I have a long-percolating post that I hope to finish soon (when everything else is finished!) on “Why String Theory Must Be Right.” Not because it actually must be right, of course; it’s an hypothesis that will ultimately have to be tested against data. But there are very good reasons to think that something like string theory is going to be part of the ultimate understanding of quantum gravity, and it would be nice if more people knew what those reasons were.
Of course, it would be even nicer if those reasons were explained (to interested non-physicists as well as other physicists who are not specialists) by string theorists themselves. Unfortunately, they’re not. Most string theorists (not all, obviously; there are laudable exceptions) seem to not deem it worth their time to make much of an effort to explain why this theory with no empirical support whatsoever is nevertheless so promising. (Which it is.) Meanwhile, people who think that string theory has hit a dead end and should admit defeat — who are a tiny minority of those who are well-informed about the subject — are getting their message out with devastating effectiveness.
The latest manifestation of this trend is this video dialogue on Bloggingheads.tv, featuring science writers John Horgan and George Johnson. (Via Not Even Wrong.) Horgan is explicitly anti-string theory, while Johnson is more willing to admit that it might be worthwhile, and he’s not really qualified to pass judgment. But you’ll hear things like “string theory is just not a serious enterprise,” and see it compared to pseudoscience, postmodernism, and theology. (Pick the boogeyman of your choice!)
One of their pieces of evidence for the decline of string theory is a recent public debate between Brian Greene and Lawrence Krauss about the status of string theory. They seemed to take the very existence of such a debate as evidence that string theory isn’t really science any more — as if serious scientific subjects were never to be debated in public. Peter Woit agrees that “things are not looking good for a physical theory when there start being public debates on the subject”; indeed, I’m just about ready to give up on evolution for just that reason.
In their rush to find evidence for the conclusion they want to reach, everyone seems to be ignoring the fact that having public debates is actually a good thing, whatever the state of health of a particular field might be. The existence of a public debate isn’t evidence that a field is in trouble; it’s evidence that there is an unresolved scientific question about which many people are interested, which is wonderful. Science writers, of all people, should understand this. It’s not our job as researchers to hide away from the rest of the world until we’re absolutely sure that we’ve figured it all out, and only then share what we’ve learned; science is a process, and it needn’t be an especially esoteric one. There’s nothing illegitimate or unsavory about allowing the hoi-polloi the occasional glimpse at how the sausage is made.
What is illegitimate is when the view thereby provided is highly distorted. I’ve long supported the rights of stringy skeptics to get their arguments out to a wide audience, even if I don’t agree with them myself. The correct response on the part of those of us who appreciate the promise of string theory is to come back with our (vastly superior, of course) counter-arguments. The free market of ideas, I’m sure you’ve heard it all before.
Come on, string theorists! Make some effort to explain to everyone why this set of lofty speculations is as promising as you know it to be. It won’t hurt too much, really.
Update: Just to clarify the background of the above-mentioned debate. The original idea did not come from Brian or Lawrence; it was organized (they’ve told me) by the Smithsonian to generate interest and excitement for the adventure of particle physics, especially in the DC area, and they agreed to participate to help achieve this laudable purpose. The fact, as mentioned on Bloggingheads, that the participants were joking and enjoying themselves is evidence that they are friends who respect each other and understand that they are ultimately on the same side; not evidence that string theory itself is a joke.
It would be a shame if leading scientists were discouraged from participating in such events out of fear that discussing controversies in public gave people the wrong impression about the health of their field.
Thomas:
Am I correct that QJT is currently unable to predict the value of any parameter in the Standard Model (or any plausible extension thereof)? If so, it seems to me that it’s intellectual status is no different than that of string theory. I have no problem with you or anyone else working on it, or on anything else that you believe is worth pursuing. If anyone asked me my opinon of it, I would say “It doesn’t look promising to me, but I’ve certainly been wrong before.”
I agree that violations of Gauss’s law are not mathematically inconsistent; they are inconsistent with experiment. Isn’t that supposed to be the ultimate test?
I would be very interested in seeing a concrete proposal for replacing the Higgs field of the Standard Model with an anomalous gauge symmetry.
Chris:
You complain that “there was no interest whatsoever” when William Shaw prospoed a 4D string theory. Perhaps that’s because he never wrote a paper on it, but only gave a talk on it (at DAMTP) in 2004, and first posted the slides for this talk on his personal web site (with no link from his home page) in January of this year (that is, three months ago). Furthermore, in a quick perusal of the slides, I could not find any discussion of the quantum Virasoro algebra for his twistor-based theory; only the classical Virasoro algebra is treated (as far as I could tell). It is the anamoly in the quantum Virasoro algebra that fixes the critical dimension. It seems extremely likely to me that his theory (if it’s consistent at all) is anomalous in all but 26 dimensions. If he (and you) want string theorists to pay attention, then I believe it is incumbent upon him to (1) write a paper demonstrating that the quantum Virasoro algebra is anomaly free in 4D, and (2) to submit the paper to the arXiv and to a journal. If he writes that paper, and it contains an argument for 4D consistency that I can follow and think is not obviously wrong, I hereby volunteer to act as endorser for the arXiv.
Meanwhile, I think it is grossly unfair of you to complain that this obscurely posted set of slides has been ignored by the string community for the past three months.
As for Occam’s razor, if you have a framework for fundamental physics that is closer to making a definite prediction for a parameter of the Standard Model (or a plausible extension thereof) than is string theory, please let us know.
Paolo:
You are correct that “building a ST-like theory is an immense task” that “has required 20+ years of work of top level physicists.” You say that, therefore, I “cannot seriously ask for ‘something alternative’.”
I believe it is fair to ask for an alternative from those who insist that string theory is not worth pursuing, who say that it has “failed” and that all this work should simply be abandoned. Abandoned in favor of what, exactly? That’s what I want to know.
Mark:
“Abandoned in favor of what, exactly? That’s what I want to know.”
Well, my understaning is that, if you scientists had a good alternative to String Theory (something that is beatiful, predictive, mathematically consistent, full developed and so on), you will be eager to throw ST to the dogs.
But the problem is exactly this: from one side, you have a complex, rich theory with 20+ years of development behind it, that right now is not fully developed yet, and has no predictive ability; from the other, you have very little (ok, you have LQG, but I understand that also LQG has lots of problems).
So, it is safe to say that if you throw away LQG, you are back to square one.
You can’t ask for an alternative to ST. The only alternative for ST is rolling up your sleeves, and going back to the blackboard.
Peter:
I would be happy to hear an explanation of how a nontrivial representation of the gauge algebra in QED can avoid looking like a violation of Gauss’ Law (or how such a violation could be made consistent with experiment).
I believe that a key point in the public debate on string theory must be, “What are the alternatives?” and “What predictions do these alternatives make?” So exploring the proposed alternatives in a public forum seems to me like a good thing to do.
Concerning your other point, relevant discussion can be found at http://asymptotia.com/2006/10/27/more-scenes-from-the-storm-in-a-teacup-v/#comment-3045
“As for Occam’s razor, if you have a framework for fundamental physics that is closer to making a definite prediction for a parameter of the Standard Model (or a plausible extension thereof) than is string theory, please let us know.” – Mark Srednicki
The basic standard model has only 19 parameters, the minimal supersymmetric (stringy) version of standard model has 125 tunable parameters! If you want to make string theory get closer to predicting something, start by dropping supersymmetry, which just drags it further from reality.
The impression you are making is that string theory is going toward making predictions, when it’s actually going in the other direction. Supersymmetry is not science because there’s no experimental evidence that the standard model forces should all need to unify at 10^16 GeV, the Planck scale is dimensional analysis and hasn’t been empirically observed, nor is there any evidence for supersymmetric partners. (It’s like supergravity: start with the guess that there are gravitons, then invent an 11-d universe for those imaginary, unobserved gravitons. The framework that’s closer to reality is the 4-d GR and SM based on observables: LQG, representation theory, whatever.)
Paolo:
When I was a grad student in the late 70s, a friend (not a physicist) said to me that he had heard that the great unsolved problem of physics was combining quantum mechanics and general relativity; he wanted to know if people were working on this.
My answer was, not so much. The problem, I said, is you needed a good idea. You couldn’t just go into work in the morning, sit at your desk, scrunch up your forehead, and say “Now I will think about quantum gravity!” You needed an idea, something concrete to pursue. Yes, I said, some people were working on certain ideas, but none of them appeared all that promising, so mostly people were working on more tractable issues.
So, yes, you can roll up your sleeves and go to your blackboard, but what will you write on it?
Mark,
I understand that I am playing the easy part of the game. The problem, as you correcttly describe, is what you write on the blackboard.
My answer is “I have no idea”. But this is not my problem. I don’t know if it is your problem.
If you feel that ST is not working really, you should get some people in front of a blackboard, and then say: “ST is a mess! LQG is a mess! Every theory we are working on is a complete mess! Now, what we want to do?”
Perhaps you could ask people like Peter and say “Ok, people, now you are in charge, which theorem are we going to demonstrate tomorrow?”. Or, in general, collecting people from different approaches and trying to come up with a new idea.
Perhaps you should try to work on something simpler. Perhaps the unification that ST promised, long time ago, was too much. If you have to explore the whole space of theories in one step, it could be impossible (even if ST could be perfectly reasonable).
Perhaps you should burn a picture of professor Witten, as some sort of rite in order to free yourself of his too much powerful influence 🙂
But I really don’t know.
The real point is if you consider a worthwile task to spend time to find a new idea.
anon:
You write, “The framework that’s closer to reality is the 4-d GR and SM based on observables: LQG, representation theory, whatever.”
I’m not sure exactly what framework you mean. Of course we must end up with 4d GR and the Standard Model (or whatever extension of it, if any, turns up at the LHC) as the low-energy limit. The question is, what happens in the high-energy limit? String theory is being criticized for not providing a definitive answer. This is a valid criticism, but it also applies to ALL proposed “alternatives”. I put “alternatives” in quotes because most of them are so undeveloped that the relevant questions can’t even be asked. I personally see extremely limited prospects for development of any of these alternatives to the point that the questions can be asked. Spin-foam LQG has the best prospects, in my view, though I would still bet heavily against it working out, and it presently has nothing whatever to say about the parameters of the Standard Model.
As for representation theory, there seems to me to be no issue in the abelian case: nontrivial representations look like violations of Gauss’ Law, and these are ruled out experimentally (though I would be happy to be pointed to a source that explains why this is wrong).
Mark,
Your idea that the way to deal with criticism of string theory is not to make the case for it, but to attack its critics remains remarkably consistent over time. On the physics issue, I’m not talking about QED, but about chiral gauge theories. You’re well aware of this, since you supposedly read the third post on my blog, more than three years ago, about exactly this topic. At the time you (like other string theory advocates in the blogosphere) thought the best way to deal with a string theory critic was to attack them as ignorant of basic facts about physics:
“I came to your web site because I was told that you are a critic of string theory, and I wanted to see what you had to say about it. What I find is appalling ignorance. You really ought to spend some time learning some physics before you attack it. I recommend starting with Weinberg’s three-volume text on quantum field theory…
…I’m horrified that the ideas of someone as ignorant as you can be so widely circulated without serious rebuttal. You haven’t gotten serious rebuttal because the serious people have better things to do with their time. If you want to learn some physics first, fine. Unless and until you do, please stop trying to tell funding agencies (all staffed with people far more knowledgable than you are) how to best use their meager sums.”
Do you care at all why, as Sean Carroll notes, string theorists are losing the battle in the public marketplace of ideas? One big reason is because of behavior like this of yours and of other string theory advocates on various blogs. You’re doing far more damage to the cause of string theory than I or Smolin ever will. Go right ahead and keep it up.
Peter:
So you agree that only the trivial representation of the U(1) of QED is allowed?
I made my point in the context of QED only for simplicity. In the nonabelian case, non-trivial representations will violate (as far as I can see) the nonabelian version of Gauss’ Law. In terms of scattering amplitudes, this will lead to extra tadpoles in Feynman diagrams that can almost certainly be ruled out by precision electroweak data (unless the tadpoles conspire in some way; once again, I would be happy to be pointed to a detailed explanation of how this is supposed to work).
Three years ago, you appeared to be confused on the fact that, in four spacetime dimensions, a left-handed Weyl field is a hermitian conjugate of a right-handed Weyl field. I found this rather surprising, though I admit that I expressed my surprise in overly strong language.
Mark,
I’ve never said anything anywhere about this being a physically interesting question in QED. Whatever you may think, I am well aware of basic facts about quantum field theory (such as basic facts about Weyl fields, and that Gauss’s law is the expression of infinitesimal gauge invariance). If you had bothered to read the posting three years ago on my blog before launching into insults based on your conviction that a string theory critic must be an ignorant fool, you’d see that my speculation related to our imperfect NON-PERTURBATIVE understanding of CHIRAL gauge theories. Feynman diagram arguments can’t address this issue.
I don’t think this is the place for a serious discussion of the issue of possible holes in our understanding of nonperturbative chiral gauge theories, and, based on your previous behavior, I personally don’t believe you have any interest in such a discussion. Again, I seriously urge you to reflect on the point Sean Carroll is making in his posting here: string theory advocates are losing this debate because they can’t defend string theory research in a positive way. If you want to continue this losing tactic, go right ahead. As for your claim that three years ago, you were just expressing “surprise”, well, you’re continuing to insult my intelligence.
Dear Mark,
I have an “off debate” question. (But these blogs have also “outreach” purposes, I suppose.) Is it possible to explain to a lay person why only the trivial representation of the U(1) of QED is allowed. (I feel with all the reading of popular descriptions that I may well be only a few sentences from understanding at least in some level this matter.) I will be very very thankful for an explanation.
Gina, off-debate answer…when quantizing higher spin fields in a Lorentz invariant theory one has to somehow eliminate the negative norm states one encounter (those are states that have negative probabilities, something not so easy to interpret). The only known way to do it is to impose local gauge invariance- since those negative norm state can have arbitrary spacetime dependence, the symmetry imposed to eliminate them must also be local. This must be a symmetry of everything, including the observables. In other words everything should be invariant (aka the trivial representation) under gauge transformation.
I don’t see anything specific to QED, or to perturbation theory, in this argument, but I am definitely going to stay off-debate.
It just isn’t the case the existing understanding of 4-d GR and the Standard Model must be the complete and correct at low energy: there are a lot of outstanding questions about both, to be resolved better before these theories are properly understood in the low energy limit. Issues include electroweak symmetry breaking, dark energy (70% of universe), dark matter (25% of universe). If you assume that the existing understanding of the universe at low energy is complete and accurate, when it accounts for only 5% of the mass-energy and there is no confirmed theory of how mass arises or how electroweak symmetry breaks, then you’re speculating. You might as well be basing your final theory on Maxwell’s aether just because his equations for electromagnetism are well checked.
Peter:
When I brought up representation theory, I was not referring to your post of three years ago (which does not mention representation theory at all), but rather to your recent talk in Orlando, where you wrote (full set of slides at http://www.math.columbia.edu/~woit/orlando.pdf ):
“There are two main technical difficulties associated with properly defining QED and doing calculations with it:
“1) Fields carry degrees of freedom for every point in space-time, thus an infinite number. Naive calculations are plagued by infinite results. Properly handling this is known as ‘renormalization’ and was not carried out for QED until the late 1940s.
“2) The group of gauge symmetries is infinite dimensional. To this day, the representation theory of this group is not understood. Physicists generally believe this doesn’t matter, that only the ‘trivial’ representation matters, not the rest. In other words, one just needs to understand the ‘gauge-invariant’ part of the theory.”
I don’t know how to reconcile this with your claim above that “I’ve never said anything anywhere about this being a physically interesting question in QED.”
Once again, the only meaning I know how to attach to a “nontrivial representation of the gauge group” involves violations of Gauss’ Law (in both the abelian and nonabelian cases, irrespective of whether the matter is chiral) that would definitely show up in perturbation theory. If you know how to attach some other meaning to this phrase, I would appreciate being pointed to the explanation.
As for “defend[ing] string theory research in a positive way”, I gave my take on the current situation above. To briefly recap: The Standard Model has lots of ugly features that ANY more fundamental theory will have to explain. To me, this makes “landscape” style explanations more plausible. No framework other than string theory comes anywhere close to explaining any feature of the Standard Model; every one of these alternative frameworks is significanlty less predictive than string theory. (In string theory, if we knew full details of a compactification that reproduces the Standard Model, we could in principle predict scattering amplitudes at superplanckian energies.) Finally, as Lee Smolin anticipates, there could well be deep surprises remaining that totally change the picture, and (as Lee concurs) this could well happen within the string framework.
So, whatever the current demerits of string theory, it’s the best theory we have. I don’t know how to think up a better one. I’m very much open to hearing of a better one from someone else, but I don’t expect to; I think quantum gravity is a very constrained problem, and that string theory is very likely to be the only solution to this problem. If it is the only solution, abandoning it now would be a foolish thing to do.
Gina:
Tell me how much physics you know (popular books, high school, 1st-year college, college major, 1st year grad school, PhD) and I will attempt an explanation at that level. (I would say that Moshe’s answer is PhD level.)
anon:
I was implicitly including dark matter as something that might show up at the LHC (eg, as the lightest supersymmetric particle), but of course it might not; it might be condensed axions or some other moduli. Then we’d have to make sure our low-energy theory includes a suitable field. (Axions might be found in the current searches.) I’m assuming dark energy is the cosmological constant, anthropically selected from the landscape to be small. All this might result in enough uncertainty to prevent predictions at high energies, but we won’t know without a lot more work.
Dear Moshe and dear Mark
Many thanks for the explanation, Moshe. A few sentences away from some understanding was a little optimistic on my part. Based on the popular reading (+ a college degree with some physics +math and some later academic experience of little relevance) what you wrote sounds appealing and not completely cryptic but indeed also somewhat above my head. I will be happy to learn more.
BTW, Is the negative norm state that force you to forget all other representations similar to the unpleasant gadgets that force you dimensions 26 (10) for strings? (I vaguely recall the term negative norm states from that story as well.)
Mark,
I am not qualified to comment on the merits of William Shaw’s work on 4D superstrings – the reason for mentioning it was just that he had felt that the talk at DAMTP had not gone down well at all. This I found surprising. I should point out, though, that the slides have been available on his web site for some years, although not always in a very “friendly” format (i.e. very large PDF files).
I have tried to, and will continue to try to accurately reflect my own perception of the issues, and their possible solutions, on my web site. Yes, it would be very nice to explain SM parameters, but establishing a mathematical framework free of inconsistencies has to be done first.
Mark Srednicki on Apr 10th, 2007 at 4:34 pm
anon:
I was implicitly including dark matter as something that might show up at the LHC (eg, as the lightest supersymmetric particle), but of course it might not; it might be condensed axions or some other moduli. Then we’d have to make sure our low-energy theory includes a suitable field. (Axions might be found in the current searches.) I’m assuming dark energy is the cosmological constant, anthropically selected from the landscape to be small. All this might result in enough uncertainty to prevent predictions at high energies, but we won’t know without a lot more work.
Mark, thanks for your reply in the string theory context. The idea that supersymmetric partners, whose masses aren’t theoretically predicted, might be dark matter is very neat: a typical “heads I win, tails you lose” type forecast. There’s a catch 22 when you defend string theory by saying a lot more work needs to be done on it to see what it really says about dark energy, dark matter, etc. Henry Kissinger replied, when someone told him the Vietnam War was like a bottomless pit for money and human lives: “Every pit has a bottom.” It’s weird that sometimes people believe so much in things that they have a convenient answer to any objection. Any complaint that string theory is a complete failure as physics is responded to by saying it just needs a lot more work. Plus more funding…
Gina, I think I may have been a little optimistic in claiming I could explain it at any level, but here goes …
The three components E_i, i=x,y,z, of an electric field the three components B_i of a magnetic field can be expressed in terms of the three components A_i of a vector potential via E_i = -(d/dt)A_i and B_x = (d/dy)A_z, B_y = (d/dz)A_x, B_z = (d/dx)A_y. (Note to cognoscenti: I am working in the partial gauge A^0=0.) The vector potential that yields particular electric and magnetic fields is not unique; it can be changed from A_i to A_i + (d/dx_i)g, where g is any function of x_i (and not time), without changing E_i and B_i.
This is a gauge transformation.
Without any charged matter around, Gauss’ Law is (d/dx_i)E_i = 0, where the repeated i is summed. Physically, it means that electric flux is conserved; electric field lines don’t end. (With charged matter, electric field lines can end on charges, and Gauss’ Law is modified to (d/dx_i)E_i – rho = 0, where rho is the electric charge density.)
In the quantum theory, because E_i = -(d/dt)A_i, E_i and A_i obey equal-time canonical commutation relations of the form [E_i(x,t),A_j(x’,t)] = i delta_{ij}delta^3(x-x’).
Now we define G, the generator of gauge transformations, given by the integral over all space of g(x)(d/dx_i)E_i(x). Then we have the commutators [G,A_i(x)] = i(d/dx_i)g(x), which (except for the factor of i) is the added term in the gauge transformation. This is why G is called the generator of gauge transformations.
If Gauss’s law holds for a quantum state |s>, then G|s>=0. This is equivalent (in the usual meaning of the terminology) to the statement that |s> is in the trivial representation of the gauge group. In any nontrivial representation, G|s> would not vanish, which is only possible if Gauss’ Law is violated.
All this still works when we add charged matter and/or go to the nonabelian theory, but the formula for G changes.
The relation of this explanation to Moshe’s is highly nontrivial. And yes, negative norm states of a single string are what force you to 26 (or 10) dimensions. (My version of the explanation also has a counterpart in string theory, where one tries to prove the Lorentz algebra, and finds that it only works in the critical dimension.)
Mark,
In the next page after the one you quote, I refer to the problem of gauge symmetry as being solved in QED, and later solved in 1967 for Yang-Mills theory. I don’t anywhere say anything about the lack of understanding of gauge group representations in QED leading to a physical problem in that theory.
The theory where the question is more interesting is the standard model. There, if you throw away the electroweak sector, pure QCD is non-chiral, and the lattice appears to provide a satisfactory non-perturbative formulation, with gauge invariance achieved by integrating over the gauge degrees of freedom. In the continuum, even in this theory, there are poorly understood aspects of gauge invariance outside of perturbation theory (e.g. Gribov ambiguities).
If you add in the electroweak sector, the theory is chiral and lattice regularizations are problematic (there are claims that overlap fermions solve this but the situation seems unclear to me, and I haven’t seen anyone able to do actual non-perturbative SM lattice calculations). Chiral theories generically are anomalous gauge symmetry, but we understand how anomalies cancel in perturbation theory in the SM case. The fact that we don’t have a well-understood non-perturbative formulation of exactly this theory that would allow a clear understanding of how gauge symmetry works there seems to me an interesting problem, and it is not unreasonable to notice that it is exactly in this case that we need the Higgs mechanism to deal with the gauge symmetry properties of the vacuum state.
In 1+1 d one can actually hope to understand exactly what is going on since one does know what states look like as representations of the gauge group, and the representation theory mostly determines what goes on. Various interesting phenomena occur here, I’ve written about some of them and their relation to twisted K-theory and the Freed-Hopkins-Teleman theorem, but there is still a lot I don’t understand about this situation, hope some day to get sorted out and written down. Maybe some of it will give some insight into higher dimensional theories.
Your claim that string theory is better than alternatives because other theories are “less predictive” is just not true. String theory explains nothing about the SM and makes no predictions. You don’t even know the string scale or string coupling (who says it is small enough to make string perturbation theory valid?), so you don’t even have the predictions you claim about super-planckian scattering amplitudes.
If string theorists want to believe, after a few bong hits, that somehow magically a compactification that describes the world will be found, will be predictive, get tested and verified, they’re welcome to do so, although they should admit there is no argument about why this should happen. If others want to believe, after some other, possibly lesser, number of bong hits, that better understanding of gauge symmetry will explain the Higgs mechanism and allow computation of Yukawas, that a dynamical triangulation will produce the right kind of particle states, or that a 4d consistent string theory that explains the SM will be found, they also should be welcome to do so. I don’t think one can sensibly claim one of these is more likely to work out than the other. The only distinguishing characteristic of the string compactification unification scheme at the present time is that it has received vastly more attention, a much greater amount of work has gone into it, and its problems (the landscape..) are now rather well understood. This is an argument for encouraging people to try to do other things.
*”If you add in the electroweak sector, the theory is chiral and lattice **regularizations are problematic (there are claims that overlap fermions solve this **but the situation seems unclear to me, and I haven’t seen anyone able to do **actual non-perturbative SM lattice calculations).”
The computational cost of doing calculations involving fermions, chiral or otherwise, is well known, and it is only recently that there has been sufficient computing power to go beyond the quenched approximation in QCD simulations, for example.
But this is different from the claim that we do not know, in principle, how to define chiral gauge theories on the lattice. What is it specifically about the overlap fermion proposal that you believe is incorrect? Is there a publication that I can refer to for your analysis?
Thanks for your assistance!
Peter, it has always seemed to me that the difficulty with chiral fermions on a lattice is really an issue of the topology we choose when going to finite volume. If we knew a good regulator on the sphere, for instance, we presumably wouldn’t have any difficulty nonperturbatively defining a chiral gauge theory on the sphere. So I don’t see it as being really a feature of the theory itself that makes it problematic, I see it as a bug in the choice of regulator. I would think that one direction to go in for trying to nonperturbatively define chiral gauge theories would be trying to construct a theory on the fuzzy sphere and showing that the continuum (zero-fuzziness) limit is good.