On Flavor Symmetry in Lattice Quantum Chromodynamics

TL;DR

This paper explores the relationship between flavor symmetry implementation in lattice QCD and singularity theory in complex algebraic geometry, revealing geometric structures underlying fermion flavors and their charges.

Contribution

It establishes a novel connection between Creutz's flavor method in lattice QCD and toric singularities in complex geometry, providing geometric insights into fermion flavors.

Findings

Creutz flavors relate to toric singularities of Kahler manifolds.

Naive fermions in QCD$_{2N}$ have flavors at poles of 2-spheres with specific quantum charges.

Creutz flavors in the Karsten-Wilczek model connect to conifold singularity resolutions.

Abstract

Using a well established method to engineer non abelian symmetries in superstring compactifications, we study the link between the point splitting method of Creutz et al of refs [1,2] for implementing flavor symmetry in lattice QCD; and singularity theory in complex algebraic geometry. We show amongst others that Creutz flavors for naive fermions are intimately related with toric singularities of a class of complex Kahler manifolds that are explicitly built here. In the case of naive fermions of QCD$_{2N}$, Creutz flavors are shown to live at the poles of real 2-spheres and carry quantum charges of the fundamental of $[SU(2)]^{2N}$. We show moreover that the two Creutz flavors in Karsten-Wilczek model, with Dirac operator in reciprocal space of the form $i\gamma_1 F_1+i\gamma_2 F_2 + i\gamma_3 F_3+\frac{i}{\sin \alpha}\gamma_4 F_4$, are related with the small resolution of conifold singularity that live at $\sin \alpha =0$. Other related features are also studied.