## March 11, 2005

### 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 $\mathrm{Rep}\left(Q\right)$ of representations of the quiver, with objects being functors

(1)$f:C\to K-\mathrm{Vect}$

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 $\mathrm{Rep}\left(Q\right)$ 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 $\mathrm{Rep}\left(Q\right)$ 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 $\mathrm{Rep}\left(Q\right)$ 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 $n×n$-matrices of elements of $\mathrm{Rep}\left(Q\right)$. 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 $\mathrm{Rep}\left(Q\right)$ does not have additive inverses (its decategorification gives $ℕ$-valued functions instead of $ℤ$-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 $\mathrm{RepI}\left(Q\right)$ 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 ${\mathrm{GL}}_{\mathrm{VectI}}\left(n\right)$ of $n×n$-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 ${\mathrm{GL}}_{n}\left(V\right)$.

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 $\mathrm{RepI}\left(Q\right)$ 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 ${\mathrm{GL}}_{\mathrm{VectI}}\left(n\right)$.

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.

Posted at March 11, 2005 3:09 PM UTC

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### Re: Quiver reps and vector 2-bundles

The derived category includes more than just brane/anti-brane information. Rather than just having something Z_2 there’s at least a Z_6 and probably a full Z-grading (see Douglas’s papers.) In particular, in the derived category, you have a shift functor [n] for all values n in Z.

Posted by: Aaron on March 11, 2005 5:14 PM | Permalink | Reply to this

### Derived categories

Thanks. What I wrote is probably better related to K-theory than to derived categories, if at all.

I am beginning to get the idea of derived categories, but I am still confused about many details.

In hep-th/0011017 Douglas discusses that grading issue on pp. 4 and then pp. 8.

Branes are graded there by the phase of their BPS central charge, normalized so as to be integral. Branes and anti-branes then differ by an odd difference in that phase.

Is that grading the same as the ghost number grading that Aspinwall has in hep-th/0403166?

Here are some other questions:

I am confused by the $\mathrm{Ext}$-notation all over the place. What does $\mathrm{Ext}$ denote, really?? (See, that’s how far off I still am…)

Is it right that the fact that the objects of the derived category $D\left(C\right)$ are complexes in the category $C$ accounts for the fact that, in the case of sheaves, these complexes represent bound states of D-branes when the complexes are not split, as opposed to just several stacks of D-branes without any strings connecting them?

On sps you wrote:

It turns out that derived category of quiver representations is equivalent to the derived category of coherent sheaves you started with.

Let’s see, I’d like to have a more heuristic understanding of this. The derived category of quiver reps has complexes of quiver reps as objects. This should mean that I can think of a D-brane configuration as a complex of quiver reps.

Hm. From the deconstruction picture I know that a quiver rep describes a condensate of certain fermion pairs in the bifundamental which at the same time seems to be sort of a Matrix-theory description of a space. Probably that’s not directly the picture we need here, though. Er, any ideas?

I hope to be able to ask better questions soon.

Posted by: Urs Schreiber on March 11, 2005 6:13 PM | Permalink | PGP Sig | Reply to this

### Re: Derived categories

For the grading, check out eqn (192) in Aspinwall.

Ext is a derived functor which takes a lot of machinery to define. But, if A and B are vector bundles then

Ext^i(A,B) = H^i(A^* (x) B)

I’m not sure I understand the rest of your questions, but, yes objects in the derived category can represent derived states (in the B-model – in general you have to worry about stability).

For the quivers I’m interested in, the picture is D-branes at a singularity. You get the quiver because the brane because marginally unstable to decay to the ‘fractional branes’ which are the nodes of the quiver (roughly). The arrows in the quiver represent strings stretching between the fractional branes.

Posted by: Aaron Bergman on March 11, 2005 6:25 PM | Permalink | Reply to this

### Re: Derived categories

Thanks again, very helpful. Gotta run now. More later.

Posted by: Urs Schreiber on March 11, 2005 6:30 PM | Permalink | PGP Sig | Reply to this

### Re: Quiver reps and vector 2-bundles

that. Not only concerning charges, but also eg about where the
branes are located. Eg a D0 brane at a certain point on a CY is
described as a different object than a brane at a different location.
So the category also contains info about moduli, in contrast to crude
constructions based on charges, such as K-theory.

That’s why the LG formulation (of B-branes, say) is so useful - it
provides a concrete TFT realization of the relevant features
of the category, in which the objects and maps can be given concrete (moduli-dependent) representations in terms of LG fields.

Posted by: WL on March 12, 2005 9:07 AM | Permalink | Reply to this

### Re: Quiver reps and vector 2-bundles

Actually the derived category contains even much more info than that. Not only concerning charges, but also eg about where the branes are located.

Ok, I think I roughly understand how that works thanks to the example on pp. 38 of Aspinwall’s hep-th/0403166. Also from the discussion on p. 60 on how K-theory is reobtained from $D\left(X\right)$ it is manifest that the latter contains much less information than the former.

Now, is it true that the derived category picture knows all there is to know about the branes? In particular, when including the stability discussion of section 6 of Aspinwall, is it true that derived categories give a complete description of ‘physical’ D-branes?

That’s why the LG formulation (of B-branes, say) is so useful

Er, I might have a blackout here, so apologies if this is a dumb question, but what do you mean by ‘LG’?

Posted by: Urs Schreiber on March 13, 2005 6:02 PM | Permalink | PGP Sig | Reply to this

### Re: Quiver reps and vector 2-bundles

>Now, is it true that the derived category picture knows all there is to know about the branes? In particular, when including the stability discussion of section 6 of Aspinwall, is it true that derived categories give a complete description of ‘physical’ D-branes?

These are though questions. At least for topological branes it seems that the category captures the relevant information (plz don’t ask for a precise definition of “relevant”); this can be shown for simple cases such as minimal models or the elliptic curve. What this implies for physical branes, well I don’t know…

> I might have a blackout here, so apologies if this is a dumb question, but what do you mean by ‘LG’?

Means Landau-Ginzburg. Well this was just to promote in what I am interested in ;-) But indeed I believe that the boundary LG formulation is as close as one can get to a physical representation of the derived category. Before I viewed this category stuff as quite esotheric of little practical use. But now by using LG models one can translate these abstract notions into very concrete physical quantities such as matrix valued chiral fields and superpotentials, and actually do calculations with them. This is much more than just counting maps or computing K-charges (ie, intersection numbers).

Posted by: WL on March 13, 2005 7:04 PM | Permalink | Reply to this

### Landau-Ginzburg

Means Landau-Ginzburg.

Ah, I see. I guess I should have a look at your hep-th/0305133.

Posted by: Urs Schreiber on March 13, 2005 8:12 PM | Permalink | PGP Sig | Reply to this

### Re: Landau-Ginzburg

IIRC, these LG categories, while triangulated, are not equivalent to the usual derived category. I think that, for example, they have T^2 = id where T is the translation functor.

Posted by: Aaron Bergman on March 13, 2005 8:18 PM | Permalink | Reply to this

### Re: Landau-Ginzburg

I guess it should be possible to ‘divide out’ the full derived category by the action of ${T}^{2}$? Maybe there is a forgetful functor

(1)$D\left(𝒞\right)\to \mathrm{LG}\left(𝒞\right)$

?

Posted by: Urs Schreiber on March 13, 2005 8:28 PM | Permalink | Reply to this

### Re: Landau-Ginzburg

>IIRC, these LG categories, while triangulated, are not equivalent to the usual derived category. I think that, for example, they have T^2 = id where T is the translation functor.

Thats certainly true in the simplest case, which is minimal models.
But for tensor products of those (which include CY’s), one can
generate longer sequences (I think Diaconescu et al have described
that), and also in the original papers of Orlov there are statements
in this direction, so there is no fundamental obstacle. But indeed
the full story hasn’t yet been worked out. Linear sigma models
would be a good way to attack this issue (actually there is forthcoming
work on this).

Posted by: WL on March 14, 2005 5:43 PM | Permalink | Reply to this

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