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Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc

Sheldon Katz and Chiu-Chu Melissa Liu

Geometry & Topology Monographs 8 (2006) 1–47

DOI: 10.2140/gtm.2006.8.1

arXiv: math.AG/0103074


In this paper, we present foundational material towards the development of a rigorous enumerative theory of stable maps with Lagrangian boundary conditions, ie stable maps from bordered Riemann surfaces to a symplectic manifold, such that the boundary maps to a Lagrangian submanifold. Our main application is to a situation where our proposed theory leads to a well-defined algebro-geometric computation very similar to well-known localization techniques in Gromov–Witten theory. In particular, our computation of the invariants for multiple covers of a generic disc bounding a special Lagrangian submanifold in a Calabi–Yau threefold agrees completely with the original predictions of Ooguri and Vafa based on string duality. Our proposed invariants depend more generally on a discrete parameter which came to light in the work of Aganagic, Klemm, and Vafa which was also based on duality, and our more general calculations agree with theirs up to sign.

Reproduced by kind permission of International Press from: Advances in Theoretical and Mathematical Physics 5 (2002) 1–49


bordered Riemann surfaces, open string theory, Gromov–Witten invariants, large N duality

Mathematical Subject Classification

Primary: 14N35

Secondary: 32G81, 53D45


Received: 22 January 2002
Published: 22 April 2006

Sheldon Katz
Departments of Mathematics and Physics
University of Illinois at Urbana-Champaign
Illinois 61801
Department of Mathematics
Oklahoma State University
Oklahoma 74078
Chiu-Chu Melissa Liu
Department of Mathematics
Harvard University
Massachusetts 02138