A little essay on
The use of non-abelian differential cohomology
in fundamental physics
Modern fundamental physics is all about gauge theory. The familiar electromagnetic field is an abelian gauge field in first nontrivial degree. But in general, the fundamental force fields are non-abelian; Yang-Mills fields and the field of gravity (in first order formulation) are non-abelian gauge fields of degree 1. Analyzing their relationship deeper reveals a whole space of fundamental physics-like structures, whose study goes by the name string theory. In that space one finds higher analogs of abelian gauge fields, such as the B-field in the next higher degree, the C-field in yet one degree higher, or the RR-fields in every even or in every odd degree. It was eventually understood (Fr00) that, mathematically, these abelian higher gauge fields all are cocycles in abelian differential cohomology (HoSi05), a differential refinement of abelian cohomology. The latter has a cousin, nonabelian cohomology. There is an evident way in which the low degree but nonabelian Yang-Mills gauge fields and the field of gravity may analogously be understood as degree-1 cocycles in nonabelian differential cohomology. This suggests an evident question:
Is there a useful notion of general nonabelian differential cohomology; and does it describe anything like higher degree nonabelian gauge fields in fundamental physics?
I point out in the following that, in fact, nonabelian differential cohomology permeates fundamental physics in general, and string theory in particular – without often having been recognized as such.
Around 2002 Witten pointed out (Wi02) that the worldvolume theory of several M5-branes must involve a 2-form gauge field that should live in both higher as well as nonabelian differential cohomology: a higher analog of a Yang-Mills gauge field. Back then it was clear that identifying this would be important for understanding Yang-Mills theory itself (see JaWi), namely its S-duality. Today this is even more so, with the relation (Wi09) of this system to geometric Langlands duality and Khovanov homology (Wi11) roughly understood.
When I set out on my PhD thesis, which was concerned with general structures potentially relevant to this question, I once asked a senior string theorist why not more people would be interested in this, or working on identifying this. He said the reason was that there was no idea how to identify the mathematics of the nonabelian 2-form theory. These days the topic is all hot again (see the Strings2011 talk Mo11 or the upcoming Simons Center program), and motivated by recent progress (BL,G, ABJM) on understanding the M2-brane, proposals for how to proceed with the nonabelian higher gauge field on the 5-brane are appearing again (SSW11, HHM11).
It took me a few years from my thesis to have all the mathematical concepts and tools set up to seriously attack the question of the nonabelian 2-form field on the 5-brane. I don’t see evidence that without such concepts from a general theory of nonabelian differential cohomology there is a reasonable chance to achieve this. With Fiorenza and Sati, we have just now made available a preprint (FSS12) on this question (see also the talk slides Sa12, US11), arguing for the correct description, using the new tools of nonabelian differential cohomology that we have developed over the years.
This may serve as occasion to say a bit more in reply to questions like the following.
Is nonabelian differential cohomology really needed?
Does it appear naturally in physics?
It is interesting to note how these questions are related. A wide-spread impression is probably still that nonabelian differential cocycles don’t play much of a role at all, beyond the degree-1 case of Yang-Mills fields and of gravity. It is interesting to observe that – as discussed in a moment – the opposite is true: there are whole subfields of theoretical physics where people are all concerned with (naive aspects of) higher nonabelian cocycles. What is instead true is that these are still not widely recognized as such, due to lacking appreciation of the necessary theory.
This may be plausible already from general considerations. Since Yang-Mills theory (unitary principal connections) and general relativity (ISO-Cartan connections) are both described by nonabelian cohomology – it would seem to be a strange state of affairs if, of all structures, these two would have to be dealt with separately, not fitting into a joint framework with abelian higher gauge fields such as the B-field or the RR-fields, in particular since all these fields interplay with each other in string theory. Sometimes it is argued that, since the Yang-Mills theory appearing on D-branes is to be understood as living in differential K-theory, it becomes abelian after all. But the same is not true for gravity, and not for, say, the gauge field in the heterotic string, so it does not quite serve as an answer.
On the other hand, in (SSS09, FSS10) we have shown that the nonabelian gauge field and gravity of the heterotic string naturally unify with the B-field to nonabelian higher cocycles called twisted String-2-connections. Using this we constructed the full moduli stack of background field configurations that satisfy the Green-Schwarz cancellation of quantum anomalies, where the nonabelian gauge field, the field of gravity and the abelian B-field all interplay. The analog statement holds for the magnetic dual of the theory, which involves nonabelian cocycles of degree 6.
Given that the Green-Schwarz mechanism is at the heart of string theory (the “first superstring revolution” (Schwarz07)), evidently nonabelian differential cohomology is at the heart of string theory.
Moreover, we show in (FSS12) that the smooth moduli 3-stack of the supergravity C-field naturally arises as a further refinement of this twisted nonabelian differential cohomology. Given that, together with the above M5-brane system, this is at the heart of the conjectured “M-theory” UV-completion of 11-dimensional supergravity (the ”second superstring revolution”), evidently nonabelian differential cohomology is crucial also here.
Once one recognizes this, and the mathematics used to describe it, one sees that elsewhere there are whole subfields where people have been using aspects of higher nonabelian cohomology without recognizing it as such.
There is, for instance, the supergeometric formulation of supergravity action functionals by D'Auria-Fré. They observed that a structure which they tried to interpret as “soft group manifolds” and described in terms of dg-algebras is a structure at the very heart of the construction of supergravity action functionals. Comparing to our theory, we see that (SSS09, Sch12), after some translation, what these authors secretly found are that supergravity fields are naturally understood as higher nonabelian differential cocycles in supergeometry. Or rather, with their toolbox they find the subclass of those whose underlying instanton sectors (whose underlying classes in ordinary nonabelian cohomology) are trivial.
There is a plethora of consequences of this which are to be investigated, but which can only be recognized when seen as what they are in nonabelian differential cohomology. For instance a general principle of higher nonabelian cohomology is that families of structures for some higher gauge group are given by -higher gerbes, which are classified by cocycles in the higher automorphism infinity-group of . Therefore the automorphism Lie 7-algebra of the supergravity Lie 6-algebra that D'Auria-Fré (secretly) construct 11-dimensional supergravity from should contain crucial information about the theory. Indeed, it turns out (S12) that its Lie 7-algebra contains in its degree 0 the super Lie algebra that has become famous as the M-theory Lie algebra. A wealth of structures in supergravity and string theory are known to be controled by this, hundreds of articles revolve around this super Lie algebra alone. Here we find that it is but a low-degree shadow of the full automorphism Lie 7-algebra of 11-dimensional supergravity. Lots of things to be investigated here.
A similar situation occurs in the field that has come to be known as AKSZ theory. Originated by Kontsevich, Schwarz and others, this has been and increasingly is playing a central role in the dg-algebraic physics that has its roots in homological mirror symmetry and whose relevance is boosted by the formal overlap with the BV-theory for quantization of gauge theories.
There are two separate aspects to this, which have not always been cleanly separated in the literature. On the one hand is the fact that the fields of AKSZ sigma-models have the interpretation of nonabelian differential cocycles. This has been fairly clear and has been fairly clearly recognized in the literature, but also here only for the same special case as before for D'Auria-Fré theory: only for configurations whose underlying ordinary cohomology class is trivial. Recently there have been attempts to guess what a globally non-trivial AKSZ field configuration would be. But due to the high gauge freedom, guessing these structures is rarely successful. In (FRS11) we point out that with the systematic theory of higher nonabelian differential cohomology applied, all this can be derived.
The first non-classical target space object in the AKSZ hierarchy is the Courant Lie 2-algebroid. This carries a canonical higher analog of a prequantum line bundle which is a generalization of the nonabelian String 2-group mentioned before (US12). The question of finding finite-dimensional presentations for the Lie 2-groupoid Lie integrating has recently found a lot of attention (LS11). The special case of the standard Courant algebroid is the object that the whole theory of generalized complex geometry revolves around. In string theory this controls the T-duality and U-duality symmetries and describes generalized Calabi-Yau spaces for compactification models. Moreover, under Lie integration all these exceptional generalized geometries underlying higher dimensional supergravity are naturally encoded in terms of twisted smooth nonabelian cohomology (US12, section 4.4.3).
This and its generalization to a full theory of higher symplectic geometry we have shown in (FRS11) to naturally sit inside nonabelian differential cohomology as a part of higher Chern-Weil theory: the graded symplectic form of the symplectic Lie n-algebroids considered in AKSZ theory, and more generally the n-plectic forms on more general symplectic infinity-groupoids may all naturall be interpreted as invariant polynomials on the underlying infinity-Lie algebroids in higher generalization of classical non-abelian Chern-Weil theory.
As a special case, this leads us to the second aspect of AKSZ theory, which is sometimes conflated with the first. This is the fact that an AKSZ -model is given by a very particular action functional on these higher gauge fields. This class of action functionals has been found from certain nice dg-algebraic properties that it has. But the dg-algebras appearing here are in the end just a model for some intrinsic higher stacks. We showed in (FRS11) that these action functionals have a deeper origin in a higher analog of Chern-Weil theory. More generally, we show in ( SSS09, FSS10) that every higher analog of an invariant polynomial on a higher analog of a Lie algebra canonically induces an action functional on higher nonabelian gauge-fields. The AKSZ -models are but the special subclass where the invariant polynomial happens to be binary and non-degenerate.
We show that when one scans the space of higher invariant polynomials beyond those satisfying this restriction, one finds, among the classes of higher nonabelian gauge field theories that appear this way, a whole zoo of systems that are known by other means in the literature. For instance the action functional of bosonic string field theory is of also this form. The 7-dimensional Chern-Simons theory on nonabelian 2-form fields is an example, and many others. Also theories for discrete higher gauge groups such as the Yetter model are all unified as special cases of higher Chern-Weil theory on higher differential cohomology.
All these action functionals on higher nonabelian cocycles are in a way of “higher Chern-Simons type”. We are currently working on an article (slides) which shows that to all of these is associated the corresponding sigma-model of “higher WZW type”. Examples of these turn out to be for instance the superbrane action functionals of Green-Schwarz type, on which a large parts of superstring theory rests. These are in fact induced by those very same higher super Lie algebras that secretly appeared in the D'Auria-Fre formulation of supergravity. The point of view of higher nonabelian differential cohomology is providing structural insights here that were invisible before. People are beginning to investigate these (H11).
In view of these classes of examples, I have come to wonder about a question converse to the sometimes-heard “Is there any natural example of nonabelian differential cohomology in physcis?” In view of the above, I am led to wonder if we are seeing accumulating evidence justifying the question: “Is there much in fundamental physics that does not originate in natural constructions in nonabelian differential cohomology?”
But even if we accept that higher analogs of Yang-Mills theory play a role in physics, is there really new mathematical structure at work here which cannot be all absorbed into abelian differential cohomology, after all?
This question is best answered guided by classical insight into bare (non-differential) abelian and nonabelian cohomology.
There, nonabelian cohomology takes place in the homotopy category of topological spaces, or more generally in the homotopy theory of higher stacks. On the other hand, abelian cohomology takes place in the stable homotopy category.
These two systems reflect on each other. Two classical facts describe how abelian cohomology serves to understand a good chunk of nonabelian cohomology: first, all homotopy groups beyond the first are abelian. And closely related to this: every space has a Postnikov decomposition, where each but the first step is classified by abelian cohomology.
It implies that we may always decompose higher nonabelian cocycles into tuples consisting of a nonabelian 1-cocycle and a higher abelian cocycle on the extension classified by this.
For instance, a good bit of the literature on (nonabelian) String-2-bundles describes them as abelian circle 2-bundles (bundle gerbes) on the total spaces of (nonabelian) Spin-principal bundles, satisfying some conditions.
This is very useful. Often in the literature one sees this way an abelian cohomology set up with a nonabelian datum added somewhere by hand. But one cannot understand String-2-bundles in terms of just abelian cohomology on Spin bundles. Where do the extra conditions come from? What are their higher generalizations? For instance what is the analogous description of Fivebrane-6-bundles, which may be decomposed as abelian 6-bundles over abelian 2-bundles over nonabelian 1-bundles? What are the correct morphisms and higher morphisms of these structures? What are twisted String-2-bundles (analogous to twisted bundles?), twisted Fivebrane-6-bundles, etc.?
Partial guesswork is possible and has been done, sometimes successfully, sometimes not. But all such question need nonabelian cohomology for a systematic understanding and have a systematic answer there. Once understood, everything can of course be decomposed into its components again, if desired and if useful for explicit computations.
This is closely related to the notion of twisted cohomology as represented by parameterized spectra, which is also naturally understood as a part of nonabelian cohomology. Twisted cohomology with coefficients in some abelian is all controled by the fiber sequence of nonabelian coefficient objects , where is the (nonabelian, in general) twisting group.
The twisted nonabelian 2-form gauge fields that we identify (FSS12) on the M5-brane are controled by twisted cohomology of this form. Again, some aspects of these have been guessed before (AJ), and we could derive that guess as correct from the general theory. But there are more subtle issues that are hard to guess. Certainly as we go to higher degrees, any pedestrian component approach is doomed. For the magnetic dual twisted Fivebrane 6-cocycles that we will discuss in (…) a systematic theory of higher nonabelian differential cohomology seems to be the only way to make any progress.
In conclusion, nonabelian differential cohomology turns out to be quite ubiquitous, once one knows what to look for. In existing literature it secretly appears reduced to its trivial sectors, or decomposed into its components, or otherwise unrecognizable. Identifying it provides answers to pressing open questions, gives unexpected relations between existing theories, and points to a wealth of further structures previously unrecognized and now waiting to be explored.