[Math] Stable homotopy theory and physics

Question

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.

Answer 1

There's an interesting application of stable homotopy theory to condensed-matter physics, and it makes heavy use of integral and torsion information, contradicting your 4^th assumption.

Within the general program of understanding topological phases of matter, condensed-matter theorists are interested in symmetry-protected topological phases (SPT phases). Approximately speaking, these are systems which have interesting topological behavior in the presence of a symmetry, but become trivial when that symmetry is broken. Their classification has gradually gotten more homotopical:

• Kitaev uses real and complex $K$-theory to classify topological insulators and superconductors. Freed-Moore later generalized this to twisted equivariant $K$-theory.
• Kapustin and Kapustin-Thorngren-Turzillo-Wang use $\mathit{MSpin}\wedge\mathit{BG}$ to classify fermionic SPT phases with symmetry group $G$.
• Freed and Freed-Hopkins classify SPTs as homotopy classes of maps $$[\mathit{MTH}, \Sigma^{n+1}I_{\mathbb Z}],$$ where $\mathit{MTH}$ is the Madsen-Tillmann spectrum (a kind of Thom spectrum) for the symmetry type $H$ and $I_{\mathbb Z}$ is the Anderson dual of the sphere. The derivation uses some equivariant stable homotopy theory, and the calculations use the Adams spectral sequence.

In all of these examples, the torsion information is essential: a $\mathbb Z/2$ classification means there's a phase which is nontrivial, but such that two copies of it stacked together can be continuously deformed to a trivial phase. Examples of such phases have come up in condensed-matter theory and are expected to display this behavior in experiments.

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There are other places integral information is, well, integral: for example, if an electron moves in a loop $\ell$ around a magnetic monopole, the value of the action depends on the winding number of $\ell$, producing discrete invariants.