Kac-Moody Extensions of 3-Algebras and M2-branes

We study the 3-algebraic structure involved in the recently shown M2-branes worldvolume gauge theories. We first extend an arbitrary finite dimensional 3-algebra into an infinite dimensional 3-algebra by adding a mode number to each generator. A unique central charge in the algebra of gauge transformations appears naturally in this extension. We present an infinite dimensional extended 3-algebra with a general metric and also a different extension with a Lorentzian metric. We then study ordinary finite dimensional 3-algebras with different signatures of the metric, focusing on the cases with a negative eigenvalue and the cases with a zero eigenvalue. In the latter cases we present a new algebra, whose corresponding theory is a decoupled abelian gauge theory together with a free theory with global gauge symmetry, and there is no negative kinetic term from this algebra.Comment: v3: Appendix A added proving an identity; minor corrections and typos fixed. v4: slight refinement in section 2.1, no other change. 18 pages, Late

T4T^4 fibrations over Calabi-Yau two-folds and non-Kahler manifolds in string theory

We construct a geometric model of eight-dimensional manifolds and realize them in the context of type II string theory. These eight-manifolds are constructed by non-trivial $T^{4}$ fibrations over Calabi-Yau two-folds. These give rise to eight-dimensional non-Kahler Hermitian manifolds with $SU(4)$ structure. The eight-manifold is also a circle fibration over a seven-dimensional $G_{2}$ manifold with skew torsion. The eight-manifolds of this type appear as internal manifolds with $SU(4)$ structure in type IIB string theory with $F_{3}$ and $F_{7}$ fluxes. These manifolds have generalized calibrated cycles in the presence of fluxes.Comment: 22 page

Relation between large dimension operators and oscillator algebra of Young diagrams

The operators with large scaling dimensions can be labelled by Young diagrams. Among other bases, the operators using restricted Schur polynomials have been known to have a large $N$ but nonplanar limit under which they map to states of a system of harmonic oscillators. We analyze the oscillator algebra acting on pairs of long rows or long columns in the Young diagrams of the operators. The oscillator algebra can be reached by a Inonu-Wigner contraction of the $u(2)$ algebra inside of the $u(p)$ algebra of $p$ giant gravitons. We present evidences that integrability in this case can persist at higher loops due to the presence of the oscillator algebra which is expected to be robust under loop corrections in the nonplanar large $N$ limit.Comment: 21 page