Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc
type: Item
instance of: scholarly article
title: Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc
zbMATH Open document ID: 1026.32028
author: Sheldon Katz
author: Chiu-Chu Melissa Liu
published in: Advances in Theoretical and Mathematical Physics
publication date: 2002-10-22T00:00:00Z
review text: 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 …
Mathematics Subject Classification ID: 32G99
Mathematics Subject Classification ID: 14N35
Mathematics Subject Classification ID: 32J99
Mathematics Subject Classification ID: 53D45
zbMATH DE Number: 1818579
zbMATH Keywords: enumerative geometry of stable maps
zbMATH Keywords: bordered Riemann surfaces
zbMATH Keywords: Lagrangian boundary conditions
reviewed by: Yong-Geun Oh
MaRDI profile type: Publication
DOI: 10.4310/ATMP.2001.V5.N1.A1
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description: scientific article; zbMATH DE number 1818579