Marcos Mariňo gave a nice talk about his and their work on open strings in topological string theory.
Incidentally, his T-shirt today contained four Marxes but only one of them was a criminal, namely Karl. The remaining three were comedians, I was explained. Nevertheless, the slogan said "Son Marxista" - "They are Marxists".
It is always good to meet this entertaining Marxist who is bothered that his favorite wing of politics - the left wing - is represented by the annoying politically correct people in the U.S. who don't like jokes, freedom of speech, and who self-consistent opinions.
In the topological A-model, one wants to calculate the partition sum "F(t)" as a function of the Kähler moduli "t". It is the sum over all holomorphic curves weighted by "exp(-A)" where "A" is the area.
However, one can also introduce Lagrange D-branes into this background. You obtain new open string degrees of freedom besides the complexified Kähler closed string fields. The open string fields include the imaginary part that encodes the Wilson lines around one-cycles inside the Lagrange three-cycles.
The partition sum may be efficiently rewritten using the topological vertex but this only works in the large volume limit. For many other purposes, one needs a better method to calculate. Marcos has looked at the B-model that is the mirror to the A-model above. The introduction of the open string degrees of freedom leads to a thickening - and smooth regularization - of the toric diagram that makes it obvious that many transitions - movement of D-branes from one line of the toric diagram into another line - is actually non-singular. This regularization arises because of the worldsheet instantons.
Marcos also introduced a chiral boson living on the Riemann surface inside his local Calabi-Yau three-fold and thus derived, at the level of physics rigor, a powerful recursion relation for the genus-g contribution to the partition sum. This contribution can be re-expressed as a function of contributions from diagrams that have either a smaller number of handles (genus) or a smaller number of holes. This recursion relation has a natural counterpart in the matrix models but he can use it even for geometries that don't have a known matrix model description. Their work is thus a generalization of the Dijkgraaf-Vafa techniques.
A mathematical kind of a rigorous proof would probably be based on the observation that they can rigorously derive the holomorphic anomaly equation from their formalism. The difference of their result from the right result must thus be holomorphic. With a few observations about the asymptotic behavior in different limits, one could argue that the difference is zero: their result is correct.
One of the moral punch lines of the talk is that the open string degrees of freedom can be viewed as more fundamental degrees of freedom than the closed string ones in topological string theory. Equivalently, all contributions to partition sums can be constructed out of the disk and cylinder contributions. This includes closed worldsheets which opens the possibility that the same method could be used in bosonic cubic string field theory to calculate purely closed string scattering.
Because Marcos' approach was named a residue technique, I was interested whether it is morally analogous to the method how we showed the equivalence of the connected and disconnected twistor prescriptions in "CP(3/4)". The answer remains somewhat ambiguous.