The String Coffee Table

The talk consisted of two parts:

I) Introduction to the concept of derived categories of coherent sheaves.

II) Equivalences of categories versus isomorphism of target spaces.

The main result presented (theorem 0.1 in Huybrechts’ paper mentioned above) says, in terms of the physics side of the dictionary, that two K3 surfaces are indistinguishable as target spaces for the topological string if and only if there are complexified Kähler classes on them which make them indistinguishable for the physical string.

The following are the notes that I took in the talk (I have included some links and some personal comments, set in italics). The notes on part II) are given in a seperate entry.

I) Introduction to the concept of derived categories of coherent sheaves.

In the following, $X$ denotes an algebraic variety ($\to$), more precisely, a smooth projective variety sitting in some projective space. Later we restrict attention to $X$ being a K3 surface ($\to$).

Denote by $\mathrm{Coh}(X)$ the category of coherent sheaves ($\to$) on $X$.

Examples for coherent sheaves on $X$ are

i) Holomorphic vector bundles ($\to$) on $X$ are coherent sheaves (or rather, their sheaves of sections are). (This corresponds to stacks of space-filling geometric branes.)

ii) If $C \subset X$ is a holomorphic curve, then its structure sheaf $O_C$ is a coherent sheaf. (This would be a lower-dimensional geometric brane.)

iii) For every point $x\in X$, the skyscraper sheaf $k(x)$ is a coherent sheaf. (This is the sheaf whose fiber over $U$ is nonempty precisely if $x \in U$). (These skyscrapersheaves encode 0-branes, localized at points (we do not consider a temporal deriction in this sort of game).))

iv) Let $E$ be a vector bundle on $X$, or rather its sheaf of sections, and let $E \to k(x)$ be the obvious projection onto the skyscraper sheaf at $x$, then

is a torsion free coherent sheaf.

For $E$ trivial this is the “ideal sheaf” (with “ideal” in the sense of ideal of a ring) $I_x$, namely the sheaf of holomorphic functions vanishing at the point $x$.

(This corresponds to a space filling stack of D-branes having reacted with an anti-0-brane at position $x$.)

Once we understand these examples, we understand all coherent sheaves, in the sense that every coherent sheaf can be constructed by performing direct sums, quotients, kernels, etc. (etc?) of the sheaves in the above examples.

Next, we pass from $\mathrm{Coh}(X)$ to $D^b(\mathrm{Coh}(X))$, its derived category ($\to$, $\to$).

This works as follows.

Denote by $K^b(X) := K^\bullet(\mathrm{Coh}(X))$ the category of bounded complexes of coherent sheaves. Its objects are bounded complexes, its morphisms are chain homotopy classes of chain maps between morphisms.

(BTW, what would happen if we did not divide out by chain homotopies? If we even considered the full 2-category of complexes, chain maps and chain homotopies?)

(Given any complex, we can compute its cohomology in every degree. We can regard the collection of the cohomologies as a complex themselves, with all differentials being trivial. Given a chain map between two complexes, we get a chain map between the corresponding trivial complexes of cohomologies. Hence cohomology is really a functor from the category of complexes of coherent sheaves to to the category of complexes of abelian groups. If a morphism in $K^b(X)$ is not an isomorphism, but maps to an isomorphism under this functor, then we call it a “quasi isomorphism”.)

The derived category $D^b(X) := D^b(\mathrm{Coh}(X))$ is like $K^b(X)$, but with all quasi-isomorphisms regarded as true isomorphisms.

This is a special case of “localization of a category” ($\to$).

Now, $D^b(X)$ is no longer abelian ($\to$) - but it is triangulated ($\to$)!

In fact, every derived category of any abelian category is triangulated. This is hence not a special property of coherent sheaves. On the other hand $D^b(X)$ is actually $\mathbb{C}$-linear triangulated (which hopefully means the obvious thing).

A category being triangulated means in particular that it carries the following two crucial structures.

i) There is a shift endomorphism

(for the case of derived categories $E[1]$ is the same complex as $E$, but shifted in degree to the left, by one unit, i.e. $E[1]^n = E^{n+1}$)

and

ii) there are “distinguished triangles”, which are diagrams

As an example: every “mapping cone” ($\to$) gives rise to such a triangle.

(See math.AG/0001045 for an explanation of the mapping cone and why it has the name it has. The triangle obtained from the mapping cone of the map $E \overset{f}{\to} F$ between “geometric branes” $E$ and $F$ corresponds to the D-brane reaction induced by $f$ when this is regarded as a tachyon condensate of strings stretching between brane $E$ and brane $F$.)

We also need to mention something called a t-structure ($\to$) on a triangulated category. This is a collection of subcategories with some properties. For derived categories, such subcategories include $D^{\leq 0}(X)$ and $D^{\geq 0}(X)$, the subcategories of complexes concentrated in non-positive or in non-negative degree, respectively.

The intersection of these two subcategories is called the core of $D^b(X)$. Evidently, this is naturally identified with the original category $\mathrm{Coh}(X)$.

Continue reading part II of these notes.