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

Question

I'm trying to get motivated in learning the Atiyah-Singer index theorem. In most places I read about it, e.g. wikipedia, it is mentioned that the theorem is important in theoretical physics. So my question is, what are some examples of these applications?

Answer 1

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 $L(u_0 + v) = 0 + L'|_{u_0}(v) + ... =: D(v)$ 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.

Answer 2

Eric and others have given good answers as to why one expects the index theorem to arise in various physical systems. One of the earliest and most important applications is 't Hooft's resolution of the $U(1)$ problem. This refers to the lack of a ninth pseudo-Goldstone boson (like the pions and Kaons) in QCD that one would naively expect from chiral symmetry breaking. There are two parts to the resolution. The first is the fact that the chiral $U(1)$ is anomalous. The second is the realization that there are configurations of finite action (instantons) which contribute to correlation functions involving the divergence of the $U(1)$ axial current. The analysis relies heavily on the index theorem for the Dirac operator coupled to the $SU(3)$ gauge field of QCD. For a more complete explanation see S. Coleman's Erice lectures "The uses of instantons." There are also important applications to S-duality of $N=4$ SYM which involve the index theorem for the Dirac operator on monopole moduli spaces.

Answer 3

First let me explain what the index in question refers to. If the math gets too full of jargon let me know in the comments.

In physics we are often interested in the spectrum of various operators on some manifolds we care about. Eg: the Dirac operator in 3+1 spacetime. In particular the low-energy long distance physics is contained in the zero modes (ground states).

Now what the "index" measures, for the Dirac operator $D$ and a given manifold $M$, is the difference between the number of left-handed zero modes and the number of right-handed zero modes. More technically:

$$ ind\,D = dim\,ker\,D - dim\,ker\,D^{+} $$

where $D$ is the operator in question; $ker\,D$ is the kernel of $D$ - the set of states which are annihilated by $D$; and $ker\,D^{+}$ is the kernel of its adjoint. Then, as you can see, $ind\,D$ counts the difference between the dimensionalities of these two spaces. This number depends only on the topology of $M$.

In short, the ASI theorem relates the topology of a manifold $M$ to the zero modes or ground states of a differential operator $D$ acting on $M$. This is obviously information of relevance to physicists.

Perhaps someone else can elaborate more on the physical aspects.

The best reference for this and other mathematical physics topics, in my opinion, is Nakahara.

Answer 4

In the case of a Dirac operator, the index is the (signed) excess dimension of the space of vacuum modes of one chirality w/r/t the other: i.e., the number of anomalous “ghost” states in a chiral field theory.

Anomalies arise when the classical/quantum symmetry correspondence breaks down under renormalization (a global anomaly could be responsible for quark mass in QCD; resolving the local chiral anomaly in the SM accounts for quarks and leptons; resolving it in superstring theory fixes the gauge group [to either SO(32) or E8 x E8], and the resolution of a conformal anomaly fixes the dimension of spacetime and the fermion content). When trying to turn string theory into actual physics, one asks

• Can it explain three generations of chiral fermions?
• Can it explain the experimental results on proton decay?
• Can it explain the smallness of the electron mass?
• Can it explain [things about the cosmological constant]?

and AST helps to answer these questions.