Eulerian digraphs and toric Calabi-Yau varieties

We investigate the structure of a simple class of affine toric Calabi-Yau varieties that are defined from quiver representations based on finite eulerian directed graphs (digraphs). The vanishing first Chern class of these varieties just follows from the characterisation of eulerian digraphs as being connected with all vertices balanced. Some structure theory is used to show how any eulerian digraph can be generated by iterating combinations of just a few canonical graph-theoretic moves. We describe the effect of each of these moves on the lattice polytopes which encode the toric Calabi-Yau varieties and illustrate the construction in several examples. We comment on physical applications of the construction in the context of moduli spaces for superconformal gauged linear sigma models.Comment: 27 pages, 8 figure

Non-associative gauge theory and higher spin interactions

We give a framework to describe gauge theory on a certain class of commutative but non-associative fuzzy spaces. Our description is in terms of an Abelian gauge connection valued in the algebra of functions on the cotangent bundle of the fuzzy space. The structure of such a gauge theory has many formal similarities with that of Yang-Mills theory. The components of the gauge connection are functions on the fuzzy space which transform in higher spin representations of the Lorentz group. In component form, the gauge theory describes an interacting theory of higher spin fields, which remains non-trivial in the limit where the fuzzy space becomes associative. In this limit, the theory can be viewed as a projection of an ordinary non-commutative Yang-Mills theory. We describe the embedding of Maxwell theory in this extended framework which follows the standard unfolding procedure for higher spin gauge theories.Comment: 1+49 pages, LaTeX; references and clarifying remarks adde

Half-BPS M2-brane orbifolds

Smooth Freund-Rubin backgrounds of eleven-dimensional supergravity of the form AdS_4 x X^7 and preserving at least half of the supersymmetry have been recently classified. Requiring that amount of supersymmetry forces X to be a spherical space form, whence isometric to the quotient of the round 7-sphere by a freely-acting finite subgroup of SO(8). The classification is given in terms of ADE subgroups of the quaternions embedded in SO(8) as the graph of an automorphism. In this paper we extend this classification by dropping the requirement that the background be smooth, so that X is now allowed to be an orbifold of the round 7-sphere. We find that if the background preserves more than half of the supersymmetry, then it is automatically smooth in accordance with the homogeneity conjecture, but that there are many half-BPS orbifolds, most of them new. The classification is now given in terms of pairs of ADE subgroups of quaternions fibred over the same finite group. We classify such subgroups and then describe the resulting orbifolds in terms of iterated quotients. In most cases the resulting orbifold can be described as a sequence of cyclic quotients.Comment: 51 pages; v3: substantial revision (20% longer): we had missed some cases, but the paper now includes a check of our results via comparison with extant classification of finite subgroups of SO(4

Metric Lie 3-algebras in Bagger-Lambert theory

We recast physical properties of the Bagger-Lambert theory, such as shift-symmetry and decoupling of ghosts, the absence of scale and parity invariance, in Lie 3-algebraic terms, thus motivating the study of metric Lie 3-algebras and their Lie algebras of derivations. We prove a structure theorem for metric Lie 3-algebras in arbitrary signature showing that they can be constructed out of the simple and one-dimensional Lie 3-algebras iterating two constructions: orthogonal direct sum and a new construction called a double extension, by analogy with the similar construction for Lie algebras. We classify metric Lie 3-algebras of signature (2,p) and study their Lie algebras of derivations, including those which preserve the conformal class of the inner product. We revisit the 3-algebraic criteria spelt out at the start of the paper and select those algebras with signature (2,p) which satisfy them, as well as indicate the construction of more general metric Lie 3-algebras satisfying the ghost-decoupling criterion.Comment: 38 page

Open G2 Strings

We consider an open string version of the topological twist previously proposed for sigma-models with G2 target spaces. We determine the cohomology of open strings states and relate these to geometric deformations of calibrated submanifolds and to flat or anti-self-dual connections on such submanifolds. On associative three-cycles we show that the worldvolume theory is a gauge-fixed Chern-Simons theory coupled to normal deformations of the cycle. For coassociative four-cycles we find a functional that extremizes on anti-self-dual gauge fields. A brane wrapping the whole G2 induces a seven-dimensional associative Chern-Simons theory on the manifold. This theory has already been proposed by Donaldson and Thomas as the higher-dimensional generalization of real Chern-Simons theory. When the G2 manifold has the structure of a Calabi-Yau times a circle, these theories reduce to a combination of the open A-model on special Lagrangians and the open B+\bar{B}-model on holomorphic submanifolds. We also comment on possible applications of our results.Comment: 55 pages, no figure

On the Lie-algebraic origin of metric 3-algebras

Since the pioneering work of Bagger-Lambert and Gustavsson, there has been a proliferation of three-dimensional superconformal Chern-Simons theories whose main ingredient is a metric 3-algebra. On the other hand, many of these theories have been shown to allow for a reformulation in terms of standard gauge theory coupled to matter, where the 3-algebra does not appear explicitly. In this paper we reconcile these two sets of results by pointing out the Lie-algebraic origin of some metric 3-algebras, including those which have already appeared in three-dimensional superconformal Chern-Simons theories. More precisely, we show that the real 3-algebras of Cherkis-Saemann, which include the metric Lie 3-algebras as a special case, and the hermitian 3-algebras of Bagger-Lambert can be constructed from pairs consisting of a metric real Lie algebra and a faithful (real or complex, respectively) unitary representation. This construction generalises and we will see how to construct many kinds of metric 3-algebras from pairs consisting of a real metric Lie algebra and a faithful (real, complex or quaternionic) unitary representation. In the real case, these 3-algebras are precisely the Cherkis-Saemann algebras, which are then completely characterised in terms of this data. In the complex and quaternionic cases, they constitute generalisations of the Bagger-Lambert hermitian 3-algebras and anti-Lie triple systems, respectively, which underlie N=6 and N=5 superconformal Chern-Simons theories, respectively. In the process we rederive the relation between certain types of complex 3-algebras and metric Lie superalgebras.Comment: 29 pages (v4: really final version to appear in CMP. Example 7 has been improved.