# Anomaly Cancellation

Back in the early 80s, nobody was much interested in string theory. Some wrote it off as inconsistent nonsense. How wrong they were! With a stroke of genius Michael Green and John Schwarz confounded the critics. But how was it done?

First off we’ll need to understand the problem. Our best theory of nature at small scales is provided by the Standard Model. This describes forces as fields, possessing certain symmetries. In particular the mathematical description endows the force fields with an extra redundant symmetry.

The concept of adding redundancy appears absurd at first glance. But it actually makes it much easier to write down the theory. Plus you can eliminate the redundancy later to simplify your calculations. This principle is known as adding gauge symmetry.

When we write down theories, it’s easiest to start at large scales and then probe down to smaller ones. As we look at smaller things, quantum effects come into play. That means we have to make our force fields quantum.

As we move into the quantum domain, it’s important that we don’t lose the gauge symmetry. Remember that the gauge symmetry was just a mathematical tool, not a physical effect. If our procedure of “going quantum” destroyed this symmetry, the fields would have more freedom than they should. Our theory would cease to describe reality as we see it.

Thankfully this problem doesn’t occur in the Standard Model. But what of string theory? Well, it turns out (miraculously) that strings do reproduce the whole array of possible force fields, with appropriate gauge symmetries. But when you look closely at the small scale behaviour, bad things happen.

More precisely, the fields described by propagating quantum strings seem to lose their gauge symmetry! Suddenly things aren’t looking so miraculous. In fact, the string theory has got too much freedom to describe the real world. We call this issue a gauge anomaly.

So what’s the get out clause? Thankfully for string theorists, it turned out that the naive calculation misses some terms. These terms are exactly right to cancel out those that kill the symmetry. In other words, when you include all the information correctly the anomaly cancels!

The essence of the calculation is captured in the image below. Any potential gauge anomaly would come from the interaction of $6$ particles. For concreteness we’ll focus on open strings in Type I string theory. The anomalous contribution would be given by a $1$-loop effect. Visually that corresponds to an open string worldsheet with an annulus.

We’d like to sum up the contributions from all (conformally) inequivalent diagrams. Roughly speaking, this is a sum over the radius $r$ of the annulus. It turns out that the terms from $r\neq 0$ exactly cancel the term at $r = 0$. That’s what the pretty picture above is all about.

But why wasn’t that spotted immediately? For a start, the mathematics behind my pictures is fairly intricate. In fact, things are only manageable if you look at the $r=0$ term correctly. Rather than viewing it as a $1$-loop diagram, you can equivalently see it as a tree level contribution.

Shrinking down the annulus to $r=0$ makes it look like a point. The information contained in the loop can be viewed as inserting a closed string state at this point. (If you join two ends of an open string, they make a closed one)! The relevant closed string state is usually labelled $B_{\mu\nu}$.

Historically, it was this “tree level” contribution that was accidentally ignored. As far as I’m aware, Green and Schwarz spotted the cancellation after adding the appropriate $B_{\mu\nu}$ term as a lucky guess. Only later did this full story emerge.

My thanks to Sam Playle for an informative discussion on these matters.