2. 28 Sep 2022, Geometry seminar, Université libre de BruxellesTitle: Associatives in twisted connected sum G_2 manifolds
Abstract: G_2 manifolds are Riemannian manifolds whose holonomy contained in the exceptional Lie group G_2 and associatives inside them are some 3 dimensional calibrated submanifolds which play a crucial role for defining several enumerative theories of the G_2 manifolds. This motivates us to construct many examples of associatives. The most effective method to date of constructing G_2 manifolds is the twisted connected sum construction which glues a matching pair of asymptotically cylindrical (ACyl) G_2 manifold or ACyl Calabi-Yau 3-fold. In this talk we present a method to construct closed rigid (unobstructed) associatives in the twisted connected sum G_2-manifolds by gluing ACyl associatives in ACyl G_2-manifolds under a hypothesis which can be interpreted as a transverse Lagrangian intersection condition. We rewrite the gluing hypothesis for ACyl associatives obtained from ACyl holomorphic curves or ACyl special Lagrangian 3-folds in ACyl Calabi-Yau 3-folds. This helps us to construct many new associatives which are diffeomorphic to S^3, RP^3 or RP^3#RP^3.
1. 13 Sep 2022, Special Holonomy in Geometry, Analysis, and Physics: Progress and Open Problems, Simons center for Geometry and PhysicsTitle: Deformations and gluing of asymptotically cylindrical associatives
Abstract: Given a matching pair of asymptotically cylindrical (Acyl) G_2 manifolds the twisted connected sum construction produces a one parameter family of closed G_2 manifolds. We describe when we can construct closed rigid associatives in these closed G_2 manifolds by gluing suitable pairs of Acyl associatives in the matching pair of Acyl G_2 manifolds. The hypothesis and analysis in the gluing theorem requires some understanding of the deformation theory of Acyl associatives which will also be discussed. At the end we will describe examples of closed associatives coming from Acyl holomorphic curves or special Lagrangians.
Abstract: In this talk, I will discuss the proposal of Doan and Walpuski to define an invariant of G_2-manifolds based on counting associative submanifolds with Seiberg-Witten monopoles on them. Their proposal was motivated by the analysis of various transitions that can change the number of associative submanifolds. In the most general case, the invariant they propose are Floer homology groups. Reference: Doan-Walpuski, arXiv:1712.08383