### E. Sharpe on Derived Categories

#### Posted by urs

Eric Sharpe watched my feeble attempts (I,II) to say something about derived categories and he decided to help out. Here are his remarks. Also check out his article

E. Sharpe

Derived Categories and D-Branes

Encyclopedia of mathematical physics

With kind permission, Eric’s comments:

**[begin of forwarded message]**

On a different subject, I was reading some of your older blog entries, and discovered you had an interest in derived categories. :-) [I’m beginning to think I ought to read people’s blogs more often, there’s often good info there.]
Just for fun, I’ve attached the proofs of an article I wrote for the
*Encyclopedia of mathematical physics* on derived categories in physics.

Richard Thomas has already written an excellent article on the mathematics, so I tried to concentrate on the physics. [I’m not entirely happy with how it turned out, but there was a tight page limit, and squeezing everything into place was a bit tricky.] I’m not planning to put it on hepth.

The history of derived categories in physics is a bit unusual.
Back in ‘93-‘94, *before* D-branes, Kontsevich made a conjecture for an interpretation of mirror symmetry in terms of derived categories. At the time, nobody had a clue what it could mean physically. In ‘95, D-branes were introduced, and in ‘98-‘99, antibranes and their K-theory interpretation came to the fore.
In hepth/9902116, I conjectured that the derived categories that Kontsevich was writing about, should be interpreted in terms of brane/antibrane systems, and that’s the basic picture that people believe today.

After all, in some sense a derived category of sheaves is a refinement of ${K}^{0}$. Very roughly, given a virtual bundle $E-F$, imagine adding a gauge field to each of E and F. Really, what you get more nearly is a Grothendieck group, of which the derived category is a further refinement, but for the purposes of this email, I’ll ignore the distinction, and claim that a derived category is roughly ${K}^{0}$ + gauge fields.

This also means that there’s a potential problem here that’s closely related to one appearing in stacks. A given (isomorphism class of) object in a derived category can have multiple presentations in the form of complexes of locally-free sheaves. The conjecture is that (boundary) renormalization group flow on the worldsheet realizes localization on quasi-isomorphisms, which can’t be directly checked, so instead one performs lots of indirect tests…

At the time, my paper was pretty much ignored. [I was a young guy, and there weren’t any exciting predictions coming out of it, aside from an interpretation of Kontsevich’s conjecture.] A few years later, Douglas wrote up the same picture and also introduced the notion of pi-stability, which took off. The Aspinwall-Lawrence paper a few months later came up with a nice picture of why ${d}^{2}=0$, the property defining the complex, should appear physically: giving tachyons vevs deforms the BRST operator, so when we demand that the new BRST operator square to zero (a necessary but not sufficient condition for the B model to continue to make sense), you get the condition ${d}^{2}=0$, ie, you get complexes.

This still wasn’t very satisfying because, although one could certainly map complexes to brane/antibrane/tachyon systems, the converse was less pretty: why should a set of tachyons between branes & antibranes be describable as maps between neighboring elements in a chain, ie, as a complex? Why can’t you have maps between non-neighboring elements? Why must everything align as complex?

Calin Lazaroiu observed that, in general, they won’t. The tachyons can realize something more complicated than an ordinary complex, and demanding that the (new) BRST operator square to zero imposes a correspondingly more complicated condition. Moreover, the resulting picture was already known in the math community: it’s the Bondal-Kapranov enhancement of derived categories, which mathematically behave better than ordinary derived categories.

There were (& are) still lots of open questions at this point. For example: massless states. For years now, people had assumed that when D-branes are described by sheaves, the massless states between the branes should be counted by $\mathrm{Ext}$ groups, but nobody had ever succeeded in writing down vertex operators. Aspinwall-Lawrence had an argument in the massive nonconformal theory for why that should be true, but their argument was a bit fast, and could only apply to branes/antibranes wrapped on the entire space. (To apply it to more general cases involves assumptions about the behavior of RG flow, which have to be checked….)

The first paper to figure out how to assign vertex operators (really, RR states) in the open string CFT to $\mathrm{Ext}$ group elements was written by myself and Sheldon Katz a few years ago, and the story is somewhat involved.

For example: ordinarily massless states are counted by (bundle-valued) differential forms, but $\mathrm{Ext}$ groups have no realization, so how can it be made to work? The answer is that worldsheet physics realizes a spectral sequence; boundary conditions on the open string coming from the Chan-Paton factors realize the spectral sequence.

The Freed-Witten anomaly also plays an important role – the precise D-brane/sheaf dictionary is slightly more complicated than people had previously assumed. There are lots of subtleties; what emerges is a much more clear picture of what works and why than could be obtained from the UV picture of Aspinwall-Lawrence.

There are lots of other questions – eg, to which sheaves can one associate D-branes, and how? What D-brane can be assigned to the structure sheaf of a nonreduced scheme, for ex? This particular question was answered in some work I did with Sheldon & Ron Donagi, turns out the nonreduced scheme structure corresponds to nilpotent Higgs fields on the D-brane. Ordinarily nilpotent Higgs vevs aren’t allowed b/c of D-terms, with some important exceptions:

a) in a TFT, where D-terms don’t appear

b) in orbifolds – those nilpotent Higgs fields are the reason why D-branes on orbifolds see a resolution of the quotient space.

If you turn the paper on its head, you can get some very efficient computations of $\mathrm{Ext}$ groups between fairly ugly sheaves, by turning the Ext group computation into a physics computation, but that’s another story.

There’s a review article I wrote, hep-th/0307245, with more details, if you’re interested. :-)

A few more references:

The paper I wrote with Sheldon on explicitly seeing Ext groups via open string CFT physics is hepth/0208104.

The paper with Sheldon and Ron Donagi finding D-brane interpretations for more sheaves than just, pushforwards of vector bundles, is hepth/0309270. There was an older paper I wrote with Tomas Gomez making the same observation, but the one with Sheldon & Ron does a better job of explaining and justifying the dictionary.

**[end of forwarded message]**