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