Vincent Bouchard

What? Physics? Why would physics have anything to do with Hurwitz numbers? Interesting question, isn't it?

Well, it turns out that physics – in particular, string theory – does indeed have much to say about Hurwitz numbers, and enumerative geometry in general. In this appendix I will try to explain why physics has deep connections with enumerative invariants such as Hurwitz numbers. I will not be precise; nor will I state explicit results or theorems (in fact, there is not a single equation in this appendix!). Rather, my goal is simply to convey some of the fascinating ideas behind the connection between string theory and enumerative geometry. Hopefully, by the end of the appendix, you will find these relations interesting enough to delve into the literature, where you can find precise conjectures and theorems!

Physical Mathematics

For many physicists, mathematics is seen as a tool: a language for building models of nature. However, in the last forty years or so, a fascinating new research area has flourished: using physics as a tool to further our understanding of mathematics. To some pure mathematicians, this statement may sound like an abomination. But let me try to convince you that physics indeed has much more to say about mathematics itself than one may expect.

While this interconnection between physics and mathematics is certainly not new (historically, physics and mathematics have always been intimately related), it has been very successful in recent years. So successful that it has been given its own name: Physical Mathematics (Moore, 2014). In the particular case where the corner of physics studied is string theory, it is also sometimes called String-Math1. The idea is simple but far-reaching: use the complex structural properties of physical theories to discover new connections between different areas of mathematics.

One of the most useful tools that physicists have at their disposal is physical dualities. Roughly speaking, physicists are interested in constructing mathematical models that explain the universe (and provide predictions that can be tested in experiments, following the scientific method). But sometimes, it happens that more than one mathematical model provides the same observable quantities (or, at least, observables that are in a one-to-one relationship, and/or perhaps only in some appropriate limit). When this is the case, from a physics standpoint both models are valid physical descriptions. We say that these models are dual.