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 which has the following strong finiteness properties: namely and are finite for any objects . Then one can define an algebra, called the Hall algebra of , which has a basis given by isomorphism classes of objects of and whose structure constants are the number of subobjects of which are isomorphic to and whose quotient is isomorphic to .
The main source of interest of Hall algebras for me is the Ringel-Green theorem which states that if you start with a quiver , then the Hall algebra of the category of representation of over a finite field is isomorphic to the upper half of the quantum group corresponding to at the parameter .
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 is the category of coherent sheaves on a Calabi-Yau 3-fold.
Let us begin by reprasing the above definition. Let be the groupoid of objects of (ie the same category but we throw away all morphisms which are not isomorphisms). Then we have a subgroupoid of 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 to . The “groupoid” nature of this becomes evident when we do the pushforward.
Now, let us generalize. We will need that our abelian category comes equipped with some additional structure, namely there should be a moduli stack of objects . 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 , it is the stack whose points are coherent sheaves on , flat over .
Then we have a substack of 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 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 .
Joyce states that when is the category of representations of a quiver, the Hall algebra 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 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 . Once Joyce proves that there is push-forward and pullback for these stack functions, he defines the Hall algebra of stack functions in a similar way.
In the case of quiver representations, he shows that 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 to a twisted group algebra of in the case where is the category of coherent sheaves on a Calabi-Yau 3-fold and how one can use this to say something about DT invariants.