At various points in my life, I have held the following beliefs:
1) Stable homotopy theory is "easy" rationally, and "interesting" integrally.
2) The spectrum of topological modular forms (TMF) is an object that stable homotopy theorists are trying hard to understand integrally.
3) TMF has many connections to physics.
4) The mathematics relevant to physics is a "rational" story, and does not care much about integral or torsion aspects.
Taken together, this set of beliefs is evidently inconsistent. But I do not possess the knowledge, especially in physics, to know which one is incorrect (I would suspect the last one). I would be grateful if someone can clarify the situation. Thank you.
EDIT (8/31/17): I am grateful to the comments and answer. It seems that the problem indeed lies with (4). But I would love an example explaining a connection between physics and the integral aspects of the study of TMF.