Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc
In this paper, the authors attempt to lay the mathematical framework of a theory of enumerative geometry of stable maps form bordered Riemann surfaces with Lagrangian boundary conditions. According to the work by Fukaya, Oh, Ohta and Ono (Kyoto University preprint, 2000), and also to some physical calculations carried out by various physicists, e.g., a work by M. Aganagic, A. Klemm and C. Vafa (preprint, 2001), there is an ambiguity in counting contributions of multiple covered discs, and the seeking enumerative invariants are not intrinsic to the geometry and necessarily depend on additional parameters. More precisely, unlike the case of closed string moduli space, the naive open string moduli space does not carry a canonical fundamental class. There are essentially two obstructions to the existence of the canonical fundamental class of the disc moduli space: one is non-existence of an orientation on the moduli space and the other existence of the codimension one boundary. The orientation problem has been thoroughly studied and shown to be related to the spin structure of the Lagrangian submanifold by Fukaya, Oh, Ohta and Ono (op. cit.; 2000). The implication of the latter obstruction is that the Euler number of the obstruction bundle will depend on the choice of a non-vanishing section on the boundary of the moduli space whose homotopy class, however, is not unique. It is shown by Fukaya, Oh, Ohta and Ono that only the total sum of the instanton contributions would make sense in general. The main example that the present authors study is the case where there is an \(U(1)\)-action that preserves the Lagrangian submanifold. This is the example that \textit{H. Ooguri} and \textit{C. Vafa} [Nucl. Phys. B 577, 419-438 (2000; Zbl 1036.81515)] studied in relation to computing the multiple cover contribution of some particular disc: the ambient space \(X\) is a Calabi-Yau threefold which admits an anti-holomorphic involution \(A\) with the special Lagrangian \(L\) as its fixed locus, and \(C\) a smooth rational curve in \(X\) which is preserved by \(A\) and \(C\cap L = S^1\). Assuming the normal bundle of \(C \subset X\) is \(N = N_{C/X} = \mathcal O_{{\mathbb\;P}^1}(-1) \oplus \mathcal O_{{\mathbb\;P}^1}(-1)\) which itself is a non-compact Calabi-Yau, the counting problem makes sense. Based on some physical arguments, Ooguri and Vafa (op. cit.; 2000) computed the multiple cover contributions of the disc that is the half of \(C\). The main result of the paper under review proposes a rigorous mathematical argument to verify the above mentioned Ooguri-Vafa's formula. This \(S^1\) action allows the authors to choose a suitable choice of a non-vanishing section on the boundary of the multiple cover disc moduli space whose associated Euler numbers of the obstruction bundle gives rise to the multiple cover Ooguri-Vafa's formula. There is a work by Li and Song (preprint 2001) with a slightly different approach that carried out a similar verification of the same Ooguri-Vafa's formula. The paper under review also provides an elegant exposition of the Maslov index for bordered Riemann surfaces of arbitrary genus.
scientific article; zbMATH DE number 1818579