Abelian-Higgs and Vortices from ABJM: towards a string realization of AdS/CMT

We present ans\"{a}tze that reduce the mass-deformed ABJM model to gauged Abelian scalar theories, using the fuzzy sphere matrices $G^\alpha$. One such reduction gives a Toda system, for which we find a new type of nonabelian vortex. Another gives the standard Abelian-Higgs model, thereby allowing us to embed all the usual (multi-)vortex solutions of the latter into the ABJM model. By turning off the mass deformation at the level of the reduced model, we can also continuously deform to the massive $\phi^4$ theory in the massless ABJM case. In this way we can embed the Landau-Ginzburg model into the AdS/CFT correspondence as a consistent truncation of ABJM. In this context, the mass deformation parameter $\mu$ and a field VEV $$ act as $g$ and $g_c$ respectively, leading to a well-motivated AdS/CMT construction from string theory. To further this particular point, we propose a simple model for the condensed matter field theory that leads to an approximate description for the ABJM abelianization. Finally, we also find some BPS solutions to the mass-deformed ABJM model with a spacetime interpretation as an M2-brane ending on a spherical M5-brane.Comment: 43 pages, latex, explanations added in the introduction, end of section 4, and on page 2

The fuzzy S^2 structure of M2-M5 systems in ABJM membrane theories

We analyse the fluctuations of the ground-state/funnel solutions proposed to describe M2-M5 systems in the level-k mass-deformed/pure Chern-Simons-matter ABJM theory of multiple membranes. We show that in the large N limit the fluctuations approach the space of functions on the 2-sphere rather than the naively expected 3-sphere. This is a novel realisation of the fuzzy 2-sphere in the context of Matrix Theories, which uses bifundamental instead of adjoint scalars. Starting from the multiple M2-brane action, a U(1) Yang-Mills theory on R^{2,1} x S^2 is recovered at large N, which is consistent with a single D4-brane interpretation in Type IIA string theory. This is as expected at large k, where the semiclassical analysis is valid. Several aspects of the fluctuation analysis, the ground-state/funnel solutions and the mass-deformed/pure ABJM equations can be understood in terms of a discrete noncommutative realisation of the Hopf fibration. We discuss the implications for the possibility of finding an M2-brane worldvolume derivation of the classical S^3 geometry of the M2-M5 system. Using a rewriting of the equations of the SO(4)-covariant fuzzy 3-sphere construction, we also directly compare this fuzzy 3-sphere against the ABJM ground-state/funnel solutions and show them to be different.Comment: 60 pages, Latex; v2: references added; v3: typos corrected and references adde

D-branes Wrapped on Fuzzy del Pezzo Surfaces

We construct classical solutions in quiver gauge theories on D0-branes probing toric del Pezzo singularities in Calabi-Yau manifolds. Our solutions represent D4-branes wrapped around fuzzy del Pezzo surfaces. We study the fluctuation spectrum around the fuzzy CP^2 solution in detail. We also comment on possible applications of our fuzzy del Pezzo surfaces to the fuzzy version of F-theory, dubbed F(uzz) theory.Comment: 1+42 pages, 9 figures v2: references added v3: statements on the structure of the Yukawa couplings weakened. published versio

Cartan-Weyl 3-algebras and the BLG Theory I: Classification of Cartan-Weyl 3-algebras

As Lie algebras of compact connected Lie groups, semisimple Lie algebras have wide applications in the description of continuous symmetries of physical systems. Mathematically, semisimple Lie algebra admits a Cartan-Weyl basis of generators which consists of a Cartan subalgebra of mutually commuting generators H_I and a number of step generators E^\alpha that are characterized by a root space of non-degenerate one-forms \alpha. This simple decomposition in terms of the root space allows for a complete classification of semisimple Lie algebras. In this paper, we introduce the analogous concept of a Cartan-Weyl Lie 3-algebra. We analyze their structure and obtain a complete classification of them. Many known examples of metric Lie 3-algebras (e.g. the Lorentzian 3-algebras) are special cases of the Cartan-Weyl 3-algebras. Due to their elegant and simple structure, we speculate that Cartan-Weyl 3-algebras may be useful for describing some kinds of generalized symmetries. As an application, we consider their use in the Bagger-Lambert-Gustavsson (BLG) theory.Comment: LaTeX. 34 pages.v2. deleted some distracting paragraphs in the introduction to bring more out the main results of the paper. typos corrected and references adde

The unwarped, resolved, deformed conifold: fivebranes and the baryonic branch of the Klebanov-Strassler theory

We study a gravity solution corresponding to fivebranes wrapped on the $S^2$ of the resolved conifold. By changing a parameter the solution continuously interpolates between the deformed conifold with flux and the resolved conifold with branes. Therefore, it displays a geometric transition, purely in the supergravity context. The solution is a simple example of torsional geometry and may be thought of as a non-K\"ahler analog of the conifold. By U-duality transformations we can add D3 brane charge and recover the solution in the form originally derived by Butti et al. This describes the baryonic branch of the Klebanov-Strassler theory. Far along the baryonic branch the field theory gives rise to a fuzzy two-sphere. This corresponds to the D5 branes wrapping the two-sphere of the resolved conifold in the gravity solution.Comment: 41 pages, 7 figure

Causal structures and causal boundaries

We give an up-to-date perspective with a general overview of the theory of causal properties, the derived causal structures, their classification and applications, and the definition and construction of causal boundaries and of causal symmetries, mostly for Lorentzian manifolds but also in more abstract settings.Comment: Final version. To appear in Classical and Quantum Gravit