String Dualities, Higgs Bundles and $G_2$ Geometry

Abstract: Dualities in string/M theory often provide novel perspectives for deformation problems in geometry. In one such instance, involving Large $N$ duality in the B-model, one can construct a family of ALE-fibered Calabi-Yau threefolds over a Riemann surface $S_g$ ($g \geq 2$) whose Jacobian integrable system is isomorphic to the $G$-Hitchin system over $S_g$, where $G$ is the compact real form associated to the ALE type via the McKay correspondence. I will explain a different physical framework, involving M-theory/Type IIA duality, that gives an analogous construction of ALE-fibered $G_2$-manifolds parametrized by spectral covers of certain "smooth Higgs bundles" over a $3$-manifold. I will explain how this theory connects with Donaldson's theory of Kovalev-Lefschetz fibrations and how it presents a window for applying algebro-geometric techniques to moduli problems in $G_2$-geometry. Time permitting, I will comment on a second algebro-geometric model for the moduli space of (complexified) $G_2$-structures derived from SYZ Mirror Symmetry for the Type IIA geometry.

HEP - theorymathematical physicsalgebraic geometrydifferential geometrysymplectic geometry

Audience: researchers in the topic

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