For exceptions in the context of string theory, see for example [2, 3, 4, 5].

The additional element generating $\text{Spin}(d)$ is a $2\pi$ rotation, often denoted $(-1{)}^F$ since it changes the sign of all spinor fields.

This makes it impossible to satisfy $\mathrm{\Psi }=C\mathrm{\Psi }$, and is the reason Majorana (self-conjugate) spinors do not exist in 6D.

In verifying these signs it helps to use explicit spinor indices. See for example [15] page 70.

The factor of $i$ is included to make $i\overline{\mathrm{\Psi }}\overline{\mathrm{\Gamma }}\mathrm{\Psi }$ real.

In checking the behavior under $R_5$, it is important that, as shown in (29), the spinor is periodic on the covering torus.

This is a simple example of building an invariant by averaging over a group of transformations. It is important that ${\mathrm{\Psi }}^K$ satisfies the Dirac equation and that $K^2=1$. Note that while $\mathrm{\Psi }+{\mathrm{\Psi }}^K$ satisfies the Klein bottle boundary conditions, the other linear combination $\mathrm{\Psi }-{\mathrm{\Psi }}^K$ violates them.

This solves the 6D Dirac equation because $\overline{\gamma }{\psi }_2$ satisfies $i{\gamma }^{\mu }{\partial }_{\mu }\overline{\gamma }{\psi }_2(x)=-m\overline{\gamma }{\psi }_2(x)$.

To verify this, note that $\alpha$ changes sign under $n_4\to -n_4$.

The relation is $-i{e}^{-i\alpha }(-1{)}^{{\ell }_5}({\mathrm{\Psi }}_2+{\mathrm{\Psi }}_2^K){|}_{{\ell }_4,{\ell }_5}=({\mathrm{\Psi }}_1+{\mathrm{\Psi }}_1^K){|}_{-{\ell }_4,{\ell }_5}$.

We could make them look completely identical if we had built the ${\mathrm{pin}}^{-}$ structure using $-i{\mathrm{\Gamma }}^4\overline{\mathrm{\Gamma }}$.

This yields a time-ordered correlator. See for example [28], pages 23 and 76.

We have not found boundary conditions that involve charge conjugation and define a ${\mathrm{pin}}^{-}$ structure.