Non-Abelian Conifold Transitions and N=4 Dualities in Three Dimensions
Kentaro Hori, Hirosi Ooguri, Cumrun Vafa
TL;DR
This work shows that nonabelian $N=2$ Higgsing in four dimensions can be realized geometrically as conifold-type transitions in type II string compactifications on Calabi–Yau threefolds, and that, upon circle compactification, the resulting $N=4$, $d=3$ dualities arise from perturbative string symmetries like T-duality and mirror symmetry. By using local CY models and ADE singularity resolutions, the authors derive explicit 3d dual pairs, including $U(k)$ and $Sp(k)$ theories with matter, and connect these to the Higgs/Coulomb branch structure via extremal transitions. They further show that some duals are non-Lagrangian and verify a conjecture that ordinary gauge theories can be dual to compactifications of exceptional tensionless string theories in three dimensions. The results provide a unifying geometric framework for 3d $N=4$ dualities, bridging geometric engineering, mirror symmetry, and tensionful/non-tensionless string sectors with perturbative string symmetries. The approach highlights the potential ubiquity of dual descriptions across dimensions and suggests a broader class of non-Lagrangian dualities in quantum field theory.
Abstract
We show how Higgs mechanism for non-abelian N=2 gauge theories in four dimensions is geometrically realized in the context of type II strings as transitions among compactifications of Calabi-Yau threefolds. We use this result and T-duality of a further compacitification on a circle to derive N=4, d=3 dual field theories. This reduces dualities for N=4 gauge systems in three dimensions to perturbative symmetries of string theory. Moreover we find that the dual of a gauge system always exists but may or may not correspond to a lagrangian system. In particular we verify a conjecture of Intriligator and Seiberg that an ordinary gauge system is dual to compactification of Exceptional tensionless string theory down to three dimensions.