On Flavor Symmetry in Lattice Quantum Chromodynamics

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.