The String Coffee Table

I have just returned from my two week visit of John Baez’s group at UC Riverside, which was extremely nice and productive. I hope will be able to reveal some new insights soon.

Now I am about to go on vacation in Sweden. In March I will return from Östersund directly to Berlin to attend the annual DPG meeting where I give a short talk on 2-bundles and nonabelian strings.

Similar talks are due end of March at Problemi Attuali di Fisica Teorica in Vietri/Italy by kind invatation by Paolo Aschieri and end of April at the math or hep seminar at LMU in Munich, thanks to Branislav Jurčo and Ivo Sachs.

This is the rough schedule of the Vietri conference for March 24:

R.Zucchini - ‘A sigma model field theoretic realization of Hitchin’s generalized complex geometry’

M.Zabzine - ‘Generalized complex geometry and supersymmetry’

F.Bonechi - ‘Non perturbative Poisson Sigma model and Geometric quantization’

J.Michelsson - ‘Families of index theorem in supersymmetric WZW model and twisted K theory’

J.Evslin - ‘What K theory does and doesn’t classify’ (see also this)

U.Schreiber - ‘Nonabelian Strings’

T.Strobl - ‘Algebroid Yang-Mills Theories’

Many people seem to be interested in nonabelian strings and categorified gauge theory. If everything works out I’ll visit Zoran Skoda in Bonn in April or March to talk about these things.

In this entry here I want to develop some notes on categorification of Not necessarily Commutative Geometry by means of spectral triples in the sense of Connes, and, eventually and if possible, relations of this to superstrings, loop space geometry and maybe generalized cohomology.

This entry will be work in progress for a while since I plan to develop these notes as I go along.

The original reason for developing these notes in the public is actually the following: I usually type all my notes on my notebook computer. Currently I am visiting John Baez and his group at UC Riverside. It seems that the screen of my notebook computer did not survive the security checks at the airport, unfortunately, so I am left without my crucial thinking tool. This weblog must now serve as a substitute.

But maybe it is an interesting experiment in its own right. I’d be grateful for anyone interested in accompanying me on my quest for seeing more of the big picture of which ordinary NCG, superstrings, loop spaces, Dirac operators, 2-bundles, etc. are jigsaw pieces. Collaboration on papers not excluded.

I had been thinking and talking about related ideas for a long time already. But it was not before last Sunday when I was sipping hot chocololate at Starbucks late at night that something occurred to me which suddenly made all the pieces fall in place.

The crucial catalyst was John Baez’s remark that what I was talking about seemed to call for the 2-Hilbert spaces, $2-\mathbb{C}^*$-algebras and the categorification of the Gelfand-Naimark theorem he discussed in [1]. Meditating over that paper I realized that it should be possible to start with a bosonic 2-$\mathbb{C}^*$-algebra and form its abstract differential 2-calculus by throwing in a formal $\mathbf{d}$-functor. By an analogous procudure one can obtain nice ordinary spectral triples, as I have discussed together with Eric Forgy in [2].

Therefore I propose that a 2-spectral triple should be a triple

consisting of an 2-algebra $A$, a 2-Hilbert space $H$ on which this algebra acts functorially and on which a 2-Dirac operator $D$ is represented.

My first step shall be to develop in detail a simple but interesting example for such a 2-spectral triple, namely one describing discrete superstrings. Depending on how that works out there are many directions to follow and things to work out.

Over on sci.physics.strings Kea and I seem to have found a lot of very interesting things to talk about, related to NCG, categories and - I hope - categorified NCG, but it becomes quite of a stretch to consider this discussion on-topic for sps. Therefore I’ll reply to Kea’s latest post here at the Coffee Table.

In his reference frame, Lubos reports on a seminar by Reall on higher dimensional black holes and black rings and mentions that they violate a possible no hair theorem.

In addition to what he says there, even in 4D, if you go beyond Einstein-Maxwell by including non-abelian gauge groups, there are different solutions whose difference decays exponentially if you go to infinity and thus are not distinguished by ADM type charges. Thus, in Einstein-YM, there is hair even in 4D.

I would like to kick off a discussion of what we should make of this. Is the no hair theorem just a coincidence of a small class of theories or does it have a fundamental meaning or relevance (as for example cosmic censorship, a violation of which would have bad consequences for predictability)?

If you want to discuss a larger class of theorems, you can follow V. GATES, Empty KANGAROO, M. ROACHCOCK, and W.C. GALL in Stuperspace where they mention

“a black hole has no hair (the `Fuzzy Wuzzy’ theorem), you can’t comb the hair on a billiard ball, you can’t lasso a basketball, you can’t peel an orange without breaking the skin, you can’t make an omelet without breaking eggs, you can’t push a rope, you can’t roller-skate in a buffalo herd, and you can’t take a shower in a parakeet’s cage.”

Point particles are described by nonabelian gauge theory. We can lift this by stringifying/categorifying and realize the point particles as boundaries of strings described by abelian 2-bundles. We can lift this again to M-theory where the strings become membranes attached to 5-branes. By all what is known these membranes are described by abelian 3-bundles with their boundaries coupled to nonabelian 2-bundles.

On the physics side we went up from 0-branes to 1-branes to 2-branes. On the math side we went from 0-categories (sets) to 1-categories to 2-categories, i.e. from groups to 2-groups to 3-groups.

This is curious for the following reason: We know that on the physics side there is no further step. While there do exist $p$-branes with $p \gt 2$ these are not fundamental like the F-string and the M2 are, and – more importantly for the above pattern – there is no abelian 3-brane whose boundary would be a nonabelian membrane (as far as I am aware at least).

So what happens when we increase on the math side the dimension ever more? Can we get gauge theories for abelian 15-branes whose boundaries are nonabelian 14-branes?

I think the answer is: ‘NO’. More precisely, I think the following is true:

- 1-gauge theory admits arbitrary gauge groups $G$

- 2-gauge theory admits gauge groups coming from crossed modules subject to the constraint of vanishing fake curvature

- 3-gauge theories admit gauge groups coming from crossed modules of crossed modules which are fake flat at the level of 2-morphism and abelian at the level of 3-morphisms

- necessarily all higher $(p \gt 3)$-gauge theories are abelian at the $p$-morphism level

This looks interesting, because it sort of solves the above puzzle: The membrane is the endpoint in a chain of ‘stringifications’ and the gauge theory it couples to is an endpoint in a chain of categorifications in the sense that beyond it no nonabelian gauge theory is possible.

Here is a more precise statement of what I am saying together with what I think is the proof for it.

Recently I had reported work on 2-connections in 2-bundles. There are several loose ends that still need to be tied up, though. For instance I claimed that one of the important points of hep-th/0412325 is the conception of global nonabelian surface holonomy. But the formula for computing this from the global nonabelian 2-connection has not made it into that, already long, paper. So I’ll present it here. In fact, not a formula, but a picture.

The definition and consistency proof of global nonabelian 2-holonomy is a rather enchanting exercise in ‘category LEGO’, also known as diagrammatic reasoning.

Instead of writing down formulas it proves to be way more elegant to draw diagrams and fit them together like LEGO blocks. In fact, this is not just more elegant. This ‘higher dimensional algebra’ reduces the definition and consistency of global 2-holonomy from something totally impenetrable in terms of formulas into an easy LEGO game. (If you don’t believe this and feel bored, try to rederive the following in terms of formulas.)

So to get a feeling for what I am talking about first consider global line holonomy in an ordinary fiber bundle.

This involves partitioning your manifold in open sets $U_i$ and proceeding as indicated in this picture:

Paths $\gamma_i$ get mapped to group elements $W(\gamma_i)$. Note how the point $x$ is sort of ‘blown up’ under the holonomy map, being sent to a nontrivial group element $g_{ij}(x)$.

Under gauge transformations the local holonomy transforms under what I shall call here a 1-dimensional onion move for reasons to become clear shortly:

Gauge invariance of the global holonomy is a consequence of cancelling of adjacent ‘onion slices’:

Here and in the following $\tilde{(\cdot)}$ is the gauge transformed version of $(\cdot)$.

Note how it is sufficient for proving global gauge invariance to prove that every piece of holonomy transforms under gauge transformations only by being conjugated at its ‘boundary’ while the ‘internal’ conjugations mutually cancel.

That shall suffice as introduction. Now consider a principal 2-bundle. (For convenience, restrict attention to the special case where base 2-space has only identity morphisms and where the structure 2-group is strict.)

The usual identities between transition functions become 2-arrows in the 2-group:

From this one reads off the gauge transformation of the transition 2-morphism $f$ by looking at the 2-dimensional onion move

Now put a 2-connection on this 2-bundle. A gauge transformation on a piece of local 2-holonomy (which is a 2-morphism in a 2-group) is a ‘2-conjugation’:

In particular the transition from one patch $U_i$ to another one $U_j$ has been shown to be of this form:

But a local gauge transformation is completely analogous:

Note that 2-conjugation is a 2-similarity transformation and respects 2-compositon:

This means that, just as in the ordinary bundle holonomy discussed above, it is sufficient for proving the gauge invariance of some quantity to demonstrate that any piece of it under gauge transformations gets 2-conjugated only from the periphery.

We shall be particularly interested in the surface part of the 2-conjugation of 2-holonomy:

This is the onion skin $a_{ij}$ in the 2-conjugation law for the transition law of local 2-holonomy drawn above. (I realize that the resolution of the graphics is a little too low for the labels to be easily readable. You can find a better readable pdf file here.)

By performing a 2-transition first in one gauge and then in the other, one can read off the gauge transformation law for the $a_{ij}$ by combining all of the above diagrams iteratively into the following :

See why I call this ‘onion moves’?

By the magic of diagrammatic math we can simply read off the sub-onion move which describes the gauge transformation of $a_{ij}$:

Ok, so much for the derivation of 2-gauge transformation in a 2-bundle with 2-connection. Now let’s define global 2-holonomy.

First triangulate the surface whose holonomy is to be computed. This can always be done in such a way that the resulting graph is trivalent and that every face comes to lie in a single $U_i$, every edge in a double overlap and every vertex in a triple overlap (just as in the prescription for 2-holonomy in an abelian gerbe, which we will recover as a special case of our non-abelian surface holonomy).

Certainly we have to assign local 2-holonomy (computable in terms of path space holonomy) to faces contained in a single $U_i$. Then, there is only one candidate 2-group element to be assigned to edges in double overlaps, namely that $a_{ij}$. Furthermore there is only one candidate 2-group element to be assigned to vertices in triple overlaps, namely $f_{ijk}$. Finally, there is only one way to glue all them together.

Hence category-LEGO logic tells us that the definition of global 2-holonomy must be as follows:

There is no other choice, hence this must be right.

As a first check, those who know the well-known definition of global surface holonomy in an abelian gerbe and who recall how we derived that the above symbols are related to the gerbe cocylce data will see at a glance that this diagram correctly reproduces the gerbe holonomy formula in the case where the 2-group involved is abelian.

But we should directly prove that the above is a well defined prescription for global 2-holonomy in the general (non-abelian) case. It’s easy using diagrams:

First consider computing global surface holonomy in the gauge $\tilde{(\cdot)}$:

Now express all the $\tilde{(\cdot)}$ objects here in terms of the original gauge, by using the onion moves derived above on each piece:

One sees that one can cancel 2-morphism $a$ against their reversed version in six places, which have been shaded in the diagram. Doing so yields:

Here we also turned some arrows around by ‘whiskering’ so that now 2-morphisms $p$ cancel against their reverses:

What is left is the 2-holonomy in the original gauge, 2-conjugated on the periphery. As discussed above, this is sufficient for proving full gauge invariance, since these 2-conjugations will cancel against their counterparts at the other vertices and edges.

(Of course in the end, when computing 2-holonomy of a closed surface, two unpaired edges will be left, the source and the target edge. Just like for line holonomy, they must be paired by a suitable trace operation.)

So this is how global nonabelian surface holonomy works in a principal 2-bundle with 2-connection. As a special case this reproduces the well-known formula for global abelian 2-holonomy as derived in the context of abelian gerbes.