Majorana modes and the first Stiefel–Whitney class

Question

1. The first Stiefel–Whitney class of a vector bundle is an element in the first cohomology group of the base space. Namely, the first Stiefel–Whitney class for a vector bundle $E$ over a $d$-dimensional manifold $M^d$ is called $w_1(E) \in H^1(M^d;\mathbb{Z}/2\mathbb{Z})$.

2. The Majorana modes are certain "fermions" whose creation and annihilation operators are the same $\hat{\gamma}=\hat{\gamma}^\dagger$.

For physical reasons, in the context of 2+1 dimensional $p \pm ip$ superconductors, people have a vague impression that Majorana zero modes in the 2+1 dimensional space-time may be related a nontrivial generator in the cohomology group $H^1(\mathbb{Z}/2\mathbb{Z};\mathbb{Z}/2\mathbb{Z})$, where the first $\mathbb{Z}/2\mathbb{Z}$ can be viewed as $\mathbb{Z}/2\mathbb{Z}$-gauge fields, and the second $\mathbb{Z}/2\mathbb{Z}$ has something to do with the orientation of manifold. This may be similar to the context of the first Stiefel–Whitney class, that has something to do with the orientability of the based manifold.

In contrast, the second Stiefel–Whitney class $w_2(E) \in H^2(M^d;\mathbb{Z}/2\mathbb{Z})$ has something to do with the spin structure of the bundle $E$, which is suitable for defining spinors, including the Dirac spinors. Naively, the Dirac (complex) fermion is a pair of Majorana (real) fermions, say in 2+1 dimensional spacetime.

A side remark is that $H^1(\mathbb{Z}/2\mathbb{Z};\mathbb{Z}/2\mathbb{Z})$ also occurs in the calculation of the 3rd spin bordism group $\Omega_3^{Spin}(B \mathbb{Z}/2\mathbb{Z})$ for the classifying space $B \mathbb{Z}/2\mathbb{Z}$.

So, my question is the following.

In mathematics (references/literature are welcome), do the following objects appear in a unified context?

1. Majorana modes/fermions;

2. the first Stiefel—Whitney classes $w_1(E) \in H^1(M^d;\mathbb{Z}/2\mathbb{Z})$;

3. the cohomology group $H^1(\mathbb{Z}/2\mathbb{Z};\mathbb{Z}/2\mathbb{Z})$.

What are the math principles and structures behind?

Answer 1

An early paper on this connection is Dirac and Majorana Spinors on Non-Orientable Riemann Surfaces (1987). More recently, The Stiefel--Whitney theory of topological insulators (2016) relates the existence of an unpaired Majorana zero mode, i.e., geometrically a conical singularity created by the intersection of edge states, to the first Stiefel–Whitney class of a Pfaffian line bundle over the momentum space. (See proposition 1 on page 15.) A follow-up article is Noncommutative topological $\mathbb{Z}_2$ invariant.

I might add that a Majorana zero mode is an altogether different, and more interesting/useful, object than the Majorana fermion mentioned in the OP. Any self-adjoint fermion operator $\gamma$ is a Majorana operator, and any fermion operator $a$ can be decomposed into a pair of Majorana operators, $a=\gamma_1+i\gamma_2$, so the self-adjointness is not a particularly significant restriction. A Majorana zero mode satisfies in addition the requirement that it commutes with the Hamiltonian, $H\gamma=\gamma H$, which is a severe restriction that is only realised approximately in any physical system. As a consequence, when there exist $n$ Majorana zero modes the Hamiltonian has a $2^n$-degenerate ground state manifold, which is potentially useful as a storage of quantum information. The storage is referred to as "protected", because any term in the Hamiltonian that couples only to a single Majorana operator cannot lift the degeneracy. In a physical system this protection is realized by keeping the Majorana zero modes spatially separated.