New conjecture on exact Dirac zero-modes of lattice fermions

TL;DR

This paper proposes a conjecture linking the maximum number of fermion species on a lattice to the sum of Betti numbers of the underlying manifold, supported by examples on various topologies.

Contribution

It introduces a novel conjecture connecting lattice fermion species doubling to topological invariants, specifically Betti numbers, and explores potential proof strategies.

Findings

Maximal fermion species on torus matches Betti number sum (16).

Hyperball and product spaces also show agreement with Betti sums.

Discretized hypersphere exhibits species count equal to Betti number sum.

Abstract

We propose a new conjecture on the relation between the species doubling of lattice fermions and the topology of manifold on which the fermion action is defined. Our conjecture claims that the maximal number of fermion species on a finite-volume and finite-spacing lattice defined by discretizing a $D$-dimensional manifold is equal to the summation of the Betti numbers of the manifold. We start with reconsidering species doubling of naive fermions on the lattices whose topologies are torus ($T^{D}$), hyperball ($B^D$) and their direct-product space ($T^{D} \times B^{d}$). We find that the maximal number of species is in exact agreement with the sum of Betti numbers $\sum^{D}_{r=0} \beta_{r}$ for these manifolds. Indeed, the $4D$ lattice fermion on torus has up to $16$ species while the sum of Betti numbers of $T^4$ is $16$. This coincidence holds also for the $D$-dimensional hyperball and their direct-product space $T^{D} \times B^{d}$. We study several examples of lattice fermions defined on discretized hypersphere ($S^{D}$), and find that it has up to $2$ species, 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 fermion species and the summation of Betti numbers. We discuss a program for proof of the conjecture in terms of Hodge theory and spectral graph theory.