Gross-Neveu Models, Nonlinear Dirac Equations, Surfaces and Strings

TL;DR

This paper explores the deep connections between integrable systems, geometric surface embeddings, and solutions in Gross-Neveu models, revealing how these mathematical structures underpin inhomogeneous phases and string theory mappings.

Contribution

It introduces a geometric perspective using the Weierstrass spinor representation to explain the integrable systems underlying Gross-Neveu models and their relation to classical string solutions.

Findings

Integrable hierarchies govern the condensate equations in Gross-Neveu models.

Time-dependent solutions relate to classical string solutions in AdS3.

The geometric approach may extend to higher-dimensional models.

Abstract

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 Schr\"odinger 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.