Theorem. Free and antiperiodic Dirac kernels with internal multiplicity [fdip-009Q]
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  • November 1, 2026

For the covariant Dirac data, take the trivial connection \(U_j(x)=I_F\) and write \(D_0\) for its operator. If \(e\) is the number of even periods, then \[\dim _{\mathbb C}\ker D_0=q\,2^{d+e}.\] There are exactly \(2^e\) zero momentum blocks, each of dimension \(q\,2^d\). If instead every coordinate has the fixed scalar antiperiodic boundary condition, the corresponding operator \(D^-\) satisfies \[\dim _{\mathbb C}\ker D^- =\begin {cases} q\,2^d,&\text {all }N_j\text { are odd},\\ 0,&\text {some }N_j\text { is even}. \end {cases}\] The antiperiodic statement applies to the specified scalar boundary condition on \(S\otimes F\). Interpreting its seam transport as a \(G\)-connection requires \(-I_F\in \rho (G)\); otherwise it is an external scalar twist, or can be supplied by an additional \(\mathbb Z_2\) factor.

Proof. 

  • November 1, 2026

Tensor the complete normalized finite Fourier basis with \(S\otimes F\). For \(p_j=2\pi r_j/N_j\), \(0\le r_j<N_j\), the block is \[\begin {aligned} D_0(p)&=i\sum _j\Gamma _j\sin p_j,\\ D_0(p)^2&=-s(p)I_{S\otimes F},\\ s(p)&=\sum _j\sin ^2p_j. \end {aligned}\] The square follows from the Clifford anticommutators; the internal factor is the identity. If \(s(p)>0\), the inverse is \(-D_0(p)/s(p)\). If \(s(p)=0\), every sine vanishes and the entire block is zero. A coordinate has only \(p_j=0\) when \(N_j\) is odd, and also \(p_j=\pi \) when it is even. The \(2^e\) choices and complete Fourier decomposition give the asserted dimension. Setting \(q=1\) recovers Proposition [fdip-000T].

For the antiperiodic box, the complete basis from Example [fdip-006J] instead has \(p_j=(2r_j+1)\pi /N_j\). Tensoring with \(S\otimes F\) gives the same matrix symbol and square. In the interval \(0<p_j<2\pi \), the only possible zero is \(p_j=\pi \). The equation \(2r_j+1=N_j\) has one allowed solution exactly when \(N_j\) is odd. Thus there is one zero block if every period is odd and none otherwise. Its dimension is \(q\,2^d\). These scalar seam operators commute in different coordinates; the calculation asserts no such property for arbitrary links.

For two coordinates the explicit comparison is \[\begin {array}{c|ccc} (N_1,N_2)&(4,4)&(4,3)&(3,3)\\\hline \dim \ker D_0&16q&8q&4q\\ \dim \ker D^-&0&0&4q \end {array}\] These are dimensions over \(\mathbb C\) in the chosen reducible module \(S\), with internal multiplicity \(q\). They count exact zero vectors on a finite grid, not physical species or chirality.