Abstract

Compactifications with fluxes and branes motivate us to study various enumerative invariants of Calabi-Yau manifolds. In this paper, we study non-perturbative corrections depending on both open and closed string moduli for a class of compact Calabi-Yau manifolds in general dimensions. Our analysis is based on the methods using relative cohomology and generalized hypergeometric system. For the simplest example of compact Calabi-Yau fivefold, we explicitly derive the associated Picard-Fuchs differential equations and compute the quantum corrections in terms of the open and closed flat coordinates. Implications for a kind of open-closed duality are also discussed.

Highlights

  • More general enumerating problem in the framework of mirror symmetry

  • We studied the open and closed string deformations for a class of Calabi-Yau hypersurfaces in general dimensions

  • To formulate the open mirror symmetry and extract the generating functions for topological invariants, we employed two methods using 1) the Gauss-Manin system for the relative cohomology group and 2) the GKZ system associated with the generalized hypergeometric functions

Read more

Summary

Disk enumeration and open mirror symmetry

We describe the disk enumeration problem on complex odd dimensional. Topological observables Ohi of the A-model are defined from the hyperplane class h of Xm+2 ⊂ CPm+1 This n-point function can be regarded as a generating function of open Gromov-Witten invariants associated with the holomorphic disks ending on Lm+2.1 Note that (2.4) is nonzero only if n pi. The integral (2.9) is regarded as the on-shell disk partition function whose expression in the vicinity of the large complex structure point is given by [29]. Where q = e2πit and z(q) is the inverse mirror map derived from (2.17) Utilizing this formula and a prescription of [37], it is possible to find exact Kahler potential of the Kahler moduli space for Calabi-Yau fivefolds with one Kahler parameter. No2pd−en1 ≡ no2pd−en1(h)/(2d − 1) is a positive integer topological invariant which is called Ooguri-Vafa invariant [40]

Off-shell formalism via relative periods
Open string moduli and relative cohomology
The extended Griffith-Dwork algorithm
Example: septic Calabi-Yau fivefold
On-shell open Gromov-Witten invariants
Off-shell formalism via generalized hypergeometric system
Generalized hypergeometric system for period integrals
Extended GKZ system for relative period integrals
On-shell disk partition function
Off-shell Picard-Fuchs equations
Open-closed duality in higher dimensions
Conclusions and discussions
A Direct calculation in the A-model via localization
Counting holomorphic spheres
B Monodromy analysis
C On-shell disk two-point function
D Solutions to the extended GKZ system
Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call