On F-Algebroids and Dubrovin's Duality

In this note we introduce the concept of F-algebroid, and give its elementary properties and some examples. We provide a description of the almost duality for Frobenius manifolds, introduced by Dubrovin, in terms of a composition of two anchor maps of a unique cotangent F-algebroid.Comment: 13 pages; v2 has small changes, it has improved exposition. Revised version to appear in: "Archivum Mathematicum

Fermions via spinor-valued one-forms

Spinor-valued one-forms (Rarita-Schwinger fields) are normally used in the context of supergravity, where they describe spin 3/2 particles (gravitinos). Indeed, when decomposed into irreducible representations of the Lorentz group such a field contains both a spin 1/2 and a spin 3/2 component, and the Rarita-Schwinger Lagrangian is designed to make only the spin 3/2 propagate. We point out that the opposite construction is also possible, and give a spinor-valued one-form field Lagrangian that describes a propagating spin 1/2 particle.Comment: 23 pages, no figure

The homotopy type of the topological cobordism category

We define a cobordism category of topological manifolds and prove that if $d \neq 4$ its classifying space is weakly equivalent to $\Omega^{\infty -1} MTTop(d)$, where $MTTop(d)$ is the Thom spectrum of the inverse of the canonical bundle over $BTop(d)$. We also give versions with tangential structures and boundary. The proof uses smoothing theory and excision in the tangential structure to reduce the statement to the computation of the homotopy type of smooth cobordism categories due to Galatius-Madsen-Tillman-Weiss.Comment: 61 pages, 9 figures. Minor correction

2D sigma models and differential Poisson algebras

We construct a two-dimensional topological sigma model whose target space is endowed with a Poisson algebra for differential forms. The model consists of an equal number of bosonic and fermionic fields of worldsheet form degrees zero and one. The action is built using exterior products and derivatives, without any reference to any worldsheet metric, and is of the covariant Hamiltonian form. The equations of motion define a universally Cartan integrable system. In addition to gauge symmetries, the model has one rigid nilpotent supersymmetry corresponding to the target space de Rham operator. The rigid and local symmetries of the action, respectively, are equivalent to the Poisson bracket being compatible with the de Rham operator and obeying graded Jacobi identities. We propose that perturbative quantization of the model yields a covariantized differential star product algebra of Kontsevich type. We comment on the resemblance to the topological A model.Comment: 20 page

PLASTIC POLLUTION’S AFFECT ON CORAL REEFS

Plastic pollution has been an increasing problem in the world. This is getting worse every year and it is having an impact in our world specifically in our oceans. A big thing in the ocean is coral reefs, coral reefs end up dying by diseases and debris caused from plastic pollution. Coral reefs are important in the lives of humans and fish so if we do nothing about it many will suffer the consequences. With coral reefs gone many fish suffer from change and do not adapt quick enough and later end up dying. Plastic pollution stays on top of shallow ocean water where coral reefs are normally located and tend to heat up the water which causes coral bleaching (Sigler, 2014) when coral “bleaches” it turns white and because very weak and diseases get into the reefs and could possibly kill it (Begter, 2014)