/sci/ - Science & Math

>>10337176
Noether's theorem (or generalized Noether's theorem) states that for any generator [math]\tau^a[/math] of the compact Lie group [math]G[/math] for which the Lagrangian [math]L\in \Omega^p(\mathcal{J},M)[/math] (as a differential p-form on the jet bundle of sufficient regularity) is invariant under, there exists a conserved current [math]J^a[/math] satisfying [math]\operatorname{div}J^a = 0[/math], and whose charge [math]Q^a = \int_V J^a[/math] is a constant of the motion.
(Topological) K theory is the study of isomorphism classes of vector bundles. Since for each vector bundle [math]E\rightarrow M[/math] there exists [math]n \in \mathbb{Z}[/math] such that [math]E \oplus (n) \rightarrow M[/math] is trivial, there exists a notion of "addition" [math]+[/math] (distinct from the Whitney sum [math]\oplus[/math], but no unique "subtraction") on the Abelian monoid [math]A(M)[/math] of equivalence classes of vector bundles on [math]M[/math]. By quotienting out the relation [math][E\oplus F] - [E] + [F][/math] for vector bundles [math]E,F[/math] on [math]M[/math] we get an Abelian group [math]K(M)[/math], called the K group of [math]M[/math].