New conjecture on exact Dirac zero-modes of lattice fermions

We propose a new conjecture on the relation between the exact Dirac zero-modes of free and massless lattice fermions and the topology of the manifold on which the fermion action is defined. Our conjecture claims that the maximal number of exact Dirac zero-modes of fermions on finite-volume and finite-spacing lattices defined by a discretizing torus, hyperball, their direct-product space, and hypersphere is equal to the summation of the Betti numbers of their manifolds if several specific conditions on lattice formulations are satisfied. We start with reconsidering exact Dirac zero-modes of naive fermions on the lattices whose topologies are a torus, hyperball, and their direct-product space (T^D × B^d). We find that the maximal number of exact zero-modes of free Dirac fermions is in exact agreement with the sum of Betti numbers $\sum r=0D{\beta }_r$ for these manifolds. Indeed, the 4D lattice fermion on a torus has up to 16 zero-modes while the sum of Betti numbers of T⁴ is 16. This coincidence holds also for the D-dimensional hyperball and their direct-product space T^D × B^d. We study several examples of lattice fermions defined on a certain discretized hypersphere (S^D), and find that it has up to two exact zero-modes, which is the same number as the sum of Betti numbers of S^D. From these facts, we conjecture the equivalence of the maximal number of exact Dirac zero-modes and the summation of Betti numbers under specific conditions. We discuss a program for proof of the conjecture in terms of Hodge theory and spectral graph theory.