CMSA Geometry and Mathematical Physics Seminar: Skein valued curve counts and associatives

We consider Lagrangians in the cotangent bundle of a 3-manifold that give branch covers the 0-section. A circle bundle over the cotangent bundle that degenerates over the Lagrangian gives an A_n-fibration over the base 3-manifold. We equip this A_n fibration with a circle invariant positive 3-form for which fibers are co-associative and show that circle invariant associatives correspond (with first order data) to holomorphic curves in the cotangent bundle with boundary on the Lagrangian. Counts of curves with Lagrangian boundary condition, by the values of their boundary in the skein module of the boundary condition, are deformation invariant and this way we obtain deformation invariant counts of circle invariant associatives taking values in the skein module of the fixed-point locus.  We consider an adiabatic limit of the fibration where the Lagrangian collapses onto the zero section. Here holomorphic curves converge to flow graphs that are directly related to the flow graphs in adiabatic co-associative fibrations studied by Donaldson and Scaduto. Furthermore, sufficiently close to the adiabatic limit, the positive 3-form (which is not closed) is tamed by another positive closed 3-form. The talk reports on joint work with Esfahani, Shende, and Wang.