Mirror Symmetries for Brane Configurations and Branes at Singularities
Mina Aganagic, Andreas Karch, Dieter Lust, Andre Miemiec
TL;DR
This work develops a local mirror framework for non-compact Calabi–Yau singularities probed by D3-branes, using two T-dual NS-brane configurations to map geometries to brane setups. It identifies two families, the orbifolded conifold ${ m C}_{kl}$ and the generalized conifold ${ m G}_{kl}$, which are related by a conifold transition and realized respectively as brane boxes and brane intervals; their mirror relation exchanges complex and Kähler data through brane configurations. A key result is the generalization of the brane box rules to include diamonds at intersections, enabling correct reading of the gauge theories on D3 probes and recovering orbifold theories on baryonic branches. The analysis connects D3/D4/D5 setups to D6 branes wrapping 3-cycles in the mirror, and extends to M-theory via domain walls and G2 3-cycles, offering a unified view of interval-box duality, conifold transitions, and their gauge-theory content. The findings provide a concrete, calculable bridge between geometric transitions in non-compact Calabi–Yau spaces and their brane realizations, with potential implications for non-perturbative dynamics and geometric engineering in string/M-theory.
Abstract
We study local mirror symmetry on non-compact Calabi-Yau manifolds (conifold type of singularities) in the presence of D3 brane probes. Using an intermediate brane setup of NS 5-branes `probed' by D4 resp. D5 branes, we can explicitly T-dualize three isometry directions to relate a non-compact Calabi-Yau manifold to its local mirror. The intermediate brane setup is the one that is best suited to read off the gauge theory on the probe. Both intervals and boxes of NS 5-branes appear as brane setups. Going from one to the other is equivalent to performing a conifold transition in the dual geometry. One result of our investigation is that the brane box rules as they have been discussed so far should be generalized. Our new rules do not need diagonal fields localized at the intersection. The old rules reappear on baryonic branches of the theory.