# Stable homotopy theory and physics

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.

• I doubt that 4) is true. Aug 30, 2017 at 15:19

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 4th 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.

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.

• While this is a story I find awesome, I'm not sure if it addresses the OPs TMF-related concerns. And actually I think the OP's question is an important one: what physical relevance does the integral information in tmf have? Aug 30, 2017 at 16:13
• @DylanWilson it absolutely doesn't address the TMF-related concerns, but as I read it, OP's question, especially point 4, was more general. Unfortunately, I don't know anything about TMF, so I can't answer the more specific question :/ Aug 30, 2017 at 16:24
• Thank you for the answer, it is very helpful. I indeed had in mind something along the lines of what Dylan writes, and have edited the question to reflect this. Aug 31, 2017 at 13:52
• There is the idea of stacking together of 576 free fermions replacing that two copies of a phase deforming to a trivial phase. Oct 12, 2018 at 21:45
• @AHusain Could one read about it somewhere? Oct 13, 2018 at 4:57