On Dirac Zero Modes in Hyperdiamond Model

Using the SU(5) symmetry of the 4D hyperdiamond and results on the study of 4D graphene given in "Four Dimensional Graphene" (L.B Drissi, E.H Saidi, M. Bousmina, CPM-11-01, Phys. Rev. D (2011)), we engineer a class of 4D lattice QCD fermions whose Dirac operators have two zero modes. We show that generally the zero modes of the Dirac operator in hyperdiamond fermions are captured by a tensor {\Omega}_{{\mu}}^{l} with 4\times5 complex components linking the Euclidean SO(4) vector {\mu}; and the 5-dimensional representation of SU(5). The Bori\c{c}i-Creutz (BC) and the Karsten-Wilzeck (KW) models as well as their Dirac zero modes are rederived as particular realizations of {\Omega}_{{\mu}}^{l}. Other features are also given. Keywords: Lattice QCD, Bori\c{c}i-Creutz and Karsten-Wilzeck models, 4D hyperdiamond, 4D graphene, SU(5) Symmetry.Comment: LaTex, 28 pages, To appear in Phys Rev

Four Dimensional Graphene

Mimicking pristine 2D graphene, we revisit the BBTW model for 4D lattice QCD given in ref.[5] by using the hidden SU(5) symmetry of the 4D hyperdiamond lattice H_4. We first study the link between the H_4 and SU(5); then we refine the BBTW 4D lattice action by using the weight vectors \lambda_1, \lambda_2, \lambda_3, \lambda_4, \lambda_5 of the 5-dimensional representation of SU(5) satisfying {\Sigma}_i\lambda_i=0. After that we study explicitly the solutions of the zeros of the Dirac operator D in terms of the SU(5) simple roots \alpha_1, \alpha_2, \alpha_3, \alpha_4 generating H_4; and its fundamental weights \omega_1, \omega_2, \omega_3, \omega_4 which generate the reciprocal lattice H_4^\ast. It is shown, amongst others, that these zeros live at the sites of H_4^\ast; and the continuous limit D is given by ((id\surd5)/2) \gamma^\muk_\mu with d, \gamma^\mu and k_\mu standing respectively for the lattice parameter of H_4, the usual 4 Dirac matrices and the 4D wave vector. Other features such as differences with BBTW model as well as the link between the Dirac operator following from our construction and the one suggested by Creutz using quaternions, are also given. Keywords: Graphene, Lattice QCD, 4D hyperdiamond, BBTW model, SU(5) Symmetry.Comment: LaTex, 26 pages, 1 figure, To appear in Phys Rev

Extremal Black Attractors in 8D Maximal Supergravity

Motivated by the new higher D-supergravity solutions on intersecting attractors obtained by Ferrara et al. in [Phys.Rev.D79:065031-2009], we focus in this paper on 8D maximal supergravity with moduli space [SL(3,R)/SO(3)]x[SL(2,R)/SO(2)] and study explicitly the attractor mechanism for various configurations of extremal black p- branes (anti-branes) with the typical near horizon geometries AdS_{p+2}xS^{m}xT^{6-p-m} and p=0,1,2,3,4; 2<=m<=6. Interpretations in terms of wrapped M2 and M5 branes of the 11D M-theory on 3-torus are also given. Keywords: 8D supergravity, black p-branes, attractor mechanism, M-theory.Comment: 37 page

't Hooft lines of ADE-type and Topological Quivers

We investigate 4D Chern-Simons theory with ADE gauge symmetries in the presence of interacting Wilson and 't Hooft line defects. We analyse the intrinsic properties of these lines' coupling and explicate the building of oscillator-type Lax matrices verifying the RLL integrability equation. We propose gauge quiver diagrams Q$_{G}^{\mu }$ encoding the topological data carried by the Lax operators and give several examples where Darboux coordinates are interpreted in terms of topological bi-fundamental matter. We exploit this graphical description $\left( i\right)$ to give new results regarding solutions in representations beyond the fundamentals of $sl_{N}$, $so_{2N}$ and $e_{6,7}$, and $\left( ii\right)$ to classify the Lax operators for simply laced symmetries in a unified E$_{7}$ CS theory. For quick access, a summary list of the leading topological quivers Q$$ is given in the conclusion section [Figures 29.(a-e), 30.(a-d) and 31.(a-d)].Comment: LaTeX, 74 pages, 32 figure