The String Coffee Table

This is the transcript, continued where I left off last time (again, there are some personal comments, set in italics):

II) Equivalences of categories versus isomorphism of target spaces.

Suppose we have two varieties $X$ and $Y$ which are isomorphic. It follows easily that then their categories of coherent sheaves are equivalent:

It is a theorem by Gabriel that this condition is actually also necessary, i.e. that varieties are isomorphic precisely if their coherent sheaves are equivalent:

(The “$\simeq$” on the right hand side denotes $\mathbb{C}$-linear equivalence of categories.)

The idea behind this theorem is the following:

In every abelian category we have the notion of “simple objects”, which are those that do not have proper subobjects (which here should be equivalent to saying that their endomorphism spaces are ismorphic to $\mathbb{C}$).

The simple objects of $\mathrm{Coh}(X)$ are precisely the skyscraper sheaves $k(x)$, for all $x \in X$. The equivalence of categories $\mathrm{Coh}(X) \simeq \mathrm{Coh}(Y)$ induces a bijection between its simple objects

hence a bijection between the points of $X$ and $Y$.

(Notice that this fails for complex $X$ and $Y$. We still get a bijection at the level of sets of points, but not a holomorphic one.)

The big question now is what happens to the above theorem when we pass to derived categories.

There are really two questions:

1) What is the equivalence classed coming from the equivalence relation

(where on the right we have $\mathbb{C}$-linear exact equivalence of derived categories, sending simple objects to simple objects, distinguished triangles to distinguished triangles, etc. (etc.?))?

2) What are the autoequivalences

of the derived category of coherent sheaves of $X$?

Some well known results concerning this question were found by Bondal and Orlov. They found that question 1) above has no interesting answer whenever the first Chern class of $X$ is nonvanishing, $\mathrm{ch}(X) \neq 0$. For these cases we simply have

and hence passing to derived categories has little effect as far as intrinsic characterization of target spaces is concerned.

There are some other, mildly interesting cases (for instance elliptic surfaces). And then there is a truly interesting case. This turns out to be the one where our spaces are Calabi-Yau.

(From the purely mathematical point of view this must be rather unexpected. This surprise is best understood from the physical point of view, of course. Even though not visible in the formal language used, the very fact that stability conditions on triangulated categories are interesting at all can be traced back to the fact that some sort of supersymmetry plays a role in the background. Calabi-Yaus are just those targets with enhanced supersymmetry.)

For Calabi-Yau targets, one knows a relatively easy result for $\mathrm{dim}(X) = 1$, $X$ an elliptic curve. Here one finds

For $\mathrm{dim}(X)=2$ we can have either that $X$ is an abelian surface of the form $\mathbb{C}^2/\Gamma$, or a K3 surface.

For the case of K3s Mukai and Orlov found that

$\bullet$ if $X$ is a K3 and $D^b(X)\simeq D^b(Y)$, then $Y$ must be a K3.

$\bullet$ for both $X$ and $Y$ K3s we have that they are “derived equivalent”, $D^b(X) \simeq D^b(Y)$, precisely if there exists an isomorphism

which is an isometry with respect to the metric given by the pairing coming from the intersection product.

(There was a little more discussion on an equivalent way to formulate the isometry condition, but I am afraid I cannot reproduce a coherent account using my notes.)

There is also a geometric reformulation of this condition: if $X$ and $Y$ are K3 surfaces, then they are derived equivalent precisely if

$\bullet$ either $X \simeq Y$

$\bullet$ or if $Y$ is the moduli space of stable vector bundles on $X$ (for some well-known definition of stable which I’d need to check before I reproduce it).

(I wonder if there is a nice way to understand this. A priori the second condition looks a little weird. After all, it says that K3s are “the same” (in the derived sense) as their own moduli spaces of stable vector bundles over them. Does anyone have an intuition for why this is the way it is? )

Now, the main result of Daniel Huybrecht to be presented in this talk is to give an alternative characterization of derived equivalence of K3 surfaces, one that makes use of the notion of stability in derived categories.

The statement is that, for every choice of complexified Kähler class $B + i\omega$, there is a certain derived subcategory

such that $X$ is derived equivalent to $Y$ precisely if there exist $B+i\omega$ on $X$ and $B'+i\omega'$ on $Y$ such that

In more detail, $A_X(\mathrm{exp}(B + i \omega))$ is precisely the derived category of stable (BPS) branes (inside the collection of all topological branes), which are stable with respect to the stability condition “induced” by $B+i\omega$.

More precisely, the stability condition comes from a map

from the Grothedieck group of $D^b(X)$ (essentially the group completed decategorification of $D^b(X)$) to complex numbers, which associates to each class of brane configurations the “central charge” given by the formula

(The term in the square root is the Todd class ($\to$).)

There is also a more explicit description of $A_X(\mathrm{exp}(B + i \omega))$: it is the subcategory of $D^b(X)$ consisting of complexes concentrated in the first two degrees

(probably meaning that, as opposed to the topological string, a physical anti-anti-brane is a brane, i.e. replacing the $\mathbb{Z}$-grading of topological branes with a $\mathbb{Z}_2$-grading (?)) such that

$\bullet$ the kernel of $f$ is torsion free and

In order to compare this to the notation used by Bridgeland in his papers, note that the BPS brane category called $A_X(\cdots)$ here corresponds to the category called

by Bridgeland.

(That’s the end of my notes. There might be typos in the formulas. I am grateful for all comments and corrections.)