According to Atiyah, Hitchin, and Singer, Proc. R. Soc. Lond. A. 362 (1978) 425-461 (see also Differential Topology and Quantum Field Theory, by Charles Nash (Academic 1991),

Just as

Consider the k = 2 Spin(8) Instanton over S4. It double-covers the S4.

- Since S7 has the twisted product fibration reduction S7 = S3 x S4, and
- since S3 has the twisted product fibration reduction S3 = S1 x S2, and
- since G2 has the twisted product fibration reduction G2 = SU(3) x S6:

28-dim Spin(8) can be factored into the twisted product fibration reduction

S7 x G2 x S7 = S3 x S4 x G2 x S3 x S4 = S1 x S2 x S4 x SU(3) x S6 x S3 x S4 =

Let one of the double-coverings of the S4 correspond to the 4-dim Internal Symmetry Space that remains after dimensional reduction of SpaceTime from 8-dim to 4-dim, and let the 12-dimensional Lie algebra

be the Standard Model gauge group represented on the 4-dim Internal Symmetry Space to produce gauge fields by local action on the 4-dim Physical SpaceTime.

Let the other of the double-coverings of the S4 correspond to the 4-dim Physical SpaceTime that remains after dimensional reduction of SpaceTime from 8-dim to 4-dim, and let the 16 dimensions of spheres

correspond to the following actions on the the 4-dim Physical SpaceTime:

- S6 corresponds to the 6-dim Spin(4) of Rotations and Lorentz Boosts (since S6 has an almost-complex structure by which it can be identified with the (i,j,k) and (I,J,K) subspaces of the Octonions with basis (1,i,j,k,E,I,J,K), and since each of the 3-dim the (i,j,k) and (I,J,K) subspaces correspond to S3 = SU(2));
- one S4 corresponds to Translations;
- the other S4 corresponds to Special Conformal Transformations; and
- S2 corresponds to the Complex
Riemann Sphere, and therefore to:
- one U(1) around the equator of the Riemann Sphere; and
- one Dilatation Scale Transformation.

Therefore, the 16 dimensions of spheres S6 x S4 x S4 x S2 correspond to

one local U(1)

and

- 6-dim Lorentz Boost and Rotations plus
- 4-dim Translations plus
- 4-dim Special Conformal Transformations plus
- 1-dim Dilatation Scale Transformation

so that they correspond to

plus

or, taken together, to

- D4 Spin(8)
- Ghosts and BRST
- Weyl Symmetric Polynomials (Casimirs) for some Groups related to the D4-D5-E6-E7-E8 VoDou Physics Model

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 -> QP2 (where QP2 = quaternionic projective 2-space), decompose the "structure" of Spin(8) further to get four S3 spheres, two S4 spheres, and one QP2.

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 QP2.

In the D4-D5-E6-E7-E8 VoDou Physics Model,

- one S1 sphere corresponds to the Electromagnetic 1-dim U(1);
- one S1 sphere and one S2 sphere correspond to the Weak Force 3-dim SU(2) = S3;
- one QP2 corresponds to the Color Force 8-dim SU(3); and
- two S1 spheres, three S2 spheres, and two S4 spheres correspond to the 16-dim U(4) that contains a U(1) for Complex propagator phase and a 15-dim Conformal SU(2,2) = Spin(2,4) that gives Gravity plus the Higgs mechanism.

Note that the last two correspondences are not simple isomorphisms, but are non-trivial transformations.

Note further that the Automorphism group of QP2 is the 35-dim simple group PGL(3,Q), the projective general linear group of 3-dim Quaternionic space.

Cohomology is related to BRST symmetry and to ghosts. Ghosts and/or GraviPhotons might be useful in constructing star-gate worm-holes.

The D4-D5-E6-E7-E8 VoDou Physics model has a -+++++++ (1,7) spacetime with a (1,3) Minkowski physical spacetime and a compact (0,4) CP2 internal symmetry space.

The Clifford algebra Cl(1,7) = Cl(0,8) = Cl(8,0) = M(16,R) = 16x16 real matrices (note that Cl(7,1) = Cl(2,6) = M(8,Q) = 8x8 quaternion matrices with 4x8 = 32-dimensional left ideal column vector spinors decomposable into two 4x4 = 16-dimensional half-spinors) with 1x16 = 16-dimensional left ideal column vector spinors decomposable into two 1x8 = 8-dimensional half-spinors, and with a triality automorphism between the 8-dimensional Cl(1,7) vectors and its half-spinors. Cl(1,7) graded structure

The gauge group the (1,7) spacetime base manifold is 28-dimensional Spin(1,7) coming from the bivector Lie algebra of the Cl(1,7) Clifford algebra.

The Lagrangian of the D4-D5-E6-E7-E8 VoDou Physics model has:

- a Yang-Mills type term;
- a scalar term;
- a spinor fermion Dirac operator term; and
- at the quantum level, a gauge-fixing term; and
- at the quantum level, a ghost term.

To try to understand the BRST ghost structure of the D4-D5-E6-E7-E8 VoDou Physics model, consider the paper hep-th/0201124 in which J. W. van Holten says:

"The central idea of the BRST construction is to identify the solutions of the constraints with the cohomology classes of a certain nilpotent operator, the BRST operator W. ... to bring out the central ideas of the BRST construction as clearly as possible, here we discuss theories with bosonic symmetries only. ...... the Yang-Mills fields ...[are]... described by the Lie-algebra valued one-form

A = dx^a A_u^a T_a, and the ghosts ...[are]...described by the Lie-algebra valued grassmann variable

c = c^a T_a ...

[The 28-dimensional Spin(1,7) D4 gauge group T_a has 28
components 1__<__ a __<__ 28, so it would be conventional
to say that there are 28 ghosts. However, note that for D4 the T_a
are bivectors of Cl(1,7) and can be written as T_a = d_u G_v where G
is a ghost potential and 1 __<__ u,v __<__ 8 so that G
has 8 components. Note that such a ghost potential representation is
natural only for Lie algebras of type B and D, real orthogonal
transformations representable by antisymmetric real
matrices.]

... As the lagrangean is a scalar function under space-time transformations, it is better suited for the development of a manifestly covariant formulation of gauge-fixed BRST-extended dynamics of theories with local symmetries, including Maxwell-Yang-Mills theory ... The procedure follows quite naturally the steps ...

- a. Start from a gauge-invariant lagrangean L_0(q;q^dot).
- b. For each gauge degree of freedom (each gauge parameter), introduce a ghost variable c^a ; by definition these ghost variables carry ghost number N_g[c^a] = +1. Construct BRST transformations dW X for the extended configuration-space variables X = (q^i; c^a), satisfying the requirement that they leave L_0 invariant (possibly modulo a total derivative), and are nilpotent: dW^2 X = 0.
- c. Add a trivially BRST-invariant set of terms to the action, of the form dW PSI for some anti-commuting function PSI (the gauge fermion).
The last step is to result in an effective lagrangean L_eff with net ghost number N_g[L_eff ] = 0. To achieve this, the gauge fermion must have ghost number N_g[PSI] = -1. However, so far we only have introduced dynamical variables with non-negative ghost number: N_g[q^i; c^a] = (0,+1).

To solve this problem we introduce anti-commuting anti-ghosts b_a , with ghost number N_g[b_a] = -1.

The BRST-transforms of these variables must then be commuting objects a_a, with ghost number N_g[a] = 0.

In order for the BRST-transformations to be nilpotent, we require

dW b_a = i a_a , dW a_a = 0 , which indeed trivially satisfy dW^2 = 0 ...

... local gauge symmetries are encoded in the BRST-transformations.

- First, the BRST-transformations of the classical variables correspond to ghost-dependent gauge transformations.
- Second, the closure of the algebra of the gauge transformations (and the Poisson brackets or commutators of the constraints), as well as the corresponding Jacobi-identities, are part of the condition that the BRST transformations are nilpotent.
... the closure of the classical gauge algebra does not necessarily guarantee the closure of the gauge algebra in the quantum theory, because it may be spoiled by anomalies. Equivalently, in the presence of anomalies there is no nilpotent quantum BRST operator, and no local action satisfying ...[BRST invariance]... A particular case in point is that of a Yang-Mills field coupled to chiral fermions, as in the electro-weak standard model. ...

... the BRST construction can be mapped to a standard cohomology problem on a principle fibre bundle with local structure M x G , where M is the space-time and G is the gauge group viewed as a manifold ...

... for SU(2) = SO(3) ... the anomaly ... vanishes identically, as is true for any orthogonal group SO(N); in contrast the anomaly does not vanish identically for SU(N), for any N

>3. In that case it has to be anulled by cancellation between the contributions of chiral fermions in different representations of the gauge group G ...".

Garcia-Compean, Lopez-Romero, Rodriguez-Segura, and Socolovskymin hep-th/9408003 review the mathematical structure of BRST symmetry. They say:

"... [Bonora and Cotta-Ramusino give] ... a geometric interpretation of ... Lie algebra cohomology in terms of the vertical part of the De Rham exterior derivative on the space of irreducible connections ... however we believe that an interpretation in terms of the topological (Eilenberg-Steenrod) cohomologies of the relevant spaces of the bundle (Lie group, total space and base space) should be more conclusive towards establishing a relationship between quantum mechanics and topology ...".

H. S. Yang and B. H. Lee in hep-th/9503204 describe the relationship between the cohomology of the compact Lie algebra of a Lagrangian gauge theory and the BRST cohomology. They say:

"... we identify the ghost field with the Cartan-Maurer form on an infinite-dimensional Lie group G_oo - the group of gauge transformation - and the BRST generator Q with the coboundary operator s on its Lie algebra G ... Let P be a principal bundle with a structure group G (a compact Lie group with the invariant inner product de ned on its Lie algebra g) over a differentiable manifold M (flat Minkowski space or Euclidean space R^n). The gauge transformation group G_oo - an automorphism of P - and its Lie algebra G can be identified with the set of C^oo-functions on M taking values in the structure group G and its Lie algebra g, respectively. ...... the group invariant structure of constrained system can be described by the Lie algebra cohomology induced by the BRST generator Q in the algebra of invariant polynomials on G with the generalized Poisson bracket ... Consider any physical system with gauge transformation group G1 and its compact Lie algebra G with N generators G_a ...

... we identify the ghost field n(x) with a left-invariant Cartan- Maurer form on the group G_oo ... with respect to "exterior derivative" s for forms n(x) on G_oo. ...

... one can identify the antighost p_a with the Cartan-Maurer form with respect to the "exterior derivative" s+ as well. ...

... the Lie algebra cohomology constructed here is quite different from the BRST cohomology ...[described in some other papers]... In the two cohomologies, the role of ghost fields is quite different and each inner product to obtain Hodge theory is defined by the definitely different schemes. It can be shown ... that there is no paired singlet in the BRST cohomology so that higher cohomologies with nonzero ghost number vanish as long as the asymptotic completeness is assumed. Therefore the ghost number characterizing cohomology classes in this paper has different meaning from the ghost number of state space. ... In QCD, there are nontrivial cohomologies H^p(su(3); R) for p = 0, 8 and p = 3, 5 and they are, respectively, related to each other by the Poincare duality. ...".

L. O'Raifeartaigh, in his book Group Structures of Gauge Theories (Cambridge 1986) says:

"... if j^a_u = PSIbar gamma_5 gamma_u sigma^a PSI is an axial fermion current which is conserved at the classical level ( d_u j^a_u = 0) then, after second quantization it is no longer is no longer conserved but satisfies the equationd_u j^a_u = (e^2 / 24 pi^2) T^a_bc(f) G^bc(A_u) , where the T^a_bc , which are symmetric in b and c, are constants that depend only on the group and the fermion representation f ... and the G^bc(A_u) are pseudo-scalars that depend only on the gauge potentials. ... whenever ... the ABJ ... Adler ... Bell and Jackiw ... anomaly ... is not zero it destroys the renormalizability of the theory ... because the current (J_u) which is conserved and the current (j_u) that couples to the gauge fields in the Lagrangian do not coincide ... Thus, in order to preserve renormalizability one must arrange that the coefficients T^a_bc in the anomaly vanish ...[which is known as]... the ABJ condition ... setting ... [t]he coefficients T^a_bc ... equal to zero, one obtains

T^a_bc = L^a_bc - R^a_bc = 0 , (L,R)^a_bc = tr sigma^a_L,R { sigma^b_L,R sigma^c_L,R} = 2 g^abc I_3(L,R) ,

where sigma_L,R are the generators of the left- and right-handed fermion respectively, g^abc are the invariant symmetric tensors used to construct the third-degree Casimir ... C_3 ... and I_3(L,R) are the third-degree indices ... I3(j) = dim(j) C_3(j) / dim(F) C_3(F) = dim(j) C_3(j) / (g^abc g_abc) ... It is clear that the ABJ conditions will be automatically satisfied if the fermion representations are such that C_3 = 0 , and such representations are called 'safe'. Similarly, groups for which all representations have C_3 = 0 are called 'safe' ... this includes all simple groups except SU(n) ...[for n

>3]...[and Spin(6) = SU(4)]... Furthermore, if the left- and right-handed fermion assignments are 'vectorlike', i.e. sigma_L and sigma_R are equivalent, sigma^a_L = U sigma^a_R U^(-1) , then evidently C_3^L = C_3^R and the anomaly vanishes. ... For the U(1) groups T^a_bc reduces to tr sigma^3_L - tr sigma^3_R and thus the ABJ condition requires that the sum of the cubes of the U(1) charges for the left- and right-handed fermions be the same ... in the standard U(2) electroweak model ... where the SU(2) subgroup of U(2) is safe ... the U(1) subgroup requires the number of quark and lepton families to be equal. ...".

Blaga, Moritsch, Schweda, Sommer, Tataru, and Zerrouki in hep-th/9409046 describe the BRST cohomology for gravity using the Ashtekar variables.

Ghosts are described by Mike Guidry in his book Gauge Field Theories (John wiley & Sons 1991) (at pages 153-154):

"... The only path integral that is easy to evaluate is a Gaussian ... However, in many cases of interest the path integral cannot be coerced into a form that approximates a Gaussian path integral ... Physically, the role of fermions is to impose constraints in path integrals; for actual fermions one example of such a constraint is the Pauli principle ... As another example, the ... Faddeev-Popov ghost fields (spinless fermion fields) ... play the role of "negative degrees of freedom," with the minus signs coming to cancel the spurious boson degrees of freedom associated with unphysical longitudinal polarizations that propagate on internal lines of gauge field diagrams. ... in gauges haunted by ghosts the ghosts will be essential to the unitarity of the S-matrix. ...".

Ghosts are described by Bailin and Love in their book Introduction to Gauge Field Theory (IOP Publishing 1993) (at page 125):

"... In the Abelian case of QED ... There is no coupling of the ghost fields to the gauge field, and the ghosts simply contribute to a multiplicative constant, which may be absorbed into the normalization of the generating functional. Thus, in the covariant gauges ... there is no need to introduce Fadeev-Popov ghosts into QED. However, the ghosts play an important role in the non-Abelian case of QCD. ... non-covariant gauges ... referred to as axial gauges ... have the advantage that Fadeev-Popov ghosts decouple from gauge fields even in the non-Abelian case. However, there is the (more than) compensating disadvantage that the gauge field propagator turns out to be very complicated in these gauges. ...".

Geoffrey Dixon, in his book Division Algebras: Octonions, Quaternions, Complex Numbers, and the Algebraic Design of Physics (Kluwer 1994), says with notation such that R_p,q is the Clifford algebra of the vector space R^p,q of real p+q dimensional vectors with signature (p,q):

"... Define N_p,q = R_p+1,q = R_q+1,p = N_q,p This is what [he] refer[s] to as the nilpotent Clifford algebra of the pseudo-orthogonal space R^p,q. Note that unlike the ordinary Clifford algebra case, where R_p,q and R_q,p are seldom isomorphic, N_p,q is always isomorphic to N_q,p. ...".

I would prefer a definition of the nilpotent Clifford algebra of the pseudo-orthogonal space R^p,q as being Nil(p,q) = R_p+1,q+1 = Cl(p+1,q+1) so that Nil(1,7) = Cl(2,8) = M(32.R) = Cl(1,9) . Then, Spin(1,7) D4 of the fundamental local Clifford algebra Cl(1,7) = M(16,R) of the D4-D5-E6-E7-E8 VoDou Physics Model would sit inside the SL(2,O) = Spin(1,9) D5 that includes 8-dimensional (1,7) spacetime and the restriction of its related Cl(1,9) = Nil(1,7) to the (1,7) vector space might be physically interpreted as an algebra of the Nilpotent BRST traansformations. Of the 45 generators of Spin(1,9), 28 correspond to generators of Spin(1,7). The remaining 17 correspond to a Lie Sphere 8-complex-dimensional symmetric space Spin(1,9) / Spin(1,7) x U(1) whose Shilov boundary is 8-real-dimensional RP1xS7 whose basis vectors correspond to Lie Algebra generators of the Ghosts of the D4-D5-E6-E7-E8 VoDou Physics Model.

I think that my definition may be more in line with the BRST interpretation for osonic string F-theory that is described by Jose M Figueroa-O'Farrill, in his paper F-theory and the universal string theory, hep-th/9704009:

"... the BRST cohomology of ... a bosonic string propagating in a 28-dimensional pseudo-euclidean space with signature ( 26, 2 ). ... agrees with that of an underlying bosonic string propagating in a 26-dimensional Minkowski subspace perpendicular to v and not containing v, provided that we identify states whose momenta differ by a multiple of v ... The BRST operator is invariant under the subgroup of the ( 26 + 2 ) pseudo-euclidean group of motions which preserves the null vector v. This is nothing but the ( 25 + 1 ) conformal group, which does not act linearly in Minkowski spacetime but does on the larger space.Symmetries of the BRST operator induce symmetries in the cohomology, hence we would expect that the spectrum should assemble itself into representations on the conformal group. We know that the physical spectrum of the bosonic string only possesses ( 25 + 1 ) Poincare covariance, so what happens to the special conformal transformations?

... bosonic ghosts ... have a (countably) infinite number of inequivalent vacua which can be understood as the momenta in one of two auxiliary compactified dimensions introduced by the bosonisation procedure. The picture changing operator interpolates between these different vacua, commuting with the BRST operator and thus introducing an infinite degeneracy in the cohomology. ...

.. the special conformal transformations ... change the picture. By definition a picture-changing operator is a BRST invariant operator which changes the picture, whence the special conformal transformations are picture-changing operators.

A remarkable fact of this treatment is that the appearance of the lorentzian torus is very natural. In other words, by enhancing the gauge principle on the worldsheet to incorporate the extra U(1) gauge invariance we are forced to reinterpret bosonic stringvacua corresponding to propagation on a given manifold M, as propagation in a manifold which at least locally is of the form M x T 2 where T 2 is the lorentzian torus corresponding to the bosons ... This theory is precisely the F-theory introduced ...[ by Vafa in hep-th/9602022 ]... except that there the compactness of the extra two coordinates was an ad hoc assumption. ...".

Such a BRST picture may be consistent with the ideas of Peter Rowlands in his paper A defragmented Dirac equation describes a Nilpotent Dirac Operator whose "... nilpotent spinor ... can [be] consider[ed] ... as a mathematical object ina 10-dimensional phase space ...". Peter Rowlands's physics model ( see physics/0110092 and quant-ph/0301071 ) may not only be related to the D4-D5-E6-E7-E8 VoDou Physics Model, but may be even more closely related to the physics model of Geoffrey Dixon ( see hep-th/9303039 , hep-th/9902050 , 10Dnew.pdf, and his 7 Stones web site ).

- E8:
- degrees - 2, 8, 12, 14, 18, 20, 24, 30; note that 1, 7, 11, 13, 17, 19, 23, and 29 are all relatively prime to 30.
- type - 3, 15, 23, 27, 35, 39, 47, 59; center = Z1 = 1 = trivial

- D8 Spin(16):
- degrees - 2, 4, 6, 8, 10, 12, 14, 8
- type - 3, 7, 11, 15, 19, 23, 27, 15; center = Z2 + Z2

- E7:
- degrees - 2, 6, 8, 10, 12, 14, 18
- type - 3, 11, 15, 19, 23, 27, 35; center = Z2

- D6 Spin(12):
- degrees - 2, 4, 6, 8, 10, 6
- type - 3, 7, 11, 15, 19, 11; center = Z2 + Z2

- E6:
- degrees - 2, 5, 6, 8, 9, 12
- type - 3, 9, 11, 15, 17, 23; center = Z3

- D5 Spin(10):
- degrees - 2, 4, 6, 8, 5
- type - 3, 7, 11, 15, 9; center = Z4

- D4 Spin(8):
- degrees - 2, 4, 6, 4
- type - 3, 7, 11, 7; center = Z2 + Z2

- F4:
- degrees - 2, 6, 8, 12; note that 1, 5, 7, and 11 are all relatively prime to 12.
- type - 3, 11, 15, 23; center = Z1 = 1 = trivial

- B4 Spin(9):
- degrees - 2, 4, 6, 8
- type - 3, 7, 11, 15; center = Z2

- G2:
- degrees - 2, 6; note that 1 and 5 are both relatively prime to 6.
- type - 3, 11; center = Z1 = 1 = trivial

- A2 SU(3):
- degrees - 2, 3
- type - 3, 5; center = Z3

- D3 Spin(6) = A3 SU(4):
- degrees - 2, 4, 3
- type - 3, 7, 5; center = Z4

The quadric Q8 = Spin(10)/Spin(8)xU(1) is in CP9 in C10.

Q8 has projective rank 4, and so is a cone over a smooth quadric Q'4 in CP3 in C4.

Q'4 is RP1xS3 spacetime .

16-Real-dimensional, 8-Complex-dimensional Q8 is the complex domain of type BDI whose Silov boundary RP1xS7 is the 8-Real-dimensional spacetime of the D4-D5-E6 model.

If physical spacetime is required to be a quadric that is smooth (i.e., has maximal projective rank and nondegenerate associated bilinear form), then physical spacetime should be defined by the 4-Real-dimensional smooth quadric Q'4 of projective rank 4 in CP3 in C4, over which Q8 in CP9 in C10 is the cone.

(Note that dimC(M) = rank(M) x projective rank(M) for even dimensional BDI manifolds M (Fauntleroy).)

The smooth quadric Q'4 should be the Minkowski spacetime of twistor theory, and so be the Silov boundary of the 8-Real-dimensional, 4-Complex-dimensional Klein quadric Q*4 in CP5 in C6.

In twistor theory, the Klein quadric Q*4 is complexified compactified Minkowski spacetime, and also complexified compactified Euclidean spacetime.

(Note: If the topology of compactified Euclidean spacetime is taken to be S4, then it is topologically distinct from compactified Minkowski spacetime RP1xS3. Therefore, in the D4-D5-E6-E7-E8 VoDou Physics model, the topology of compactified Euclidean spacetime is taken to be RP1xS3. This involves a non-standard definition of Euclidean spacetime, but is consistent, allows Weyl unitary trick Wick rotations t -> it, and gives realistic physics.)

The smooth quadric Q'4 over which the quadric Q8 is a cone should be related to the Cayley calibration 4-form on the octonions.

The quadrics are related to Lie spheres, Legendre submanifolds, and contact transformations, while the calibrations are related to Lagrangian submanifolds.

The smooth quadric Q'4 should be related to the totally geodesic embedding of CP4 in Q8 in CP9 in C10 defined by

(x0,x1,x2,x3,x4) -} (x0,ix0,x1,ix1,x2,ix2,x3,ix3,x4,ix4).

To construct Penrose Twistors, begin with CMk, Complexified Compactified Minkowski Space, which is the 4-Complex-dimensional Klein quadric Q*4 in CP5 in C6 , and then remove the structure at infinity to get down to 4-Complex-dimensional (non-compact) space C4.

CMk | Q2 = C4 = R8 __________|__________ | | | | QP1 CP3 Mk R4

Topologically, 2-Quaternionic-dimensional Q2 and 4-Complex-dimensional C4 and Complexified Minkowski and 8-Real-dimensional Euclidean R8 are isomorphic,

and they all have as subspaces:

6-Real-dimensional Complex Projective 3-space CP3

4-Real-dimensional Minkowski space Mk

4-Real-dimensional Real Euclidean 4-space R4

**Twistor Space T over Mk is defined to be C4 with a Hermitian
quadratic form with signature ++--. T is a representation space for
SU(2,2). **

**Since SU(2,2) = Spin(4,2), Twistors are ****Conformally****
invariant.**

**T is topologically isomorphic to C4. **

**Projective Twistor Space PT is CP3, the set of Complex lines
through the origin in C4. **

QP1 = S7 / Sp(1) = S7 / S3 = S4 is the set of Quaternionic lines through the origin in Q2.

Each Quaternionic line is a copy of Q = C2, which contains CP1 = S2, the set of Complex lines through the origin in C2. Therefore

The points of CMk represent lines of PT. This is the **Klein
correspondence**, which dates back to Grassmann.
Later, Lie noted that oriented
spheres in C3 could be represented by lines in CP3 with consistently
oriented contact between spheres in C3 being represented by meeting
lines in CP3. This is the Lie Sphere
Geometry.

Lie Sphere Geometry of the SU(2,2) = Spin(2,4) correspondence is described by E. V. Ferapontov in math.DG/0104034, in which he says:

"... the Lie sphere map ... associates the 6-vector { y0, y1, y2, y3, y4, y5 } with hexaspherical coordinates ... which obey the relation- y0^2 + y1^2 + y2^2 + y3^2 + y4^2 - y5^2 = 0 This equation defines the so-called Lie quadric. Thus, with any sphere ... in E3 we associate a point on the Lie quadric. The Lie sphere map linearises the action of the Lie sphere group which is a group of contact transformations in E3 generated by conformal transformations and normal shifts. In hexaspherical coordinates, the action of the Lie sphere group coincides with the linear action of SO(4,2) which preserves the Lie quadric ...

... in the Plucker coordinates ... consider a line l in P3 passing ... With the line l we associate a point ... in projective space P5 with ... homogeneous coordinates ... The coordinates ... satisfy the well-known quadratic Plucker relation p01 p23 + p02 p31 + p03 p12 = 0 ... Hence, we arrive at the well-defined Plucker coorrespondence .... between lines in P3 and points on the Plucker quadric in P5. Plucker correspondence plays an important role in the projective differential geometry of surfaces and often sheds some new light on those properties of surfaces which are not 'visible' in P3 but acquire a precise geometric meaning only in P5. ...

... the so-called Lie sphere frame ..[is]... canonically associated with a surface in Lie sphere geometry ... the action of the Lie sphere group in E3 induces linear transformations of the coordinates of... the normalized vector ...[corresponding to]... the Dirac equation ... with ... coefficients ... that ... are Lie-invariant ... Since this linear action should necessarily preserve the Lie quadric, we arrive at the well-known isomorphism of the Lie sphere group and SO(4, 2). Thus, the normalization linearises the action of the Lie sphere group ...

... consider a space C4 equiped with the pseudo-Hermitian scalar product of the signature (2,2) ... and define the wedge product ... The pseudo-Hermitian scalar product ... in C4 induces the pseudo-Hermitian scalar product ... in /\2(C4) ... this pseudo-Hermitian scalar product ... is of the signature (4,2). ...".

Penrose and Rindler show, in section 7.4 of volume 2 of Spinors and Space-Time, that twistors are a natural way to represent

and

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Fauntleroy, Projective Ranks of Hermitian Symmetric Spaces, Math. Intell. 15 (1993) 27-32 and NCSU preprint

Harris, Algebraic Geometry, Springer (1992)

Cecil, Lie Sphere Geometry, Springer (1992)

Ward and Wells, Twistor Geometry and Field Theory, Cambridge (1990)

Penrose and Rindler, Spinors and Space-Time, 2 vols., Cambridge (1986, 1988)

Harvey, Spinors and Calibrations, Academic (1990)

Arnold, Mathematical Methods of Classical Mechanics, 2nd ed, Springer (1989)

Nash and Sen, Topology and Geometry for Physicists, Academic (1983)

Salzmann et.. al., Compact Projective Planes, Walter de Gruyter (1995)

Wells, Complex Geometry in Mathematical Physics, Les Presses de L'Universite de Montreal (1982)

Humphreys, Reflection Groups and Coxeter Groups, Cambridge (1990)

Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, AMS (2001)

Mimura and Toda, Topology of Lie Groups, I and II, AMS (1991)

Kane, The Homology of Hopf Spaces, Elsevier North-Holland (1988)

......