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

Sheldon Katz, Chiu Chu Melissa Liu

Research output: Contribution to journalArticlepeer-review

Abstract

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 welldefined algebrogeometric computation very similar to wellknown 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.

Original languageEnglish (US)
Pages (from-to)1-47
Number of pages47
JournalAdvances in Theoretical and Mathematical Physics
Volume5
Issue number1
DOIs
StatePublished - Jan 2001

ASJC Scopus subject areas

  • Mathematics(all)
  • Physics and Astronomy(all)

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