Introduction to my book “Advanced Differential Geometry for Theoreticians”

G.Sardanashvily, Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory (Lambert Academic Publishing, Saarbrucken, 2013)  #

In contrast with quantum field theory, classical field theory can be formulated in a strict mathematical way by treating classical fields as sections of smooth fibre bundles. This also is the case of time-dependent non-relativistic mechanics on fibre bundles over R. This book aim to compile the relevant material on fibre bundles, jet manifolds, connections, graded manifolds and Lagrangian theory. The book is based on the graduate and post graduate courses of lectures given at the Department of Theoretical Physics of Moscow State University (Russia). It addresses to a wide audience of mathematicians, mathematical physicists and theoreticians. It is tacitly assumed that the reader has some familiarity with the basics of differential geometry.

Contents

1 Geometry of fibre bundles: 1.1 Fibre bundles, 1.2 Vector and affine bundles, 1.3 Vector fields, 1.4 Exterior and tangent-valued forms.

2 Jet manifolds: 2.1 First order jet manifolds, 2.2 Higher order jet manifolds, 2.3 Differential operators and equations, 2.4 Infinite order jet formalism.

3 Connections on fibre bundles: 3.1 Connections as tangent-valued forms, 3.2 Connections as jet bundle sections, 3.3 Curvature and torsion, 3.4 Linear and affine connections, 3.5 Flat connections, 3.6 Connections on composite bundles.

4 Geometry of principal bundles: 4.1 Geometry of Lie groups, 4.2 Bundles with structure groups, 4.3 Principal bundles, 4.4 Principal connections, 4.5 Canonical principal connection, 4.6 Gauge transformations, 4.7 Geometry of associated bundles, 4.8 Reduced structure.

5 Geometry of natural bundles: 5.1 Natural bundles, 5.2 Linear world connections, 5.3 Affine world connections.

6 Geometry of graded manifolds: 6.1 Grassmann-graded algebraic calculus, 6.2 Grassmann-graded differential calculus, 6.3 Graded manifolds, 6.4 Graded differential forms.

7 Lagrangian theory: 7.1 Variational bicomplex, 7.2 Lagrangian theory on fibre bundles, 7.3 Grassmann-graded Lagrangian theory, 7.4 Noether identities, 7.5 Gauge symmetries.

8 Topics on commutative geometry: 8.1 Commutative algebra, 8.2 Differential operators on modules, 8.3 Homology and cohomology of complexes, 8.4 Differential calculus over a commutative ring, 8.5 Sheaf cohomology, 8.6 Local-ringed spaces.

30 Years of “The Gauge Treatment of Gravity”

Thirty years of our pioneer article: D.Ivanenko, G.Sardanashvily, "The Gauge Treatment of Gravity", Physics Reports, 94 (1983) 1-45, where a gravitational field (a pseudo-Riemannian metric) is described as a classical Higgs field responsible for spontaneous breakdown of space-time symmetries in accordance with the geometric Equivalence Principle.

References:

WikipediA: Higgs field (classical)

WikipediA: Gauge gravitation theory

G.Sardanashvily, Classical gauge gravitation theory, Int. J. Geom. Methods Mod. Phys. 8 (2011) 1869-1895.

What is a classical Higgs field

Spontaneous symmetry breaking, a vacuum Higgs field, a Higgs boson are quantum phenomena. A vacuum '''Higgs field''' is responsible for spontaneous symmetry breaking the gauge symmetries of fundamental interactions and provides the Higgs mechanism of generating mass of elementary particles. However, no adequate mathematical model of this Higgs vacuum has been suggested in the framework of quantum gauge theory, though somebody treats it as sui generis a condensate by analogy with that of Cooper pairs in condensed matter physics.

At the same time, classical gauge theory admits comprehensive geometric formulation where gauge fields are represented by connections on principal bundles. In this framework, spontaneous symmetry breaking is characterized as a reduction of the structure group G of a principal bundle P -> X to its closed subgroup H. By the well-known theorem, such a reduction takes place if and only if there exists a global section h of the quotient bundle P/G -> X. This section is treated as a classical Higgs field.

A key point is that there exists a composite bundle P -> P/G -> X where P -> P/G is a principal bundle with the structure group H. Then matter fields, possessing an exact symmetry group H, in the presence of classical Higgs fields are described by sections of some composite bundle E -> P/G -> X, where E -> P/G is some associated bundle to P -> P/G. Herewith, a Lagrangian of these matter fields is gauge invariant only if it factorizes through the vertical covariant differential of some connection on a principal bundle P -> P/G, but not P -> X.

An example of a classical Higgs field is a classical gravitational field identified with a pseudo-Riemannian metric on a world manifold X. In the framework of gauge gravitation theory, it is described as a global section of the quotient bundle FX/O(1,3) ->  X where FX is a principal bundle of the tangent frames to X with the structure group GL(4,R).

WikipediA

Lectures on integrable Hamiltonian systems

G.Sardanashvily, Lectures on integrable Hamiltonian systems, arXiv: 1303.5363 

Abstract. We consider integrable Hamiltonian systems in a general setting of invariant submanifolds which need not be compact. For instance, this is the case a global Kepler system, non-autonomous integrable Hamiltonian systems and integrable systems with

time-dependent parameters.

Introduction

The Liouville -- Arnold theorem for completely integrable systems, the Poincar\'e -- Lyapounov -- Nekhoroshev theorem for partially integrable systems and the Mishchenko -- Fomenko theorem for the superintegrable ones state the existence of action-angle coordinates around a compact invariant submanifold of a Hamiltonian integrable system which is a torus. However, it is well known that global extension of these action-angle coordinates meets a certain topological obstruction.

Note that superintegrable systems sometimes are called non-commutative (or non-Abelian) completely integrable systems.

In these Lectures, we consider integrable Hamiltonian systems in a general setting of invariant submanifolds which need not be compact. These invariant submanifolds are proved to be diffeomorphic to toroidal cylinders. A key point is that a fibred manifold whose fibres are diffeomorphic either to a compact manifold or an Euclidean space is a fibre bundle, but this is not the case of toroidal cylinders.

In particular, this is the case of non-autonomous integrable Hamiltonian systems and Hamiltonian mechanics with time-dependent parameters.

It may happen that a Hamiltonian system on a phase space Z falls into different integrable Hamiltonian systems on different open subsets of Z. For instance, this is the case of the Kepler system. It contains two different globally superintegrable systems on different open subsets of a phase space Z. Their integrals of motion form the Lie algebras so(3) and so(2,1) with compact and non-compact invariant submanifolds, respectively.

Geometric quantization of completely integrable and superintegrable Hamiltonian systems with respect to action-angle variables is considered. The reason is that, since a

Hamiltonian of an integrable system depends only on action variables, it seems natural to provide the Schrodinger representation of action variables by first order differential

operators on functions of angle coordinates.

Throughout the Lectures, all functions and maps are smooth, and manifolds are real smooth and paracompact. We are not concerned with the real-analytic case because a paracompact real-analytic manifold admits the partition of unity by smooth functions. As a consequence, sheaves of modules over real-analytic functions need not be acyclic that is essential for our consideration.