The arXiv today has a very comprehensive survey of a conventional point of view on where “Physical Mathematics” is in 2022 and where it is going, written by a group of six authors. “Physical Mathematics” is a term popularized by one of them, Greg Moore (see here and here, with some commentary here), and it’s an expansion of a Snowmass white paper. A separate paper by Nikita Nekrasov covering the material listed in Section 10 is advertised as forthcoming with the title “The Ghosts of Past and Future Ideas and Inspirations on Interface of Physics and Mathematics”.
The term “Physical Mathematics” is a play on the more conventional name of “Mathematical Physics” to describe work being done at the intersection of math and physics. In its usage by Moore et al. it refers to a point of view on the relation of math and physics which heavily emphasizes certain specific topics that have been worked on intensively during the last four decades. These topics mostly have roots in seminal ideas of Witten and his collaborators, and involve calculational methods developed in quantum field theory and string theory research. The huge volume of this research is reflected in the fact that the survey reference section contains 62 pages giving 1276 separate references. A major problem for anyone taking up an interest in this field has been the sheer scale and complexity of all this work, and this survey should be helpful in providing an overview.
While some of these 1276 papers could equally well be simply characterized as “Mathematics”, it’s hard to describe exactly what makes a lot of the rest “Physical Mathematics” rather than “Physics”. Part of the answer is that these are not physics papers because they don’t answer a question about physics. A striking aspect of the survey is that while a lot of it is about QFT, the only mention at all of the QFT that governs fundamental physics (the standard model) is in a mention of one paper relevant to some supersymmetric extensions of the SM. The only other possible connection to fundamental physics I noticed was about the landscape/swampland, something only a vanishingly small number of people take seriously.
Also striking is the description of the relation of this field to string theory: while much of it was motivated by attempts to understand what string/M-theory really is, section 3.1 asks “What Is The Definition Of String Theory And M-Theory?” and answers with a doubly-boxed
We don’t know.
This is a fundamental question on which relatively little work is currently being done, presumably because nobody has any good new ideas.
In the background of this entire subject is the 1995 conjecture that there is a unique M-theory which explains various dualities as well as providing a unified fundamental theory. After nearly 30 years of fruitless looking for this, the evidence is now that there is no such thing, and maybe the way forward is to abandon the M-theory conjecture and focus on other ways of understanding the patterns that have been found.
I share a faith in the existence of deep connections between math and physics with those doing this kind of research. But the sorts of directions I find promising are very different than the ones being advertised in this survey. More specifically, I’m referring to:
- the very special chiral twistor geometry of four-dimensions (no twistors in the survey)
- the subtle relation of Euclidean and Minkowski signature (only a mention of the recent Kontsevich-Segal paper in the survey)
- the central nature of representation theory in quantum physics and number theory (very little representation theory in the survey)
Looking back at Greg Moore’s similar 2014 survey, I find that significantly more congenial, with a more promising take on future directions (in particular he emphasizes the role of geometric representation theory).