Here is a non-rigorous, non-technical attempt at an answer:
If CL(n) is the Clifford algebra
of an n-dimensional vector space,
then
Cl(n) is a graded algebra of total dimension 2^n
and
the grade k part of Cl(n), denoted by Cl(n)k,
has dimension n! / k! (n-k)!
The Hodge star * map is an automorphism of Cl(n) onto Cl(n)
that
is an isomorphism between Cl(n)k and Cl(n)(n-k)
The grade of the image of the Hodge star * map is called the 8-grade.
FOR EXAMPLE:
----------------
16-DIMENSIONAL Cl(4)
has graded dimensions:
grade 0 1 2 3 4
dimension 1 4 6 4 1
*-grade 4 3 2 1 0
The Hodge star * map takes the 6-dimensional bivector space into itself.
Therefore, the 6-dimensional bivector space splits into
two 3-dimensional parts that are interchanged by the Hodge star * map.
If the 4-dimensional vector space of Cl(4) is Minkowski space,
the two 3-dimensional bivector spaces are
the Lie algebras of rotations and boosts.
If the 4-dimensional vector space of Cl(4) is Euclidean space,
the two 3-dimensional bivector spaces are
two copies of SU(2) = Spin(3).
Let F be a bivector form. So is *F
F is self-dual if F = *F
and
F is anti-self-dual if F = -*F
Let mn be lower indices and MN be upper indices for F.
Then *Fmn = (1/2) e(mnab) FAB
The INTEGRAL of the trace of F /\ *F over the vector space of Cl(4)
is the (negative of) the Yang-Mills action for
a pure gauge SU(2) gauge field theory over the vector space of Cl(4).
If the vector space of Cl(4) is S4,
every self-dual connection of index 1
reduces to the connection SU(2) = Spin(3).
An SU(2) = Spin(3) bivector 2-vector space
acts as a transitive transformation group
of the symmetric space Spin(3) / Spin(2) = S2
and S2 x S2 is a 4-dimensional space with quaternionic structure.
----------------
256-DIMENSIONAL Cl(8)
has graded dimensions:
grade 0 1 2 3 4 5 6 7 8
dimension 1 8 28 56 70 56 28 8 1
*-grade 8 7 6 5 4 3 2 1 0
The Hodge star * map takes the 70-dimensional 4-vector space into itself.
Therefore, the 70-dimensional 4-vector space splits into
two 35-dimensional parts that are interchanged by the Hodge star * map.
256-dimensional Cl(8) can be represented by a 16x16 real matrix algebra.
The numbers refer to the grade in Cl(8) of the matrix entry.
0 2 2 2 2 2 2 2 7 5 5 5 5 5 5 5
4 4 2 2 2 2 2 2 5 7 5 5 5 5 5 5
4 4 4 2 2 2 2 2 5 5 7 5 5 5 5 5
4 4 4 4 2 2 2 2 5 5 5 7 5 5 5 5
4 4 4 4 4 2 2 2 5 5 5 5 7 5 5 5
4 4 4 4 4 4 2 2 5 5 5 5 5 7 5 5
4 4 4 4 4 4 4 2 5 5 5 5 5 5 7 5
4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 7
*
1 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4
3 1 3 3 3 3 3 3 6 4 4 4 4 4 4 4
3 3 1 3 3 3 3 3 6 6 4 4 4 4 4 4
3 3 3 1 3 3 3 3 6 6 6 4 4 4 4 4
3 3 3 3 1 3 3 3 6 6 6 6 4 4 4 4
3 3 3 3 3 1 3 3 6 6 6 6 6 4 4 4
3 3 3 3 3 3 1 3 6 6 6 6 6 6 4 4
3 3 3 3 3 3 3 1 6 6 6 6 6 6 6 8
Important Notation Notes:
The two blocks of the form
0 2 2 2 2 2 2 2
4 4 2 2 2 2 2 2
4 4 4 2 2 2 2 2
4 4 4 4 2 2 2 2
4 4 4 4 4 2 2 2
4 4 4 4 4 4 2 2
4 4 4 4 4 4 4 2
4 4 4 4 4 4 4 4
are more symbolic than literal. They mean that:
the 28 entries labelled 2 correspond
to the antisymmetric part of an 8x8 matrix;
the 35 entries labelled 4 correspond
to the traceless symmetric part of an 8x8 matrix; and
the 1 entry labelled 0 corresponds
to the trace of an 8x8 matrix.
A more literal, but more complicated, representation
of the graded structure of those two blocks is:
0 2,4 2,4 2,4 2,4 2,4 2,4 2,4
2,4 4 2,4 2,4 2,4 2,4 2,4 2,4
2,4 2,4 4 2,4 2,4 2,4 2,4 2,4
2,4 2,4 2,4 4 2,4 2,4 2,4 2,4
2,4 2,4 2,4 2,4 4 2,4 2,4 2,4
2,4 2,4 2,4 2,4 2,4 4 2,4 2,4
2,4 2,4 2,4 2,4 2,4 2,4 4 2,4
2,4 2,4 2,4 2,4 2,4 2,4 2,4 4
However, in the more literal representation,
the entries are not all independent. The more symbolic
representation is a more accurate reflection of the
number of independent entries of each grade.
The two blocks of the form
1 3 3 3 3 3 3 3
3 1 3 3 3 3 3 3
3 3 1 3 3 3 3 3
3 3 3 1 3 3 3 3
3 3 3 3 1 3 3 3
3 3 3 3 3 1 3 3
3 3 3 3 3 3 1 3
3 3 3 3 3 3 3 1
can be taken more literally, as they mean that:
the 8 entries labelled 1 correspond
to the diagonal part of an 8x8 matrix; and
the 56 entries labelled 3 correspond
to the off-diagonal part of an 8x8 matrix.
The conventions of the above Notation Notes are
used from time to time in my papers and web pages.
The Hodge star * map acts like central symmetry of the Cl(8) 16x16 matrix.
The spinor space of Cl(8) is a 1x16 column vector.
It reduces to two mirror image 1x8 column vectors,
the +half-spinor space and the -half-spinor space.
+
+
+
+
+
+
+
+
-
-
-
-
-
-
-
-
The +half-spinor space is acted on by the elements of Cl(8) of
grade 0 1 2 3 4
dimension 1 8 28 56 35
while
the -half-spinor space is acted on by the elements of Cl(8) of
grade 4 5 6 7 8
dimension 35 56 28 8 1
Since the Hodge star * map interchanges the two sets of
elements of Cl(8), the Hodge star * map interchanges
the +half-spinor space and the -half-spinor space.
That is why the two half-spinor spaces
are mirror images of each other.
Let F be a bivector form. *F is a 6-vector form.
F is self-dual if F /\ F = *(F /\ F)
and
F is anti-self-dual if F /\ F = -*(F /\ F)
Let mn be lower indices and MN be upper indices for F.
Then *Fmn = (1/2) e(mnabwxyz) FABWXYZ
The INTEGRAL of the trace of F /\ *F over the vector space of Cl(8)
is the (negative of) the action for
a pure gauge Spin(8) gauge field theory over the vector space of Cl(8).
A Spin(8) bivector 2-vector space
acts as a transitive transformation group
of the symmetric space Spin(8) / Spin(7) = S7
and S7 x RP1 is an 8-dimensional space with octonionic structure.
----------------
D4-D5-E6 MODEL
AFTER DIMENSIONAL REDUCTION TO 4-DIMENSIONAL SPACETIME:
Dimensional reduction of vector spacetime from 8 to 4 dimensions
is done in the D4-D5-E6 model by fixing an associative 3-form
and a coassociative 4-form.
Since the Hodge star * map takes 3-forms into 5-forms,
dimensional reduction removes from the Lagrangian any
terms involving Cl(8) elements of
grade 3 4 5
dimension 56 70 56
leaving only terms of
grade 0 1 2 6 7 8
dimension 1 8 28 28 8 1
Also, since the space spanned by the coassociative 4-form
is reduced from spacetime
(It forms an internal symmetry space for the gauge groups)
the grade of 6-, 7-, and 8-vectors are reduced by 4,
the dimension of 1-vectors is reduced to 4, and
the dimension of 3-vectors (formerly 7-vectors) is reduced to 4.
The resulting structure is
grade 0 1 2 3 4
dimension 1 4 56 4 1
NOW, IN THE RESULTING STRUCTURE, THE HODGE STAR * MAP
IS DERIVED FROM THE Cl(8) HODGE STAR MAP.
Let F be a bivector form. *F is a 2-vector form.
F is self-dual if F = *F
and
F is anti-self-dual if F = -*F
Let mn be lower indices and MN be upper indices for F.
Then *Fmn = (1/2) e(mnab) FAB
The INTEGRAL of the trace of F /\ *F over the 4-dim vector space
is the (negative of) the action for
a pure gauge Spin(8) gauge field theory over the 4-dim vector space.
However, a Spin(8) bivector 2-vector space is too big to act
as a transitive transformation group of a symmetric space
of the form Spin(8) / G = M
where the dimension of M is 4 or less.
(Maximal subgroup of Spin(8) is Spin(7).)
An SU(3) subgroup of Spin(8)
acts as a transitive transformation group
of the symmetric space SU(3) / S(U(2)xU(1)) = CP2
and CP2 is a 4-dimensional space with quaternionic structure.
An SU(2) subgroup of Spin(8)
acts as a transitive transformation group
of the symmetric space SU(2) / U(1) = S2
and S2 x S2 is a 4-dimensional space with quaternionic structure.
A U(1) subgroup of Spin(8)
acts as a transitive transformation group
of the symmetric space U(1) = S1
and S1 x S1 x S1 x S1 = T4 is a 4-dimensional space
with quaternionic structure.
A U(4) subgroup of Spin(8)
has 12-dimensional rank-2 coset space Spin(8) / U(4) = M12
M12 corresponds to SU(3) x SU(2) x U(1).
U(4) = Spin(6) x U(1) has subgroup Spin(6).
Spin(6) acts as the conformal group over
the 4-dimensional space RP1 x S3
that is the Shilov boundary of
the bounded complex homogeneous domain corresponding to
the Hermitian symmetric space Spin(6) / Spin(4) x U(1).
A Spin(5) subgroup of Spin(6)
acts as a transitive transformation group
of the symmetric space Spin(5) / Spin(4) = S4
and S4 is a 4-dimensional space with quaternionic structure.
The 5-dimensional coset space Spin(6) / Spin(5)
represents the scale and conformal degrees of freedom
of the Higgs mechanism.
Spin(5) produces gravity
by the MacDowell-Mansouri mechanism.
If the vector space is S4,
every self-dual connection of index 2
is contained in the connection Spin(8).
Spin(8) contains
BOTH
Spin(5) gravity that acts on
4-dim associative spacetime
AND
SU(3) x SU(2) x U(1) that acts on
4-dim coassociative internal symmetry space.
----------------
If the vector space is S4,
every self-dual connection of index 3
is contained in the connection E8.
E8 contains the global structure of
the 3-fermion-generation D4-D5-E6 model.
----------------
References:
Atiyah, Hitchin, and Singer,
Self-Duality in Four-Dimensional Riemannian Geometry,
Proc. R. Soc. Lond. A362 (1978) 425-461.
Gockeler and Schucker,
Differential Geometry, Gauge Theories, and Gravity,
Cambridge (1987)
Grossman, Kephart, and Stasheff,
Solutions to Yang-Mills Field Equations in Eight Dimensions
and the Last Hopf Map,
Commun. Math. Phys. 96 (1984) 4531-437
Nash and Sen,
Topology and Geometry for Physicists,
Academic Press (1983)
Thirring (translated by Harrell)
Classical Dynamical Systems and Classical Field Theory
vols. 1 and 2 of A Course in Mathematical Physics (2nd edition)
Springer-Verlag (1992)
Tony Smith's Home Page ......