30 research outputs found
Iterated Differential Forms I: Tensors
We interpret tensors on a smooth manifold M as differential forms over a
graded commutative algebra called the algebra of iterated differential forms
over M. This allows us to put standard tensor calculus in a new differentially
closed context and, in particular, enriches it with new natural operations.
Applications will be considered in subsequent notes.Comment: 9 pages, extended version of the published note Dokl. Math. 73, n. 2
(2006) 16
Iterated Differential Forms II: Riemannian Geometry Revisited
A natural extension of Riemannian geometry to a much wider context is
presented on the basis of the iterated differential form formalism developed in
math.DG/0605113 and an application to general relativity is given.Comment: 12 pages, extended version of the published note Dokl. Math. 73, n. 2
(2006) 18
On the category of Lie n-algebroids
Lie n-algebroids and Lie infinity algebroids are usually thought of
exclusively in supergeometric or algebraic terms. In this work, we apply the
higher derived brackets construction to obtain a geometric description of Lie
n-algebroids by means of brackets and anchors. Moreover, we provide a geometric
description of morphisms of Lie n-algebroids over different bases, give an
explicit formula for the Chevalley-Eilenberg differential of a Lie n-algebroid,
compare the categories of Lie n-algebroids and NQ-manifolds, and prove some
conjectures of Sheng and Zhu [SZ11].Comment: 29 pages, to appear in Journal of Geometry and Physic
The noncommutative geometry of Yang-Mills fields
We generalize to topologically non-trivial gauge configurations the
description of the Einstein-Yang-Mills system in terms of a noncommutative
manifold, as was done previously by Chamseddine and Connes. Starting with an
algebra bundle and a connection thereon, we obtain a spectral triple, a
construction that can be related to the internal Kasparov product in unbounded
KK-theory. In the case that the algebra bundle is an endomorphism bundle, we
construct a PSU(N)-principal bundle for which it is an associated bundle. The
so-called internal fluctuations of the spectral triple are parametrized by
connections on this principal bundle and the spectral action gives the
Yang-Mills action for these gauge fields, minimally coupled to gravity.
Finally, we formulate a definition for a topological spectral action.Comment: 14 page
Morita base change in Hopf-cyclic (co)homology
In this paper, we establish the invariance of cyclic (co)homology of left
Hopf algebroids under the change of Morita equivalent base algebras. The
classical result on Morita invariance for cyclic homology of associative
algebras appears as a special example of this theory. In our main application
we consider the Morita equivalence between the algebra of complex-valued smooth
functions on the classical 2-torus and the coordinate algebra of the
noncommutative 2-torus with rational parameter. We then construct a Morita base
change left Hopf algebroid over this noncommutative 2-torus and show that its
cyclic (co)homology can be computed by means of the homology of the Lie
algebroid of vector fields on the classical 2-torus.Comment: Final version to appear in Lett. Math. Phy
Symmetries in Classical Field Theory
The multisymplectic description of Classical Field Theories is revisited,
including its relation with the presymplectic formalism on the space of Cauchy
data. Both descriptions allow us to give a complete scheme of classification of
infinitesimal symmetries, and to obtain the corresponding conservation laws.Comment: 70S05; 70H33; 55R10; 58A2
Smooth manifolds and observables
This textbook demonstrates how differential calculus, smooth manifolds, and commutative algebra constitute a unified whole, despite having arisen at different times and under different circumstances. Motivating this synthesis is the mathematical formalization of the process of observation from classical physics. A broad audience will appreciate this unique approach for the insight it gives into the underlying connections between geometry, physics, and commutative algebra. The main objective of this book is to explain how differential calculus is a natural part of commutative algebra. This is achieved by studying the corresponding algebras of smooth functions that result in a general construction of the differential calculus on various categories of modules over the given commutative algebra. It is shown in detail that the ordinary differential calculus and differential geometry on smooth manifolds turns out to be precisely the particular case that corresponds to the category of geometric modules over smooth algebras. This approach opens the way to numerous applications, ranging from delicate questions of algebraic geometry to the theory of elementary particles. Smooth Manifolds and Observables is intended for advanced undergraduates, graduate students, and researchers in mathematics and physics. This second edition adds ten new chapters to further develop the notion of differential calculus over commutative algebras, showing it to be a generalization of the differential calculus on smooth manifolds. Applications to diverse areas, such as symplectic manifolds, de Rham cohomology, and Poisson brackets are explored. Additional examples of the basic functors of the theory are presented alongside numerous new exercises, providing readers with many more opportunities to practice these concepts