Fourier modes give index zero for the periodic flat even torus

First classify the characters of $\mathbb{T}=\mathbb{R}\mathrm{/}\mathbb{Z}$. If $\chi :\mathbb{T}\to S^1$ is a continuous character, let $f(t)=\chi ([t])$. By continuity at $0$, choose $\delta >0$ so that on $\mathrm{\mid }t\mathrm{\mid }<2\delta$ the values of $f$ lie in the arc with a continuous argument $\theta (t)\in (-1\mathrm{/}8,1\mathrm{/}8)$, normalized by $f(t)=e^{2\pi i\theta (t)}$. Whenever $\mathrm{\mid }s\mathrm{\mid },\mathrm{\mid }t\mathrm{\mid },\mathrm{\mid }s+t\mathrm{\mid }<\delta$, the character identity gives $\theta (s+t)-\theta (s)-\theta (t)\in \mathbb{Z}$; this difference has absolute value less than $3\mathrm{/}8$, so it is zero. Define $\mathrm{\Theta }(u)=N\theta (u\mathrm{/}N)$ for any positive integer $N$ with $\mathrm{\mid }u\mathrm{/}N\mathrm{\mid }<\delta$. If $N,N^{\mathrm{\prime }}$ both work, repeated local additivity gives $N\theta (u\mathrm{/}N)=N{N}^{\mathrm{\prime }}\theta (u\mathrm{/}(N{N}^{\mathrm{\prime }}))=N^{\mathrm{\prime }}\theta (u\mathrm{/}{N}^{\mathrm{\prime }})$, so $\mathrm{\Theta }$ is well-defined. Choosing $N$ large enough for $s,t,s+t$ shows $\mathrm{\Theta }(s+t)=\mathrm{\Theta }(s)+\mathrm{\Theta }(t)$. Near zero $\mathrm{\Theta }=\theta$, so $\mathrm{\Theta }$ is continuous, and the continuous additive equation gives $\mathrm{\Theta }(t)=\alpha t$ for some $\alpha \in \mathbb{R}$. For arbitrary $t$, choose $N$ with $\mathrm{\mid }t\mathrm{/}N\mathrm{\mid }<\delta$; then $f(t)=f(t\mathrm{/}N{)}^N=e^{2\pi i\alpha t}$. Since $f(1)=1$, $\alpha \in \mathbb{Z}$. Conversely each $k\in \mathbb{Z}$ defines ${\chi }_k([t])=e^{2\pi ikt}$, so $\widehat{\mathbb{T}}=\mathbb{Z}$. Finite-product duality now identifies the frequencies of ${\mathbb{T}}^{2m}$ with ${\mathbb{Z}}^{2m}$, and compact-group Plancherel makes these characters a complete orthonormal basis.

The quotient ${\mathbb{R}}^{2m}\mathrm{/}{\mathbb{Z}}^{2m}$ has a global parallel coframe, so the Levi–Civita connection and the periodic spin connection have zero local connection matrix. For each $k\in {\mathbb{Z}}^{2m}$, the Fourier character $e_k(x)=e^{2\pi ik\cdot x}$ satisfies $D(e_k\otimes s)=2\pi i{e}_k\otimes c(k)s$. For $k\mathrm{\ne }0$, $c(k{)}^2=-\mathrm{\mid }k{\mathrm{\mid }}^{2}I$, so $c(k)$ is invertible.

By the completeness in [F3], every $L^2$ spinor is determined by its Fourier coefficients. For $\psi \in H^1$, testing each weak derivative against a Fourier character and integrating over a fundamental cube gives $\widehat{D\psi }(k)=2\pi ic(k)\widehat{\psi }(k)$; the opposite boundary faces cancel by periodicity. Thus $D\psi =0$ forces every nonzero-frequency coefficient to vanish by step 1.2, since $c(k)$ is invertible there; the zero-frequency coefficient is arbitrary. The same argument on the two graded summands shows that $\ker D^{+}$ consists exactly of constant elements of $S^{+}$ and $\ker (D^{+}{)}^{\ast }=\ker D^{-}$ consists exactly of constants in $S^{-}$.

By [F2], ${\dim }_{\mathbb{C}}{S}^{+}={\dim }_{\mathbb{C}}{S}^{-}=2^{m-1}$. Hence the Fredholm index supplied by [F4] is $\dim \ker D^{+}-\dim \ker (D^{+}{)}^{\ast }=2^{m-1}-2^{m-1}=0$. For $m=1$ both dimensions are one; the zero-dimensional torus is outside the stated range $m\ge 1$. The half-spin dimensions use the exact algebra suppliers in [A1].