Open string instantons and relative stable morphisms

Abstract

We show how topological open string theory amplitudes can be computed by using relative stable morphisms in the algebraic category. We achieve our goal by explicitly working through an example which has been previously considered by Ooguri and Vafa from the point of view of physics. By using the method of virtual localization, we successfully reproduce their results for multiple covers of a holomorphic disc, whose boundary lies in a Lagrangian submanifold of a Calabi-Yau 3-fold, by Riemann surfaces with arbitrary genera and number of boundary components. In particular we show that, in the case we consider, there are no open string instantons with more than one boundary component ending on the Lagrangian submanifold. e-print archive: http://xxx.laiil.gov/abs/hep-th/0103100

Discrete symmetries of the superpotential and calculation of disk invariants

“…Besides being of mathematical interest, these amplitudes capture exactly certain terms in the effective N = 1 four dimensional theory [5]. The calculation of disk amplitudes has been carried out both using mirror symmetry [1,2] and directly using localization techniques similar to the ones used for calculating closed string Gromov-Witten invariants [6,8,7]. These calculations have verified the integrality of open string invariants defined in [9].…”

Open-string Gromov-Witten invariants: calculations and a mirror “theorem”

“…The technology for finding mirror manifolds [3] more than one boundary component in [24]. These formulas have been verified through localization calculations in [25] and [21]. Recently, Aganagic, Klemm and Vafa [2] expanded this mirror procedure ("B-model") in a number of new examples, and were able to make integer predictions of the invariants.…”

“…The authors of both [21] and [25] find an algebraic moduli space in which the space of stable maps from Riemann surfaces with boundary embeds.…”

Twisted complex geometry

“…When the Lagrangian submanifold is the real part of an anti-holomorphic involution, Katz and Liu [5] observe that these holomorphic curves on the disc can be identified with real holomorphic curves on the double CP 1 of the disc. As a consequence, they are able to verify partially the mathematical formulae of [9]; see also Li and Song [6]. However when the real part of the anti-holomorphic involution is empty, the method of Katz and Liu does not work.…”

“…Indeed the following result is a little stronger. Shuguang Wang [6] (2) It is clear from the definitions of h* = h* and h from (1).…”