Saturday, July 23, 2005

Matrix Factorizations

The first day of talks has finally come to an end. There were five talks in all today at a total time of four hours, with long coffee breaks (total: 6 coffees and 2 biscuits, it's amazing I can keep my hands steady enough to type). This afternoon's talks were all concerned with matrix factorisations. The first two were given by Matthias Gaberdiel under the title "Matrix factorisations and D-branes" (a good reference for this talk is here) and the final talk of the day was by Daniel Roggenkamp and entitled "Permutation branes and linear matrix factorisations". There was a significant overlap in these two talks, so I will only describe the first here.

Matthias Gaberdiel was interested in the string's viewpoint of the Calabi-Yau manifold. He argued that from a microscopic (C.F.T.) point of view that Calabi-Yau compactifictions are best understood at the Gepner point. A class of branes located at this point were found by Andreas Recknagel and Schomerus which preserved the full chiral symmetry of the theory. Gaberdiel called these RS branes (he said, at one point, that the abbreviation was an amalgum of Ramond-Ramond and Neveu-Schwarz but dropping an R and an N). However these RS branes do not account for all the D-brane (RR) charges. The case study used for the talk was the quintic superpotential: W=(x_1)^5+...(x_5)^5=0. In this case Gaberdiel told us that the RS branes only accounted for a 25-dimensional sublattice of the full RR charge lattice, and in particular that the D0 brane on the quintic was not described by any of the RS boundary states. The aim of his talk, he said, was to construct the fundamental branes using that give rise to the full RR charge lattice.

The route taken was that initiated by Maxim Kontsevich who proposed a connection between Landau-Ginsburg models and supersymmetric B-type D-branes. It transpired that in order to make the appropriate action with an additional Lansdau-Ginsburg (superpotential, W) F-term invariant under a SuSy variation that an extra term proportional to E must be added such that W=EJ. I am told a clear exposition of this, and some actual definitions of these terms can be found in "Landau-Ginzburg Realization of Open String TFT" by Ilka Brunnera, Manfred Herbsta, Wolfgang Lerchea and Bernhard Scheunera. This is the matrix factorization referred to in the talk's title.

The argument proceeded that since Landau-Ginsburg models are related to N=2 minimal models and that these are the basic unit of the Gepner model description, that there should be a correspondence between them. Gaberdiel told us that this relationship is well known for the single minimal model, but in cases more general than this little is known. The correspondence suggested was then used to tease the D0 brane out of the quintic, the arguments leading to this can be read here. Furthermore the process could be generalised to other Gepner models and D-branes, thus allowing the full charge lattice to be uncovered from fundamental branes in some other cases, as well as the quintic. Gaberdiel took us through a "baby" example and talked to us about product theory and permutation branes (I would give a definition but I'm afraid I haven't quite understood their essence yet, but the literature in the links above should give a clear picture). However there are many areas still left to be investigated in this field, for while the quintic is an example of a Gepner model where the RS-branes generate a vector space of RR-charges and permutation branes generalise this and generate the full RR-charge lattice, there are cases amongst the (147) Gepner models where neither RS-branes nor permutation branes produce the full RR-charge lattice. Furthermore to identify the permutation branes one cornerstone involved considering certain preferred roots of unity. If the preferred roots (the notion of the preferred roots are derived from an index, m, associated with a U(1) charge in a certain N=2 bosonic coset model, again see the above links) of unity were all consecutive then the state being considered was judged to be a permutation brane, however in any other case the association between the model and the field theoretic content remains to be understood.

After dinner tonight they held a wine reception for us in the JCR of Grey College, where we are staying, and now we are thinking of hitting the town. If there are no posts tomorrow you can make a good guess that I hit the town a little too hard...


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