{3,3,2} is the group of the 24 quaternions
{+/-1, +/-i, +/-j, +/-k, (+/-1 +/-i +/-j +/-k)/2}, which are the 24 vertices of a 24-cell.
{3,3,2} is also the finite group SL(2,3).
The McKay corresponding Lie group is E6 with Lie algebra E6.
THIS IS THE GROUP CONTAINING THE STRUCTURE OF THE D4-D5-E6 MODEL OF PHYSICS.
No further expansions are considered physically relevant, because:
to go by Z2 expansion to {4,3,2}=Binary Octahedral group, with McKay corresponding Lie group E7, does not lead to a group of the form SL(2,n) (It is a subgroup of order 7 in SL(2,7).); and
to try to go by Z5 expansion to {5,3,2}=Binary Icosahedral group=SL(2,5), with McKay corresponding Lie group E8, is not based on expansion from a normal subgroup, because the Icosahedral group (5,3,2) is simple.
D5 does not appear in the sequence A0-A1-A3-D4-E6 because D5 is the McKay corresponding Lie group of {3,2,2} of order 12, and {3,2,2} does not correspond to a regular polytope, such as the polyhedron, the dipolyhedron, or the tetrahedron.
References: Coxeter, Complex Regular Polytopes, 2nd ed (Cambridge (1991)); Green, Schwarz, and Witten, Superstring Theory, vol. 1, p. 285 Cambridge (1987)).
How Does the McKay Correspondence Work? Let n+1 be the dimension of the center of the group algebra of the finite group. There are n conjugacy classes, other than the identity, of the finite group. The McKay correspondence is that their columns in the character table are the eigenvectors of the extended Cartan matrix of the corresponding rank n Lie algebra. The n column eigenvectors define an n-dimensional vector space that is the root vector space of the Lie algebra. In the case of finite group cyclic Z(n+1) - A(n) - SU(n+1) Lie algebra, the center of the group algebra is the entire algebra, and the n vectors, plus the origin, define an n-simplex the symmetries of which form the symmetric group S(n+1) that is the Weyl group of the A(n) Lie Algebra SU((n+1). For the dicyclic groups - D Lie algebras, and the binary tetrahedral, octahedral, and icosahedral groups - E6, E7, and E8, the nontrivial relations of the finite group algebra define root vector spaces, and therefore Weyl groups, that are more complicated than a simplex, or a symmetric group. In alg-geom/9411010, Ito and Reid extend the McKay correspondence beyond finite subgroups of SU(2) or SL(2,C) to SL(3,C). Their examples 1,2,3 contain the SL(2,C) McKay singularities D4, D5, and E6 corresponding to the D4-D5-E6 physics model.
4. Division Algebras and Spinors:
DIXON considers the division algebras R, C, H, and O to be fundamental.What follows is my understanding of his work. Since my understanding may be incomplete and/or wrong, I encourage interested people to read Dixon's book and papers.
Dixon starts with the 64-dimensional real tensor product T = R x C x H x O.
He notes that the factor R is redundant for a real tensor product,
and that T = C x H x O.
Dixon then considers a division algebra to be the spinor space acted upon by the Clifford algebra of adjoint actions of the division algebra on itself.
The algebra of left-adjoint actions of C on C is CL = C = Cl(0,1).
The spinor space of Cl(0,1) is C.
The algebra of right-adjoint actions of C on C is CR = C = CL.
The combined left-right adjoint actions of C on C is CA = C = CL = CR.
The adjoint actions are not enough to get all R(2) actions on the spinor space C, so:
Add the 4 actions:
Identity(x) = x; Conjugate(x) = x*; i(x) = ix; and i*(x) = ix*.
The last 3 actions are outside all the adjoint structures, and so must be added by forming the tensor product R(2) x C = C(2). Expansion by R(2) takes the real Euclidean 1-dimensional space of Cl(0,1) = C to
the complexification of Cl(1,1) Minkowski 2-dimensional spacetime.
The algebra of left-adjoint actions of H on H is HL = H = Cl(0,2).
The spinor space of Cl(0,2) is H.
The algebra of right-adjoint actions of H on H is HR = H, but HR =/= HL.
The combined left-right adjoint algebra HA = R(4) = Cl(3,1).
The action HR must be included to get all R(4) actions on the spinor space H.
Since HR is inside the adjoint structure, it need not be added in by a tensor product as in the complex case of R(2) x C = C(2).
Since HR = H is outside Cl(0,2), it can be regared as the SU(2) generated by an outer automorphism symmetry of spinor space between +half-spinor space and -half-spinor space.
The algebra of left-adjoint actions of O on O is OL = R(8) = Cl(0,6).
The spinor space of Cl(0,6) is O.
Since OL = R(8), no R(2) or outer automorphism symmetry need be added to get R(8) actions on the spinor space O.
Define PL = CL x HL = C(2) = Cl(3,0).
The spinor space of PL is P = C x H = P+ + P-, where P+ and P- are each copies of the Pauli algebra and invariant under PL.
Note that the outer automorphism symmetry of HR acts on {P+,P-} as an SU(2) doublet.
Add in the R(2) factor from the case of the complex division algebra
to form R(2) x PL = R(2) x C(2) = C(4) = C x Cl(3,1) = Dirac algebra.
Expansion by R(2) takes the real Euclidean 3-dimensional space of Cl(3,0) to
the complexification of Cl(3,1) Minkowski 4-dimensional spacetime.
The spinor space of R(2) x PL is P2.
As P decomposes into P+ and P-, so does P2 decompose into P2+ and P2-, each of which is a 4-complex-dimensional Dirac spinor.
The algebra of left-adjoint actions of T on T is TL = CL x HL x OL = PL x OL.
Since PL = C(2) and OL = R(8), TL = C(2) x R(8) = C(16).
At this point, I will depart from discussing Dixon's model to compare his division algebra structures to the structures of the D4-D5-E6 model.
The algebra of left-adjoint actions of T on T is TL = CL x HL x OL = PL x OL.
Since PL = C(2) and OL = R(8), TL = C(2) x R(8) = C(16) = C x R(16) = C x Cl(0,8).
TL = C x Cl(0,8) is complexification of the Clifford algebra Cl(0,8) of the D4 Lie algebra Spin(0,8). Cl(0,8) = Cl(1,7), so the Euclidean and Minkowski 8-dimensional spacetimes are related by Wick rotation.
D4 is Lie algebra of the gauge group Spin(0,8).
At this point, Dixon's model differs from the D4-D5-E6 model:
Here, Dixon adds in the R(2) factor from the case of the complex division algebra to get:
R(2) x TL = R(2) x C x R(16) = C x R(32) = C x Cl(10,0) = C x Cl(1,9), which is the complexification of the Clifford algebras Cl(0,10) and Cl(1,9) of the D5 Lie algebras Spin(10) (compact) and Spin(1,9) (non-compact).
Dixon then has a 10-dimensional spacetime similar to string theory.
In the D4-D5-E6 model construction, no extra R(2) is added because the purpose of the R(2) is to get to complexified 2- or 4-dimensional spacetime from real 1- or 3-dimensional space.
The complexified D4 Clifford algebra C x Cl(0,8) = C x Cl(1,7) already has complexified 8-dimensional spacetime, and C x Cl(0,8) = C x Cl(,7) can be considered to be the expansion by R(2) of the real Euclidean 7-dimensional space of Cl(7,0) = C(8).
In the D4-D5-E6 model, D5 combines the compact gauge group Spin(0,8) with complexified 8-dimensional spacetime and a U(1) related to the complexification.
Another difference between Dixon's model and the D4-D5-E6 model is that Dixon takes the spinor space of TL = C x R(16) to be T = C x H x O, which is 64-dimensional and decomposes into T+ and T-, each of which is 16-complex-dimensional. Dixon then takes T+ and T- to be the + and - half-spinor spaces of the complexification of his Cl(1,9).
In the D4-D5-E6 model, the full spinor space of TL = C x R(16) is taken to be C x R16, which is 16-complex-dimensional.
The C x R16 decomposes into two 8-complex-dimensional + and - half-spinor spaces of complexified Cl(0,8) = Cl(1,7), which, along with a U(1) related to the complexification, can be added to the D5 to construct E6.
The outer automorphism HR spinor symmetry interchanges the half-spinor spaces. In the special case of D4-D5-E6, it extends by triality to interchange the half-spinor spaces and vector spacetime.
Octonionic representations of Clifford algebras such as
Cl(0,8), including triality and the "opposite algebra" relationship between the +half-spinor fermion particle and -half-spinor fermion antiparticle representations of Cl(0,8), and
Cl(0,6) used in the D4-D5-E6 model
have been described by Schray and Manogue. Their algebraic structures are similar to the X-product of Cederwall and Preitschopf and a later paper of Dixon.
The for a given unit (norm = 1) octonion X, the X-product of two octonions A and B is given by (AX)(XtB), where t denotes transpose. The nonassociativity of octonion multiplication means that the X-product is non-trivial. It can be used to define the parallelizing torsion of the 7-sphere, which varies with position on the 7-sphere. It cannot be used to define the structure constants of a 7-sphere Lie algebra product [A,B] because such structure "constants" are not constant, but vary with position on the 7-sphere (unlike the cases of the 1-sphere and the 3-sphere).
IN TERMS OF THE E8 LATTICE DISCRETE VERSION OF THE D4-D5-E6 MODEL:
The 240 elements of the orbit of the permutation group S7 of the 7 imaginaries of the octonion algebra correspond to the discrete octonionic algebra representation of the 240 vertices near the origin of the 8-dimensional E8 spacetime lattice.
The 240 vertices form a 4-complex-dimensional (8-real-dimensional) Witting polytope, with 240 complex 0-cells (vertices), 2160 complex 1-cells, 2160 complex 2-cells, and 240 complex 3-cells (faces of 6 real dimensions).
If w is the cube root of unity in the complex plane, then the 240 vertices are 24 of the form
(X, 0, 0, 0),
(0, X, 0, 0),
(0, 0, X, 0), and
(0, 0, 0, X),
where X = +/- i w^a sqrt(3) and a is in {0,1,2}, and
216 of the form
(0, +/- w^a, -/+ w^b, +/- w^c),
( -/+ w^b, 0, +/- w^a, +/- w^c),
(+/- w^a, -/+ w^b, 0, +/- w^c), and
(-/+ w^a, -/+ w^b, -/+ w^c, 0)
where a,b,c are in {0,1,2}.
In real 8-dimensional coordinates, the 240 vertices can be taken to be
16 of the form
+/- ea , where a is in {0,1,2,3,4,5,6,7}, and
224 of the form
(+/- ea, +/- eb, +/- ec, +/- ed) / 2,
where abcd is one of;
0123 4567 0145 2367 0246 1357 0347
1256 0167 2345 0257 1346 0356 1247
(see Coxeter, Regular Polytopes, Dover 1973 and
Coxeter, Regular Complex Polytopes, 2nd ed, Cambridge 1991.)
The 480 elements of the orbit of the group that is the product of the permutation group S7 of the 7 imaginaries of the octonion algebra and the group of reflections of the 7 imaginaries that are consistent with octonionic multiplication correspond to
the discrete octonionic algebra representation of the 240 vertices near the origin of the 8-dimensional fermion particle +half-spinor E8 lattice (The 240 vertices form a 4-complex-dimensional (8-real-dimensional) Witting polytope.) and
the discrete octonionic algebra representation of a second set of 240 vertices near the origin of the 8-dimensional fermion antiparticle -half-spinor E8 lattice (The Witting polytope is self-dual, and the second set of 240 vertices form another 4-complex-dimensional (8-real-dimensional) Witting polytope that is dual to the first Witting polytope.).
In real 8-dimensional coordinates, the 240 dual vertices can be taken to be
112 of the form
(+/- ea +/- eb) , where a,b are in {0,1,2,3,4,5,6,7}, and
128 of the form
(+/- e0 +/- e1 +/- e2 +/- e3 +/- e4 +/- e5 +/- e6 +/- e7) / 2,
where the number of - signs is odd.
Schray and Manogue use the Z2P2 (lines in (Z2)^3 projective space of triples of Z2={0,1}) to define octonion multiplication. If the 7 imaginary octonions are denoted by {e1,...,e7}, 1=(100), 2=(110), 3=(010), 4=(111), 5=(011), 6=(001), and 7=(101) and the real octonion 1=(000) corresponds to the empty set, then Z2P2 can be represented by their figure 1:
giving octonion multiplication by ei ei = -1 and eA eB = eC = -eB eA for ABC collinear in Z2P2, and cyclic identities for ABC collinear in Z2P2.
From their point of view, the algebra and "opposite algebra" describe spinors of opposite chirality, which is consistent with their D4-D5-E6 model interpretation as representations of fermion particles and fermion antiparticles.
5. Simple Non-Morse Germs and D4-D5-E6:
Only for 5 or fewer control parameters are all catastrophe germs simple (Gilmore, p. 32).As Gilmore (p. 15) says, "The germ resides between the early [Taylor series] terms which are killed off by the control parameters and the later terms which are killed off by a coordinate transformation."
The number L of variables with vanishing eigenvalue at a non-Morse critical point must be such that L(L+1)/2 is less than K, the number of control parameters, so that L is at most 2 for K at most 5.
For L = 2 and K = 3, the simple non-Morse catastrophe germ is of type D4.
For L = 2 and K = 4, the simple non-Morse catastrophe germ is of type D5.
For L = 2 and K = 5, the simple non-Morse catastrophe germ is of type D6 or E6.
Section 4 of chapter 7 of Gilmore describes the diagrammatic representations of catastrophe germs D4, D5, D6, and E6, showing how they are related to the Coxeter-Dynkin diagrams of the Lie algebras D4, D5, D6, and E6.
Gilmore (pp. 640-641) also analyzes the D4, D5, D6, and E6 catastrophes in terms of the Yang Hui (Pascal) triangle of the terms of Taylor series in 2 variables.
Since the number of variables L = 2, the germs D4, D5, and E6 can be described in terms of surfaces in R^3. In fact, D4, D5, and E6 correspond to umbilics of surfaces in R^3.
Chapter 12 of Porteous, with extensive discussion and nice illustrations, shows the correspondences:
D4 with elliptic (star or monstar) and hyperbolic (star or lemon) umbilics;
D5 with parabolic umbilics; and
E6 with perfect umbilics.
References:
Catastrophe Theory for Scientists and Engineers, Robert Gilmore, Dover 1993 republication of Wiley 1981 edition;
Geometric Differentiation, Ian Porteous, Cambridge 1994.
6. E8 LATTICE, D4 LATTICE, and FEYNMAN CHECKERBOARD:
The D4-D5-E6 model is the natural Feynman Checkerboard theory based on the octonionic 8-real-dimensional E8 lattice with 4-complex-dimensional Witting polytope vertex figure, reduced to a 4-real-dimensional D4 lattice with 24-cell vertex figure.
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