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.

with commentary:

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).

Disclaimer: I am neither a professional physicist nor a professional mathematician. But writing as an outsider with some technical background, it strikes me that people in this new field of “physical mathematics” have stumbled on something wonderful: physical mathematics doesn’t sound like it is constrained either by the exacting rigor of logic (as in “mathematics”) or the crucible of experiment (as in “physics”).

But such constraints in mathematics and in physics are precisely the things that make both fields so hard, as well as so useful. I imagine that centuries of effort in both fields led people to conclude that such constraints *were* essential. Does “physical mathematics” indeed lack the constraints above? Does it have new (“emergent”?) constraints not shared by physics or math? Or is it essentially without constraint? Is it essential for an intellectual area with scientific aspirations to be bound by rigorous logic or experiment?

Brathmore,

The problem is that among the 1276 papers there’s everything from the fully rigorous to the highly incoherent. Typically though, calculations are being done based upon some list of assumptions about how the objects involve behave (e.g. you don’t have a rigorously defined QFT, but do have a conjectural part of a theory with conjectural properties). One thing that makes the subject difficult is that these assumptions are rarely explicitly stated. There are constraints: your assumptions have to be internally consistent, otherwise you’ll get nonsense.

The best of this work does capture non-trivial new deep ideas about mathematics, even if not completely understood. A good example would be Witten’s work on Chern-Simons which won him the Fields medal. One thing that was clear was that he had his hands on a powerful new set of topological invariants, which he could compute in many cases. His QFT methods for getting these were somewhat ad hoc and not clearly consistent, but the result was compelling.

@Brathmore,

Jaffe and Quin wrote an article (“Theoretical mathematics”: Toward a cultural synthesis of mathematics and theoretical physics) motivated by exactly the question of how theoretical-physics-style speculation in mathematics can be sufficiently constrained to avoid bad consequences. They advocated basically for a balance between speculation and rigorous proof, lumping not just physicsy stuff but any kind of conjecturing together under speculation.

This article is perhaps most famous for a brief comment they made criticizing Thurston’s proof of the geometrization theorem for Haken manifolds as an example of insufficiently rigorous work that damages the field, inspiring Thurston to write a rebuttal in the form of his beautiful “On Proof and Progress in Mathematics”.

A shout0ut and second re Kontsevich – Segal …

Because no one has linked to it, here’s the Kontsevich and Segal paper:

Wick rotation and the positivity of energy in quantum field theory, arXiv:2105.10161.Graeme Seagal presents his “rival” version to the standard Wightman axiomatisation of QFT in the following video (answering questions by Alain Connes & Nigel Higson among others).

At 10’30” he starts to explain how does Wick rotation fits to this new axiomatisation and exposes the most important idea of his recent paper with Maxim Kontsevitch at 37’06”.

Waiting for Nikita Nekrasov future paper on his panorama of physical mathematics, we can relax and learn about his personal trajectory at the interface of theoretical physics & mathematics with a short video interview (from 2018?) at IHES I don’t think Peter has ever mentioned.

One can hear and interpret his slight hesitation & correction while defining string theory at 16’39”.

Less anecdotal, more objective and relevant piece of info I think for Peter’s blog post: Nekrasov answer to the question “how do you make peace with the difficulty in your field to make predictions that are experimentally testable…?” at 23’40”. It reveals IMHO the human rather than epistemic necessity of “physical mathematics” as a kind of “Compagnons du Devoir” craft’s guild name for these bold & brilliant (& sometimes almost stateless) people who have engaged in the conflation of gauge theory, general relativity, supergravity, bosonic strings, supergraviton & Mp branes, IKKT model, BFSS Matrix model, 2D quantum gravity, gauge/gravity duality, AdS/CFT correspondance… hoping to build a cathedral (have a seat at the wedding feast at Planck scale?)

In the 2014 post Peter linked one of the questions he asked was, “What is QFT?” At the bottom of page 4 in the survey there is reference to another Snowmass paper that may be of interest to readers of this blog, titled “The Quest to Define QFT”. It gives a go at answering this question and reviews various axiomatic approaches to defining what is QFT in 11 pages with 531 references.

https://arxiv.org/abs/2203.08053

Glimpsing at the survey paper it seems to describe some of the efforts at mixing math with explorations which are inspired and motivated by physics. The Simons foundation seems very fond of such efforts and many of its collaborations (and the physics and geometry center at Stony brook) mix mathematicians, physicists and some computer scientists. The current survey seems strongly related to the rather recent global categorical symmetries collaboration and the survey also mentions the Special holonomy collaboration, both of which mix mathematicians and physicists in the general sense of the word (people who learned physics, think using the methods and jargon that they have learned, sit in physics departments and whose papers are not in the form of lemma, lemma, theorem…, trying to avoid the question of whether these people are actually doing physics which is very popular here). Math has profited immensely from such mixing of people and ideas and sometimes actual physical studies of the real world also profited.

Waiting for the sometimes related surveys of the Homological mirror symmetry, localization of waves, Hidden symmetries and fusion energy, New structures in low dimensional topology (essentially mathematicians but strongly inspired by physicists), extreme wave phenomenon based on symmetries collaborations to provide a broader view of physical mathematics.