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September 9, 2003

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=2N=2 σ\sigma-model with target space a Calabi-Yau 3-fold, XX. The closed-string sector of the topological theory consists of a finite number of states corresponding to infinitesimal deformations of the complex structure of XX. Naively, a D-brane in this theory consists of a holomorphic submanifold CXC\subset X with a holomorphic vector bundle, VV on it (with certain properties).

If you want to be fancy, you can call that data a (particular example of a) coherent sheaf on XX (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 𝒢\mathcal{G}” is the ext group Ext(,𝒢)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)D^b(X), the Bounded Derived Category of Coherent Sheaves on XX 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:

(1)0 1 2 n1 n0 0 \to \mathcal{F}_1 \to\mathcal{F}_2 \to \cdots \to\mathcal{F}_{n-1} \to \mathcal{F}_n \to 0

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 𝒢\mathcal{G}, the space of morphisms between those two objects in the derived category is again Ext(,𝒢)Ext(\mathcal{F},\mathcal{G}). So the “familiar” D-branes certainly form a subcategory of D b(X)D^b(X). But no one has constructed the D-branes corresponding to more general objects in D b(X)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 /2\mathbb{Z}/2 grading, corresponding to D-branes and anti-D-branes. Correspondingly, one can construct a category in which we identify the objects AA and A[2]A[2] (the same complex, AA, shifted to the left by two units) and whose morphisms are

(2)Mor(A,B)= nMor D b(X)(A,B[2n]) Mor(A,B) = \bigoplus_n Mor_{D^b(X)} (A, B[2n])

(with the obvious composition law). Or maybe we should consider a /6\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)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)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)D^b(X) satisfies the Moore-Segal axioms.

That’s great! A “physical” explanation for why D b(X)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.

Posted by distler at September 9, 2003 11:23 PM

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