Categorified Symplectic Geometry and the String Lie 2-Algebra

Multisymplectic geometry is a generalization of symplectic geometry suitable for n-dimensional field theories, in which the nondegenerate 2-form of symplectic geometry is replaced by a nondegenerate (n+1)-form. The case n = 2 is relevant to string theory: we call this 2-plectic geometry. Just as the Poisson bracket makes the smooth functions on a symplectic manifold into a Lie algebra, the observables associated to a 2-plectic manifold form a "Lie 2-algebra", which is a categorified version of a Lie algebra. Any compact simple Lie group G has a canonical 2-plectic structure, so it is natural to wonder what Lie 2-algebra this example yields. This Lie 2-algebra is infinite-dimensional, but we show here that the sub-Lie-2-algebra of left-invariant observables is finite-dimensional, and isomorphic to the already known "string Lie 2-algebra" associated to G. So, categorified symplectic geometry gives a geometric construction of the string Lie 2-algebra.Comment: 16 page

Linear Algebra and Analytic Geometry for Physical Sciences

This book originates from a collection of lecture notes that the first author prepared at the University of Trieste with Michela Brundu, over a span of fifteen years, together with the more recent one written by the second author. The notes were meant for undergraduate classes on linear algebra, geometry and more generally basic mathematical physics delivered to physics and engineering students, as well as mathematics students in Italy, Germany and Luxembourg. The book is mainly intended to be a self-contained introduction to the theory of finite-dimensional vector spaces and linear transformations (matrices) with their spectral analysis both on Euclidean and Hermitian spaces, to affine Euclidean geometry as well as to quadratic forms and conic sections. Many topics are introduced and motivated by examples, mostly from physics. They show how a definition is natural and how the main theorems and results are first of all plausible before a proof is given. Following this approach, the book presents a number of examples and exercises, which are meant as a central part in the development of the theory. They are all completely solved and intended both to guide the student to appreciate the relevant formal structures and to give in several cases a proof and a discussion, within a geometric formalism, of results from physics, notably from mechanics (including celestial) and electromagnetism

Gauging the Carroll Algebra and Ultra-Relativistic Gravity

It is well known that the geometrical framework of Riemannian geometry that underlies general relativity and its torsionful extension to Riemann-Cartan geometry can be obtained from a procedure known as gauging the Poincare algebra. Recently it has been shown that gauging the centrally extended Galilei algebra, known as the Bargmann algebra, leads to a geometrical framework that when made dynamical gives rise to Horava-Lifshitz gravity. Here we consider the case where we contract the Poincare algebra by sending the speed of light to zero leading to the Carroll algebra. We show how this algebra can be gauged and we construct the most general affine connection leading to the geometry of so-called Carrollian space-times. Carrollian space-times appear for example as the geometry on null hypersurfaces in a Lorentzian space-time of one dimension higher. We also construct theories of ultra-relativistic (Carrollian) gravity in 2+1 dimensions with dynamical exponent z<1 including cases that have anisotropic Weyl invariance for z=0.Comment: 27 page

Linking geometry and algebra with GeoGebra

GeoGebra is a software package and is so named because it combines geometry and algebra as equal mathematical partners in its representations. At one level, GeoGebra can be as a dynamic geometry system like other, commercially available, software. But this is only part of the story. Another window (the algebra part of GeoGebra) provides an insight into the relationship between the geometric aspects of figures and their algebraic representations. Here each equation or set of coordinates can be edited in the algebra window and the figure instantly changes. What is more, an equation (or a function) can be typed into the space at the foot of the GeoGebra interface and the corresponding geometric representation will appear in the geometry window. Perhaps utilising GeoGebra could inspire a change from regular forms of enrichment/ extension activity to things that need high level thinking, and things that pupils may find themselves wanting to follow-up outside school lessons