The equations of motion, or the equations of instantons, or solitons, or Einstein's equations, or just about any equations in physics, are differential equations.  In many cases, we are interested in the space of solutions of a differential equation.  If we write the total (possibly nonlinear) differential equation of interest as $L(u) = 0,$ we can linearize near a solution $u_0,$ i.e. write $u = u_0 + v$ and expand $L(u_0 + v) = 0 + L&#39;|_{u_0}(v) + ... =: D(v)$ to construct a linear equation $D(v)=0$ in the displacement $v.$

A linear differential equation is like a matrix equation.  Recall that an $n\times m$ matrix $M$ is a map from $R^n$ to $R^m$, and $dim(ker(M)) - dim(ker(M^*)) = n-m,$ independent of the particular matrix (or linear transformation, more generally).  This number is called the "index."  In infinite dimensions, these numbers are not generally finite, but often (especially for elliptic differential equations) they are, and depend only on certain "global" information about the spaces on which they act.

The index theorem tells you what the index of a linear differential operator ($D,$ above) is.  You can use it to calculate the dimension of the space of solutions to the equation $L(u)=0.$  (When the solution space is a manifold [another story], the dimension is the dimension of the tangent space, which the equation $D(v)=0$ describes.)  It does not tell you what the actual space of solutions is.  That's a hard, nonlinear question.

Where is the Atiyah-Singer index theorem used in physics?
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