hep-th 9302044
arXiv:abs
arXiv:pdf
local:pdf
local:txt
|
Finite Lorentz Transformations, Automorphisms, and Division Algebras
|
Manogue, Corinne A. |
Schray, Jörg |
|
We give an explicit algebraic description of finite Lorentz transformations
of vectors in 10-dimensional Minkowski space by means of a parameterization in
terms of the octonions. The possible utility of these results for superstring
theory is mentioned. Along the way we describe automorphisms of the two highest
dimensional normed division algebras, namely the quaternions and the octonions,
in terms of conjugation maps. We use similar techniques to define $SO(3)$ and
$SO(7)$ via conjugation, $SO(4)$ via symmetric multiplication, and $SO(8)$ via
both symmetric multiplication and one-sided multiplication. The
non-commutativity and non-associativity of these division algebras plays a
crucial role in our constructions.
|
Comment: 24 pages, Plain TeX, 2 figures on 1 page submitted separately as
uuencoded compressed tar file |
|
hep-th 9502009
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Seven Dimensional Octonionic Yang-Mills Instanton and its Extension to an Heterotic String Soliton
|
Nicolai, Hermann |
Gunaydin, Murat |
|
We construct an octonionic instanton solution to the seven dimensional
Yang-Mills theory based on the exceptional gauge group $G_2$ which is the
automorphism group of the division algebra of octonions. This octonionic
instanton has an extension to a solitonic two-brane solution of the low energy
effective theory of the heterotic string that preserves two of the sixteen
supersymmetries and hence corresponds to $N=1$ space-time supersymmetry in the
(2+1) dimensions transverse to the seven dimensions where the Yang-Mills
instanton is defined.
|
Comment: 7 pages, Latex document. This is the final version that appeared in
Phys. Lett. B that includes an extra paragraph about the physical properties
of the octonionic two-brane. We have also put an addendum regarding some
related references that were brought to our attention recently |
|
hep-th 9601072
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Seven-Sphere and the Exceptional N=7 and N=8 Superconformal Algebras
|
Ketov, Sergei V. |
Gunaydin, Murat |
|
We study realizations of the exceptional non-linear (quadratically generated,
or W-type) N=8 and N=7 superconformal algebras with Spin(7) and G_2 affine
symmetry currents, respectively. Both the N=8 and N=7 algebras admit unitary
highest-weight representations in terms of a single boson and free fermions in
8 of Spin(7) and 7 of G_2, with the central charges c_8=26/5 and c_7=5,
respectively. Furthermore, we show that the general coset Ans"atze for the N=8
and N=7 algebras naturally lead to the coset spaces SO(8)xU(1)/SO(7) and
SO(7)xU(1)/G_2, respectively, as the additional consistent solutions for
certain values of the central charge. The coset space SO(8)/SO(7) is the
seven-sphere S^7, whereas the space SO(7)/G_2 represents the seven-sphere with
torsion, S^7_T. The division algebra of octonions and the associated triality
properties of SO(8) play an essential role in all these realizations. We also
comment on some possible applications of our results to string theory.
|
Comment: 50 pages, LaTeX, macros included; revised: a few missing factors of
1/2 added and two references modified; the version to appear in Nucl. Phys. B |
|
hep-th 9410202
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Octonion X-product orbits
|
|
The octonionic X-product gives the octonions a flexibility not found in the
other real division algebras. The pattern of that flexibility is investigated
here.
|
Comment: 15 pages |
|
physics 9703033
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Hypercomplex Group Theory
|
|
Due to the noncommutative nature of quaternions and octonions we introduce
barred operators. This objects give the opportunity to manipulate appropriately
the hypercomplex fields. The standard problems arising in the definitions of
transpose, determinant and trace for quaternionic and octonionic matrices are
immediately overcome. We also investigate the possibility to formulate a new
approach to Hypercomplex Group Theory (HGT). From a mathematical viewpoint, our
aim is to highlight the possibility of looking at new hypercomplex groups by
the use of barred operators as fundamental step toward a clear and complete
discussion of HGT.
|
Comment: 18 pages, RevTex, PACS numbers: 02.10.Tq/Vr, 02.20.-a/Qs |
|
funct-an 9701004
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| A New Definition of Hypercomplex Analyticity
|
Stefano, De Leo |
Pietro, Rotelli |
|
Complex analyticity is generalized to hypercomplex functions, quaternion or
octonion, in such a manner that it includes the standard complex definition and
does not reduce analytic functions to a trivial class. A brief comparison with
other definitions is presented.
|
Comment: 9 pages, LaTex, 1991 Mathematics Subject Classifications: 11R52,
30G35, 46S20 |
|
hep-th 0008063
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Conformal and Quasiconformal Realizations of Exceptional Lie Groups
|
Gunaydin, M. |
Koepsell, K. |
Nicolai, H. |
|
We present a nonlinear realization of E_8 on a space of 57 dimensions, which
is quasiconformal in the sense that it leaves invariant a suitably defined
``light cone in 57 dimensions. This realization, which is related to the
Freudenthal triple system associated with the unique exceptional Jordan algebra
over the split octonions, contains previous conformal realizations of the lower
rank exceptional Lie groups on generalized space times, and in particular a
conformal realization of E_7 on a 27 dimensional vector space which we exhibit
explicitly. Possible applications of our results to supergravity and M-Theory
are briefly mentioned.
|
Comment: 21 pages, 1 figure. Revised version. Connection between SU(8) and
SL(8,R) bases clarified; formulas corrected; references added |
|
hep-th 9411063
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Octonion X-Product and E8 Lattices
|
|
In this episode, it is shown how the octonion X-product is related to E8
lattices, integral domains, sphere fibrations, and other neat stuff.
|
Comment: 8 pages, latex, no figures |
|
hep-th 9501007
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| OCTONIONS: E_{8} LATTICE TO \Lambda_{16}
|
|
I present here another example of a lattice fibration, a discrete version of
the highest dimensional Hopf fibration: $S^{7}\longrightarrow S^{15}
\longrightarrow S^{8}$.
|
Comment: 7 pages, latex, no figures |
|
hep-th 9503053
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| OCTONION XY-PRODUCT
|
|
The octonion X-product changes the octonion multiplication table, but does not
change the role of the identity. The octonion XY-product is very similar, but
shifts the identity as well. This will be of interest to those applying th
octonions to string theory.
|
Comment: 8 pages, Latex |
|
math 9807133
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Finding Octonionic Eigenvectors Using Mathematica
|
Manogue, Corinne A. |
Dray, Tevian |
|
The eigenvalue problem for 3x3 octonionic Hermitian matrices contains some
surprises, which we have reported elsewhere. In particular, the eigenvalues
need not be real, there are 6 rather than 3 real eigenvalues, and the
corresponding eigenvectors are not orthogonal in the usual sense. The
nonassociativity of the octonions makes computations tricky, and all of these
results were first obtained via brute force (but exact) Mathematica
computations. Some of them, such as the computation of real eigenvalues, have
subsequently been implemented more elegantly; others have not. We describe here
the use of Mathematica in analyzing this problem, and in particular its use in
proving a generalized orthogonality property for which no other proof is known.
|
Comment: LaTeX2e, 22 pages, 8 PS figures (uses included PS prolog; needs
elsart.cls and one of epsffig, epsf, graphicx) |
|
gr-qc 9704048
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Should Metric Signature Matter in Clifford Algebra Formulations of Physical Theories?
|
Pezzaglia Jr., William M. |
Adams, John J. |
|
Standard formulation is unable to distinguish between the (+++-) and (---+)
spacetime metric signatures. However, the Clifford algebras associated with
each are inequivalent, R(4) in the first case (real 4 by 4 matrices), H(2) in
the latter (quaternionic 2 by 2). Multivector reformulations of Dirac theory by
various authors look quite inequivalent pending the algebra assumed. It is not
clear if this is mere artifact, or if there is a right/wrong choice as to which
one describes reality. However, recently it has been shown that one can map
from one signature to the other using a "tilt transformation" [see P. Lounesto,
"Clifford Algebras and Hestenes Spinors", Found. Phys. 23, 1203-1237 (1993)].
The broader question is that if the universe is signature blind, then perhaps a
complete theory should be manifestly tilt covariant. A generalized multivector
wave equation is proposed which is fully signature invariant in form, because
it includes all the components of the algebra in the wavefunction (instead of
restricting it to half) as well as all the possibilities for interaction terms.
|
Comment: 12 pages, latex, no figures, Summary of talk at the Special Session
on Octonions and Clifford Algebras Algebras, at the 1997 Spring Western
Sectional Meeting of the American Mathematical Society, Oregon State
University, Corvallis, OR, 19-20 April 1997.
ftp://www.clifford.org/clf-alg/preprints/1995/pezz9502.latex |
|
hep-th 9408165
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Representations of Clifford Algebras and its Applications
|
|
A real representation theory of real Clifford algebra has been studied in
further detail, especially in connection with Fierz identities. As its
application, we have constructed real octonion algebras as well as related
octonionic triple system in terms of 8-component spinors associated with the
Clifford algebras $C(0,7)$ and $C(4,3)$.
|
Comment: 30 pages, UR1377 |
|
hep-th 9906065
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Matrix Representation of Octonions and Generalizations
|
|
We define a special matrix multiplication among a special subset of $2N\x 2N$
matrices, and study the resulting (non-associative) algebras and their
subalgebras. We derive the conditions under which these algebras become
alternative non-associative and when they become associative. In particular,
these algebras yield special matrix representations of octonions and complex
numbers; they naturally lead to the Cayley-Dickson doubling process. Our matrix
representation of octonions also yields elegant insights into Dirac's equation
for a free particle. A few other results and remarks arise as byproducts.
|
Comment: 18 printed pages |
|
hep-th 0206213
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Kaluza-Klein bundles and manifolds of exceptional holonomy
|
Tomasiello, Alessandro |
Petrini, Michela |
Minasian, Ruben |
Kaste, Peter |
|
We show how in the presence of RR two-form field strength the conditions for
preserving supersymmetry on six- and seven-dimensional manifolds lead to
certain generalizations of monopole equations. For six dimensions the string
frame metric is Kaehler with the complex structure that descends from the
octonions if in addition we assume F^{(1,1)}=0. The susy generator is a gauge
covariantly constant spinor. For seven dimensions the string frame metric is
conformal to a G_2 metric if in addition we assume the field strength to obey a
selfduality constraint. Solutions to these equations lift to geometries of G_2
and Spin(7) holonomy respectively.
|
Comment: LaTeX, 13 pages |
|
hep-ph 9501252
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Gravity and the Standard Model with 130 GeV Truth Quark from D4-D5-E6 Model using 3x3 Octonion Matrices.
|
|
The D4-D5-E6 model of gravity and the Standard Model with a 130 GeV truth quark
is constructed using 3x3 matrices of octonions. The model has both continuum
and lattice versions. The lattice version uses HyperDiamond lattice structure.
|
Comment: 108 pages, latex. Revised with respect to lattice structure to use
HyperDiamond lattice. Added references. Some minor revisions. |
|
hep-th 9308128
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| A Ten-Dimensional Super-Yang-Mills Action with Off-Shell Supersymmetry
|
|
After adding seven auxiliary scalar fields, the action for ten-dimensional
super-Yang-Mills contains an equal number of bosonic and fermionic non-gauge
fields. Besides being manifestly Lorentz and gauge-invariant, this action
contains nine spacetime supersymmetries whose algebra closes off-shell.
Octonions provide a convenient notation for displaying these symmetries.
|
Comment: 6 pages plain Tex, KCL-TH-93-11 (a clarifying comment is made
concerning the statistics of the supersymmetry transformation) |
|
hep-th 9401047
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Octonions and Binocular Mobilevision
|
|
This paper is devoted to an interaction of 2 objects: the 1st of them is
octonions, the classical structure of pure mathematics, the 2nd one is
Mobilevision, the recently developped technique of computer graphics. Namely,
it is shown that the binocular Mobilevision maybe elaborated by use of the
octonionic colour space - the 7-dimensional extension of the classical one,
which includes a strange overcolour besides two triples of ordinary ones
(blue,green, red for left and right eyes).
Contents.
I. Interpretational geometry, anomalous virtual realities, quantum projective
field theory and Mobilevision:(1.1. Interpretational geometry; 1.2. Anomalous
virtual realities; 1.3. Colours in anomalous virtual realities; 1.4. Quantum
projective field theory; 1.5. Mobilevision).
II. Quantum conformal and q_R-conformal field theories, an infinite
dimensional quantum group and quantum field analogs of Euler-Arnold
top:(2.1. Quantum conformal field theory; 2.2. Lobachevskii algebra, the
quantization of the Lobachevskii plane; 2.3. Quantum q_R-conformal field
theory; 2.4. An infinite dimensional quantum group; 2.5. Quantum-field
Euler-Arnold top and Virasoro master equation).
III. Octonionic colour space and binocular Mobilevision:(3.1. Quaternionic
description of ordinary colour space; 3.2. Octonionic colour space and
binocular Mobilevision).
|
Comment: 9p AMSTEX, (revised version: printing formats are changed,
typographi- cal errors are excluded) |
|
hep-th 0102049
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| On Fermions in Compact momentum Spaces Bilinearly Constructed with Pure Spinors
|
|
It is shown how the old Cartan's conjecture on the fundamental role of the
geometry of simple (or pure) spinors, as bilinearly underlying euclidean
geometry, may be extended also to quantum mechanics of fermions (in first
quantization), however in compact momentum spaces, bilinearly constructed with
spinors, with signatures unambiguously resulting from the construction, up to
sixteen component Majorana-Weyl spinors associated with the real Clifford
algebra $\Cl(1,9)$, where, because of the known periodicity theorem, the
construction naturally ends. $\Cl(1,9)$ may be formulated in terms of the
octonion division algebra, at the origin of SU(3) internal symmetry.
In this approach the extra dimensions beyond 4 appear as interaction terms in
the equations of motion of the fermion multiplet; more precisely the directions
from 5$^{th}$ to 8$^{th}$ correspond to electric, weak and isospin interactions
$(SU(2) \otimes U(1))$, while those from 8$^{th}$ to 10$^{th}$ to strong ones
SU(3). There seems to be no need of extra dimension in configuration-space.
Only four dimensional space-time is needed - for the equations of motion and
for the local fields - and also naturally generated by four-momenta as
Poincar\'e translations.
This spinor approach could be compatible with string theories and even
explain their origin, since also strings may be bilinearly obtained from simple
(or pure) spinors through sums; that is integrals of null vectors.
|
Comment: 55 pages Latex |
|
hep-th 0105313
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| An N=8 Superaffine Malcev Algebra and Its N=8 Sugawara
|
Carrion, H. L. |
Toppan, F. |
Rojas, M. |
|
A supersymmetric affinization of the algebra of octonions is introduced. It
satisfies a super-Malcev property and is N=8 supersymmetric. Its Sugawara
construction recovers, in a special limit, the non-associative N=8 superalgebra
of Englert et al. This paper extends to supersymmetry the results obtained by
Osipov in the bosonic case.
|
Comment: 10 pages, LaTex |
|
hep-th 0107158
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| From the Geometry of Pure Spinors with their Division Algebras to Fermion's Physics
|
|
The Cartan's equations definig simple spinors (renamed pure by C. Chevalley)
are interpreted as equations of motion in momentum spaces, in a constructive
approach in which at each step the dimesions of spinor space are doubled while
those momentum space increased by two. The construction is possible only in the
frame of geometry of simple or pure spinors, which imposes contraint equations
on spinors with more than four components, and the momentum spaces result
compact, isomorphic toinvariant-mass-spheres imbedded in each other, since the
signatures appear to be unambiguously defined and result steadily lorentzian;
up to dimension ten with Clifford algebra Cl(1,9), where the construction
naturally ends. The equations of motion met in the construction are most of
those traditionally postulated ad hoc for multicomponent fermions. The 3
division algebras: complex numbers, quaternions and octonions appear to be
strictly correlated with this spinor geometry, from which they appear to
gradually emerge in the construction, where they play a basic role for the
physical interpretation. In fact they seem then to be at the origin of
electroweak and strong charges, of the 3 families and of the groups of the
standard model. In this approach there seems to be no need of higher
dimensional (>4) space-time, here generated merely by Poincare translations,
and dimensional reduction from Cl(1,9) to Cl(1,3) is equivalent to decoupling
of the equations of motion.
|
Comment: 42 pages Latex |
|
hep-th 9407179
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Octonionic representations of Clifford algebras and triality
|
Manogue, Corinne A. |
Schray, Jörg |
|
The theory of representations of Clifford algebras is extended to employ the
division algebra of the octonions or Cayley numbers. In particular, questions
that arise from the non-associativity and non-commutativity of this division
algebra are answered. Octonionic representations for Clifford algebras lead to
a notion of octonionic spinors and are used to give octonionic representations
of the respective orthogonal groups. Finally, the triality automorphisms are
shown to exhibit a manifest $\perm_3 \times SO(8)$ structure in this framework.
|
Comment: 33 pages |
|
hep-th 9703162
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Octonions and Super Lie algebra
|
|
We discuss how to represent the non-associative octonionic structure in terms
of the associative matrix algebra using the left and right octonionic
operators. As an example we construct explicitly some Lie and Super Lie
algebra. Then we discuss the notion of octonionic Grassmann numbers and explain
its possible application for giving a superspace formulation of the minimal
supersymmetric Yang-Mills models.
|
Comment: RevTex file, 12 pages, to be published in Int. J. of Mod. Phys. A;
Some references added |
|
hep-th 0210132
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Octonions, G_2 Symmetry, Generalized Self-Duality and Supersymmetries in Dimensions D \le 8
|
Nishino, Hitoshi |
Rajpoot, Subhash |
|
We establish N=(1,0) supersymmetric Yang-Mills vector multiplet with
generalized self-duality in Euclidian eight-dimensions with the original full
SO(8) Lorentz covariance reduced to SO(7). The key ingredient is the usage of
octonion structure constants made compatible with SO(7) covariance and
chirality in 8D. By a simple dimensional reduction together with extra
constraints, we derive N=1 supersymmetric self-dual vector multiplet in 7D with
the full SO(7) Lorentz covariance reduced to G_2. We find that extra
constraints needed on fields and supersymmetry parameter are not obtained from
a simple dimensional reduction from 8D. We conjecture that other self-dual
supersymmetric theories in lower dimensions D =6 and 5 with respective reduced
global Lorentz covariances such as SU(3) \subset SO(6) and SU(2) \subset SO(5)
can be obtained in a similar fashion.
|
Comment: 14 pages, no figures. Very minor corrections in eq. (2.14), and in
the last paragraph in Concluding Remarks |
|
gr-qc 9806060
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| An Octonionic Geometric (Balanced) state Sum Model
|
|
We propose a new 4D state sum model, related to the balanced model, which is
constructed using the octonions, or equivalently, triality. An effective
continuum physical theory constructed from this model coupled to the balanced
model would have a non-vanishing cosmological constant, chiral asymmetry, and a
gauge group related to the octonions.
|
Comment: 9 pages, Latex |
|
hep-th 0002155
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Ring Division Algebras, Self-Duality and Supersymmetry
|
|
We argue that once octonions are formulated as soft Lie algebras, they may be
safely used and the non-associativity can be overcame. The necessary points
are: (a) Fixing the direction of action by introducing the \delta operator. (b)
Closing the \delta algebra by using structure functions f_{ijk} (\phi). (c)
Representation of the \delta algebra can be developed. The E or E(\phi) can be
found and their structure functions can be computed easily. There may be
different applications of soft seven sphere in physics. We have given two cases
where the ring division algebras occupies a special position. Self-duality and
Simple supersymmetric Yang-Mills theories are two promising places where soft
seven sphere prove to be useful and essential.
|
Comment: Latex,PH.D. Thesis |
|
hep-th 9710177
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| The Ring Division Self Duality
|
|
We present a simple construction of the instantonic type equation over
octonions where its similarities and differences with the quaternionic case are
very clear. We use the unified language of Clifford Algebra. We argue that our
approach is the pure algebraic formulation of the geometric based soft Lie
algebra. The topological criteria for the stability of our solution is given
explicitly to establish its solitonic property. Many beautiful features of the
parallelizable ring division spheres and Absolute Parallelism (AP) reveal their
presence in our formulation.
|
Comment: LaTeX file. various typos, equations and calculations errors fixed. A
new part is added where the incompatibility between the self duality and the
Yang-Mills equation of motion is mentioned. The conclusion is modified |
|
math-ph 9905024
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Octonionic Mobius Transformations
|
Manogue, Corinne A. |
Dray, Tevian |
|
A vexing problem involving nonassociativity is resolved, allowing a
generalization of the usual complex Mobius transformations to the octonions.
This is accomplished by relating the octonionic Mobius transformations to the
Lorentz group in 10 spacetime dimensions. The result will be of particular
interest to physicists working with lightlike objects in 10 dimensions.
|
Comment: Plain TeX, 12 pages, 1 PostScript figure included using epsf |
|
quant-ph 9503009
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Standard Model plus Gravity from Octonion Creators and Annihilators.
|
|
Octonion creation and annihilation operators are used to construct the Standard
Model plus Gravity. The resulting phenomenological model is the D4-D5-E6 model
described in hep-ph/9501252 .
|
Comment: 21 pages, latex, see http://www.gatech.edu/tsmith/home.html |
|
hep-th 0212201
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Octonionic M-theory and D=11 generalized conformal and superconformal algebras
|
Toppan, Francesco |
Lukierski, Jerzy |
|
Following [1] we further apply the octonionic structure to supersymmetric
D=11 $M$-theory. We consider the octonionic $2^{n+1} \times 2^{n+1}$ Dirac
matrices describing the sequence of Clifford algebras with signatures ($9+n,n$)
($n=0,1,2, ...$) and derive the identities following from the octonionic
multiplication table. The case $n=1$ ($4\times 4$ octonion-valued matrices) is
used for the description of the D=11 octonionic $M$ superalgebra with 52 real
bosonic charges; the $n=2$ case ($8 \times 8$ octonion-valued matrices) for the
D=11 conformal $M$ algebra with 232 real bosonic charges. The octonionic
structure is described explicitly for $n=1$ by the relations between the 528
Abelian O(10,1) tensorial charges $Z_\mu $Z_{\mu\nu}$, $Z_{\mu >... \mu_5}$ of
the $M$-superalgebra. For $n=2$ we obtain 2080 real non-Abelian bosonic
tensorial charges $Z_{\mu\nu}, Z_{\mu_1 \mu_2 \mu_3}, Z_{\mu_1 ... \mu_6}$
which, suitably constrained describe the generalized D=11 octonionic conformal
algebra. Further, we consider the supersymmetric extension of this octonionic
conformal algebra which can be described as D=11 octonionic superconformal
algebra with a total number of 64 real fermionic and 239 real bosonic
generators.
|
Comment: LateX, 13 pages, corrected some typos, version in press in PLB |
|
hep-th 0301037
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Octonions and M-theory
|
|
We explain how structures related to octonions are ubiquitous in M-theory.
All the exceptional Lie groups, and the projective Cayley line and plane appear
in M-theory. Exceptional G_2-holonomy manifolds show up as compactifying
spaces, and are related to the M2 Brane and 3-form. We review this evidence,
which comes from the initial 11-dim structures. Relations between these objects
are stressed, when extant and understood. We argue for the necessity of a
better understanding of the role of the octonions themselves (in particular
non-associativity) in M-theory.
|
Comment: 6 pages, iopart.sty, Presented at the the 24th International
Colloquium on Group-Theoretical Methods in Physics. Paris, July 15-22, 2.002 |
|
cond-mat 0306045
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| The Eight Dimensional Quantum Hall Effect and the Octonions
|
Bernevig, B. A. |
Zhang, S. C. |
Toumbas, N. |
Hu, J. P. |
|
We construct a generalization of the quantum Hall effect where particles move
in an eight dimensional space under an SO(8) gauge field. The underlying
mathematics of this particle liquid is that of the last normed division
algebra, the octonions. Two fundamentally different liquids with distinct
configurations spaces can be constructed, depending on whether the particles
carry spinor or vector SO(8) quantum numbers. One of the liquids lives on a 20
dimensional manifold of with an internal component of SO(7) holonomy, whereas
the second liquid lives on a 14 dimensional manifold with an internal component
of $G_2$ holonomy.
|
Comment: 5 pages |
|
hep-ph 9708379
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| From Sets to Quarks
|
|
From sets and simple operations on sets, a Feynman Checkerboard physics
model is constructed that allows computation of force strength constants and
constituent mass ratios of elementary particles, with a Lagrangian
structure that gives a Higgs scalar particle mass of about 146 GeV and a
Higgs scalar field vacuum expectation value of about 252 GeV, giving a tree
level constituent Truth Quark (top quark) mass of roughly 130 GeV, which is
(in my opinion) supported by dileptonic events and some semileptonic events.
See http://galaxy.cau.edu/tsmith/HDFCmodel.html and
http://www.innerx.net/personal/tsmith/HDFCmodel.html
Chapter 1 - Introduction.
Chapter 2 - From Sets to Clifford Algebras.
Chapter 3 - Octonions and E8 lattices.
Chapter 4 - E8 spacetime and particles.
Chapter 5 - HyperDiamond Lattices.
Chapter 6 - Internal Symmetry Space.
Chapter 7 - Feynman Checkerboards.
Chapter 8 - Charge = Amplitude to Emit Gauge Boson.
Chapter 9 - Mass = Amplitude to Change Direction.
Chapter 10 - Protons, Pions, and Physical Gravitons.
|
Comment: Higgs mass revised to 146 GeV, also miscellaneous additions and typo
corrections, 164 pages, LaTeX |
|
hep-th 9503189
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| An Octonionic Gauge Theory
|
Joshi., G. C. |
Lassig, C. C. |
|
The nonassociativity of the octonion algebra necessitates a bimodule
representation, in which each element is represented by a left and a right
multiplier. This representation can then be used to generate gauge
transformations for the purpose of constructing a field theory symmetric under
a gauged octonion algebra, the nonassociativity of which appears as a failure
of the representation to close, and hence produces new interactions in the
gauge field kinetic term of the symmetric Lagrangian.
|
Comment: 14 pages, Revtex. |
|
hep-th 9504040
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| OCTONIONS: INVARIANT REPRESENTATION OF THE LEECH LATTICE
|
|
The Leech lattice, $\Lambda_{24}$, is represented on the space of octonionic
3-vectors. It is built from two octonionic representations of $E_{8}$, and is
reached via $\Lambda_{16}$. It is invariant under the octonion index cycling
and doubling maps.
|
Comment: 7 pages, latex, no figures |
|
hep-th 9807107
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Matroid Theory and Supergravity
|
|
In this work, we consider matroid theory. After presenting three different
(but equivalent) definitions of matroids, we mention some of the most important
theorems of such theory. In particular, we note that every matroid has a dual
matroid and that a matroid is regular if and only if it is binary and includes
no Fano matroid or its dual. We show a connection between this last theorem and
octonions which at the same time, as it is known, are related to the Englert's
solution of D = 11 supergravity. Specifically, we find a relation between the
dual of Fano matroid and D = 11 supergravity. Possible applications to M-theory
are speculated upon.
|
Comment: 8 pages, Revtex, to appear in Rev. Mex. Fis. 1998 |
|
physics 9702031
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| "Cayley-Klein" schemes for real Lie algebras and Freudhental Magic Squares
|
Herranz, Francisco J. |
Santander, Mariano |
|
We introduce three "Cayley-Klein" families of Lie algebras through
realizations in terms of either real, complex or quaternionic matrices. Each
family includes simple as well as some limiting quasi-simple real Lie algebras.
Their relationships naturally lead to an infinite family of $3\times 3$
Freudenthal-like magic squares, which relate algebras in the three CK families.
In the lowest dimensional cases suitable extensions involving octonions are
possible, and for $N=1, 2$, the "classical" $3\times 3$ Freudenthal-like
squares admit a $4\times 4$ extension, which gives the original Freudenthal
square and the Sudbery square.
|
Comment: 6 pages, LaTeX; M.S. contribution to Group 21, Goslar 1996 |
|
hep-th 0203149
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Generalized Space-time Supersymmetries, Division Algebras and Octonionic M-theory
|
Toppan, Francesco |
Lukierski, Jerzy |
|
We describe the set of generalized Poincare and conformal superalgebras in
D=4,5 and 7 dimensions as two sequences of superalgebraic structures, taking
values in the division algebras R, C and H. The generalized conformal
superalgebras are described for D=4 by OSp(1;8| R), for D=5 by SU(4,4;1) and
for D=7 by U_\alpha U(8;1|H). The relation with other schemes, in particular
the framework of conformal spin (super)algebras and Jordan (super)algebras is
discussed. By extending the division-algebra-valued superalgebras to octonions
we get in D=11 an octonionic generalized Poincare superalgebra, which we call
octonionic M-algebra, describing the octonionic M-theory. It contains 32 real
supercharges but, due to the octonionic structure, only 52 real bosonic
generators remain independent in place of the 528 bosonic charges of standard
M-algebra. In octonionic M-theory there is a sort of equivalence between the
octonionic M2 (supermembrane) and the octonionic M5 (super-5-brane) sectors. We
also define the octonionic generalized conformal M-superalgebra, with 239
bosonic generators.
|
Comment: 14 pages, LaTeX |
|
hep-th 0212251
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Observable Algebra
|
|
A physical applicability of normed split-algebras, such as hyperbolic
numbers, split-quaternions and split-octonions is considered. We argue that the
observable geometry can be described by the algebra of split-octonions. In such
a picture physical phenomena are described by the ordinary elements of chosen
algebra, while zero divisors (the elements of split-algebras corresponding to
zero norms) give raise the coordinatization of space- time. It turns to be
possible that two fundamental constants (velocity of light and Planck constant)
and uncertainty principle have geometrical meaning and appears from the
condition of positive definiteness of norms. The property of non-associativity
of octonions could correspond to the appearance of fundamental probabilities in
four dimensions. Grassmann elements and a non-commutativity of space
coordinates, which are widely used in various physical theories, appear
naturally in our approach.
|
Comment: 11 pages, RevTeX, no figures, minor corrections, refferences added |
|
math-ph 0308020
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Vector Coherent States on Clifford algebras
|
Thirulogasanthar, K. |
Hohoueto, A. L. |
|
The well-known canonical coherent states are expressed as an infinite series
in powers of a complex number $z$ together with a positive sequence of real
numbers $\rho(m)=m$. In this article, in analogy with the canonical coherent
states, we present a class of vector coherent states by replacing the complex
variable $z$ by a real Clifford matrix. We also present another class of vector
coherent states by simultaneously replacing $z$ by a real Clifford matrix and
$\rho(m)$ by a real matrix. As examples, we present vector coherent states on
quaternions and octonions with their real matrix representations.
|
Comment: 10 pages |
|
hep-th 0112144
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| A theory of algebraic integration
|
|
In this paper we extend the idea of integration to generic algebras. In
particular we concentrate over a class of algebras, that we will call
self-conjugated, having the property of possessing equivalent right and left
multiplication algebras. In this case it is always possible to define an
integral sharing many of the properties of the usual integral. For instance, if
the algebra has a continuous group of automorphisms, the corresponding
derivations are such that the usual formula of integration by parts holds. We
discuss also how to integrate over subalgebras. Many examples are discussed,
starting with Grassmann algebras, where we recover the usual Berezin's rule.
The paraGrassmann algebras are also considered, as well as the algebra of
matrices. Since Grassmann and paraGrassmann algebras can be represented by
matrices we show also that their integrals can be seen in terms of traces over
the corresponding matrices. An interesting application is to the case of group
algebras where we show that our definition of integral is equivalent to a sum
over the unitary irreducuble representations of the group. We show also some
example of integration over non self-conjugated algebras (the bosonic and the
$q$-bosonic oscillators), and over non-associative algebras (the octonions).
|
Comment: LaTex file, 50 pages, no figures. Contribution to the Michael Marinov
Memorial Volume |
|
hep-th 9211123
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Gauging octonion algebra
|
Waldron, A. K. |
Joshi, G. C. |
|
By consireding representation theory for non-associative algebras we
construct the fundamental and adjoint representations of the octonion algebra.
We then show how these representations by associative matrices allow a
consistent octonionic gauge theory to be realised. We find that
non-associativity implies the existence of new terms in the transformation laws
of fields and the kinetic term of an octonionic Lagrangian.
|
Comment: 20 pages,latex, UM- P-92/60 |
|
hep-th 9506080
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| OCTONIONS: INVARIANT LEECH LATTICE EXPOSED
|
|
The structure of a previously developed representation of the Leech lattice,
$\Lambda_{24}$, is exposed to further light with this unified and very simple
construction.
|
Comment: 5 pages, latex, no figures |
|
hep-th 9512134
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Octonions and Supersymmetry
|
|
We apply the techniques of $S^7$-algebras to the construction of N=5-8
superconformal algebras and of S{\bf O}(1,9), a modification of SO(1,9) which
commutes with $S^7$-transformations. We discuss the relevance of S{\bf O}(1,9)
for off-shell super-Maxwell theory in D=(1,9).
|
Comment: 8 pages, latex |
|
hep-th 9604116
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Octonion X,Y-Product $G_{2}$ Variants
|
|
The automorphism group $G_{2}$ of the octonions changes when octonion
X,Y-product variants are used. I present here a general solution for how to go
from $G_{2}$ to its X,Y-product variant.
|
Comment: 5 pages, latex, no figures |
|
hep-th 9607152
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Quaternion Analysis
|
|
Quaternion analysis is considered in full details where a new analyticity
condition in complete analogy to complex analysis is found. The extension to
octonions is also worked out.
|
Comment: RevTeX file; A new analyticity condition for quaternionic polynoimals
has been given |
|
hep-th 9811069
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Algebraic Realization of Quark-Diquark Supersymmetry
|
|
Algebraic realizations of supersymmetry through SU(m,n) type superalgebras
are developed. We show their applications to a bilocal quark-antiquark or a
quark-diquark systems. A new scheme based on SU(6/1) is fully exploited and the
bilocal approximation is shown to get carried unchanged into it. Color algebra
based on octonions allows the introduction of a new larger algebra that puts
quarks, diquarks and exotics in the same supermultiplet as hadrons and
naturally suppresses quark configurations that are symmetrical in color space
and antisymmetrical in remaining flavor, spin and position variables. A
preliminary work on the first order relativistic formulation through the spin
realization of Wess-Zumino super-Poincare algebra is presented.
|
Comment: 20 pages, Latex |
|
math 0003166
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Matrix Representations of Octonions and Their Applications
|
|
As is well-known, the real quaternion division algebra $ {\cal H}$ is
algebraically isomorphic to a 4-by-4 real matrix algebra. But the real division
octonion algebra ${\cal O}$ can not be algebraically isomorphic to any matrix
algebras over the real number field ${\cal R}$, because ${\cal O}$ is a
non-associative algebra over ${\cal R}$. However since ${\cal O}$ is an
extension of ${\cal H}$ by the Cayley-Dickson process and is also
finite-dimensional, some pseudo real matrix representations of octonions can
still be introduced through real matrix representations of quaternions. In this
paper we give a complete investigation to real matrix representations of
octonions, and consider their various applications to octonions as well as
matrices of octonions.
|
Comment: 23 pages, LaTex |
|
math 0011040
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Clifford algebras obtained by twisting of group algebras
|
Albuquerque, H. |
Majid, S. |
|
We investigate the construction and properties of Clifford algebras by a
similar manner as our previous construction of the octonions, namely as a
twisting of group algebras of Z_2^n by a cocycle. Our approach is more general
than the usual one based on generators and relations. We obtain in particular
the periodicity properties and a new construction of spinors in terms of left
and right multiplication in the Clifford algebra.
|
Comment: 16 pages latex, no figures |
|
math 0011260
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| The 42 Assessors and the Box-Kites they fly: Diagonal Axis-Pair Systems of Zero-Divisors in the Sedenions' 16 Dimensions
|
|
G. Moreno's abstract depiction of the Sedenions' normed zero-divisors, as
homomorphic to the exceptional Lie group G2, is fleshed out by exploring
further structures the A-D-E approach of Lie algebraic taxonomy keeps hidden. A
breakdown of table equivalence among the half a trillion multiplication schemes
the Sedenions allow is found; the 168 elements of PSL(2,7), defining the finite
projective triangle on which the Octonions' 480 equivalent multiplication
tables are frequently deployed, are shown to give the exact count of primitive
unit zero-divisors in the Sedenions. (Composite zero-divisors, comprising all
points of certain hyperplanes of up to 4 dimensions, are also determined.) The
168 are arranged in point-set quartets along the 42 Assessors (pairs of
diagonals in planes spanned by pure imaginaries, each of which zero-divides
only one such diagonal of any partner Assessor). These quartets are
multiplicatively organized in systems of mutually zero-dividing trios of
Assessors, a D4-suggestive 28 in number, obeying the 6-cycle crossover logic of
trefoils or triple zigzags. 3 trefoils and 1 zigzag determine an octahedral
vertex structure we call a box-kite -- seven of which serve to partition
Sedenion space. By sequential execution of proof-driven production rules, a
complete interconnected box-kite system, or Seinfeld production (German for
field of being; American for 1990's television's Show About Nothing), can be
unfolded from an arbitrary Octonion and any (save for two) of the Sedenions.
Indications for extending the results to higher dimensions and different
dynamic contexts are given in the final pages.
|
Comment: 73 pages,17 figures |
|
math 0105155
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| The Octonions
|
|
The octonions are the largest of the four normed division algebras. While
somewhat neglected due to their nonassociativity, they stand at the crossroads
of many interesting fields of mathematics. Here we describe them and their
relation to Clifford algebras and spinors, Bott periodicity, projective and
Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also
touch upon their applications in quantum logic, special relativity and
supersymmetry.
|
Comment: 56 pages LaTeX, 11 Postscript Figures, some small corrections |
|
math 0106021
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Orthonormal Eigenbases over the Octonions
|
Manogue, Corinne A |
Okubo, Susumu |
Dray, Tevian |
|
We previously showed that the real eigenvalues of 3x3 octonionic Hermitian
matrices form two separate families, each containing 3 eigenvalues, and each
leading to an orthonormal decomposition of the identity matrix, which would
normally correspond to an orthonormal basis. We show here that it nevertheless
takes both families in order to decompose an arbitrary vector into components,
each of which is an eigenvector of the original matrix; each vector therefore
has 6 components, rather than 3.
|
Comment: LaTeX2e, 14 pages |
|
math 0206028
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| The Lie algebra splitg2 with Mathematica using Zorn's matrices
|
Bjerregaard, Pablo Alberca |
Gonzalez, Candido Martin |
|
We will obtain in this paper a generic expression of any element in athe Lie
algebra of the derivations of the split octonions a over an arbitrary field.
For this purpose, we will use the Zorn's matrices. We will also compute the
multiplication table of this Lie algebra.
|
Comment: Mathematica file (*.nb) |
|
math 9801141
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| A quantum octonion algebra
|
Pérez-Izquierdo, José M. |
Benkart, Georgia |
|
Using the natural irreducible 8-dimensional representation and the two spin
representations of the quantum group $U_q$(D$_4$) of D$_4$, we construct a
quantum analogue of the split octonions and study its properties. We prove
that the quantum octonion algebra satisfies the q-Principle of Local Triality
and has a nondegenerate bilinear form which satisfies a q-version of the
composition property. By its construction, the quantum octonion algebra is a
nonassociative algebra with a Yang-Baxter operator action coming from the
R-matrix of $U_q$(D$_4$). The product in the quantum octonions is a
$U_q$(D$_4$)-module homomorphism. Using that, we prove identities for the
quantum octonions, and as a consequence, obtain at $q = 1$ new
``representation theory proofs for very well-known identities satisfied by
the octonions. In the process of constructing the quantum octonions we
introduce an algebra which is a q-analogue of the 8-dimensional para-Hurwitz
algebra.
|
|
math 9802116
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Quasialgebra structure of the octonions
|
Albuquerque, H. |
Majid, S. |
|
We show that the octonions are a twisting of the group algebra of Z_2 x Z_2 x
Z_2 in the quasitensor category of representations of a quasi-Hopf algebra
associated to a group 3-cocycle. We consider general quasi-associative algebras
of this type and some general constructions for them, including quasi-linear
algebra and representation theory, and an automorphism quasi-Hopf algebra.
Other examples include the higher 2^n-onion Cayley algebras and examples
associated to Hadamard matrices.
|
Comment: 34 pages LATEX |
|
math 9810037
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| New approach to octonions and cayley algebras
|
Albuquerque, H. |
Majid, S. |
|
We announce a new approach to the octonions as quasiassociative algebras. We
strip out the categorical and quasi-quantum group considerations of our longer
paper and present here (without proof) some of the more algebraic conclusions
|
Comment: A short conference proceedings (presented at ivnonalg, Brazil, 1998)
announcement of our long paper |
|
math 9810140
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| On the projective geometry of homogeneous spaces
|
Landsberg, Joseph M. |
Manivel, Laurent |
|
We study the projective geometry of homogeneous varieties $X= G/P\subset
P(V)$, where $G$ is a complex simple Lie group, $P$ is a maximal parabolic
subgroup and $V$ is the minimal $G$-module associated to $P$. Our study began
with the observation that Freudenthal's magic chart could be derived from Zak's
theorem on Severi varieties and standard geometric constructions. Our attempt
to understand this observation led us to discover further connections between
projective geometry and representation theory.
Among other things, we calculate the variety of tangent directions to lines
on $X$ through a point and determine unirulings of $X$. We show this variety is
a Hermitian symmetric space if and only if $P$ does not correspond to a short
root. We describe the spaces corresponding to the exceptional short roots and
their unirulings using the octonions. Further calculations, in the case $X$ is
a Hermitian symmetric space, give rise to a strict prolongation property and
the appearance of secant varieties at the infinitesimal level. Our work
complements and advances that of Freudenthal and Tits, who studied homogeneous
varieties in an abstract/axiomatic setting.
|
|
math 9903128
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Z_n Quasialgebras
|
Albuquerque, H. |
Majid, S. |
|
Recently we have reformulated the octonions as quasissociative algebras
(quasialgebras) living in a symmetric monoidal category. In this note we
provide further examples of quasialgebras, namely ones where the
nonassociativity is induced by a Z_n-grading and a nontrivial 3-cocycle
|
Comment: 6 pages LATEX |
|
math-ph 0002023
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Beyond Octonions
|
|
We investigate Clifford Algebras structure over non-ring division algebras.
We show how projection over the real field produces the standard
Attiyah-Bott-Shapiro classification.
|
Comment: Latex, 6 pages; typos corrected |
|
physics 9710038
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Angular Momentum, Quaternion, Octonion, and Lie-Super Algebra osp(1,2)
|
|
We will derive both quaternion and octonion algebras as the Clebsch-Gordan
algebras based upon the su(2) Lie algebra by considering angular momentum
spaces of spin one and three. If we consider both spin 1 and 1/2
states, then the same method will lead to the Lie-super algebra osp(1,2). Also,
the quantum generalization of the method is discussed on the basis of the
quantum group $su_q(2)$.
|
Comment: 17 pages, TEX |
|
math 0207003
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Flying Higher Than a Box-Kite: Kite-Chain Middens, Sand Mandalas, and Zero-Divisor Patterns in the 2n-ions Beyond the Sedenions
|
|
Methods for studying zero-divisors (ZD's) in 2n-ions generated by
Cayley-Dickson process beyond the Sedenions are explored. Prior work showed a
ZD system in the Sedenions, based on 7 octahedral lattices ("Box-Kites"), whose
6 vertices collect and partition the "42 Assessors" (pairs of diagonals in
planes spanned by pure imaginaries, one a pure Octonion, hence of subscript <
8, the other a Sedenion of subscript > 8 and not the XOR with 8 of the chosen
Octonion). Potential connections to fundamental objects in physics (e.g., the
curvature tensor and pair creation) are suggested. Structures found in the
32-ions ("Pathions") are elicited next. Harmonics of Box-Kites, called here
"Kite-Chain Middens," are shown to extend indefinitely into higher forms of
2n-ions. All non-Midden-collected ZD diagonals in the Pathions, meanwhile, are
seen belonging to a set of 15 "emanation tables," dubbed "sand mandalas."
Showcasing the workings of the DMZ's (dyads making zero) among the products of
each of their 14 Assessors with each other, they house 168 fillable cells each
(the number of elements in the simple group PSL(2,7) governing Octonion
multiplication). 7 of these emanation tables, whose "inner XOR" of their
axis-pairs' indices exceed 24, indicate modes of collapsing from higher to
lower 2n-ion forms, as they can be "folded up" in a 1-to-1 manner onto the 7
Sedenion Box-Kites. These same 7 also display surprising patterns of DMZ
sparsity (with but 72 of 168 available cells filled), with the animation-like
sequencing obtaining between these 7 "still-shots" indicating an entry-point
for cellular-automata-like thinking into the foundations of number theory.
|
Comment: 20 pages, incl. 4 figures, 3 textual tables. V2: Fixed boneheaded
mislabeling of standard operator names on p. 4; added paragraph of
simplifying conclusions at top of p. 18 |
|
hep-th 0207216
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| The Possible Role of Pure Spinors in Some Sectors of Particle Physics
|
|
The equations defining pure spinors are interpreted as equations of motion
formulated on the lightcone of a ten-dimensional, lorentzian, momentum space.
Most of the equations for fermion multiplets, usually adopted by particle
physics, are then naturally obtained and their properties like internal
symmetries, charges, families appear to be due to the correlation of the
associated Clifford algebras, with the 3 complex division algebras: complex
numbers at the origin of U(1) and charges; quaternions at the origin of SU(2);
families and octonions at the origin of SU(3). Pure spinors instead could be
relevant not only because the underlying momentum space results compact, but
also because it may throw light on some aspects of particle physics, like:
masses, charges, constaint relations, supersymmetry and epistemology.
|
Comment: 20 pages, Latex |
|
math-ph 0212031
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Mathematics of CLIFFORD - A Maple package for Clifford and Grassmann algebras
|
Fauser, Bertfried |
Ablamowicz, Rafal |
|
CLIFFORD performs various computations in Grassmann and Clifford algebras. It
can compute with quaternions, octonions, and matrices with entries in Cl(B) -
the Clifford algebra of a vector space V endowed with an arbitrary bilinear
form B. Two user-selectable algorithms for Clifford product are implemented:
'cmulNUM' - based on Chevalley's recursive formula, and 'cmulRS' - based on
non-recursive Rota-Stein sausage. Grassmann and Clifford bases can be used.
Properties of reversion in undotted and dotted wedge bases are discussed.
|
Comment: 19 pages, see CIFFORD website atshttp://math.tntech.edu/rafal/ |
|
hep-ph 0302153
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Symmetries and Mass Predictions of the Supersymmetric Quark Model
|
|
QCD justification of SU(m/n) supergroups are shown to provide a basis for the
existence of an approximate hadronic supersymmetry. Effective Hamiltonian of
the relativistic quark model is derived, leading to hadronic mass formulae in
remarkable agreement with experiments. Bilocal approximation to hadronic
structure and incorporation of color through octonion algebra (based on
quark-antidiquark symmetry) is also shown to predict exotic diquark-antidiquark
($D-\bar{D}$) meson states. A minimal supersymmetric sceme based on $SU(3)^c
\times SU(6/1)$ that excludes exotics is constructed. Symmetries of three quark
systems and possible relativistic formulation of the quark model through the
spin realization of Wess-Zumino algebra is presented.
|
Comment: 62 pages, 3 figures, Latex |
|
hep-th 0302139
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Self-Dual Fields and Quaternion Analyticity
|
|
Quaternionic formulation of D=4 conformal group and of its associated
twistors and their relation to harmonic analyticity is presented.
Generalization of $SL(2,\cal{C})$ to the D=4 conformal group SO(5,1) and its
covering group $SL(2,\cal{Q})$ that generalizes the euclidean Lorentz group in
$R^4$ [namely $SO(3,1)\approx SL(2,\cal{C})$ which allow us to obtain the
projective twistor space $CP^3$] is shown. Quasi-conformal fields are
introduced in D=4 and Fueter mappings are shown to map self-dual sector onto
itself (and similarly for the anti-self-dual part). Differentiation of Fueter
series and various forms of differential operators are shown, establishing the
equivalence of Fueter analyticity with twistor and harmonic analyticity. A
brief discussion of possible octonion analyticity is provided.
|
Comment: 45 pages, Latex |
|
math 0303153
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Riemannian geometry over different normed division algebra
|
|
We develop a unifed theory to study geometry of manifolds with different
holonomy groups.
They are classified by (1) real, complex, quaternion or octonion number
they are defined over and (2) being special or not. Specialty is an orientation
with respect to the corresponding normed algebra A. For example, special
Riemannian A-manifolds are oriented Riemannian, Calabi-Yau, Hyperkahler and
G_2-manifolds respectively.
For vector bundles over such manifolds, we introduce (special)
A-connections. They include holomorphic, Hermitian Yang-Mills, Anti-Self-Dual
and Donaldson-Thomas connections. Similarly we introduce (special)
A/2-Lagrangian submanifolds as maximally real submanifolds. They include
(special) Lagrangian, complex Lagrangian, Cayley and (co-)associative
submanifolds.
We also discuss geometric dualities from this viewpoint: Fourier
transformations on A-geometry for flat tori and a conjectural SYZ mirror
transformation from (special) A-geometry to (special) A/2-Lagrangian geometry
on mirror special A-manifolds.
|
Comment: 45 pages. To appear in Journal of Differential Geometry |
|
hep-th 0304244
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Self-dual quaternionic lumps in octonionic space-time
|
|
The theory of self-dual bosonic lumps immersed in the Cayley-calibrated space
of octonions has a large class of exact finite quaternionic power series
solutions.
|
Comment: 7 pages |
|
hep-th 0306075
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Self-Dual Supergravity in Seven Dimensions with Reduced Holonomy G_2
|
Nishino, Hitoshi |
Rajpoot, Subhash |
|
We present self-dual N=2 supergravity in superspace for Euclidean seven
dimensions with the reduced holonomy G_2 \subset SO(7), including all
higher-order terms. As its foundation, we first establish N=2 supergravity
without self-duality in Euclidean seven dimensions. We next show how the
generalized self-duality in terms of octonion structure constants can be
consistently imposed on the superspace constraints. We found two self-dual N=2
supergravity theories possible in 7D, depending on the representations of the
two spinor charges of N=2. The first formulation has both of the two spinor
charges in the {\bf 7} of G_2 with 24 + 24 on-shell degrees of freedom. The
second formulation has both charges in the {\bf 1} of G_2 with 16 + 16 on-shell
degrees of freedom.
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Comment: 14 pages, no figures |
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hep-th 0307070
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| On the Octonionic M-algebra and Superconformal M-algebra
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It is shown that the $M$-algebra related with the $M$ theory comes in two
variants. Besides the standard $M$ algebra based on the real structure, an
alternative octonionic formulation can be consistently introduced. This second
variant has striking features. It involves only 52 real bosonic generators
instead of 528 of the standard $M$ algebra and moreover presents a novel and
surprising feature, its octonionic $M5$ (super-5-brane) sector is no longer
independent, but coincides with the octonionic $M1$ and $M2$ sectors. This is
in consequence of the non-associativity of the octonions. An octonionic version
of the superconformal $M$-algebra also exists. It is given by $OSp(1,8|{\bf
O})$ and admits 239 bosonic and 64 fermionic generators. It is speculated that
the octonionic $M$-algebra can be related to the exceptional Lie and Jordan
algebras that apparently play a special role in the Theory Of Everything.
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Comment: 14 pages, LaTex. Lecture Notes delivered at the II Summer School in
Modern Mathematical Physics, Kopaonik (YU), Sept. 2002. In the Proceedings |
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math-ph 0308045
arXiv:abs
arXiv:pdf
local:pdf
local:txt
| Multi Matrix Vector Coherent States
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Thirulogasanthar, K. |
Honnouvo, G. |
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We present a class of vector coherent states labeled by multiple of matrices
as a vector on a Hilbert space, where the Hilbert space is taken to be the
tensor product of several other Hilbert spaces. As examples vector coherent
states labeled by multiple of quaternions and octonions were given. The
resulting generalized oscillator algebra is discussed.
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Comment: 15 pages |
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