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The Nielsen-Ninomiya theorem states that for a local, Hermitian, translationally invariant lattice fermion theory in even-dimensional spacetime, the number of left-handed Weyl fermions is equal to the number of right-handed Weyl fermions.

I don't understand dimensions in this concept. This theorem is stated in some sources to be true for even dimensional systems, and in some sources to be true for odd dimensional systems. For example, it is stated to be true for 1+1 or 3+1 dimensional systems. However, its proof via differential topology is related to the Brillouin torus, which corresponds to 2+1 dimensions.

I'm confused, please help me.

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  • $\begingroup$ From where do you get the notion that the Brillouin torus implies 2+1 dimensions? The Nielsen and Nimoniya paper considers momentum space to be an $S_1 \times S_1 \times S_1$ hypertorus. $\endgroup$
    – Anyon
    Commented Dec 8 at 22:36

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