Question

I'm looking for good introductory references for Donaldson-Thomas theory and Pandharipande-Thomas theory. I know a bit about Gromov-Witten theory, but almost nothing about Donaldson-Thomas and Pandharipande-Thomas. Are there some canonical (or good non-canonical) references for Donaldson-Thomas theory and Pandharipande-Thomas theory? References that assume knowledge of Gromov-Witten theory are fine.

Answer 1

Here's a few that I found back when I was considering doing enumerative geometry:

Gromov-Witten Theory and Donaldson-Thomas 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.

Gromov-Witten, Gopakumar-Vafa, and Donaldson-Thomas invariants of Calabi-Yau 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.

Answer 2

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 3-folds, 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 3-dimensional CY manifold X with a holomorphic volume form $W$.

Statement. On the space of smooth 2-dimesnional surfaces in X there is a natural (possibly multi-valued) 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 3-manifold M whose boundary is $S-S_1$. 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 3-fold can be seen as finding the number of critical points of a functional.

Answer 3

This survey by Pandharipande and Thomas appeared today on arXiv:

13/2 ways to count curves

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 Gromov-Witten theory, Copakumar-Vafa / BPS invariants, Donaldson-Thomas 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.