The String Coffee Table

You all know that I have been thinking about 2-holonomy a lot, lately. Hence of course a paper by E. Akhmedov which appeared today

E. Akhmedov, Towards the Theory of Non-Abelian Tensor Fields I

attracted my attention with its abstract, which reads

We present a triangulation-independent area-ordering prescription which naturally generalizes the well known path ordering one. For such a prescription it is natural that the two–form ‘connection’ should carry three ‘color’ indices rather than two as it is in the case of the ordinary one-form gauge connection. To define the prescription in question we have to define how to exponentiate a matrix with three indices. The definition uses the fusion rule structure constants.

I have just read through this paper and I think the idea is what I am going to summarize in the following. My presentation is a little different from E. Akhmedov’s in that I take his last remark right before the conclusions as the starting point and motivate the construction from there.

[Update: Some discussion of these issues is taking place here.]

This time I will sketch the key steps in the argument that demonstrates that D-branes on CY with strings between them are described by the derived category of coherent sheaves on that space. In other words, I review Aspinwall’s review in a way that makes the key step memorizable.

Part I and II of this series of entries dealt with elements of derived categories and derived functors on them. Now it is time to begin discussing some applications. Here I start talking about the derived category of quiver representations. Actually it turns out that I just review a small number of very elementary facts and soon end up speculating about how 2-bundles show up here.

I spent yesterday sitting on the beach at Vietri and reading Weibel, ‘An introduction to homological algebra’, trying to understand the details of derived functors. It takes many, many pages to really define them in detail, but here I try to summarize the key steps.

Today’s sessions in Vietri are about quantum computation. Most of them go into more and different detail than I currently care about, but one had an interesting provocative theme related to high energy physics.

Next week the Istituto Internazionale per gli alti Studi Scientifici hosts the conference Problemi Attuali di Fisica Teorica in Vietri sul Mare (Italy). As I have mentioned before, I’ll give a talk on ‘nonabelian strings’ concerned with
2-bundles, 2-connections and 2-holonomy and the relation to nonabelian gerbes.

Here are the slides: VietriTalk.ps.

(Special thanks go once again to Eric Forgy for having provided the picture that I display on the title slide.)

Here is the abstract:

Intersections of membranes with stacks of M5-branes are expected to give rise to a dimensional generalization of nonabelian gauge theory from point particles to strings. This can be described in the language of nonabelian gerbes. A gerbe is a generalization of the concept of a sheaf of sections of a fiber bundle obtained by a method known as categorification. Applying the same method to the more direct description of bundles in terms of projection maps and local trivializations yields the concept of a 2-bundle. We describe the theory of 2-bundles in general, how gerbes are obtained from 2-bundles and how global nonabelian surface holonomy can be defined using 2-bundles. This should play a role both for action principles of nonabelian strings as well as for elliptic cohomology.

I am still trying to learn about derived categories, mostly using

P. Aspinwall: D-Branes on Calabi-Yau Manifolds hep-th/0403166 (2004).

For my own good here I’ll try to review a couple of key ideas. Corrections are welcome.

I am currently trying to learn about quivers and derived categories. I hope to have something more substantive to say/ask soon, but right now I would like to clarify a statement I made on s.p.s. in conversation with Aaron Bergman, which was probably not well formulated, but which should have some relation to quivers/derived categories, which I would like to understand.

So the simple observation is this:

Given a quiver $Q$ with associated graph category $C_Q$ the category $\mathrm{Rep}(Q)$ of representations of the quiver, with objects being functors

and morphisms being natural transformations between these, is not just an abelian category but actually a (strict) monoidal category with the product functor being given the vertex-wise and edgewise tensor product.

Hence $\mathrm{Rep}(Q)$ is actually a 2-algebra. As emphasized in HDA2, 2-algebras of this kind should be thought of as a categorified version of the algebra of complex-valued functions on some space.

I mention this because it suggests to look at finitely generated projective (2-)modules of the 2-algebra $\mathrm{Rep}(Q)$ and address them as vector 2-bundles. Thinking in terms of deconstruction we can think of the quiver as a discretized 2-space which should be the base 2-space of these vector 2-bundles.

So these modules of $\mathrm{Rep}(Q)$ are spaces of 2-sections of a 2-bundle whose typical fiber is like $\mathrm{Vect}^n$.

Let’s be naïve, assume the continuum limit and demand that our bundle is locally (2-)trivializable. The transition 2-maps will be something like $n\times n$-matrices of elements of $\mathrm{Rep}(Q)$. Restricting to the special case that these transition functions involve only identity maps associated to edges (that’s the assumption you need to make to get gerbes from 2-bundles!) and imposing the obvious conditions on them leaves us with precisely the vector 2-bundles studied by Bass, Dundas & Rognes.

One important point of the whole derived category business is that anti-D-branes are correctly included into the picture. In a vaguely related form precisely this aspect arises here.

Since $\mathrm{Rep}(Q)$ does not have additive inverses (its decategorification gives $\mathbb{N}$-valued functions instead of $\mathbb{Z}$-valued ones) the above mentioned transition 2-maps are not really transition 2-maps, since they are not invertible! BD&R in their section 3, discusss the abelian group completion, which amounts to throwing in formal additive inverses.

If we think of the vector spaces sitting over vertices as the Chan-Paton spaces of the stack of D-branes at that point, as in the derived category picure, then this amounts to accounting for anti-D-branes.

So let $\mathrm{VectI}$ be the ‘group completion’ of $\mathrm{Vect}$ by inclusion of formal additive inverses and let $\mathrm{RepI}(Q)$ be the 2-algebra of representations of $Q$ in $\mathrm{VectI}$ instead of $\mathrm{Vect}$.

I believe there is an obvious and honest strict 2-group $\mathrm{GL}_{\mathrm{VectI}}(n)$ of $n\times n$-matrices with entries in $\mathrm{VectI}$. Restricting it to the sub-2-group with all morphisms the identity and then restricting again to the ‘semi-2-group’ with only ‘non-negative’ entries should give (unless I am mixed up) what BD&R call $\mathrm{GL}_n(V)$.

Does anyone see why BD&R use this instead of the full invertible (2-)group for the transition functions of their 2-bundle? I might have to think harder, but it seems to be that the finitely generated projective ‘2-modules’ over $\mathrm{RepI}(Q)$ are honest locally trivializable 2-bundles with typical 2-fiber $\mathrm{VectI}^n$ and (invertible as it should be) transition 2-maps taking values in $\mathrm{GL}_{\mathrm{VectI}}(n)$.

The point is that once we have these honest 2-bundles we know how they gives rise to nonabelian gerbes, to connection, curving and 2-holonomy, etc. Maybe their cohomology is even closer to elliptic cohomology than that of the bundles considered by BD&R??

I have the strong feeling that all this has a tight connection to derived catorical description of D-branes, but before speculating about that at this point I will continue familiarizing myself with this stuff a little more.