Closed Sub-Monodromy Problems, Local Mirror Symmetry and Branes on Orbifolds

TL;DR

This paper investigates D-branes on orbifold Calabi-Yau threefolds using local mirror symmetry, revealing similarities with del Pezzo models and uncovering the influence of background B-fields on physical properties.

Contribution

It demonstrates that orbifold models share Picard-Fuchs equations with del Pezzo models and analyzes the effects of B-fields and monodromies using Meijer G-functions, linking mirror symmetry to number theory.

Findings

Picard-Fuchs equations match those of del Pezzo models

Background B-field affects physical properties of orbifold models

Exponential growth of Gromov-Witten invariants relates to Dirichlet L-functions

Abstract

We study D-branes wrapping an exceptional four-cycle P(1,a,b) in a blown-up C^3/Z_m non-compact Calabi-Yau threefold with (m;a,b)=(3;1,1), (4;1,2) and (6;2,3). In applying the method of local mirror symmetry we find that the Picard-Fuchs equations for the local mirror periods in the Z_{3,4,6} orbifolds take the same form as the ones in the local E_{6,7,8} del Pezzo models, respectively. It is observed, however, that the orbifold models and the del Pezzo models possess different physical properties because the background NS B-field is turned on in the case of Z_{3,4,6} orbifolds. This is shown by analyzing the periods and their monodromies in full detail with the help of Meijer G-functions. We use the results to discuss D-brane configurations on P(1,a,b) as well as on del Pezzo surfaces. We also discuss the number theoretic aspect of local mirror symmetry and observe that the exponent which governs the exponential growth of the Gromov-Witten invariants is determined by the special value of the Dirichlet L-function.