A few weeks ago I recorded a podcast with Robinson Erhardt, which has now appeared as String Theory and the Crisis in Physics. We mainly talk about the current situation of string theory in physics and the history of how things have gotten to this point, topics familiar to readers of this blog.

Some other items:

- A Frequently Asked Question from students is for a good place to learn about the geometry used in gauge theory, i.e. the theory of connections and curvature for principal and vector bundles. Applied to the case of the frame bundle, this also gives a way of understanding the geometry of general relativity. One reference I’m aware of is
*Gauge Fields, Knots and Gravity*, by John Baez and Javier P. Muniain, but I’d love to hear other suggestions. These could be more mathematical, but in a form physicists have a fighting chance at reading, or more from the physics point of view. - For new material from mathematicians lecturing about quantum field theory, see Pavel Etingof’s course notes, and Graeme Segal’s four lectures at the ICMS (available in this youtube playlist).
- If you want to understand the mindset of the young string theory true believer these days, stringking42069 is back.
- There’s something called “Plectics Laboratories” which has been hosting mainly historical talks from leading physicists and mathematicians. For past talks, see their youtube channel. For a series upcoming September 23-27, see here.
- The IAS is hosting an ongoing Workshop on Quantum Information and Physics. One topic is prospects for future wormhole publicity stunts based on quantum computer calculations, see here. At the end of the talk, Maldacena raises the publicity stunt question (he calls it a “philosophical question”) of whether you can get away with claiming that you have created a black hole when you do a quantum computer simulation of one of the models he discussed.
- Ananyo Bhattacharya at Nautilus has an article on the role of physics in creating new math. While there is a lot there to point to, recent years have not seen the same kind of breakthroughs Witten and Atiyah were involved in during the 1980s and 90s. I’m hoping for some progress the other way, that new ideas from mathematics will somehow help fundamental theoretical physics out of its doldrums.

Nakahara!

https://www.amazon.com/Geometry-Topology-Physics-Graduate-Student/dp/0852740956

Peter,

In addition to Baez and Muniain, I have enjoyed “Differential Geometry, Gauge Theories and Gravity” by Göckeler and Schücker, as well as the classic “Geometrical Methods of Mathematical Physics” by Schutz.

The Eguchi, Gilkey and Hanson review is also very readable.

Kind Regards

I did a quick review in https://arxiv.org/abs/1511.00388v3

Some more physics-oriented ones:

Y. Choquet-Bruhat, C. Dewitt-Morette, and M. Dillard-Bleick, Analysis, Manifolds and

Physics, Part I: Basics.

Y. Choquet-Bruhat and C. Dewitt-Morette, Analysis, Manifolds and Physics. Part II: 92

Applications.

Some more mathematical ones:

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Volume I. Interscience Tracts in Pure and Applied Mathematics.

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Volume II. Interscience Tracts in Pure and Applied Mathematics.

Concerning the first point, I would suggest Hamilton’s Mathematical Gauge Theory.

Geometry of Physics by Frankel has somewhat higher mathematical standards.

The Eguchi, Gilkey and Hanson review mentioned above refers to T. Eguchi, P.B. Gilkey and A. J. Hanson, “Gravitation, Gauge Theories and Differential Geometry”, Physics Reports 66 (6) (1980) pp. 213-393.

Chris Isham’s book, Modern Differential Geometry for Physicists, World Scientific 1991 is very approachable.

Robert Hermann’s book, Gauge Fields and Cartan-Ehresmann Connections, Math Sci Press 1975 is a good summary once you’ve mastered the material in the first two items.

Two books by Gregory Naber, “Topology, Geometry and Gauge fields: Foundations” and “Topology, Geometry and Gauge fields: Interactions” keep a quite elementary approach, but go quite far and will be a good foundation for a more advanced reading.

I’m seeing lots of great suggestions above for covering the geometry of gauge theory. Over the years I’ve compiled some of my notes covering this and other topics into a personal wiki. People who enjoy a casual but mathematical wiki format may find it helpful. Here are three notes in it, of increasing sophistication:

https://deferentialgeometry.org/#connection%20%5B%5BClifford%20vector%20bundle%5D%5D%20%5B%5BEhresmann%20principal%20bundle%20connection%5D%5D

Humble request: Could you stop giving air to Twitter?

When you share a link to a tweet and you don’t have an account, you can only see that tweet. Maybe. More often than not you just see a screen that says “Something went wrong. Retry?” It won’t work if you retry.

You might well say, “Well then go get an account.” I used to have one but I deleted it when the terms of service got much more invasive. I have felt better about that decision since the list of owners came out.

Moreover, Twitter is already inaccessible from Brazil. Surely there will be other countries. There is a great tradition of physics in Brazil: Bohm, Lattes, et al. Do you really want to cut them out from what you want to say about physics?

Cynicism,

Twitter is pretty much just a sewer of idiocy. Almost always when I link to something there it’s to point to some kind of exceptional idiocy from string theorists. If you’re a big fan of this kind of thing, maybe it would justify getting a twitter account, maybe not.

Another reason for me to look at Twitter is that there are a few accounts that sometimes provide links to something interesting I didn’t know about. In those cases what I put on the blog is not the Twitter source but the link.

i second that twitter request. i got an error or login request or some bs.

zzz,

Sorry, but if you want to experience the stringking phenomenon, you have to take the plunge into the Twitter sewer. I’m pointing to it rather than ignoring it because it gives some weird sort of insight into what is going on among some string theorists as they face the collapse of their subject.

The whole thing reminds me a bit of a Hunter Thompson saying: “when the going gets weird, the weird turn pro.”

There is no such thing as Twitter.

John Baez,

Just because billionaires change names of things doesn’t mean I have to use their new name. I’m boycotting using not only “X”, but “Alphabet”, “Meta” and “SLMath”.

Dear Peter,

indeed it would be nice if you would consider strengthening your boycott of using the name X to a complete boycott of linking to Twitter/X altogether… even if a sewage, I would think that linking to it still gives it relevance and encourages people to use it, which ultimately legitimizes Musk’s take on it. And too bad if this makes us miss interesting sociological phenomena like stringking.

Back to the question about references: on top of the many excellent ones already given, and for the specifics of the frame bundle approach to GR: the first few chapters of Chandra’s mathematical theory of black holes are very nice and pedagogical.

Quite mathematical, but always keeping physics in sight, is

Shlomo Sternberg, Curvature in Mathematics and Physics, Dover, 2012.

Re. the podcast (and the one with Jaimungal) I have to say that it was nice getting a somewhat nuanced view of your take on the current state of physics (much of which rings true for me). As on old retired geezer who happily surfed the nonlinear dynamics fad and was able to parlay that into a happy, bottom feeding career in applied mathematical physics I have no skin in the game, but nonetheless found how pervasive the internal political influence of the string theorists, a little surprising. When I was in grad school in the late eighties, taking GR and QFT classes things seemed a little more wide open (though part of that may have been the result of a pretty strong crossover between cosmologists and high energy theorists in my department). Anyway that’s all a rather long winded way of asking if there’s a lot in Not Even Wrong that wasn’t covered in these podcasts. I’m a pretty slow reader so at this age have to pick wisely, and though I’m sure I’d enjoy Not Even Wrong, if there’s not much more than you expressed in the podcasts, it sounds like maybe I should go get up to speed on spinors instead (and any recommendations there are welcome).

Karl Young,

As far as the string theory controversy goes, the two recent podcasts give a pretty good idea of my side of that, so if that’s all you care about, you can live without reading the book. There is however, much material on other topics there, in particular a lot about the relation of math and physics, nothing to do with string theory.

Not much in the book about spinors, I’d agree that learning about those is a more fruitful pursuit than thinking about string theory. Hard to find good readable references on the topic. I’ve written some blog entries, should find time to write more and should create a category here so people could find them.

Below I will put my two cents (literally!) on good references for learning the basics of (geometric) Gauge Theory:

i) before embarking on the project resulting in his monumental “Foundations of differential geometry” with Kobayashi, Nomizu wrote a booklet entitled “Lie groups and differential geometry”, which I still find the best (mathematically oriented) introduction to principal bundles, connections, holonomy, etc;

ii) to those who have mastered Nomizu’s little gem, I recommend reading Bleecker’s “Gauge theory and variational principles”, a slightly thicker and (essentially) self-contained account of the fundamentals of gauge theory, with emphasis on its variational aspects and including elegant discussions on spinors and on the unification of gauge fields and gravitation usually associated to Kaluza and Klein. Despite being too mathematically oriented on the whole, this jewel is filled here and there with insightful comments on the emergency of gauge invariance a more physically inclined reader may find fruitful.

Physics students wanting to understand more about the geometry of gauge theories might find the following texts useful:

1. 2004 A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry by Peter Szekeres

2. 2013 Mathematical Physics: A Modern Introduction to Its Foundations by Sadri Hassani

3. 2017 Differential Geometry and Mathematical Physics: Part I. Manifolds, Lie Groups and Hamiltonian Systems by Gerd Rudolph and Matthias Schmidt

4. 2017 Differential Geometry and Mathematical Physics: Part 2. Fibre Bundles, Topology and Gauge Fields by Gerd Rudolph and Matthias Schmidt

Physics students prepared to read mathematics literature (at the level of Choquet-Bruhat & Dewitt-Morett) may find useful content in

1. 1991 Modern Geometry: Part I: The Geometry of Surfaces, Transformation Groups, and Fields by B. A. Dubrovin, A. T. Foment, and S. P. Novikov

2. 1985 Modern Geometry: Part II: The Geometry and Topology of Manifolds by B. A. Dubrovin, A. T. Foment, and S. P. Novikov

3. 2017 Riemannian Geometry and Geometric Analysis Seventh Edition Jurgen Jost

Thanks for posting the Etingof notes. It looks like a very interesting take on the subject (algebraic combinatorics view?), but that could be a reflection of my ignorance.

Do you know of any similar books/notes that take a more elementary approach along these lines?

I think “Differential Geometry: Connections, Curvature, and Characteristic Classes” by Loring W. Tu is quite accessible. It’s the sequel to his much liked “An Introduction to Manifolds”.

That one here,

Helga Baum, Eichfeldtheorie (Gauge field theory)

is in german but mathematically sound and rather simple:

https://www.amazon.de/Eichfeldtheorie-Differentialgeometrie-Faserb%C3%BCndeln-Springer-Lehrbuch-Masterclass/dp/3642385389

some of it is here on this pdf

https://www.mathematik.hu-berlin.de/~baum/Skript/KompaktkursHFB-05.pdf

For those who want a simple book on gauge theory that is easy to read, from a typical german math department which is not as high profile like Bonn….