By Mikio Furuta

The Atiyah-Singer index theorem is a impressive outcome that enables one to compute the gap of options of a linear elliptic partial differential operator on a manifold when it comes to only topological info concerning the manifold and the emblem of the operator. First proved by way of Atiyah and Singer in 1963, it marked the start of a very new path of analysis in arithmetic with kinfolk to differential geometry, partial differential equations, differential topology, K-theory, physics, and different components. The author's major aim during this quantity is to provide a whole facts of the index theorem. The model of the evidence he chooses to offer is the single in keeping with the localization theorem. the necessities comprise a primary path in differential geometry, a few linear algebra, and a few proof approximately partial differential equations in Euclidean areas.

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**Example text**

17 (the index theorem on R with actions). trace(rv| K er(V + A)) —trace(rv^| Ker(—V + A*)) = trace T. A similar result also holds on S1. The details are left to readers. E x e r c is e 1. Formulate the index theorem on S 1 with actions and prove it. 4. T h e m o d 2 In d ex T h e o re m in D im en sion 1. Let A(x) be a smooth mapping with values in skew symmetric complex matri ces of size r. We consider a differential equation (V + A ) f = 0 for a Cr-valued function f(x). ( 1) The case over S1.

Mr {C) be a smooth mapping with values in complex matrices of size r. Here, we assume that A (x ) is periodic with the period R: A{x + R) = A{x). For Cr-valued functions f( x ) and g(x), we consider the following two linear differential equations: ( 1 . 2) df(x) dx dg(x ) dx + A (x )f(x ) = 0 + A(x)*g(x) = 0 Here A*(x) is the adjoint matrix of A(x). We, in practice, study differential equations on S 1 = [0, R\/(0 R). If we do not require the periodicity condition, the solution unique ly exists for a given initial value at a point.

A typical example is the case that V = ^2k(qk)2/2, which corre sponds to oscillation of a spring following the Hoock’s law. In quan tum mechanics, the corresponding system is called the h arm on ic oscillator. The harmonic oscillator is not only simple in V, but also has basic significance in quantum mechanics. Because the algebraic structure (the Heisenberg algebra) behind the harmonic oscillator is used in formulation of annihilation and creation of particles in quantum field theory, in which an arbitrary number of particles can appear.