This summer I was expecting to be working mainly on an extended field theory and geometry project as well as pushing toward the conclusion of a study on double sigma models. But somewhat unexpectedly I’ve found myself also working on some interesting things at the interface of number theory and physics. It has to do with infinite series and smooth asymptotics. The motivation is almost entirely stringy, sparked first by an observation from my supervisor; but the early phases of research have since deepened and expanded the focus in a few intriguing directions. My hope is that we might find a canonical class of regulators, but that will undoubtedly prove to be a difficult end. I have started writing a post on some of the background material, which hopefully I’ll have finished and be able to share sooner rather than later.
I have enjoyed the opportunity to dive back into an area that I find fascinating and could probably spend years thinking about. Some of the stuff I’m currently looking at reminds me of this paper on Jensen polynomials that I was playing with a couple of years ago, particularly the behaviour of the growth conditions. I still find this paper super interesting and continue to dabble when I have time, hoping that the project and the calculations I had in mind may formalise. Additionally, as I think I might have alluded in my blog post at the time, there seems to also be a rather nice connection with the Hermite polynomials and what we observe with the harmonic oscillator in quantum mechnics. This is not a new observation, but I do think it is curious. Incidently, I came across a comment by Raphael J.F. Berger a couple weeks ago which sums it up quite well. Another one for the future, perhaps.
There is absolutely no surprise that new connections with number theory have appeared and continue to appear in string theory. There is some very deep and cool things going on here, which we’re only beginning to glimpse. String amplitudes and automorphic representations; modular forms and moonshine; zeta values, renormalisation, and stringy geometry; are just a few examples. In fact, I recently discovered that the Isaac Newton Institute for Mathematical Sciences is currently hosting a programme on new connections in number theory and physics, and there have already been a number of interesting presentations of stringy flavour.
For the interested reader, I wanted to quickly share a few other resources that may stimulate further curiousity. There is a nice article by Matilde Marcolli on number theory in physics that offers a brief survey and status report, along with many references. In particular, pay attention to the conference papers from the Winter School in Les Houches, France. A fantastic encyclopeadic resource by Matthew R. Watkins also offers an invaluable database. And then there is the open-access journal, Communications in Number Theory, which I have been digging through lately.