Open string instantons and relative stable morphisms
In the paper under review, the authors compute certain topological open string amplitudes. They work by an example which has been previously considered by \textit{H. Ooguri} and \textit{C. Vafa} from the point of view of physics, and give a mathematical proof to the Ooguri-Vafa physical results. The main ideas are to use relative stable morphisms, relative Gromov-Witten invariants, and a virtual localization formula in algebraic geometry. In section one, the authors give a brief description of the multiple cover problem considered by Ooguri and Vafa, and they write down explicitly Ooguri and Vafa's formula for topological open string amplitudes. Proving this formula by using relative stable morphisms is the main goal of this paper. In section two, the definition of the moduli space of relative stable morphisms is recalled, and a \(S^1\)-action is introduced. Then the authors describe how multiple covers of a holomorphic disc can be viewed as a problem regarding relative stable morphisms. In section three, the obstruction theory of the moduli spaces is studied, and the obstruction bundle is identified. The expected (virtual) number of maps is expressed in terms of the integration of the top Chern class of the obstruction bundle over the virtual moduli cycle. In section four, the authors investigate the localization of the virtual moduli cycle, and compute the equivariant Euler class of the virtual normal bundle to the \(S^1\)-fixed locus. In section five and six, using the virtual localization formula of \textit{T. Graber} and \textit{R. Pandharipande} [Invent. Math. 135, 487-518 (1999; Zbl 0953.14035)] the authors evaluate the above-mentioned integration and comfirm the formula of Ooguri and Vafa. The final section (section 7) is devoted to some further remarks. In particular, the authors stress the importance of using relative stable morphisms to study open string instantons.
scientific article; zbMATH DE number 1818581