Abstract: In this talk, we will discuss the relationship between SL(2,C) gauge theory and special Lagrangian submanifolds. Gauge-theoretic equations with the gauge group SL(2,C) over 3 manifolds have garnered significant interest due to their connection to new topological invariants, as proposed by Witten. The analytical aspects of these equations have also been extensively studied. Special Lagrangian submanifolds, a distinguished class of real calibrated submanifolds within Calabi-Yau manifolds, hold importance in string theory and are anticipated to play a role in mirror symmetry. In this talk, we will first survey the study of the compactness problem of the SL(2,C) character variety from gauge theory perspectives. We will then discuss the relationship between the boundary of the SL(2,C) flat connections and the multiplicity two problems of the special Lagrangian submanifolds.
