51 research outputs found
Modular realizations of hyperbolic Weyl groups
We study the recently discovered isomorphisms between hyperbolic Weyl groups and unfamiliar modular groups. These modular groups are defined over integer domains in normed division algebras, and we focus on the cases involving quaternions and octonions. We outline how to construct and analyse automorphic forms for these groups; their structure depends on the underlying arithmetic properties of the integer domains. We also give a new realization of the Weyl group W(E8) in terms of unit octavians and their automorphism group
Modular realizations of hyperbolic Weyl groups
We study the recently discovered isomorphisms between hyperbolic Weyl groups and unfamiliar modular groups. These modular groups are defined over integer domains in normed division algebras, and we focus on the cases involving quaternions and octonions. We outline how to construct and analyse automorphic forms for these groups; their structure depends on the underlying arithmetic properties of the integer domains. We also give a new realization of the Weyl group W(E8) in terms of unit octavians and their automorphism group
Galilean free Lie algebras
We construct free Lie algebras which, together with the algebra of spatial rotations, form infinite-dimensional extensions of finite-dimensional Galilei Maxwell algebras appearing as global spacetime symmetries of extended non-relativistic objects and non-relativistic gravity theories. We show how various extensions of the ordinary Galilei algebra can be obtained by truncations and contractions, in some cases via an affine Kac-Moody algebra. The infinite-dimensional Lie algebras could be useful in the construction of generalized Newton-Cartan theories gravity theories and the objects that couple to them
Unifying N=5 and N=6
We write the Lagrangian of the general N=5 three-dimensional superconformal
Chern-Simons theory, based on a basic Lie superalgebra, in terms of our
recently introduced N=5 three-algebras. These include N=6 and N=8
three-algebras as special cases. When we impose an antisymmetry condition on
the triple product, the supersymmetry automatically enhances, and the N=5
Lagrangian reduces to that of the well known N=6 theory, including the ABJM and
ABJ models.Comment: 19 pages. v2: Published version. Minor typos corrected, references
adde
Tensor hierarchies, Borcherds algebras and E11
Gauge deformations of maximal supergravity in D=11-n dimensions generically
give rise to a tensor hierarchy of p-form fields that transform in specific
representations of the global symmetry group E(n). We derive the formulas
defining the hierarchy from a Borcherds superalgebra corresponding to E(n).
This explains why the E(n) representations in the tensor hierarchies also
appear in the level decomposition of the Borcherds superalgebra. We show that
the indefinite Kac-Moody algebra E(11) can be used equivalently to determine
these representations, up to p=D, and for arbitrarily large p if E(11) is
replaced by E(r) with sufficiently large rank r.Comment: 22 pages. v2: Published version (except for a few minor typos
detected after the proofreading, which are now corrected
Three-dimensional topologically gauged N=6 ABJM type theories
In this paper we construct the conformal supergravity in three
dimensions from a set of Chern-Simons-like terms one for each of the graviton,
gravitino, and R-symmetry gauge field and then couple this theory to the
superconformal ABJM theory. In a first step part of the coupled
Lagrangian for this topologically gauged ABJM theory is derived by demanding
that all terms of third and second order in covariant derivatives cancel in the
supersymmtry variation of the Lagrangian. To achieve this the transformation
rules of the two separate sectors must be augmented by new terms. In a second
step we analyze all terms in that are of first order in covariant
derivatives. The cancelation of these terms require additional terms in the
transformation rules as well as a number of new terms in the Lagrangian. As a
final step we check that all remaining terms in which are bilinear
in fermions cancel which means that the presented Lagrangian and transformation
rules constitute the complete answer. In particular we find in the last step
new terms in the scalar potential containing either one or no structure
constant. The non-derivative higher fermion terms in that have not
yet been completely analyzed are briefly discussed.Comment: 26 pages, v.2 minor corrections, comment on relation to chiral
gravity added
Newton-Hooke/Carrollian expansions of (A)dS and Chern-Simons gravity
We construct finite- and infinite-dimensional non-relativistic extensions of the Newton-Hooke and Carroll (A)dS algebras using the algebra expansion method, starting from the (anti-)de Sitter relativistic algebra in D dimensions. These algebras are also shown to be embedded in different affine Kac-Moody algebras. In the three-dimensional case, we construct Chern-Simons actions invariant under these symmetries. This leads to a sequence of non-relativistic gravity theories, where the simplest examples correspond to extended Newton-Hooke and extended (post-)Newtonian gravity together with their Carrollian counterparts
k-Leibniz algebras from lower order ones: from Lie triple to Lie l-ple systems
Two types of higher order Lie -ple systems are introduced in this
paper. They are defined by brackets with arguments satisfying
certain conditions, and generalize the well known Lie triple systems. One of
the generalizations uses a construction that allows us to associate a
-Leibniz algebra \fL with a metric -Leibniz algebra \tilde{\fL}
by using a -linear Kasymov trace form for \tilde{\fL}. Some specific
types of -Leibniz algebras, relevant in the construction, are introduced as
well. Both higher order Lie -ple generalizations reduce to the standard
Lie triple systems for .Comment: 22 pages, no figure
Superconformal M2-branes and generalized Jordan triple systems
Three-dimensional conformal theories with six supersymmetries and SU(4)
R-symmetry describing stacks of M2-branes are here proposed to be related to
generalized Jordan triple systems. Writing the four-index structure constants
in an appropriate form, the Chern-Simons part of the action immediately
suggests a connection to such triple systems. In contrast to the previously
considered three-algebras, the additional structure of a generalized Jordan
triple system is associated to a graded Lie algebra, which corresponds to an
extension of the gauge group. In this note we show that the whole theory with
six manifest supersymmetries can be naturally expressed in terms of such a
graded Lie algebra. Also the BLG theory with eight supersymmetries is included
as a special case.Comment: 15 pages, v2 and v3: minor corrections and clarifications, references
added, v2: section 4 extended, v3: published versio
Higgsing M2 to D2 with gravity: N=6 chiral supergravity from topologically gauged ABJM theory
We present the higgsing of three-dimensional N=6 superconformal ABJM type
theories coupled to conformal supergravity, so called topologically gauged ABJM
theory, thus providing a gravitational extension of previous work on the
relation between N M2 and N D2-branes. The resulting N=6 supergravity theory
appears at a chiral point similar to that of three-dimensional chiral gravity
introduced recently by Li, Song and Strominger, but with the opposite sign for
the Ricci scalar term in the lagrangian. We identify the supersymmetry in the
broken phase as a particular linear combination of the supersymmetry and
special conformal supersymmetry in the original topologically gauged ABJM
theory. We also discuss the higgsing procedure in detail paying special
attention to the role played by the U(1) factors in the original ABJM model and
the U(1) introduced in the topological gauging.Comment: 53 pages, Late
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