Fiber sum formulas for 4-manifolds, topological modular forms and $6d\ \mathcal{N}=(1,0)$ theories

Using the relation between four manifolds and topological modular form (TMF) from the six dimensional approach, we exhibit fiber sum formulas for infinite families of smooth spin four manifolds associated to compactifications of free and interacting 6d (1,0) SCFTs. We find that even the free theories have nontrivial fiber sum formulas and their forms are sensitive to an individual theory and parameters of four-manifolds. Furthermore, we reinforce the conjecture of Stolz and Teichner by expanding its evidence.