The String Coffee Table

So the claim ist that the structure that describes D-brane configurations is a derived category of coherent sheaves. More precisely, this is one of several ways to describe this setup, as there turn out to be other, equivalent ways. For instance this derived categroy of coherent sheaves is equivalent to what is called a triangulated Fukaya categroy and also to (at least for a large number of cases) the derived category of representations of some quiver (which I mentioned already in part III). These equivalent ways to express the same category correspond to equivalent but different (“dual”) ways to look at the physics of these branes. For instance the relation between triangulated Fukaya categories and derived categories of coherent sheaves is related to mirror symmetry.

I hope to give a rough impression of how all that comes about in the following by condensing the key ideas out of Aspinwall’s review.

The proof proceedes in the steps listed below. A couple of them involve simplifying assumtions which, after the consequences have been worked out, are then removed at the end.

Step -1.

Before really beginning I have to make up for something that I should have mentioned in part II, namely the $\mathrm{Ext}$-functors. Actually, given the way how I proceeded in this series of posts, I should have a seperate post on $\mathrm{Ext}$-functors and on what are called $\mathrm{Tor}$-functors. Maybe I do that next time. Up to then the following has to suffice:

I discussed in part II how from suitable functors $F : A \to B$ one gets functors $\mathbf{Ch}(F) : \mathbf{Ch}(A) \to \mathbf{Ch}(B)$ in the obvious way, namely by just applying $F$ to an entire complex in $A$, morphism-by-morphism.

Now, one very important functor is the Hom-functor. But that is not quite ‘suitable’ in the above sense. One still would like to have a generalization of $\mathrm{Hom}$ applied to $A$ and $B$ to something which can be applied to $\mathbf{Ch}(A)$ and $\mathbf{Ch}(B)$.

However, there is a natural way in which, given objects $\mathcal{A}^\bullet \in \mathbf{Ch}(A)$ and $\mathcal{B}^\bullet \in \mathbf{Ch}(B)$ the array of groups

can be turned into a double complex. This is described in 2.7.4 of Weibel’s book, for instance. The total complex of that double complex is called the total Hom cochain complex of abelian groups. In fact, this gives rise to a functor

to the category of chain complexes (up to chain homotopy) of abelian groups.

Being a functor on $\mathbf{K}$-categories we can consider its lift to derived categories (by inverting quasi-isomorphisms), i.e. consider its derived functor

The $n$th cohomology of this functor is what is called $\mathrm{Ext}$. More precisely, one writes

That all sounds convoluted at first but once one thinks about it a little while it is all rather natural.

Step 0. To begin with, the physical setup is suitably simplified, so that any progress at all becomes possible. One considers D-branes on Calabi-Yau spaces and then switches to one of the the two topological string theories asscociated with that.

Step 1. Concentrate on the topological B-model. Consider open B-strings between manifold-like space-filling branes $E$ and $F$, i.e. between branes that can truly be regarded as manifolds with fiber bundles on them.

Call these fiber bundles also $E$ and $F$, respectively. Since the BRST operator in the B-model acts just like the antiholomorphic exterior derivative one sees that the Hilbert space $\mathcal{H}^q_{E,F}$ of B-strings of intrinsic ghost number $q$ stretching between the branes $E$ and $F$, which must be the qth cohomology group of $Q$,

is hence the $(0,q)$th Dolbeault cohomology group of our CY with respect to (p,q) forms taking values in the space of bundle homomorphisms from $E$ to $F$, i.e.

3. Steps 1. and 2. were the first piece of physics input. Now one makes use of a mathematical fact which says that this cohomology group can be algebraically reformulated in terms of a right derived functor. It is a theorem that

where

- $\mathcal{E}$ and $\mathcal{F}$ are the sheaves of sections of the bundles $E$ and $F$, repsectively

- $\mathrm{Ext}^q(\mathcal{E},\cdot)$ is the right derived functor of the functor $\mathrm{Hom}(\mathcal{E},\cdot)$ in the category of coherent sheaves

- where the category of coherent sheaves is like that of locally free sheaves (= vector bundles) but with enough cokernel objects thrown in so as to make it an abelian categrory,

i.e.

in the notation of part II.

The derivation of this could be summarized by a couple of sub-steps, which are however not strictly necessary in order to move on to step 4.

Step 4. The above tells us that topological string states between manifold-like branes are elements of $\mathrm{Ext}^q\left( \mathcal{E}, \mathcal{F} \right)$. In order to generalize this to the most general kind of brane we need some physics input again. New types of branes are obtained by deforming the theory. If $Q$ is a valid BRST operator, then so is

when the deformation operator $d$ is of ghost number one and such that

For an infinitesimal deformation $d$ is supposed to be a vertex operator so that $\{Q,d\} = 0$. This leaves us with the condition

Step 5. In order to identify the admissable set of deformation operators $d$ one makes a little analysis of the ghost number structure. A real understanding of this issue requires a look at the A-model. From that one sees that the manifold-like D-branes $\mathcal{E}$ that we had should be assigned a ghost number $n = \mu(\mathcal{E})$. If $\mathcal{E}^n$ is a manifold-like D-brane of ghost number $n$ then a more general D-brane state now looks like

Then a string state stretching from $\mathcal{E}^n$ to $\mathcal{E}^m$ has a ghost number $m-n$ in addition to its intrinsic ghost number $q$ (the Dolbeault degree), i.e. the total ghost number is

Out of all the qhost number 1 deformation operators $d$ that we could construct this way one now focuses on the subset of the form

with

i.e. those that have ‘intrinisc’ ghost number zero and correspond to strings stretching between manifold-like D-branes that differ by 1 in their ghost number. The reason for that is once again that in the end it will turn out sufficient to regard just this case.

This somewhat non-elegant physical derivation is now readily seen to result in a very elegant mathematical structure:

The condition $d^2 = 0$ says nothing but that associated to every deformation operator $d$ of the above kind is associated a (‘cochain’) complex $\mathcal{E}^\bullet$ of coherent sheaves $\mathcal{E}^n$

It is maybe worthwhile to pause a second and recall how the physics here is beginning to give rise to the structures we know from derived categories: we found that

1) The Hilbert space of topological strings between manifold-like branes happens to be exactly described by the derived functor (described in part II) of homomorphisms between the coherent sheaves associated to the two branes.

2) By deforming the BRST operator one finds that more general (not necessarily manifold-like) D-branes are described by complexes of coherent sheaf homomorphisms. These complexes are of course precisely the objects in the derived category of coherent sheaves, as described in part I.

Step 6.

Step 5. has given us a more general set of D-branes then we originally considered, i.e. than were described by the original BRST operator $Q$. Now that we have more general BRST operators $\tilde Q = Q + d$ we need to compute again the Hilbert space of strings, now with respect to that new operator, i.e. we have to compute the cohomology groups of $\tilde Q$.

Since $Q^2 = 0$ and $d^2 = 0$ and $\{Q,d\} = 0$ it follows that the cohomology of $\tilde Q$ is related to what is called the total cohomology of a double complex.

Again, there is some general abstract nonsense that tells us that the $q$th cohomology group of $\tilde Q = Q + d$ for strings stretched between D-branes described by complexes $\mathcal{E}^\bullet$ and $\mathcal{F}^\bullet$, is given by $\mathrm{Ext}(\mathcal{E}^\bullet,\mathcal{F}^\bullet)$, as described in step -1:

As with the other purely mathematical step, number 3, this could be described in more detail, which however I am currently lacking time and energy for. Instead I hurry to get to the final step:

Step 7.

Lean back and contemplate what has happened here. Up to loads of details that I have swept under the carpet this shows that topological strings with their topological branes on a Calabi-Yau $X$ are described by the derived category of coherent sheaves on $X$.

One thing among many that remains to be discussed is how this now generalizes to ‘physical’ strings (i.e. the ordinary fundamental ones which are not topological).