Musings

What’s Up in the GST

One of the topics we will be covering this semester in the Geometry and String Theory seminar, that I run jointly with Dan Freed, is recent work of Caldararu on D-branes in the topological B-model.

Recall the basic setup, which is a twisted version of the $N=2$ $\sigma$-model with target space a Calabi-Yau 3-fold, $X$. The closed-string sector of the topological theory consists of a finite number of states corresponding to infinitesimal deformations of the complex structure of $X$. Naively, a D-brane in this theory consists of a holomorphic submanifold $C\subset X$ with a holomorphic vector bundle, $V$ on it (with certain properties).

If you want to be fancy, you can call that data a (particular example of a) coherent sheaf on $X$ (by extension by zero). In this context, Eric Sharpe and collaborators have written a series of papers recently which prove by explicit calculation that the open-string spectrum stretched between “sheaf $\mathcal{F}$” and “sheaf $\mathcal{G}$” is the ext group $Ext(\mathcal{F},\mathcal{G})$.

A number of years ago, Douglas proposed that the full set of D-branes in the B-model are objects in $D^b(X)$, the Bounded Derived Category of Coherent Sheaves on $X$ and that the open string states are the morphisms of that category. The objects in the derived category are easy to describe. They are (bounded) complexes of coherent sheaves:

and the morphisms are … well, that’s a bit complicated to state. Suffice it to say that, in the special case of one-term complexes, $\mathcal{F}$ and $\mathcal{G}$, the space of morphisms between those two objects in the derived category is again $Ext(\mathcal{F},\mathcal{G})$. So the “familiar” D-branes certainly form a subcategory of $D^b(X)$. But no one has constructed the D-branes corresponding to more general objects in $D^b(X)$, much less computed the corresponding open string spectra.

There are many interesting consequences of Douglas’s conjecture, but there are many puzzling features as well. There’s a $\mathbb{Z}$-grading which doesn’t have an obvious physical interpretation. Physically, one might have a use for a $\mathbb{Z}/2$ grading, corresponding to D-branes and anti-D-branes. Correspondingly, one can construct a category in which we identify the objects $A$ and $A[2]$ (the same complex, $A$, shifted to the left by two units) and whose morphisms are

(with the obvious composition law). Or maybe we should consider a $\mathbb{Z}/6$ grading (since the closed-string sector violates ghost charge by 6 units). Or … There are lots of possibilities. And since we don’t have such a good physical handle on the (existence or properties of) the “exotic” D-branes described by $D^b(X)$, we are not in great shape to decide between them. [To be more precise, Douglas’s conjecture is that the D-branes of the topological B-model correspond to quasi-isomorphism classes of objects in $D^b(X)$. The “exotic” D-branes are the ones not quasi-isomorphic to a (direct sum of) coherent sheaves.]

This is where (I hope) Caldararu’s work comes in. The set of possible D-branes and the corresponding open strings must satisfy a definite set of axioms, the Moore-Segal Axioms of open/closed Topological Field Theory. What Caldararu shows is that $D^b(X)$ satisfies the Moore-Segal axioms.

That’s great! A “physical” explanation for why $D^b(X)$ is the right answer and, perhaps, the alternatives unsatisfactory. After we’ve studied this stuff for a while, maybe I’ll be able to post a cogent explanation for the … ahem! … masses.