of the instanton action. We show that this system displays the generic features of resurgence, with explicit quantitative relations between the fluctuations about different orders of these nonhydrodynamic modes. The imaginary part of the trans-series parameter is identified with the Stokes constant, and the real part with the freedom associated with initial conditions

Scaling relation between pA and AA collisions

We compare the flowlike correlations in high multiplicity proton-nucleus (p+A) and nucleus-nucleus (A+A) collisions. At fixed multiplicity, the correlations in these two colliding systems are strikingly similar, although the system size is smaller in p+A. Based on an independent cluster model and a simple conformal scaling argument, where the ratio of the mean free path to the system size stays constant at fixed multiplicity, we argue that flow in p+A emerges as a collective response to the fluctuations in the position of clusters, just like in A+A collisions. With several physically motivated and parameter free rescalings of the recent LHC data, we show that this simple model captures the essential physics of elliptic and triangular flow in p+A collisions. We also explore the implications of the model for jet energy loss in p+A, and predict slightly larger transverse momentum broadening in p+A than in A+A at the same multiplicity

Twisted kink crystal in the chiral Gross-Neveu model

We present the detailed properties of a self-consistent crystalline chiral condensate in the massless chiral Gross-Neveu model. We show that a suitable ansatz for the Gorkov resolvent reduces the functional gap equation, for the inhomogeneous condensate, to a nonlinear Schrödinger equation, which is exactly soluble. The general crystalline solution includes as special cases all previously known real and complex condensate solutions to the gap equation. Furthermore, the associated Bogoliubov-de Gennes equation is also soluble with this inhomogeneous chiral condensate, and the exact spectral properties are derived. We find an all-orders expansion of the Ginzburg-Landau effective Lagrangian and show how the gap equation is solved order by order

Effective action for Einstein-Maxwell theory at order RF**4

We use a recently derived integral representation of the one-loop effective action in Einstein-Maxwell theory for an explicit calculation of the part of the effective action containing the information on the low energy limit of the five-point amplitudes involving one graviton, four photons and either a scalar or spinor loop. All available identities are used to get the result into a relatively compact form.Comment: 13 pages, no figure

N -Person Dynamic Stackelberg Difference Games with Open-Loop Information Pattern

Abstract. In this paper, extensions are presented for the open-loop Stackelberg equilibrium solution of n-person discrete-time affine-quadratic dynamic games of prespecified fixed duration, concerning the number of followers and the possibility of an algorithmic disintegration. The given results extend the current state of research which is defined by the results for one leader and one follower that are given in T. Başar and G. Olsde

Electric dipole moment induced by a QCD instanton in an external magnetic field

In the chiral magnetic effect, there is a competition between a strong magnetic field, which tends to project positively charged particles to have spin aligned along the magnetic field, and a chirality imbalance which may be produced locally by a topologically nontrivial gauge field such as an instanton. We study the properties of the Euclidean Dirac equation for a light fermion in the presence of both a constant Abelian magnetic field and an SU(2) instanton. In particular, we analyze the zero modes analytically in various limits, both on R4 and on the four-torus, in order to compare with recent lattice QCD results, and study the implications for the electric dipole moment