Hall algebras and Donaldson-Thomas invariants I

I would like to tell you about recent work of Dominic Joyce and others (Bridgeland, Kontsevich-Soibelman, Behrend, Pandaripande-Thomas, etc) on Hall algebras and Donaldson-Thomas invariants.  I don’t completely understand this work, but it seems very exciting to me. This post will largely be based on talks by Bridgeland and Joyce that I heard last month at MSRI.

In this post, I will concentrate on different versions of Hall algebras. Let us start with the most elementary one. Suppose I have an abelian category $\mathcal{A}$ which has the following strong finiteness properties: namely $Hom(A,B)$ and $Ext^1(A,B)$ are finite for any objects $A, B$. Then one can define an algebra, called the Hall algebra of $\mathcal{A}$, which has a basis given by isomorphism classes of objects of $\mathcal{A}$ and whose structure constants $c_{[M], [N]}^{[P]}$ are the number of subobjects of $P$ which are isomorphic to $N$ and whose quotient is isomorphic to $M$.

The main source of interest of Hall algebras for me is the Ringel-Green theorem which states that if you start with a quiver $Q$, then the Hall algebra of the category of representation of $Q$ over a finite field $\mathbb{F}_q$ is isomorphic to the upper half of the quantum group corresponding to $Q$ at the parameter $q^{1/2}$.

The obvious question concerning Hall algebras is to come up with a framework for understanding them when the Hom and Ext sets are not finite. This is what Joyce has done and he has applied it where $A$ is the category of coherent sheaves on a Calabi-Yau 3-fold.

Let us begin by reprasing the above definition.  Let $M$ be the groupoid of objects of $\mathcal{A}$ (ie the same category but we throw away all morphisms which are not isomorphisms).  Then we have a subgroupoid $M^{(2)}$ of $M \times M \times M$ which consists of short exact sequences.  Then we can think of the Hall algebra as the convolution algebra you get from the various projections from $M^{(2)}$ to $M$.  The “groupoid” nature of this becomes evident when we do the pushforward.

Now, let us generalize.  We will need that our abelian category $\mathcal{A}$ comes equipped with some additional structure, namely there should be a moduli stack of objects $\mathcal{M}$.  This moduli stack is usually easy to construct in examples.  For quiver representations, it is simply the quotient stack corresponding to the quiver variety.  For coherent sheaves on some variety $X$, it is the stack whose $S$ points are coherent sheaves on $X \times S$, flat over $S$.

Then we have a substack $\mathcal{M}^{(2)}$ of $\mathcal{M} \times \mathcal{M} \times \mathcal{M}$ for short exact sequences.  Now we can do the same construction of convolution algebra as before except that we work with constructible functions on our stack.  This means functions on the set of $k$ points of our stack (note this is just iso classes of objects) which are constructible in the sense of algebraic geometry (ie using constant functions on substacks).  Thus we get a Hall algebra denote $CF(\mathcal{A})$.

Joyce states that when $\mathcal{A}$ is the category of representations of a quiver, the Hall algebra $CF(\mathcal{A})$ coincides with one defined by Lusztig and hence is isomorphic to the upper half of the universal envelopping algebra for the Lie algebra corresponding to the quiver.

However, for the application to DT invariants, a different (and more sophisticated) Hall algebra is required.  Joyce considers what he calls stack functions.  This is the vector space generated by isomorphism classes of stack functions over $\mathcal{M}$ modulo “motivic relations”.  The relation with the constructible function is that you can think of the characteristic function of a substack as just that substack mapping to $\mathcal{M}$.   Once Joyce proves that there is push-forward and pullback for these stack functions, he defines the Hall algebra of stack functions $SF(\mathcal{A})$ in a similar way.

In the case of quiver representations, he shows that $SF(\mathcal{A})$ is the upper half of the quantum group — here the parameter q is the “motivic variable”, corresponding to the affine line.  This links up nicely with the finite field construction.  Joyce also says that this fits with Lusztig’s perverse sheaf construction, although I don’t see how.

If I get around to writing part II of this post, I will explain how one constructs a map of algebras from $SF(\mathcal{A})$ to a twisted group algebra of $K(\mathcal{A})$ in the case where $\mathcal{A}$ is the category of coherent sheaves on a Calabi-Yau 3-fold and how one can use this to say something about DT invariants.

4 thoughts on “Hall algebras and Donaldson-Thomas invariants I”

1. Joel Kamnitzer says:

I didn’t really say anything above about what it meant to quotient by “motivic relations”. As a first approximation, you quotient by the relation of disjoint union. There there is also some business about identifying the affine line with a variable q. Maybe, someone who understands motivic stuff better can explain this a bit better.

2. Great post! this stuff seems to me incredibly exciting, like the future of representation theory is being devised..

A couple of comments:

– the positive half of the quantum group always sits inside the Hall algebra of the corresponding quiver, but this is an isomorphism only for ADE quivers. Otherwise you get Lusztig’s “composition algebra”, a subalgebra generated by characteristic functions of components of
the moduli (or so I learned from Schiffmann’s great Lectures on Hall Algebras).

– Any abelian category has a moduli stack of objects, this is not an additional structure. Given a k-linear category C (here k can be Z), we can talk about R-families of objects in C for any k-algebra R, which are objects of C tensored up with R. We can also formulate flatness as an exactness property. This gives the functor of points of the moduli stack. In fact you don’t need the category to be abelian: in a derived category (construed correctly, i.e. with an enriched/dg structure) you get a moduli stack of objects without negative self-Exts
(Lieblich) and a moduli higher-stack of all objects (Toen-Vaquie)..

Maybe a nicer way to say this is that the functor (abelian categories) to (stacks) giving the moduli of objects can be characterized as the right adjoint of the functor (stacks) to (abelian categories) sending a stack to its category of coherent sheaves.. the idea is a functor from sheaves on X to your given category A gives an X-family of objects of A, namely the images of all the skyscraper sheaves on X.

– Finally, another name to mention in your list is Toledano-Laredo (for his joint work with Bridgeland). Both of them have given great expository “GRASP” talks on related topics at UT which are available online at http://www.math.utexas.edu/~benzvi/GRASP.html

3. David –
1. Joyce also defines a composition subalgebra in the general case to be the subalgebra generated by the functions supported on the isoclasses of simple objects.

2. It is a nice, simple construction that you describe for producing a moduli stack of objects. I heard about that construction from Xinwen — I think that it will be used in his forthcoming paper with Frenkel about actions of algebraic groups on categories. Do you know of anywhere else where it is written up? Is the stack that you get this way always an algebraic (Artin) stack? (I guess I should look in those Lieblich and Toen-Vaquie papers you mention.)

In Joyce’s papers, he doesn’t use construction of a moduli stack of objects. Rather, he postulates it as extra data, which must satisfy certain axioms. I wonder if it amounts to the same thing in the end or if it differs in some examples.

Comments are closed.