The String Coffee Table

Quiver reps and vector 2-bundles

Posted by Urs Schreiber

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.