Yumoto–Misumi conjecture on lattice-fermion species and Betti numbers

Let ${\mathcal M}$ be a $D$-dimensional manifold and let ${}^{\ast}{\mathcal M}$ be a finite-volume, finite-spacing lattice obtained by discretizing it. Identify the number of fermion species with the number of Dirac zero-modes of free lattice fermions. Assume that the lattice fermion action has central difference, $\gamma_{5}$-hermiticity, four spinors, locality, and finite lattice volume. Write $\beta_r({\mathcal M})$ for the $r$-th Betti number of ${\mathcal M}$.

Yumoto–Misumi conjecture. Under these conditions, the maximal number of fermion species is

$$\operatorname{max}[{\mathcal N}({}^{\ast}{\mathcal M})]=\sum_{r=0}^{D}\beta_r({\mathcal M}),$$

where ${\mathcal N}({}^{\ast}{\mathcal M})$ denotes the number of fermion species on the lattice-discretized manifold.

This conjecture proposes that finite-volume lattice fermions detect the topology of the underlying continuum manifold, generalizing the familiar doubling counts for tori and the corresponding counts for balls and spheres. The paper discusses evidence in several geometries and suggests that a proof could follow from relating zero-modes of the squared free naive Dirac operator to the Betti numbers, but no general proof is given.