Ctrl-z, 18 years later — “Yang-Mills theory for bundle gerbes” is now retracted

tl;dr – my very first paper, co-authored with Mathai Varghese when I was a PhD student (and he was advising me), had a critically flawed assumption, which I only discovered in the second half of last year.

It was tremendously exciting for me in 2005 to have my rough ideas turned into a real draft of a real paper by Mathai, to work on the paper together a bit more before sending it off to the arXiv, for publication, and then to get it published in early 2006. That my own idea could be made into a scientific publication was a thrill. At the time my PhD supervision was a bit in flux. Students were required to have at least two supervisors (a primary and a secondary), and my primary supervisor—Michael Murray—had just taken up the role of Head of School, the School in question being newly formed from the merge of three departments. Meetings were often rescheduled or cancelled due to time constraints. My original secondary supervisor could contribute little specific to the topic I was looking at, and it was arranged I would swap Mathai in as secondary. This started a period of about 18 months, going from memory, of mostly working with him, though ultimately I moved on and found a different project for my thesis. Our joint paper was the first thing we did in short order, and I was still very fresh and inexperienced. I had moved to the maths department from the physics department, switching disciplines. My thinking was still very approximate and “physicsy” (how times have changed!). From what I can only guess are my notes going into the meeting that kicked this project off, it looks like I was hoping we could reproduce the derivation of the sourceless vacuum Maxwell equations from the appropriate Lagrangian/action, except now from the 3-form curvature of a bundle gerbe, rather than from the 2-form curvature (the Faraday tensor) of a connection on a -bundle (i.e. the EM vector potential).

Page from my notebook dated 10th August 2005, and showing a first pass at a variational calculus approach to a bundle gerbe Yang–Mills action.

The details of who did what in the paper will be left vague, but suffice it to say that I did at one point “verify” the gauge group in the paper did indeed satisfy the conditions of having a group action on our space of bundle gerbe curvings. Looking at my calculations, they are not wrong—they just took the assumption behind the definition for granted. This assumption is that there is a certain 2-form on the Banach Lie group with a reasonable-seeming “primitivity” property (which certainly holds at the level of de Rham cohomology, and even in the linearised level, at the identity element). Our paper cites no source for this assumption. Given such a 2-form, everything works fine, and my calculation in 2005 is sound.

However, as is my wont, I like to read over my past papers to keep the ideas from being completely forgotten, and particularly papers like this one that has left its original rationale dangling: can we generalise not just electromagnetism to higher-degree forms, but non-abelian Yang–Mills theory? As one can tell from the 2006 paper’s title, this was at the forefront of my mind at the time. The knowledge of non-abelian higher gauge theory was at the time not developed enough for my to attack this problem, though it was the original goal of the PhD project proposal. But in recent years, the kind of nitty-gritty down-to-earth tech has pretty much arrived, and so while I’m not planning on picking up my original goal, I do still care about supplying physicists with concrete examples to illustrate the theory.

In the process of re-reading our paper, I noticed, for the first time, the primitivity assumption as being a bit of a bold claim that it is:

…recall that the line bundle associated with the central extension … is primitive in the sense that there are canonical isomorphisms …, and there is a connection , on the line bundle , called a primitive connection, which is compatible with these isomorphisms.

Mathai and Roberts (2006), section 2.2

In the treatment by Murray and Stevenson of connections on central extensions (see section 3 at the link), one generically does not get such a primitive connection; rather there is a 1-form on the square of the quotient group measuring the failure of the compatibility of the connection with the isomorphisms. Back in 2005–06, this was not a paper I had spent time with, but in the past decade it has become a standard reference for me. And so only now does the unwarranted assumption stand out like a sore thumb. The problem is not so much that the connection fails to be compatible with the isomorphisms, but that there is in all cases that I knew of, an resulting exact 2-form, the exterior derivative of the 1-form mentioned above. To satisfy the axiom for a group action of our gauge group on the affine space of bundle gerbe curvings, the exact 2-form needed to vanish everywhere.

I was somewhat perturbed, and to gloss over the subsequent discussions of the awkward situation, I ended up writing a paper that not only showed that there was no 2-form on as we had assumed, but classifying the more general class of bi-invariant 2-forms on all (reasonable) infinite-dimensional Lie groups in terms of the topological abelianisation of the Lie algebra. In many examples of interest this more general space of 2-forms still turns out to be trivial (i.e. only the identically vanishing 2-form is bi-invariant), so certainly there can be no primitive 2-forms as we assumed in all of these cases. Thankfully, one can wring some mildly interesting examples out of this result, namely that one can classify bi-invariant forms on some infnite-dimensional structured diffeomorphism groups (in the volume-preserving and symplectomorphism cases, for instance), in terms of specific de Rham cohomology spaces of the original compact manifold. Whether this is useful or even interesting, it isn’t clear to me. But to provide a positive result out of such a negative situation helped ease the disappointment.

Whereas at the point of finding this unjustified assumption I was hoping that perhaps there was something particularly special about the group that I didn’t yet know, I was now in the position where I knew the definition using this assumption was unfixable. Since the resulting analysis of the moduli space of solutions uses this group action in multiple instances, there was no way I could be confident in the result. So I had to (with permission from Mathai) contact the journal. I wrote a note (technically, a “corrigendum”), and submitted it, with the comment that if it had to be a retraction after all, I would be ok with that. Given the amount of time I put into correcting the literature in other instances (eg the two-year process starting with this blog post, and ending with this formal erratum, and the flow-on effects from that process), I couldn’t very well leave our own mistake in the literature to cause problems down the line.

Ultimately, the journal decided that the paper really should be retracted, and now you can find the notice at the journal website. I was thinking last year throughout this process how I might approach this problem afresh with what I know now, and find a similarly concrete description of the moduli space (as opposed to as a higher stack with some universal property). While I had some fruitful ideas, I hadn’t time then to dedicate to them (I was mostly working late at night sitting up in bed in the dark, sketching notes on my tablet in dark mode!). Were I able to fix the problems easily I would have sent them to the journal and had a corrigendum published, rather than retract the paper. Certainly if people are interested, I can share my ideas as they exist so far. As far as my coauthor goes, he was content to authorise me take charge throughout this whole affair, and I think he is willing to let the matter slide. However, as a matter of personal and professional pride, it would be nice to be able to eventually rectify this error by producing a theorem analogous to the one we had once claimed.