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Abstract:
Motivated by the Gamma conjecture, we introduce a notion of numerical vectors, such as Chern character and Mukai vector, and a notion of numerical t-stabilities and numerical slope functions on triangulated categories. The study of the derived categories of Calabi-Yau manifolds leads us to a conjecture which gives a relation between numerical t-stability and Bridgelands stability on smooth varieties. And when there exists generalized twisted Mukai vectors, we also obtain the results regarding the cohomological Fourier-Mukai (FM) transforms associated to the FM ones on the level of derived categories. In some cases, these cohomological FM transforms agree with the ones on the derived categories of twisted sheaves. In the second part, we discuss geometric and topological properties of G_2 manifolds due to the Kovalev's twisted connected sum constrction. In the Kovalev limit the Ricci-flat metrics on X_{L/R} approximate the Ricci-flat G_2-metrics and we identify the universal modulus, called the Kovalevton, that parametrizes this limit. Moreover, the low energy effective theory exhibits gauge theory sectors with extended supersymmetry in this limit. The universal (semi-classical) Kähler potential of the effective N=1 supergravity action is a function of the Kovalevton and the volume modulus of the G_2-manifold. We describe geometric degenerations in X_{L/R}, which lead to non-Abelian gauge symmetries enhancements with various matter content. Studying the resulting gauge theory branches, we argue that they lead to transitions compatible with the gluing construction and provide many new explicit examples of G_2-manifolds.
