scholarly journals E6(6) exceptional Drinfel’d algebras

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Emanuel Malek ◽  
Yuho Sakatani ◽  
Daniel C. Thompson
Keyword(s):  
Tangent Bundle ◽  
Leibniz Algebra ◽  
Baxter Equation ◽  
Duality Group ◽  
Dimensional Group ◽  
Drinfel’D Double ◽  
Group Manifold ◽  
Natural Generalisation ◽  
M Theory ◽  
Selection Of

Abstract The exceptional Drinfel’d algebra (EDA) is a Leibniz algebra introduced to provide an algebraic underpinning with which to explore generalised notions of U-duality in M-theory. In essence, it provides an M-theoretic analogue of the way a Drinfel’d double encodes generalised T-dualities of strings. In this note we detail the construction of the EDA in the case where the regular U-duality group is E6(6). We show how the EDA can be realised geometrically as a generalised Leibniz parallelisation of the exceptional generalised tangent bundle for a six-dimensional group manifold G, endowed with a Nambu-Lie structure. When the EDA is of coboundary type, we show how a natural generalisation of the classical Yang-Baxter equation arises. The construction is illustrated with a selection of examples including some which embed Drinfel’d doubles and others that are not of this type.

Three-dimensional group manifold reductions of gravity

2005 ◽  
Author(s):  
Román Linares
Keyword(s):  
Three Dimensional ◽  
Dimensional Group ◽  
Group Manifold

scholarly journals Non-Abelian U -duality for membranes

2020 ◽  
Vol 2020 (7) ◽  
Author(s):  
Yuho Sakatani ◽  
Shozo Uehara
Keyword(s):  
String Theory ◽  
Isometry Group ◽  
Standard Treatment ◽  
Abelian Extension ◽  
Target Space ◽  
Membrane Theory ◽  
Drinfel’D Double ◽  
M Theory

Abstract The $T$-duality of string theory can be extended to the Poisson–Lie $T$-duality when the target space has a generalized isometry group given by a Drinfel’d double. In M-theory, $T$-duality is understood as a subgroup of $U$-duality, but the non-Abelian extension of $U$-duality is still a mystery. In this paper we study membrane theory on a curved background with a generalized isometry group given by the $\mathcal E_n$ algebra. This provides a natural setup to study non-Abelian $U$-duality because the $\mathcal E_n$ algebra has been proposed as a $U$-duality extension of the Drinfel’d double. We show that the standard treatment of Abelian $U$-duality can be extended to the non-Abelian setup. However, a famous issue in Abelian $U$-duality still exists in the non-Abelian extension.

scholarly journals M-THEORY DUALITY AND BPS-EXTENDED SUPERGRAVITY

2001 ◽  
Vol 16 (05) ◽  
pp. 1002-1011 ◽  
Author(s):  
BERNARD DE WIT
Keyword(s):  
Lorentz Invariance ◽  
Space Time ◽  
Duality Group ◽  
Toroidal Compactifications ◽  
Maximal Supergravity ◽  
Bps States ◽  
M Theory

We discuss toroidal compactifications of maximal supergravity coupled to an extended configuration of BPS states which transform consistently under the U-duality group. Under certain conditions this leads to theories that live in more than eleven space-time dimensions, with maximal supersymmetry but only partial Lorentz invariance. We demonstrate certain features of this construction for the case of nine-dimensional N=2 supergravity.

scholarly journals Extended Drinfel’d algebras and non-Abelian duality

2020 ◽  
Author(s):  
Yuho Sakatani
Keyword(s):  
General Formula ◽  
General Structure ◽  
Natural Extension ◽  
Membrane Theory ◽  
Group Manifold ◽  
M Theory

Abstract The Drinfel’d algebra provides a method to construct generalized parallelizable spaces and this allows us to study an extended T-duality, known as the Poisson–Lie T-duality. Recently, in order to find a generalized U-duality, an extended Drinfel'd algebra (ExDA), called the Exceptional Drinfel'd algebra (EDA), was proposed and a natural extension of Abelian U-duality was studied both in the context of supergravity and membrane theory. In this paper, we clarify the general structure of ExDAs and show that an ExDA always gives a generalized parallelizable space, which may be regarded as a group manifold with generalized Nambu–Lie structures. We then discuss the non-Abelian duality that is based on a general ExDA. For a coboundary ExDA, this non-Abelian duality reduces to a generalized Yang–Baxter deformation and we find a general formula for the twist matrix. In order to study the non-Abelian U-duality, we particularly focus on the En(n) EDA for n ≤ 8 and study various aspects, both in terms of M-theory and type IIB theory.

ON TYPE II SUPERSTRINGS: THE σ MODEL AND COMPACTIFICATION TO D=4

1991 ◽  
Vol 06 (22) ◽  
pp. 4009-4039 ◽  
Author(s):  
LEONARDO CASTELLANI ◽  
RICCARDO D’ AURIA ◽  
DAVIDE FRANCO
Keyword(s):  
Dimensional Reduction ◽  
Geometric Construction ◽  
Covering Space ◽  
Type Ii ◽  
Internal Target ◽  
Dimensional Group ◽  
Group Manifold ◽  
Target Manifold

We present a geometric construction of the σ model describing type II superstrings propagating on an arbitrary Mtarget. Specializing the covering space of the internal target manifold to be the nine-dimensional group manifold SU(2)3, we discuss the massless vertices both in the (4+9)-dimensional a model and in the D=4 superconformal theory, and show how they are related via dimensional reduction.

scholarly journals Jacobi-Lie T-plurality

2021 ◽  
Vol 11 (2) ◽  
Author(s):  
Jose J. Fernandez-Melgarejo ◽  
Yuho Sakatani
Keyword(s):  
Field Theory ◽  
Leibniz Algebra ◽  
Lie Derivative ◽  
Lie Bialgebra ◽  
Drinfel’D Double ◽  
Double Field Theory ◽  
Generalized Frame ◽  
Structure Constants ◽  
One To One ◽  
The Spectator

We propose a Leibniz algebra, to be called DD^++, which is a generalization of the Drinfel’d double. We find that there is a one-to-one correspondence between a DD^++ and a Jacobi–Lie bialgebra, extending the known correspondence between a Lie bialgebra and a Drinfel’d double. We then construct generalized frame fields E_A{}^M\in\text{O}(D,D)\times\mathbb{R}^+EAM∈O(D,D)×ℝ+ satisfying the algebra \hat{\pounds}_{E_A}E_B = - X_{AB}{}^C\,E_C£̂EAEB=−XABCEC, where X_{AB}{}^CXABC are the structure constants of the DD^++ and \hat{\pounds}£̂ is the generalized Lie derivative in double field theory. Using the generalized frame fields, we propose the Jacobi–Lie TT-plurality and show that it is a symmetry of double field theory. We present several examples of the Jacobi–Lie TT-plurality with or without Ramond–Ramond fields and the spectator fields.

scholarly journals U DUALITY AS GENERAL COORDINATE TRANSFORMATIONS, AND SPACE–TIME GEOMETRY

1999 ◽  
Vol 14 (31) ◽  
pp. 4915-4942 ◽  
Author(s):  
I. V. LAVRINENKO ◽  
H. LÜ ◽  
C. N. POPE ◽  
T. A. TRAN
Keyword(s):  
Space Time ◽  
The Self ◽  
Global Symmetry ◽  
Symmetry Groups ◽  
Quantum Level ◽  
Coordinate Transformations ◽  
Duality Group ◽  
Toroidal Compactifications ◽  
Arbitrary Dimensions ◽  
M Theory

We show that the full global symmetry groups of all the D-dimensional maximal supergravities can be described in terms of the closure of the internal general coordinate transformations of the toroidal compactifications of D=11 supergravity and of type IIB supergravity, with type IIA/IIB T duality providing an intertwining between the two pictures. At the quantum level, the part of the U duality group that corresponds to the surviving discretized internal general coordinate transformations in a given picture leaves the internal torus invariant, while the part that is not described by internal general coordinate transformations can have the effect of altering the size or shape of the internal torus. For example, M theory compactified on a large torus Tn can be related by duality to a compactification on a small torus, if and only if n≥3. We also discuss related issues in the toroidal compactification of the self-dual string to D=4. An appendix includes the complete results for the toroidal reduction of the bosonic sector of type IIB supergravity to arbitrary dimensions D≥3.

Cognitive gadgets and cognitive priors

2019 ◽  
Vol 42 ◽  
Author(s):  
Gian Domenico Iannetti ◽  
Giorgio Vallortigara
Keyword(s):  
Cognitive Neuroscience ◽  
Selection Of

Abstract Some of the foundations of Heyes’ radical reasoning seem to be based on a fractional selection of available evidence. Using an ethological perspective, we argue against Heyes’ rapid dismissal of innate cognitive instincts. Heyes’ use of fMRI studies of literacy to claim that culture assembles pieces of mental technology seems an example of incorrect reverse inferences and overlap theories pervasive in cognitive neuroscience.

Note on Selection of Coordinate Systems for Geodynamics

1975 ◽  
Vol 26 ◽  
pp. 395-407
Author(s):  
S. Henriksen
Keyword(s):  
Sea Level ◽  
The Other ◽  
Coordinate Systems ◽  
Mean Sea Level ◽  
The Earth ◽  
The One ◽  
Static Systems ◽  
Selection Of

The first question to be answered, in seeking coordinate systems for geodynamics, is: what is geodynamics? The answer is, of course, that geodynamics is that part of geophysics which is concerned with movements of the Earth, as opposed to geostatics which is the physics of the stationary Earth. But as far as we know, there is no stationary Earth – epur sic monere. So geodynamics is actually coextensive with geophysics, and coordinate systems suitable for the one should be suitable for the other. At the present time, there are not many coordinate systems, if any, that can be identified with a static Earth. Certainly the only coordinate of aeronomic (atmospheric) interest is the height, and this is usually either as geodynamic height or as pressure. In oceanology, the most important coordinate is depth, and this, like heights in the atmosphere, is expressed as metric depth from mean sea level, as geodynamic depth, or as pressure. Only for the earth do we find “static” systems in use, ana even here there is real question as to whether the systems are dynamic or static. So it would seem that our answer to the question, of what kind, of coordinate systems are we seeking, must be that we are looking for the same systems as are used in geophysics, and these systems are dynamic in nature already – that is, their definition involvestime.

Measurements of the Cape Astrometric Survey of the Southern Hemisphere

1978 ◽  
Vol 48 ◽  
pp. 515-521
Author(s):  
W. Nicholson
Keyword(s):  
Southern Hemisphere ◽  
Automatic Measuring ◽  
Measuring Machine ◽  
Final Selection ◽  
Selection Of

SummaryA routine has been developed for the processing of the 5820 plates of the survey. The plates are measured on the automatic measuring machine, GALAXY, and the measures are subsequently processed by computer, to edit and then refer them to the SAO catalogue. A start has been made on measuring the plates, but the final selection of stars to be made is still a matter for discussion.

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