D-Brane Probes of Special Holonomy Manifolds
Sergei Gukov, David Tong
TL;DR
The authors generalize D2-brane probe techniques to non-compact manifolds with exceptional holonomy, establishing a G2 quotient construction and exploiting 3d mirror symmetry to realize M-theory geometries as Higgs or Coulomb branches. By blending M-theory lifts, IIA/L-picture fixed loci, and IIB brane models, they derive algebraic Higgs-branch descriptions for cones over spaces such as ${\mathbb{C}P}^3$ and $SU(3)/U(1)^2$, and map geometric transitions to field-theoretic deformations. The work extends to ${\rm Sp}(2)$ and ${\rm Spin}(7)$ holonomies, offering new brane realizations, deformations, and potential vacuum structures under various flux and Chern–Simons couplings. Overall, the paper provides a unified, brane-based framework to study special-holonomy manifolds via mirror-symmetric 3d gauge theories, with clear topological and moduli-space correspondences that enrich our understanding of geometric transitions in these backgrounds.
Abstract
Using D2-brane probes, we study various properties of M-theory on singular, non-compact manifolds of G_2 and Spin(7) holonomy. We derive mirror pairs of N=1 supersymmetric three-dimensional gauge theories, and apply this technique to realize exceptional holonomy manifolds as both Coulomb and Higgs branches of the D2-brane world-volume theory. We derive a ``G_2 quotient construction'' of non-compact manifolds which admit a metric of G_2 holonomy. We further discuss the moduli space of such manifolds, including the structure of geometrical transitions in each case. For completeness, we also include familiar examples of manifolds with SU(3) and Sp(2) holonomy, where some of the new ideas are clarified and tested.