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

TL;DR

The work analyzes closed sub-monodromy problems in local mirror symmetry for D-branes on blown-up C^3/Z_m orbifolds (m=3,4,6) yielding exceptional divisors P(1,a,b) and compares them to local E6,E7,E8 del Pezzo models. Using toric data, GKZ/Picard–Fuchs equations are derived and shown to share a common PF form with L_ell ∘ Θ_z, while the presence of a nonzero B-field in orbifold cases induces distinct physical effects. Solutions are constructed via Meijer G-functions across z=0,1,∞, enabling explicit mirror maps, torus periods, and Gromov–Witten invariants, and revealing a deep arithmetic structure through Mahler measures and special L-values. The paper then maps D-brane charges to coherent sheaves on the exceptional surfaces, discusses fractional branes at orbifold points, and derives a monodromy-invariant intersection form, highlighting intricate links between local geometry, modularity, and number theory.

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.

Figures (1)

• Figure 1: The integration path $\gamma$ for $\text{G}_{3,3}^{2,2}$ and $\text{G}_{3,3}^{3,2}$.