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 $\mathcal N=6$ 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 $\mathcal N=6$ 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 $\delta L$ 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 $\delta L$ 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 $\delta L$ 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 $\ell$-ple systems are introduced in this paper. They are defined by brackets with $\ell > 3$ 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 $(2n-3)$-Leibniz algebra \fL with a metric $n$-Leibniz algebra \tilde{\fL} by using a $2(n-1)$-linear Kasymov trace form for \tilde{\fL}. Some specific types of $k$-Leibniz algebras, relevant in the construction, are introduced as well. Both higher order Lie $\ell$-ple generalizations reduce to the standard Lie triple systems for $\ell=3$.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