Inthis study Dijkgraaf and Vafa showed that the closed string partition function of the topological B-model on a Calabi-Yau of the form$uv-H(x,y)=0$ coincides with the free energy of a certain matrix model.
Then, after taking the double-scaling limit, they get an identification between the B-model partition function and the minimal string partition function. The latter is a minimal model coupled to the Liouville theory, and the equation$H(x,y)=0$ corresponds to what is known as the minimal string Riemann surface (seehttp://arxiv.org/abs/hep-th/0312170). For the$(p,q)$ minimal model (without any insertions) one gets$H(x,y)=y^p+x^q$.
There are two kinds of branes in the Liouville theory: FZZT and ZZ, where the FZZT branes are parametrized (semiclassically) by the points on the Riemann surface$H(x,y)=0$.
What are the equivalents of the FZZT and ZZ open string partition functions in the B-model?
1 Answer1
In the context of mirror symmetry, the equivalents of the FZZT and ZZ open string partition functions in the B-model correspond to certain A-branes and B-branes, respectively. A-branes in the B-model are Lagrangian cycles in the Calabi-Yau manifold, while B-branes are holomorphic submanifolds.
The open string partition function for FZZT branes in the B-model is related to the A-model topological string partition function on the corresponding Lagrangian cycles. Similarly, the open string partition function for ZZ branes in the B-model is related to the B-model topological string partition function on the associated holomorphic submanifolds.
The specific expressions for these partition functions depend on the details of the Calabi-Yau manifold and the chosen A-branes and B-branes. The mirror symmetry duality establishes a relationship between the A-model and B-model partition functions, allowing the study of open string dynamics on different types of branes in the two formulations of string theory.
