Gauge Theory on Fuzzy S^2 x S^2 and Regularization on Noncommutative R^4

We define U(n) gauge theory on fuzzy S^2_N x S^2_N as a multi-matrix model, which reduces to ordinary Yang-Mills theory on S^2 x S^2 in the commutative limit N -> infinity. The model can be used as a regularization of gauge theory on noncommutative R^4_\theta in a particular scaling limit, which is studied in detail. We also find topologically non-trivial U(1) solutions, which reduce to the known "fluxon" solutions in the limit of R^4_\theta, reproducing their full moduli space. Other solutions which can be interpreted as 2-dimensional branes are also found. The quantization of the model is defined non-perturbatively in terms of a path integral which is finite. A gauge-fixed BRST-invariant action is given as well. Fermions in the fundamental representation of the gauge group are included using a formulation based on SO(6), by defining a fuzzy Dirac operator which reduces to the standard Dirac operator on S^2 x S^2 in the commutative limit. The chirality operator and Weyl spinors are also introduced.Comment: 39 pages. V2-4: References added, typos fixe

Membranes on Calibrations

M2-branes can blow up into BPS funnels that end on calibrated intersections of M5-branes. In this quick note, we make the observation that the constraints required for the consistency of these solutions are automatic in Bagger-Lambert-Gustavsson (BLG) theory, thanks to the fundamental identity and the supersymmetry of the calibration. We use this to explain how the previous ad hoc fuzzy funnel constructions emerge in this picture, and make some comments about the role of the 3-algebra trace form in the derivation.Comment: 9 pages, no figures; references added, minor change

Non-constant Non-commutativity in 2d Field Theories and a New Look at Fuzzy Monopoles

We write down scalar field theory and gauge theory on two-dimensional noncommutative spaces ${\cal M}$ with nonvanishing curvature and non-constant non-commutativity. Usual dynamics results upon taking the limit of ${\cal M}$ going to i) a commutative manifold ${\cal M}_0$ having nonvanishing curvature and ii) the noncommutative plane. Our procedure does not require introducing singular algebraic maps or frame fields. Rather, we exploit the K\"ahler structure in the limit i) and identify the symplectic two-form with the volume two-form. As an example, we take ${\cal M}$ to be the stereographically projected fuzzy sphere, and find magnetic monopole solutions to the noncommutative Maxwell equations. Although the magnetic charges are conserved, the classical theory does not require that they be quantized. The noncommutative gauge field strength transforms in the usual manner, but the same is not, in general, true for the associated potentials. We develop a perturbation scheme to obtain the expression for gauge transformations about limits i) and ii). We also obtain the lowest order Seiberg-Witten map to write down corrections to the commutative field equations and show that solutions to Maxwell theory on ${\cal M}_0$ are stable under inclusion of lowest order noncommutative corrections. The results are applied to the example of noncommutative AdS${}^2$.Comment: 27 page

Discrete Minimal Surface Algebras

We consider discrete minimal surface algebras (DMSA) as generalized noncommutative analogues of minimal surfaces in higher dimensional spheres. These algebras appear naturally in membrane theory, where sequences of their representations are used as a regularization. After showing that the defining relations of the algebra are consistent, and that one can compute a basis of the enveloping algebra, we give several explicit examples of DMSAs in terms of subsets of sl(n) (any semi-simple Lie algebra providing a trivial example by itself). A special class of DMSAs are Yang-Mills algebras. The representation graph is introduced to study representations of DMSAs of dimension d<=4, and properties of representations are related to properties of graphs. The representation graph of a tensor product is (generically) the Cartesian product of the corresponding graphs. We provide explicit examples of irreducible representations and, for coinciding eigenvalues, classify all the unitary representations of the corresponding algebras

Noncommutative field theory on Rλ3\mathbb{R}^3_\lambda

We consider the noncommutative space $\mathbb{R}^3_\lambda$, a deformation of the algebra of functions on $\mathbb{R}^3$ which yields a foliation of $\mathbb{R}^3$ into fuzzy spheres. We first review the construction of a natural matrix basis adapted to $\mathbb{R}^3_\lambda$. We thus consider the problem of defining a new Laplacian operator for the deformed algebra. We propose an operator which is not of Jacobi type. The implication for field theory of the new Laplacian is briefly discussed.Comment: 12 pages. Conference proceedings. Presented at the workshop "Noncommutative Field theory and Gravity" Corfu 201

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

Gravity as a Gauge Theory on Three-Dimensional Noncommutative spaces

We plan to translate the successful description of three-dimensional gravity as a gauge theory in the noncommutative framework, making use of the covariant coordinates. We consider two specific three-dimensional fuzzy spaces based on SU(2) and SU(1,1), which carry appropriate symmetry groups. These are the groups we are going to gauge in order to result with the transformations of the gauge fields (dreibein, spin connection and two extra Maxwell fields due to noncommutativity), their corresponding curvatures and eventually determine the action and the equations of motion. Finally, we verify their connection to three-dimensional gravity.Comment: arXiv admin note: text overlap with arXiv:1802.0755

The Antisymmetry Betweenness Axiom and Hausdorff Continua

An interpretation of betweenness on a set satisfies the antisymmetry axiom at a point a if it is impossible for each of two distinct points to lie between the other and a. In this paper we study the role of antisymmetry as it applies to the K-interpretation of betweenness in a Hausdorff continuum X, where a point c lies between points a and b exactly when every subcontinuum of X containing both a and b contains c as well