Cutting-gluing in Heegaard-Floer theory with coefficient

Just as Heegaard-Floer (HF) theory is about Fukaya categories for symmetric product of Riemann surfaces, HF theory with coefficient is about  Fukaya categories of horizontal Hilbert scheme (possibly with fiberwise superpotential). This is used to give a symplectic realization of categorified quantum group representation. The method of cutting and gluing Riemann surface can be used to take tensor product of categorified representations. I will give the example when g=sl_2. This is work in progress, partly joint with Vivek Shende, and partly joint with  Mina Aganagic, Elise LePage, Yixuan Li.