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Let $\pi: E \rightarrow M$ be a fiber bundle over the manifold M and denote by $\Gamma(E)$ the space of smooth sections of $E$. For compact $M$ it is well known (Hamilton 1982, Part II Corollary 1.3.9 …

Let $G$ be a compact Lie group with complexification $G^c$, and consider a principal $G^c$-bundle $P^c \to M$ together with a reduction $P \subseteq P^c$ to $G$. Assume that $M$ is a Riemann surface. …

Let $P \to M$ be a principal $G$-bundle, assume as much regularity as you want (compact $G$ or compact base manifold, ect). Via parallel transport, a connection $A$ on $P$ gives rise to the holonomy m …

Other examples of Hamiltonian systems with phase space the cotangent bundle of a group are coming from lattice gauge theory. There, a configuration is a map that assigns to every edge of the lattice a …

For a Yang-Mills connection $A$, one indeed has a decomposition $$H_A\bigl(Ad P \otimes \mathbb{C}\bigr) = \bigl(gau(P)_A\bigr)_{\mathbb{C}} \oplus \bigoplus_{\lambda > 0} H_A\bigl(Ad_\lambda P\bigr), …

Let us assume that the growth condition at infinity is implemented by requiring that all fields (connections and gauge transformations) extend to a given compactification $M$ of $\mathbb R^4$. The cla …

As spinors transform with a minus sign under a full rotation, there is no (non-trivial) lift of the action of the group of diffeomorphisms to the spinor bundle (i.e. the spinor bundle is not a natural …

Let $P \to M$ be a principal $G$-bundle. The moduli space of flat connections on $P$ is, by definition, the space $\mathcal{M} = \mathcal{C}_0 / \mathcal{G}$, where $\mathcal{C}_0$ denotes the subspac …

Each of the different formalism of classical mechanics has its advantages and disadvantages. However, in the end all three frameworks tend to be equivalent, and thus the following list is very subject …

Let $ (M, \omega) $ be a symplectic manifold. The de Rham class of $\omega$ induces a homomorphism $[\omega]: H_2(M) \to \mathbb{R}$, whose image $\Gamma_{\omega} \subseteq \mathbb{R}$ is called the g …

Ad 1.: Since every vector can be decomposed in its horizontal and vertical part. Thus it is enough to consider the case a) where all vectors are horizontal (this is trivial) and b) where at least one …

A gauge transformation can be viewed as a section of $P \times_G G$, where $G$ acts via conjugation. Hence the Lie algebra $\mathfrak{gau}$ of infinitesimal gauge transformation is the space of sectio …

I am trying to get my head around the geometric formulation of Quantum Mechanics as a projective Hilbert space (see Ashtekar, http://arxiv.org/abs/gr-qc/9706069). So one identifies all the rays $\math …

There is indeed a deep relation between Lagrangian submanifolds, the Fourier transformation and microlocal analysis. This is extensively discussed in Bates and Weinstein: Lectures on the Geometry of Q …

The easiest way to see that the norm of the curvature corresponds to the energy is to consider the special case of an abelian U(1)-Yang-Mills theory (i.e. electrodynamics). If you write out the norm s …