Jordan algebras in quantum and string theory
zirkus
Here are some initial and brief comments and references about the
role of the Jordan algebras in quantum and string theory. Perhaps you
might want to further investigate some of these topics, but I plan to
examine mainly the relevant geometry, Lie cosets and the octonions
because I have already found various new examples of these in
supergravity and string theory (e.g. as you can see below).
Some have noted that the nonassociativity of the octonions seems to
forbid a Hilbert space interpretation of the exceptional Jordan
algebra J3(O). However, instead of using the common approach of states
and associative C* algebras, paper [1] uses traces with von Neumann
algebras of type I or II and then generalizes to a Jordan formulation
so that there is a JB-algebra of observables instead of the usual C*
algebra of observables, and so that there can be a Hilbert space
interpretation of J3(O) while also including the content of the usual
associative formulation.
You can see in this paper that JB algebras (which includes J3(O)) are
the direct analogues of C* algebras (for the infinite dimensional
case) and that JBW algebras are the analog of abstract von Neumann
algebras. For a different paper about exceptional Hilbert spaces based
on Jordan algebras and releated geometry see [2].
Paper [3] proposes a J3(O) with F4 background independent cubic Matrix
model (string theory), and paper [4] constructs a similar model but
uses the complexified version of J3(O) with E6. (To see how to
diagonalize a generic Jordan matrix using F4 transformations and for
more about J3(O) matrices see [5].)
The set of all 3 x 3 Hermitian matrices over Oc (the complex Cayley
numbers) is the Cartan factor C^6 and Cartan factors, C* algebras and
Hilbert spaces are each JB* triples (a Banach space whose open unit
balls are bounded symmetric domains), and dual JB* triples are called
JBW* triples - see page 18 of [6]. J3(Oc) is a Hermitian positive
Jordan triple system (JTS) - page 21 of [7].
The Jordan triple product appears in connection with infinite
dimensional holomorphy, with Hilbert spaces and with the geometrical
properties of C* algebras [8], and Jordan triple structure is
associated with complex bounded symmetric domains - page 4 of [7].
J3(O) is a JTS and JTSs are ternary algebras which are quadratic in
the sense of operads, and so JTSs can be described via operads [9].
Jordan algebras or structures are related to vertex (operator)
algebras or superalgebras and to representations of Kac-Moody and
Virasoro algebras [10]. Note that the vertex operators of string
theory also give reps of Lie algebras, Kac-Moody and Virasoro algebras
(both infinite dimensional Lie algebras) as well as reps of the
Fischer-Griess Monster group algebra.
Jordan algebras or structures are also related to QFT, CFT, SCAs and
WZW models [11]. Another paper [12] about M-theory with J3(O) and F4
makes use of the algebraic Kostant-Dirac operator, and a variant of
this operator also appears on page 12 of a paper [13] about a gerbe
based approach to supersymmetric WZW models. (Gerbes are useful in
string theory, e.g. gerbes help illuminate the geometry of mirror
symmetry of CY threefolds, help give a noncommutative description of
D-branes in the presence of topologically non-trivial background
fields, provide a geometric way to unify properties of p-form fields
with gauge symmetries etc.)
There is more about how Jordan structures are involved with physics
[14], with Hopf algebras and quantization [15], with the Jordan ring
of self adjoint operators, the Jordan operator algebras and Jordan
homomorphisms [16], with von Neumann algebras [6] and [17], Jordan
superalgebras and super triple systems [18], Jordan-Lie algebras and
(mixed) Jordan-Lie superalgebras [19], Yang-Baxter equations [20],
the Jordan related Freudenthal triple systems (FTSs) and physics [21],
JB* triples and more about J3(O) and JTSs and string theory [22],
bounded symmetric domains [23], C* algebras [24], and also see the
Jordan preprint server for even more on these topics [25].
Considering these various connections, perhaps someone would want to
look to see if Jordan structures could be related to the subfactors
that were famously studied by Vaughan Jones, e.g. it is known that
subfactors of type II von Neumann algebras can yield topological QFTs
(TQFTs). (By the way, vertex algebras and QFT and CFT in string theory
can also be described via operads.)
"JTSs are now seen to illuminate certain classically reduced phase
spaces (e. g. that of finitely many harmonic oscillators with total
angular momentum zero)." - from page 9 of [26]. Hyperkahler
manifolds and abelian varieties can yield classical Jordan algebras
[27].
For realistic theories of supergravity with N=1 in 4d, there are
various symmetric Kahler spaces [28] including E6/SO(10) x U(1) which
could be identified with the bi-octonionic plane (C x O)P^2 as shown
on page 46 of a review article [29] of the octonions. Both E6 and
SO(10) are GUT groups, and note that pairs of bioctonions yield
spinors in 10d Euclidean space and that pairs of these Euclidean
spinors are spinors in 12d Minkowski spacetime. Any 27 dimensional rep
of E6 generates a Jordan algebra if the roots and weights are
non-trivially embedded in 7 dimensional space, and also see [30].
Here is some additional commentary regarding recent papers by P.
Ramond [31] and the paper by L. Boya [32] entitled "Octonions and
M-theory", (and sometimes I will write e.g. "SO(9)" instead of
"Spin(9)").
Ramond notes that SO(9) is the little group of massive representations
in 9 + 1 dimensions, and suggests that SO(9) is also the light cone
little group (classifying the massless degrees of freedom of 11D
supergravity) in 10 + 1 dimensions. The fundamental degrees of freedom
in Matrix theory are D0-branes whose transverse rotation group is
SO(9). As paper [33] states:
"The D0-brane solution of type IIA supergravity is an example of a
more general class of extremal p-brane. These solitonic solutions of
ten- and eleven-dimensional supergravity have been much studied. One
is usually interested in purely bosonic solitons, which nevertheless
admit Killing spinors, such solutions thus being invariant under some
fraction of the 32 supersymmetries. For a single p-brane, this
fraction is one half, which leaves 16 broken supersymmetries. They
correspond to 16 zero mode fermions, the presence of which gives rise
to the entire BPS supermultiplet of 256 states. These polarization
states of the spinning p-brane fall into representations of the little
group of the respective brane so, as we have already mentioned, the
states of the spinning D0-brane match those of the eleven-dimensional
supergraviton; the little group in both cases is SO(9)."
SO(9) corresponds to the massive Kaluza-Klein states of the 11D
supergraviton as discussed in [34]. "Making note of the special
embedding of SO(9) in SO(16) where 16 --> 16, 128 --> 128 and 128' -->
44 + 84, this demonstrates that the D0-brane soliton indeed fills out
a complete (44 +128 +84)-dimensional representation of the massive
SO(9) little group [....]", and "the complete resulting 0-brane
supermultiplet is in precisely the correct (44 + 128 + 84)-
dimensional representation necessary to correspond to the massive
Kaluza-Klein states of the 11-dimensional supergraviton." (Note also
that N=2 Maxwell-Einstein supergravity theories are connected via the
Magic square described in papers such as [29].)
However, as mentioned in [33], "Since the SO(16) vacuum state is an
SO(8) singlet, the purely bosonic M2-brane is thus exact as a BPS
state, although this is not the case for the D0-brane: there is no
SO(9) singlet of SO(16), so the purely bosonic D0-brane soliton is not
an exact BPS state. Of course, the same can be said of the purely
bosonic eleven-dimensional supergraviton."
Further discussion about the chirality involved, and the presence of
the octo-octonionic plane ((O x O)P^2) can be seen on pages 16-18 of
[35].
Similarly to how so(9,1) is isomorphic to sl(2,O) where "O" is the
octonions, it is also the case [36] that so(10,2) is isomorphic to
sp(4,O). From the non-compact magic square can be obtained SO(9,1)
with real Dirac matrices and SO(10,2) which has a real 32 dimensional
representation. The group E7 is a 'supersymmetric commuting' extension
of the group SO(10,2) x SU(1,1) because E7 = SO(10,2) x SU(1,1) x 64
'commuting' SUSY (where 64 is also the dimension of the
quater-octonionic plane).
Note that E7(-25) is related to conformal transformations and
symplectic geometry and is Der M(J3(O)) where M(J3(O) is the
Freudenthal triple system of the exceptional Jordan algebra J3(O).
The group SO(9,1) is the group of collineations fixing a specific
point of the Cayley plane OP^2 and is included in E6(-26) = Coll(OP^2)
which includes subgroup F4 = Aut(J3(O)). Multiplets of SO(10) should
express the massive degrees of freedom of 11D M-theory, and F4 is
associated with spinors in 11D (a fact related to Bott periodicity).
Rays of the 2 x 2 hermitian octonionic matrices represent null
directions in the 10D Minkowski space M^10 or points on the sphere S^8
and are used to define the octonionic projective line OP^1. The plane
OP^2 is self-dual and lines in OP^2 are copies of OP^1, and points of
OP^2 correspond to trace 1 projections in J3(O).
There are 10D null branes which resolve null (or parabolic) orbifolds
[37]. These are branes that can have one fixed direction, in this
case, a null direction. Also, "the geometry of the eleven-dimensional
nullbrane is a metric product of a flat seven-dimensional euclidean
space (which can be further compactified) and a four-dimensional
locally minkowskian spacetime which can be viewed as the total space
of a circle fibration over a 3-dimensional manifold, where the radius
of the circle varies with time and reaches a minimum nonzero size R,
which is nothing but the scale in the previous discussion. These
properties seem to allow the use of string perturbation theory in
these discrete quotients, extending the original formulation for the
null orbifold given in [X]."
"The classification of smooth flat supersymmetric M-theory backgrounds
would include, in particular, the classification of crystallographic
subgroups of Spin(7) of SO(8)". The groups SO(7) and SO(8) are
associated with the imaginary octonions and the octonions, spinors in
7D and 8D Euclidean space and Lorentz transformations in 10D. The
spinor representations of SO(9,1) come from pairs of octonions. The
Lie brackets of F4 and E8 are built from maps involving SO(8) and its
three 8D irreducible representations. Speaking of triality, perhaps
the 3-forms involved with the imaginary octonions are related to
M-theory 3-forms. Does anyone know if M-theory 3-forms are related to
E8 gauge theory?
As stated on page 1 of [38], " A good starting point for addressing
these issues is the parabolic orbifold and the associated null flux
brane. These closed-string backgrounds are supersymmetric, and have no
closed time-like curves. They thus avoid many of the obvious
difficulties present in other time-dependent backgrounds. The
geometries of interest are orbifolds of flat Minkowski space R^9,1
under the action of a Poincare isometry ^T , consisting of a Lorentz
transformation ^ of SO(9,1), combined with a translation T that
commutes with ^. The null flux brane is obtained when ^ is a null
boost in SO(2,1) and T a translation by 2.pi.r in the remaining R^7."
R^7 is related to Im(O) - the seven dimensional space of all imaginary
octonions which is the smallest nontrivial representation of the group
G2 - because SO(Im(O)) = SO(7). The group G2 = Aut(O) is included in
both SO(7) and SO(8) = SO(O). The 3-form of Im(O) is given by
*(x,y,z) = <x,yz> and the real-linear transformations preserving * are
those in the group G2 (this 3-form is important in the theory of 7d
Joyce manifolds with G2 holonomy group and which look locally like the
imaginary octonions). Note that the supergravity 3-form on pages 16-18
of [35] is obtained via SO(9) creation and annihilation operators, but
I don't know if such a 3-form could be related to the 3-form
associated with the group G2.
There is a paper [39] about strings in time-dependent backgrounds
which deals with nullbranes and suggests on the bottom of page 24 that
it might be useful to do their construction with SL(2,O) as the
Lorentz group where SL(2,O) is isomorphic to SO(9,1). The algebra
so(9,1) is the structure Lie algebra of J2(O) which lies in the lower
right corner of J3(O), and Aut(J2(O)) = SO(9). The algebra J3(O) can
be built from natural operations on the scalar, vector and spinor
representaions of Spin(9) - transformations of which preserve the
Jordan product in J3(O), and Spin(9) is the group of isometries fixing
a specific point in the Cayley plane OP^2 = F4/SO(9). In my last
email, I meant to write that points of OP^2 correspond to trace 1
projections of J3(O), and that lines are projections with trace 2. On
page 25, the authors seem unaware that SO(8) is related to O, F4, E8
and J3(O).
You can see on page 13 of the first paper [37] about nullbranes that
the matrix of supersymmetric fluxbranes belongs to a Cartan subalgebra
involving so(9,1). You might also want to see the Section entitled
"Group Theory and Spinors" on page 73 of the Kaluza-Klein reduction of
M2 and M5 branes paper [40]. By the way, the little group of M2 branes
is SO(8), and the little group of M5 branes is SO(5). For more about
D0 branes and null reduction see paper [41].
In August 2001, I noticed paper [42] which says e.g. "On the other
hand, in S^2 compactified type-IIB theory which is equivalent to K3
compactification of F-theory, exceptional gauge groups arise when
particular 7-branes coincide. [...] By using gauge symmetry
enhancement on D7-branes, we can construct field theories with
exceptional group flavor symmetries as theories on D3-branes." (for
papers about how D-branes are related to nullbranes see [43]).
The so(8) triality which is involved in the exceptional Lie algebras
and the algebras O and J3(O) is also involved in paper [42] because
here there are string configurations which belong to the adjoint,
vector and spinor reps of SO(8). These configurations can be combined
to construct configurations in the 248 of E8 which lie in 9 spatial
dimensions (for a similar paper see [44]).
However, concerning F-theory, there is a new paper [45] about
exceptional groups which raises some doubt about the existence of a
12th dimension on page 36 (and also on this page raises some doubt
about the existence of a possible "bosonic M-theory"). Note that this
paper involves the traditional use of the E8/SO(16) coset (= 128
dimensional octo-octonionic plane (O x O)P^2) that occurs in 3d
supergravity which is different than the presence of the E8/SO(16)
coset in 10 or 11 dimensions that occurs on page 18 of the [35] and
also on page 8 of paper [46].
That August, I also noticed two other examples of a 'triality'. On
especially pages 16-17 of [47] you can see that a resolved M2-brane
with a transverse 8-dim space of Spin(7) holonomy is related to a
fractional D2-brane obtained from the wrapping of a D4-brane around
the S^2 in a manifold of G2 holonomy. The D2-branes have three 7-dim
G2 holonomy transverse spaces. Also, in [48] there is a version of the
quaternionic plane HP^2 which provides a compact 8-manifold naturally
containing three 7-manifolds. (Starting on page 76, it is shown that
HP^2 / U(1) = S^7 (the unit octonions, here as a manifold with SU(3)
action)).
There is a paper [49] which describes an E8 x E8 heterotic string
theory with F4 gauge symmetry in a G2 manifold compactification and
SO(9) in a Spin(7) compactification. The "tricritical Ising model" in
this paper is identified with the Cayley plane OP^2 on page 18,
although the authors do not seem aware that the F4/SO(9) coset is the
Cayley plane. This paper has a supergravity theory with F4 (= Isom
(OP^2) and = Aut(J3(O))) gauge in 3d and N = 1 multiplets, and there
is also another paper [50] about supergravity with the hyperbolic
Cayley plane F4(-20)/SO(9).
Since paper [49] also discusses a possible connection with Calabi-Yau
threefolds it reminds me of a question I had from over a year ago:
Paper [27] describes a topological realization of J3(O) and
conjectures on page 29 that this might be doable on a Calabi-Yau
threefold with Picard group of rank 27 (such that its Neron-Severi Lie
algebra is of type E7). Do you know if such a CY threefold could
exist? I was not able to find one on Ebay :-)
There are some papers [51] which discuss a possible role for the split
octonions in string theory. There is also a paper [52] entitled
"Observable Algebra" which disscusses an observable geometry via the
split octonions and suggests that the non-associativity of the
octonions might correspond to the appearance of fundamental
probabilities in physics. This approach naturally yields Grassmann
elements and noncommutativity of space coordinates.
Note that in string theory the stringy fuzziness (spacetime
uncertainty principle for both open and closed strings) appears at the
noncommutative spacetime scale [53].
A few months ago, J. Polchinski speculated at a cosmology conference
that there could be something like 10^60 solutions of the basic string
theory equations. However, I believe that M-theory must be better than
this and that it may not be merely a coincidence that I can find
various examples of coset spaces in string theory and supergravity
which could be identified with octonionic projective geometry as well
as potential examples of triality.
[1] http://arxiv.org/abs/hep-th/9304124
[2] http://arxiv.org/abs/hep-th/0302079
[3] http://arxiv.org/abs/hep-th/0104050
[4] http://arxiv.org/abs/hep-th/0110106
[5] http://arxiv.org/abs/math-ph/9910004
http://arxiv.org/abs/math/0203010
[6] http://arxiv.org/abs/math/0002016
[7] http://arxiv.org/abs/math/0301322
[8] http://arxiv.org/abs/math/0005168
[9] Jordan triples and operads, J. Alg. 231, 744-57
[10] http://arxiv.org/abs/math/0008224
http://arxiv.org/abs/math/0011243
http://arxiv.org/abs/math/9911190
Alex J. Feingold, Igor B. Frenkel, and John F. X. Ries, Spinor
Construction of Vertex Operator Algebras, Triality, and E8 , Contemp.
Math. 121, Amer. Math. Soc., Providence, Rhode Island,1991.
The exceptional Jordan algebra and the superstring, CMP 122, 393
Jordan structure of Lie and Kac-Moody algebras, L.A. Ferreira et al.,
J. Phys. A 25, 5071
Affine exceptional Jordan algebras and vertex operators, M. Gunaydin
and S.J. Hyun
[11] http://arxiv.org/abs/hep-th/9502064
http://arxiv.org/abs/math/9911190
http://arxiv.org/abs/hep-th/9703188
http://arxiv.org/abs/hep-th/9207023
Jordan algebras in conformal field theories, L.A. Ferreira et al
[12] http://arxiv.org/abs/hep-th/0112261
[13] http://arxiv.org/abs/hep-th/0206139
also see http://arxiv.org/abs/hep-th/0205233
[14] http://arxiv.org/abs/hep-th/0001024
Jordan algebra dynamics, Phys Lett B, 210, 69
Jordan algebras and flat potentials at one loop, C.A. Savoy
Symplectic bosons, fermi fields and super Jordan algebras,
Phys. Lett. B, 234, 315
An E6 x U(1) invariant quantum mechanics for a Jordan pair,
P. Truini and L.C. Biedenharn
[15] http://arxiv.org/abs/hep-th/9406099
http://arxiv.org/abs/math/9811033
http://math1.uibk.ac.at/mathematik/jordan/archive/jordan-hopf/
[16] http://arxiv.org/abs/math/0108057
http://arxiv.org/abs/math/0108059
http://arxiv.org/abs/math/0005170
[17] Implementation of Jordan isomorphisms for general von Neumann
algebras, A. Rieckers and H. Roos, Ann. Inst. Henri Poincare 150,
95-113
Complementation of Jordan triples in von Neumann algebras, B.
Iochum and C.H. Chu, Proc. Amer. Math. Soc. 108, 19-24
http://arxiv.org/abs/math/0110115
http://arxiv.org/abs/math/0201187
http://www.wkap.nl/prod/b/0-7923-4684-X
[18] http://arxiv.org/abs/math/0202194 (Section 2)
http://arxiv.org/abs/hep-th/9301050
[19] http://arxiv.org/abs/hep-th/9411172
http://arxiv.org/abs/math-ph/0110030
[20] http://arxiv.org/abs/q-alg/9708028
http://arxiv.org/abs/hep-th/9312181
[21] http://arxiv.org/abs/hep-th/0008063
http://arxiv.org/abs/math/9811056
http://arxiv.org/abs/hep-th/9502064
Application of theories of Jordan algebras and FTSs to
particles and strings, CQG 4, 227
[22] http://arxiv.org/abs/hep-th/0008063
http://arxiv.org/abs/hep-th/9407045
http://arxiv.org/abs/hep-th/9708025
http://arxiv.org/abs/hep-th/0205235
http://arxiv.org/abs/math/0002055
http://www.mri.ernet.in/~annlrepo/node14.html
Exceptional Jordan triple systems, Comm Alg 25(8), 2687
Weakly compact operators on Jordan triples, B. Iochum and
C.H. Chu
Exceptional simple Jordan algebras and Galois cohomology,
J.C. Ferrar and H.P. Petersson, Arch. Math. (Basel) 61, 517
[23] http://arxiv.org/abs/math/0206275
[24]
http://www.math.nsysu.edu.tw/u/wong/public_html/wpaa2001/abstract/wong-abstract.pdf.
http://www.math.cas.cz/mb126-1/7.html
http://arxiv.org/abs/math.fa/9604213
http://arxiv.org/abs/math/0201187
[25] http://informatik.uibk.ac.at/mathematik/jordan/index.html
[26] http://arxiv.org/abs/math/0104213
[27] http://arxiv.org/abs/alg-geom/9604014
[28] http://arxiv.org/abs/hep-th/0301005
[29] http://arxiv.org/abs/math/0105155
[30] http://arxiv.org/abs/hep-th/0205184
[31] http://arxiv.org/abs/hep-th/0301050
http://arxiv.org/abs/hep-th/0112261
http://arxiv.org/abs/hep-th/9908208
[32] http://arxiv.org/abs/hep-th/0301037
[33] http://arxiv.org/abs/hep-th/0104046
[34] http://arxiv.org/abs/hep-th/9801072
[35] http://arxiv.org/abs/hep-th/0007157
[36] K.W. Chung and A. Sudbery, Physics Letters B, volume 198 (2),
page 161 (1987).
[37] http://arxiv.org/abs/hep-th/0110170
http://arxiv.org/abs/hep-th/0208107
[38] http://arxiv.org/abs/hep-th/0210269
[39] http://arxiv.org/abs/hep-th/0206182
[40] http://arxiv.org/abs/hep-th/0208207
[41] http://arxiv.org/abs/hep-th/9903043
[42] http://arxiv.org/abs/hep-th/9802189
[43] http://arxiv.org/abs/hep-th/0211218
http://arxiv.org/abs/hep-th/0208126
http://arxiv.org/abs/hep-th/0211042
http://arxiv.org/abs/hep-th/0210269
[44] http://arxiv.org/abs/hep-th/9804210
[45] http://arxiv.org/abs/hep-th/0210178
[46] http://arxiv.org/abs/hep-th/0201062
[47] http://arxiv.org/abs/hep-th/0106177
[48] http://arxiv.org/abs/hep-th/0107177
[49] http://arxiv.org/abs/hep-th/0108219
[50] http://arxiv.org/abs/hep-th/0209106
[51] http://arxiv.org/abs/hep-th/0008063
http://arxiv.org/abs/hep-th/0109005
http://arxiv.org/abs/hep-th/0205235
Arnold Neumaier
> Here are some initial and brief comments and references about the
> role of the Jordan algebras in quantum and string theory.
There are also two papers by Elkies and Gross relating the Jordan roots
of the exceptional cone in 27D to the Leech lattice in 24D.
See
N.D. Elkies and Benedict H. Gross,
The exceptional cone and the Leech lattice,
International Math. Research Notices 1996 #14, 665-698.
N.D. Elkies and Benedict H. Gross,
Cubic rings and the exceptional Jordan algebra
Duke Math J, 109 (2001) 383-409.
http://abel.math.harvard.edu/~gross/preprints/
Arnold Neumaier
zirkus
> There are also two papers by Elkies and Gross relating the Jordan roots
> of the exceptional cone in 27D to the Leech lattice in 24D.
Thanks for reminding me about these two papers which I had meant to
look at one day, but first I need to learn more about octonionic
lattices. The Leech lattice can be related to bosonic string theory
and perhaps the work of Elkies and Gross might also be related to this
earlier email I sent about the eleven sphere S^11 that lies within the
Cayley plane OP^2 - this plane is the projective geometry of the
exceptional Jordan algebra J3(O).
"About 18 months ago, M. Atiyah sent me an email reply to say that he
and Jurgen Berndt had found that the Cayley plane OP^2 / Sp(1) = S^13.
Later, they released a paper [1] which establishes this result and
also says e.g. on page 10 that:
"For the Cayley plane, HP^2 is clearly a special orbit. The generic
orbit is just the normal
sphere bundle of HP^2 in OP^2 with fibre S^7 . This is the unit sphere
in the normal R^8
which is a representation of the isotropy group K1 = Sp(1) x Sp(2).
Note that this
representation is not the standard action on H^2 since Sp(1) acts
trivially and only Sp(2)
acts in the standard manner, so that the generic isotropy group is K =
Sp(1) x Sp(1) x 1.
By considering all the Cayley lines (8-spheres) determined by
quaternion lines we see that
the other special orbit is fibred over the dual HP^2 with 3-sphere
fibres and therefore is an
11-dimensional sphere S^11."
As John Baez writes [2], T. Puettman and A. Rigas found an 11-sphere
embedded
in OP^2 that generates the group Z/24,
"Thomas Puettmann and A. Rigas, Isometric actions on the projective
planes and embedded generators of homotopy groups [3].
The simple idea standing behind their work is that
pi_{11}(S^8) = Z/24.
In other words: the 11th homotopy group of the 8-sphere is the group
of integers mod 24. This is just a reflection of the fact that the
pi_{n+3}(S^n) = Z/24
whenever n is big enough. I touched upon the importance of this
for string theory in "week102" [4].
But it gets cooler. S^8 is just the octonionic projective line OP^1.
The octonionic projective plane, OP^2, is formed from OP^1 by gluing
on
some extra stuff. However, this extra stuff is sufficiently
high-dimensional that it doesn't affect the 11th homotopy group, so we
get
pi_8(OP^2) = Z/24
Now, what Puettman and Rigas do is find an 11-sphere *embedded*
in the octonionic projective plane that generates the group Z/24.
In fact, it's a minimal surface: there's no way to wiggle it a
bit to make the "area" less!
Could this geometrical fact have some application to M-theory? I bet
it will. Could it be a useful clue to the math linking these special
dimensions? We'll see."
I haven't bothered yet to download the Postcript software which you
need in order to read the Puettmann and Rigas paper, but I did notice
that there is another paper which discusses in a differnt way how OP^2
can be involved with homotopy groups of spheres. In "Groups of
homotopy self-equivalences and related topics", Contemporary
Mathematics vol. 274 (2001) there is a paper starting on page 57 by
Hans-Joachim Baues and Norio Iwase called "Square rings associated to
elements in homotopy groups of spheres". They describe the square
rings only in terms of primary homotopy operations on spheres, and the
algebraically described square rings are partially made up of
(separately) the complex, quaternionic and Cayley projective planes.
(For more about pi_n+k (S^n) see Toda's book "Composition Methods in
Homotopy Groups of Spheres"). Here's some more previous commentary
from J. Baez:
"n group of smooth structures on the n-sphere
0 1
1 1
2 1
3 1
4 ?
5 1
6 1
7 Z/28
8 Z/2
9 Z/2 x Z/2 x Z/2
10 Z/6
11 Z/992
12 1
13 Z/3
14 Z/2
15 Z/8128 x Z/2
16 Z/2
17 Z/2 x Z/2 x Z/2 x Z/2
18 Z/8 x Z/2
The smooth Poincare conjecture says that the ? is 1. You'll notice
the appearance of some perfect numbers in dimensions 4k-1. This
is no coincidence - you can find the explanation in a paper by
Kervaire and Milnor! In case anyone forgot: a perfect number is one
that's the sum of all its factors, and the first few are 6, 28, 496
and 8128.
Most of these numbers are lurking in the above table, because 992 =
496 x 2.
496 is also 248 + 248, which is the dimension of E8 x E8. It's also
32 x 31 / 2, which is the dimension of SO(32). These are more or less
the gauge groups for heterotic string theory, and I think these
numerical
coincidences are actually important for gravitational anomaly
cancellation!
It seems like an outrageous stroke of good luck that both the even
self-dual
lattices in dimension 16 are root lattices of Lie algebras whose
dimensions,
when multiplied by 4, equal the number of smooth structures on the
11-sphere.
In an earlier age this might be taken as a proof of the existence of
God.
There must be a good reason for it, but I have no idea what it is
[....]"
Note that the plane OP^2 appears in E8 x E8 heterotic string theory
e.g. on page 18 of [5]. Here's J. Baez again:
"Three of my favorite dimensions are 8, 11, and 24. Why?
Well, 8 is the dimension of the octonions, which are related to
special
properties of rotations in 8-dimensional space, and also Bott
periodicity: a magical phenomenon relating rotations, spinors and the
like in n dimensions to the corresponding things in n+8 dimensions.
The
"Cayley integral octonions" form a marvelous lattice which happens to
give the densest lattice packing of spheres in 8 dimensions: each
sphere
has 240 nearest neighbors. This is also the root lattice of the group
E8, which has dimension 248 = 240+8, and is the symmetry group of the
projective plane over the octooctonions: the octonions tensored with
themselves!
In short, all sorts of beautiful madness breaks loose in dimension 8.
But this madness is *tripled* in dimension 24. In this dimension,
spinors are pairs of octooctooctonions: the octonions tensored with
themselves thrice! But more importantly, this is the dimension where
Monstrous Moonshine lives. While bosonic string theory works best in
26-dimensional spacetime, two of those dimensions really come from the
fact that a string worldsheet is a 2d surface, so the real magic comes
from secret relations between 2-dimensional stuff (complex analysis)
and
the number 24.
Some of this boils down to the fact that the only specially symmetric
lattices in 2 dimensions are the square lattice and the hexagonal one,
and 4 x 6 = 24. But there's a lot more going on! For example,
there's
a marvelous lattice in 24 dimensions called the Leech lattice, which
gives the densest lattice packing of spheres in that dimension. It
also
gives rise to a lattice in 26-dimensional spacetime, and if we
cleverly
use this to compactify 26d spacetime and do bosonic string theory
there,
we get a string theory whose symmetry group is the Monster: the
largest
sporadic finite simple group! The dimensions of the irreducible
representations of the Monster are closely connected to the
coefficients
of an important function in complex analysis, called the j-function -
this connection is known as Monstrous Moonshine. [....]"
And some commentary from Tony Smith [5]:
"The Lie group Spin(8) has cohomology of degrees 3, 7, 7, and 11, so
Spin(8) looks like an S3, two S7 spheres, and an S11.
>From the Lie algebra - Lie Group - Symmetric Space point Spin(8) can
be fibred into two S7 spheres and one 14-dim G2 Lie group, and G2 has
cohomology of degrees 3 and 11.
Since the 7-sphere S7 has Hopf fibration S3 -> S7 -> S4, decompose the
"structure" of Spin(8) further to get three S3 spheres, two S4
spheres, and one S11 sphere.
Since the 11-sphere S11 has fibration S3 -> S11 -> HP^2, decompose the
"structure" of Spin(8) further to get four S3 spheres, two S4 spheres,
and one HP^2.
Note that the four S3 spheres correspond to the rank of Spin(8).
Since the 3-sphere S3 has Hopf fibration S1 -> S3 -> S2, decompose the
"structure" of Spin(8) further to get four S1 spheres, four S2
spheres, two S4 spheres, and one HP^2."
[1] http://arxiv.org/abs/math/0206135
[2] http://math.ucr.edu/home/baez/week173.html
[3] http://www.ruhr-uni-bochum.de/mathematik8/puttmann/projplane.ps
[4] http://math.ucr.edu/home/baez/week102.html
[5] http://arxiv.org/abs/hep-th/0108219
[6] http://www.innerx.net/~tsmith/coquad.html
------------
zirkus
> but I plan to
> examine mainly the relevant geometry, Lie cosets and the octonions
While I am learning more about geometry, I wanted to mention some further
references which I have just noticed and which should be relevant for what I
said earlier.
Paper [1] discusses some about M-theory branes and the octonions on pages
10-12, and there is a paper I have not yet looked at about M-theory on a G2
manifold and triality [2].
Paper [3] describes a class of Frobenius algebras associated with each of
the Jordan algebras (in the paper), and John Baez has discussed connections
between Frobenius algebras, subfactors and TQFTs [4].
Classical holomorphic functions can be given an octonionic generalization as
shown in [5], and the multiplication rules of quarks are identical to those
of the split octonions [6].
Octonionic spinors (as Grassmann, anti-commuting fields) are classified in
[7] which also talks about an octonionic M-algebra, however, this algebra is
expressed by 4d octonionic matrices which means that the algebra is outside
of the Jordan algebra scheme, and also I am skeptical about the suggestion
to go beyond the standard HLS framework for reasons including Occam's razor
and paper [8].
[1] http://arxiv.org/abs/hep-th/0212174
[2] http://arxiv.org/abs/hep-th/0212211
[3] http://arxiv.org/abs/alg-geom/9604014
[4] http://math.ucr.edu/home/baez/week174.html
[5] http://arxiv.org/abs/math/0302186
[6] http://arxiv.org/abs/hep-th/0302079