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 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.Comment: 27 pages, 1 figur

A novel widespread cryptic species and phylogeographic patterns within several giant clam species (Cardiidae: Tridacna) from the Indo-Pacific Ocean

Giant clams (genus Tridacna) are iconic coral reef animals of the Indian and Pacific Oceans, easily recognizable by their massive shells and vibrantly colored mantle tissue. Most Tridacna species are listed by CITES and the IUCN Redlist, as their populations have been extensively harvested and depleted in many regions. Here, we survey Tridacna crocea and Tridacna maxima from the eastern Indian and western Pacific Oceans for mitochondrial (COI and 16S) and nuclear (ITS) sequence variation and consolidate these data with previous published results using phylogenetic analyses. We find deep intraspecific differentiation within both T. crocea and T. maxima. In T. crocea we describe a previously undocumented phylogeographic division to the east of Cenderawasih Bay (northwest New Guinea), whereas for T. maxima the previously described, distinctive lineage of Cenderawasih Bay can be seen to also typify western Pacific populations. Furthermore, we find an undescribed, monophyletic group that is evolutionarily distinct from named Tridacna species at both mitochondrial and nuclear loci. This cryptic taxon is geographically widespread with a range extent that minimally includes much of the central Indo-Pacific region. Our results reinforce the emerging paradigm that cryptic species are common among marine invertebrates, even for conspicuous and culturally significant taxa. Additionally, our results add to identified locations of genetic differentiation across the central Indo-Pacific and highlight how phylogeographic patterns may differ even between closely related and co-distributed species