I'm looking for good introductory references for DonaldsonThomas theory and PandharipandeThomas theory. I know a bit about GromovWitten theory, but almost nothing about DonaldsonThomas and PandharipandeThomas. Are there some canonical (or good noncanonical) references for DonaldsonThomas theory and PandharipandeThomas theory? References that assume knowledge of GromovWitten theory are fine.
3 Answers
I would be very happy if such material existed!!!
But just to statisfy the first curiosity, There is a 1 hour lecture of Richard Thomas online on MSRI
Counting curves in 3folds, 2009
http://www.msri.org/communications/vmath/VMathVideos/VideoInfo/4118/show_video
I would like to add just one little thing that I know about DT and find cool. Consider a 3dimensional CY manifold X with a holomorphic volume form $W$.
Statement. On the space of smooth 2dimesnional surfaces in X there is a natural (possibly multivalued) functional F, defined by $W$. Moreover, holomorphic curves in X are exactly the critical points of the functional.
Definition of the functional. Take a surface S, and define F(S)=0, for any other surface $S_1$ homological to S consider a 3manifold M whose boundary is $SS_1$. Integrate W over M. This gives the value of F at $S_1$.
In is not hard to check that holomorphic curves are critical points of F, so couniting holomorphic curves in a CY 3fold can be seen as finding the number of critical points of a functional.

1$\begingroup$ On that note, there is also a lecture of Bridgeland from the Atiyah 80 conference available here: maths.ed.ac.uk/~aar/atiyah80.htm $\endgroup$ Dec 23, 2009 at 1:48
Here's a few that I found back when I was considering doing enumerative geometry:
GromovWitten Theory and DonaldsonThomas Theory I and II, referred to as MNOP
Maps, Sheaves and K3 Surfaces by Pandharipande which is more of an overview type paper, and I believe discusses MNOP.
GromovWitten, GopakumarVafa, and DonaldsonThomas invariants of CalabiYau threefolds by Katz, notes from a lecture where he does define everything.
Not sure entirely that this is what you're looking for, but hope it helps. Generally Katz and Pandharipande do a lot of expositing on these things.

1$\begingroup$ Thanks. I just wanted to know if there were any canonical references containing the basic background, motivation, definitions, results. Anyway, it's funny that you view these things as enumerative geometry  I tend to view them as being mathematical physics... $\endgroup$ Dec 23, 2009 at 2:38

1$\begingroup$ In these cases, enumerative geometry just happens to agree with mathematical physics. After all, physicists are the ones who are making the cool predictions of the numbers of rational curves in things...just leaves it to us to prove it. $\endgroup$ Dec 23, 2009 at 2:42
This survey by Pandharipande and Thomas appeared today on arXiv:
It should give a partial answer to your question. They give an elementary review of 6 ways to count curves and the relations between them. These approaches are based on GromovWitten theory, CopakumarVafa / BPS invariants, DonaldsonThomas theory, Stable pairs, Stable unramified maps and Stable quotients. In the appendix of the paper they give a nice review of Virtual classes.
As they say at the beginning of the paper their goal is to provide a guide for graduate students looking for an elementary route into the subject.
I wrote this since I am very excited seeing the paper and wanted to post it here for those who have not yet looked at it.