http://www.staff.science.uu.nl/~hooft101/ap.html

I suppose Gerard probably holds the record for having his name spelled wrong. Don’t think he doesn’t notice!

And speaking of “wrong”, wouldn’t it have been nice to see someone in the video just say “I was wrong”?

]]>Wow, you’re giving me memories of grad school đź™‚ That Witten paper on Morse theory was the basis of my orals, I had to present it. Honestly, could never work out what Witten was talking about until I read Roe’s book on it.

]]>Yes, I remember BRST seeming very ad hoc to me, until I realized I could think of the Faddeev-Popov ghosts as differential forms in disguise, with Q being essentially a version the exterior differentiation d constrained to the gauge directions. It’s been a while since I worked on this but I seem to remember convincing myself that the path integral over the ghost action expresses a coordinate-invariant volume form on the gauge surfaces like we would normally do with differential forms. Since then I’ve thought of BRST as about as fundamental and well-motivates as differential forms are.

I know there are cases where BRST gets a little more complicated and I am not sure how far the correspondence with differential forms can be taken.

]]>I think we agree about strategy: step back and look for new mathematical insights that may later find applications in fundamental physics. Even if you don’t get what you want for physics, you’ll learn more about deep mathematics, which is all to the good. And sure, Z2-graded mathematics may very well be part of those insights. Now that I’m wrapping up work on the book, I’m looking forward to going back to doing precisely that, thinking about Dirac cohomology.

My problem with Urs is that while he’s often doing this sort of thing, at the same time he finds it necessary to try to use this to defend the central failed research program that has dominated (and done a huge amount of damage to) theoretical physics for over 30 years. His argument starting with Z2-graded Tannaka duality ends up with the specific endpoint of an argument for supergravity, in ten dimensions (whatever you want to call the local supergroup there, that’s the one I’m referring to). I don’t think there’s a serious argument there. You can’t get to that kind of specific theory from general ideas about the relation of QM, representation theory and Tannaka duality. When you try and do it, you’re just adding in lots of unexamined assumptions and eliding distinctions that are exactly the ones you need to be looking at to figure out where this train of inference goes wrong.

Defending 10d superstring theory and supergravity as the fundamental theory while arguing that any possible actual experiments are irrelevant is very dangerous, the “Not Ever Wrong” danger I’m trying to point out. Bringing very abstract not relevant mathematical statements in to help do this is a really bad idea. I think in this year we’re going to finally see the collapse of any hope that supersymmetric extensions of the standard model will ever see a test or get experimental support. I hope the community reacts to this by challenging the assumptions that led to enthusiasm for these models, not by permanently seeking refuge in excuses (“only visible at high energy”) and dubious invocations of abstract mathematics.

]]>I still do not see how the positive mass theorem is an example. To me, it is just an example of the usefulness of spinors in differential geometry â€” which is profound, but not â€śsuperâ€ť in a technical sense.

Okay – I was trying to give examples of “supermathematics” (Z/2-graded mathematics), not supersymmetry, but I guess this was a bad example. I was in grad school in the early 1980s when Witten was starting to wow mathematicians with his new arguments for known results. His “physics proofs” of the Atiyah-Singer index theorem and positive mass theorem came out then. Both involved the Dirac operator in ways that surprised mathematicians, so I tend to mentally lump them together.

But you’re right, his argument for the positive mass theorem doesn’t use Z/2-graded ideas, at least not explicitly. Wikipedia says his argument was “inspired by positive energy theorems in the context of supergravity”, and the fourth section of Witten’s paper discusses the connection to supergravity. But he says the connection is “not altogether clear”. I don’t know if anyone ever nailed it down!

A much better example, from around the same time, would be Witten’s approach to Morse theory. I took a course on that with Raoul Bott, and at one point he said, eyebrows wagging mischievously: “So we think about this, and we get stuck. So we need to *super*think!”

Around this time I also took a course from Quillen on Witten’s proof of the Atiyah-Singer index theorem. Quillen was trying to make it rigorous using “high-school calculus” and lot of elementary superalgebra. He got scooped by Getzler.

So, I think of this as the era when mathematicians realized the importance of supermathematics.

]]>You [Urs Schreiber] are trying to derive from an extremely general abstract theorem (that Tannaka duality works for not just groups but also Z/2-graded groups) an argument for a very specific supergroup, a rather ugly one with no experimental evidence at all for it.

I don’t think he’s doing that. For starters, he didn’t mention any specific supergroup, much less a specific “rather ugly one”. Which supergroup are you talking about, anyway? Whatever it is, Urs never mentioned it.

I think the only reasonable attitude is to realize that we are missing many pieces of the puzzle, both experimental pieces and conceptual pieces: that is, ideas we need, that we don’t have yet. So, people motivated primarily by mathematical elegance should not expect their work to make contact with experiment in the next few decades, or indeed at all. The time is not ripe for that. They should instead try to do good math. Good math can be recognized independent of any applications to physics, and it has its own inherent value.

I think Urs is doing good math. He’s not spending his days working on the minimal supersymmetric extension of the Standard Model or any other specific ugly attempt to ram our current theoretical thinking down the throat of the experimental data we have now. He’s instead inventing fundamental new ideas in geometry, topology and algebra – indeed, so fundamental that breaking math into these separate subjects doesn’t do justice to his work. Some of his work is on particular theories that string theorist like (showing how they fit into a single structure, the “brane bouquet”), but a lot of his work is more general than that, and thus of more general interest. I’m talking about things like infinity-categories, infinity-topoi, and how these permit a new much more general approach to old topics like Lie groups, Lie algebras, gauge theory, prequantization and so on. These are worth studying *regardless* of any application to physics. In fact, trying to connect them prematurely to particle physics runs the risk of screwing up their natural development, by making us focus on the theories we know and not on the ones we don’t.

That’s a good question, and I cannot resist to offer a thought. Most people (including me) assume that the Standard Model (or whatever BSM model you want to replace it with) is an effective theory at large length scales of some underlying theory U. This underlying theory does not even have to be a quantum field theory. The bosonic and fermionic fields in the effective theory are effective degrees of freedom that arise in a possibly complicated way from the degrees of freedom of U. In this situation, should we expect that the effective theory allows local spacetime supersymmetry transformations that can mix the effective bosons and fermions?

I think there are essentially two possibilities we have to consider: Either there exist *many* theories U that induce effective theories with bosons and fermions that look roughly like the Standard Model; or there is essentially only one possible U with this property.

In the second case, not having any a priori information about U, I would assign a 50% prior probability (or much less, but I’ll be generous here) to the statement that the effective theory allows local spacetime supersymmetry. I.e., I would answer Urs’ question with a counterquestion: Why should this constraint *not* exist?

In the first case, however, the situation is different. If bosons and fermions can arise in many different ways from an underlying theory, we should expect that the absence of transformations that can mix them is a pretty generic property in the class of possible theories U. After all, the bosonic fields could appear effectively by a mechanism quite different from the mechanism that produces the fermions. Then we would not expect a symmetry that can mix them, because they are just quite different objects, like apples and orangutans. (In the Standard Model, the bosons and fermions do not look similar at all. I take that as a hint that they arise indeed in quite different ways from the underlying theory, whatever it may be.)

Hence, no matter whether we are in the first or (as I hope) second case or something in between, I do not find it particularly puzzling that the effective theory that describes our universe shows no sign of spacetime supersymmetry. If we would see supersymmetry, that would be an extremely strong clue about the underlying theory: it would have to induce fermions and bosons as effective degrees of freedom in very similar ways, so that the effective theory could mix them. But we do not see it. Why should anyone be surprised? For the stated reasons, I even find it very likely that supersymmetry does not occur at *any* length scale at which the effective theory is valid. And in the underlying theory, the question “Is there spacetime supersymmetry?” might conceivably not even be well-defined.

Maybe SUSY enthusiasts do not like this idea — that the bosons and fermions in our universe arise effectively from an underlying theory U in completely different ways — because they assume that then U would have to be an inelegant, complicated theory? I do not see any argument that would support this assumption.

]]>You are trying to derive from an extremely general abstract theorem (that Tannaka duality works for not just groups but also Z2-graded groups) an argument for a very specific supergroup, a rather ugly one with no experimental evidence at all for it. I just don’t see any argument at all for this.

“All groups” covers almost all of mathematics, and adding in Z2-graded groups makes this even more general. I’m a big fan of the idea that quantum mechanics is fundamentally representation theory, and (see the book I’ve been writing) I think there’s a huge amount to say about how highly non-trivial and specific basic structures in representation theory govern quantum theory. But, you can’t get something from nothing: an extremely general piece of abstraction applying to almost the entire mathematical universe cannot possibly do the job of distinguishing the very specific mathematical structure that seems to govern the physical universe.

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