Topological Vertex, String Amplitudes and Spectral Functions of Hyperbolic Geometry

We discuss the homological aspects of the connection between quantum string generating function and the formal power series associated to the dimensions of chains and homologies of suitable Lie algebras. Our analysis can be considered as a new straightforward application of the machinery of modular forms and spectral functions (with values in the congruence subgroup of $SL(2,{\mathbb Z})$) to the partition functions of Lagrangian branes, refined vertex and open string partition functions, represented by means of formal power series that encode Lie algebra properties. The common feature in our examples lies in the modular properties of the characters of certain representations of the pertinent affine Lie algebras and in the role of Selberg-type spectral functions of an hyperbolic three-geometry associated with $q$-series in the computation of the string amplitudes.Comment: Revised version. References added, results remain unchanged. arXiv admin note: text overlap with arXiv:hep-th/0701156, arXiv:1105.4571, arXiv:1206.0664 by other author

Periderm invasion contributes to epithelial formation in the teleost pharynx

The gnathostome pharyngeal cavity functions in food transport and respiration. In amniotes the mouth and nares are the only channels allowing direct contact between internal and external epithelia. In teleost fish, gill slits arise through opening of endodermal pouches and connect the pharynx to the exterior. Using transgenic zebrafish lines, cell tracing, live imaging and different markers, we investigated if pharyngeal openings enable epithelial invasion and how this modifies the pharyngeal epithelium. We conclude that in zebrafish the pharyngeal endoderm becomes overlain by cells with a peridermal phenotype. In a wave starting from pouch 2, peridermal cells from the outer skin layer invade the successive pouches until halfway their depth. Here the peridermal cells connect to a population of cells inside the pharyngeal cavity that express periderm markers, yet do not invade from outside. The latter population expands along the midline from anterior to posterior until the esophagus-gut boundary. Together, our results show a novel role for the periderm as an internal epithelium becomes adapted to function as an external surface.Agência financiadora Ghent University Research Fund - BOF24J2015001401 Cancer Prevention Research Institute of Texas - RR140077info:eu-repo/semantics/publishedVersio