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

TL;DR

This paper explores fiber sum formulas for 4-manifolds linked to 6d (1,0) theories and topological modular forms, revealing nontrivial structures and supporting the Stolz-Teichner conjecture.

Contribution

It introduces new fiber sum formulas for families of 4-manifolds derived from 6d (1,0) SCFTs, highlighting their sensitivity to individual theories and parameters.

Findings

Even free theories exhibit nontrivial fiber sum formulas

Formulas depend on specific theories and four-manifold parameters

Supports the Stolz-Teichner conjecture with expanded evidence

Abstract

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.