507 research outputs found
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.
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