# What IS the HODGE STAR * MAP?

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)

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)

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

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,